3.3.65 \(\int x^2 (a+b x)^n (c+d x^2)^2 \, dx\)
Optimal. Leaf size=232 \[ \frac {a^2 \left (a^2 d+b^2 c\right )^2 (a+b x)^{n+1}}{b^7 (n+1)}-\frac {2 a \left (a^2 d+b^2 c\right ) \left (3 a^2 d+b^2 c\right ) (a+b x)^{n+2}}{b^7 (n+2)}-\frac {4 a d \left (5 a^2 d+2 b^2 c\right ) (a+b x)^{n+4}}{b^7 (n+4)}+\frac {d \left (15 a^2 d+2 b^2 c\right ) (a+b x)^{n+5}}{b^7 (n+5)}+\frac {\left (15 a^4 d^2+12 a^2 b^2 c d+b^4 c^2\right ) (a+b x)^{n+3}}{b^7 (n+3)}-\frac {6 a d^2 (a+b x)^{n+6}}{b^7 (n+6)}+\frac {d^2 (a+b x)^{n+7}}{b^7 (n+7)} \]
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Rubi [A] time = 0.14, antiderivative size = 232, normalized size of antiderivative = 1.00,
number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used =
{948} \begin {gather*} \frac {\left (12 a^2 b^2 c d+15 a^4 d^2+b^4 c^2\right ) (a+b x)^{n+3}}{b^7 (n+3)}+\frac {a^2 \left (a^2 d+b^2 c\right )^2 (a+b x)^{n+1}}{b^7 (n+1)}-\frac {2 a \left (a^2 d+b^2 c\right ) \left (3 a^2 d+b^2 c\right ) (a+b x)^{n+2}}{b^7 (n+2)}-\frac {4 a d \left (5 a^2 d+2 b^2 c\right ) (a+b x)^{n+4}}{b^7 (n+4)}+\frac {d \left (15 a^2 d+2 b^2 c\right ) (a+b x)^{n+5}}{b^7 (n+5)}-\frac {6 a d^2 (a+b x)^{n+6}}{b^7 (n+6)}+\frac {d^2 (a+b x)^{n+7}}{b^7 (n+7)} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[x^2*(a + b*x)^n*(c + d*x^2)^2,x]
[Out]
(a^2*(b^2*c + a^2*d)^2*(a + b*x)^(1 + n))/(b^7*(1 + n)) - (2*a*(b^2*c + a^2*d)*(b^2*c + 3*a^2*d)*(a + b*x)^(2
+ n))/(b^7*(2 + n)) + ((b^4*c^2 + 12*a^2*b^2*c*d + 15*a^4*d^2)*(a + b*x)^(3 + n))/(b^7*(3 + n)) - (4*a*d*(2*b^
2*c + 5*a^2*d)*(a + b*x)^(4 + n))/(b^7*(4 + n)) + (d*(2*b^2*c + 15*a^2*d)*(a + b*x)^(5 + n))/(b^7*(5 + n)) - (
6*a*d^2*(a + b*x)^(6 + n))/(b^7*(6 + n)) + (d^2*(a + b*x)^(7 + n))/(b^7*(7 + n))
Rule 948
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIn
tegrand[(d + e*x)^m*(f + g*x)^n*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] &&
NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0] && (IGtQ[m, 0] || (EqQ[m, -2] && EqQ[p, 1] && EqQ[d, 0]))
Rubi steps
\begin {align*} \int x^2 (a+b x)^n \left (c+d x^2\right )^2 \, dx &=\int \left (\frac {\left (a b^2 c+a^3 d\right )^2 (a+b x)^n}{b^6}+\frac {2 a \left (-b^2 c-3 a^2 d\right ) \left (b^2 c+a^2 d\right ) (a+b x)^{1+n}}{b^6}+\frac {\left (b^4 c^2+12 a^2 b^2 c d+15 a^4 d^2\right ) (a+b x)^{2+n}}{b^6}-\frac {4 a d \left (2 b^2 c+5 a^2 d\right ) (a+b x)^{3+n}}{b^6}+\frac {d \left (2 b^2 c+15 a^2 d\right ) (a+b x)^{4+n}}{b^6}-\frac {6 a d^2 (a+b x)^{5+n}}{b^6}+\frac {d^2 (a+b x)^{6+n}}{b^6}\right ) \, dx\\ &=\frac {a^2 \left (b^2 c+a^2 d\right )^2 (a+b x)^{1+n}}{b^7 (1+n)}-\frac {2 a \left (b^2 c+a^2 d\right ) \left (b^2 c+3 a^2 d\right ) (a+b x)^{2+n}}{b^7 (2+n)}+\frac {\left (b^4 c^2+12 a^2 b^2 c d+15 a^4 d^2\right ) (a+b x)^{3+n}}{b^7 (3+n)}-\frac {4 a d \left (2 b^2 c+5 a^2 d\right ) (a+b x)^{4+n}}{b^7 (4+n)}+\frac {d \left (2 b^2 c+15 a^2 d\right ) (a+b x)^{5+n}}{b^7 (5+n)}-\frac {6 a d^2 (a+b x)^{6+n}}{b^7 (6+n)}+\frac {d^2 (a+b x)^{7+n}}{b^7 (7+n)}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 199, normalized size = 0.86 \begin {gather*} \frac {(a+b x)^{n+1} \left (\frac {\left (a^3 d+a b^2 c\right )^2}{n+1}+\frac {d (a+b x)^4 \left (15 a^2 d+2 b^2 c\right )}{n+5}-\frac {4 a d (a+b x)^3 \left (5 a^2 d+2 b^2 c\right )}{n+4}-\frac {2 a (a+b x) \left (a^2 d+b^2 c\right ) \left (3 a^2 d+b^2 c\right )}{n+2}+\frac {(a+b x)^2 \left (15 a^4 d^2+12 a^2 b^2 c d+b^4 c^2\right )}{n+3}+\frac {d^2 (a+b x)^6}{n+7}-\frac {6 a d^2 (a+b x)^5}{n+6}\right )}{b^7} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[x^2*(a + b*x)^n*(c + d*x^2)^2,x]
[Out]
((a + b*x)^(1 + n)*((a*b^2*c + a^3*d)^2/(1 + n) - (2*a*(b^2*c + a^2*d)*(b^2*c + 3*a^2*d)*(a + b*x))/(2 + n) +
((b^4*c^2 + 12*a^2*b^2*c*d + 15*a^4*d^2)*(a + b*x)^2)/(3 + n) - (4*a*d*(2*b^2*c + 5*a^2*d)*(a + b*x)^3)/(4 + n
) + (d*(2*b^2*c + 15*a^2*d)*(a + b*x)^4)/(5 + n) - (6*a*d^2*(a + b*x)^5)/(6 + n) + (d^2*(a + b*x)^6)/(7 + n)))
/b^7
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IntegrateAlgebraic [F] time = 0.07, size = 0, normalized size = 0.00 \begin {gather*} \int x^2 (a+b x)^n \left (c+d x^2\right )^2 \, dx \end {gather*}
Verification is not applicable to the result.
[In]
IntegrateAlgebraic[x^2*(a + b*x)^n*(c + d*x^2)^2,x]
[Out]
Defer[IntegrateAlgebraic][x^2*(a + b*x)^n*(c + d*x^2)^2, x]
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fricas [B] time = 0.43, size = 1027, normalized size = 4.43 \begin {gather*} \frac {{\left (2 \, a^{3} b^{4} c^{2} n^{4} + 44 \, a^{3} b^{4} c^{2} n^{3} + 1680 \, a^{3} b^{4} c^{2} + 2016 \, a^{5} b^{2} c d + 720 \, a^{7} d^{2} + {\left (b^{7} d^{2} n^{6} + 21 \, b^{7} d^{2} n^{5} + 175 \, b^{7} d^{2} n^{4} + 735 \, b^{7} d^{2} n^{3} + 1624 \, b^{7} d^{2} n^{2} + 1764 \, b^{7} d^{2} n + 720 \, b^{7} d^{2}\right )} x^{7} + {\left (a b^{6} d^{2} n^{6} + 15 \, a b^{6} d^{2} n^{5} + 85 \, a b^{6} d^{2} n^{4} + 225 \, a b^{6} d^{2} n^{3} + 274 \, a b^{6} d^{2} n^{2} + 120 \, a b^{6} d^{2} n\right )} x^{6} + 2 \, {\left (b^{7} c d n^{6} + 1008 \, b^{7} c d + {\left (23 \, b^{7} c d - 3 \, a^{2} b^{5} d^{2}\right )} n^{5} + 3 \, {\left (69 \, b^{7} c d - 10 \, a^{2} b^{5} d^{2}\right )} n^{4} + 5 \, {\left (185 \, b^{7} c d - 21 \, a^{2} b^{5} d^{2}\right )} n^{3} + 2 \, {\left (1072 \, b^{7} c d - 75 \, a^{2} b^{5} d^{2}\right )} n^{2} + 36 \, {\left (67 \, b^{7} c d - 2 \, a^{2} b^{5} d^{2}\right )} n\right )} x^{5} + 2 \, {\left (a b^{6} c d n^{6} + 19 \, a b^{6} c d n^{5} + {\left (131 \, a b^{6} c d + 15 \, a^{3} b^{4} d^{2}\right )} n^{4} + {\left (401 \, a b^{6} c d + 90 \, a^{3} b^{4} d^{2}\right )} n^{3} + 15 \, {\left (36 \, a b^{6} c d + 11 \, a^{3} b^{4} d^{2}\right )} n^{2} + 18 \, {\left (14 \, a b^{6} c d + 5 \, a^{3} b^{4} d^{2}\right )} n\right )} x^{4} + {\left (b^{7} c^{2} n^{6} + 1680 \, b^{7} c^{2} + {\left (25 \, b^{7} c^{2} - 8 \, a^{2} b^{5} c d\right )} n^{5} + {\left (247 \, b^{7} c^{2} - 128 \, a^{2} b^{5} c d\right )} n^{4} + {\left (1219 \, b^{7} c^{2} - 664 \, a^{2} b^{5} c d - 120 \, a^{4} b^{3} d^{2}\right )} n^{3} + 8 \, {\left (389 \, b^{7} c^{2} - 152 \, a^{2} b^{5} c d - 45 \, a^{4} b^{3} d^{2}\right )} n^{2} + 4 \, {\left (949 \, b^{7} c^{2} - 168 \, a^{2} b^{5} c d - 60 \, a^{4} b^{3} d^{2}\right )} n\right )} x^{3} + 2 \, {\left (179 \, a^{3} b^{4} c^{2} + 24 \, a^{5} b^{2} c d\right )} n^{2} + {\left (a b^{6} c^{2} n^{6} + 23 \, a b^{6} c^{2} n^{5} + 3 \, {\left (67 \, a b^{6} c^{2} + 8 \, a^{3} b^{4} c d\right )} n^{4} + {\left (817 \, a b^{6} c^{2} + 336 \, a^{3} b^{4} c d\right )} n^{3} + 2 \, {\left (739 \, a b^{6} c^{2} + 660 \, a^{3} b^{4} c d + 180 \, a^{5} b^{2} d^{2}\right )} n^{2} + 24 \, {\left (35 \, a b^{6} c^{2} + 42 \, a^{3} b^{4} c d + 15 \, a^{5} b^{2} d^{2}\right )} n\right )} x^{2} + 4 \, {\left (319 \, a^{3} b^{4} c^{2} + 156 \, a^{5} b^{2} c d\right )} n - 2 \, {\left (a^{2} b^{5} c^{2} n^{5} + 22 \, a^{2} b^{5} c^{2} n^{4} + {\left (179 \, a^{2} b^{5} c^{2} + 24 \, a^{4} b^{3} c d\right )} n^{3} + 2 \, {\left (319 \, a^{2} b^{5} c^{2} + 156 \, a^{4} b^{3} c d\right )} n^{2} + 24 \, {\left (35 \, a^{2} b^{5} c^{2} + 42 \, a^{4} b^{3} c d + 15 \, a^{6} b d^{2}\right )} n\right )} x\right )} {\left (b x + a\right )}^{n}}{b^{7} n^{7} + 28 \, b^{7} n^{6} + 322 \, b^{7} n^{5} + 1960 \, b^{7} n^{4} + 6769 \, b^{7} n^{3} + 13132 \, b^{7} n^{2} + 13068 \, b^{7} n + 5040 \, b^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^2*(b*x+a)^n*(d*x^2+c)^2,x, algorithm="fricas")
[Out]
(2*a^3*b^4*c^2*n^4 + 44*a^3*b^4*c^2*n^3 + 1680*a^3*b^4*c^2 + 2016*a^5*b^2*c*d + 720*a^7*d^2 + (b^7*d^2*n^6 + 2
1*b^7*d^2*n^5 + 175*b^7*d^2*n^4 + 735*b^7*d^2*n^3 + 1624*b^7*d^2*n^2 + 1764*b^7*d^2*n + 720*b^7*d^2)*x^7 + (a*
b^6*d^2*n^6 + 15*a*b^6*d^2*n^5 + 85*a*b^6*d^2*n^4 + 225*a*b^6*d^2*n^3 + 274*a*b^6*d^2*n^2 + 120*a*b^6*d^2*n)*x
^6 + 2*(b^7*c*d*n^6 + 1008*b^7*c*d + (23*b^7*c*d - 3*a^2*b^5*d^2)*n^5 + 3*(69*b^7*c*d - 10*a^2*b^5*d^2)*n^4 +
5*(185*b^7*c*d - 21*a^2*b^5*d^2)*n^3 + 2*(1072*b^7*c*d - 75*a^2*b^5*d^2)*n^2 + 36*(67*b^7*c*d - 2*a^2*b^5*d^2)
*n)*x^5 + 2*(a*b^6*c*d*n^6 + 19*a*b^6*c*d*n^5 + (131*a*b^6*c*d + 15*a^3*b^4*d^2)*n^4 + (401*a*b^6*c*d + 90*a^3
*b^4*d^2)*n^3 + 15*(36*a*b^6*c*d + 11*a^3*b^4*d^2)*n^2 + 18*(14*a*b^6*c*d + 5*a^3*b^4*d^2)*n)*x^4 + (b^7*c^2*n
^6 + 1680*b^7*c^2 + (25*b^7*c^2 - 8*a^2*b^5*c*d)*n^5 + (247*b^7*c^2 - 128*a^2*b^5*c*d)*n^4 + (1219*b^7*c^2 - 6
64*a^2*b^5*c*d - 120*a^4*b^3*d^2)*n^3 + 8*(389*b^7*c^2 - 152*a^2*b^5*c*d - 45*a^4*b^3*d^2)*n^2 + 4*(949*b^7*c^
2 - 168*a^2*b^5*c*d - 60*a^4*b^3*d^2)*n)*x^3 + 2*(179*a^3*b^4*c^2 + 24*a^5*b^2*c*d)*n^2 + (a*b^6*c^2*n^6 + 23*
a*b^6*c^2*n^5 + 3*(67*a*b^6*c^2 + 8*a^3*b^4*c*d)*n^4 + (817*a*b^6*c^2 + 336*a^3*b^4*c*d)*n^3 + 2*(739*a*b^6*c^
2 + 660*a^3*b^4*c*d + 180*a^5*b^2*d^2)*n^2 + 24*(35*a*b^6*c^2 + 42*a^3*b^4*c*d + 15*a^5*b^2*d^2)*n)*x^2 + 4*(3
19*a^3*b^4*c^2 + 156*a^5*b^2*c*d)*n - 2*(a^2*b^5*c^2*n^5 + 22*a^2*b^5*c^2*n^4 + (179*a^2*b^5*c^2 + 24*a^4*b^3*
c*d)*n^3 + 2*(319*a^2*b^5*c^2 + 156*a^4*b^3*c*d)*n^2 + 24*(35*a^2*b^5*c^2 + 42*a^4*b^3*c*d + 15*a^6*b*d^2)*n)*
x)*(b*x + a)^n/(b^7*n^7 + 28*b^7*n^6 + 322*b^7*n^5 + 1960*b^7*n^4 + 6769*b^7*n^3 + 13132*b^7*n^2 + 13068*b^7*n
+ 5040*b^7)
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giac [B] time = 0.22, size = 1750, normalized size = 7.54
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^2*(b*x+a)^n*(d*x^2+c)^2,x, algorithm="giac")
[Out]
((b*x + a)^n*b^7*d^2*n^6*x^7 + (b*x + a)^n*a*b^6*d^2*n^6*x^6 + 21*(b*x + a)^n*b^7*d^2*n^5*x^7 + 2*(b*x + a)^n*
b^7*c*d*n^6*x^5 + 15*(b*x + a)^n*a*b^6*d^2*n^5*x^6 + 175*(b*x + a)^n*b^7*d^2*n^4*x^7 + 2*(b*x + a)^n*a*b^6*c*d
*n^6*x^4 + 46*(b*x + a)^n*b^7*c*d*n^5*x^5 - 6*(b*x + a)^n*a^2*b^5*d^2*n^5*x^5 + 85*(b*x + a)^n*a*b^6*d^2*n^4*x
^6 + 735*(b*x + a)^n*b^7*d^2*n^3*x^7 + (b*x + a)^n*b^7*c^2*n^6*x^3 + 38*(b*x + a)^n*a*b^6*c*d*n^5*x^4 + 414*(b
*x + a)^n*b^7*c*d*n^4*x^5 - 60*(b*x + a)^n*a^2*b^5*d^2*n^4*x^5 + 225*(b*x + a)^n*a*b^6*d^2*n^3*x^6 + 1624*(b*x
+ a)^n*b^7*d^2*n^2*x^7 + (b*x + a)^n*a*b^6*c^2*n^6*x^2 + 25*(b*x + a)^n*b^7*c^2*n^5*x^3 - 8*(b*x + a)^n*a^2*b
^5*c*d*n^5*x^3 + 262*(b*x + a)^n*a*b^6*c*d*n^4*x^4 + 30*(b*x + a)^n*a^3*b^4*d^2*n^4*x^4 + 1850*(b*x + a)^n*b^7
*c*d*n^3*x^5 - 210*(b*x + a)^n*a^2*b^5*d^2*n^3*x^5 + 274*(b*x + a)^n*a*b^6*d^2*n^2*x^6 + 1764*(b*x + a)^n*b^7*
d^2*n*x^7 + 23*(b*x + a)^n*a*b^6*c^2*n^5*x^2 + 247*(b*x + a)^n*b^7*c^2*n^4*x^3 - 128*(b*x + a)^n*a^2*b^5*c*d*n
^4*x^3 + 802*(b*x + a)^n*a*b^6*c*d*n^3*x^4 + 180*(b*x + a)^n*a^3*b^4*d^2*n^3*x^4 + 4288*(b*x + a)^n*b^7*c*d*n^
2*x^5 - 300*(b*x + a)^n*a^2*b^5*d^2*n^2*x^5 + 120*(b*x + a)^n*a*b^6*d^2*n*x^6 + 720*(b*x + a)^n*b^7*d^2*x^7 -
2*(b*x + a)^n*a^2*b^5*c^2*n^5*x + 201*(b*x + a)^n*a*b^6*c^2*n^4*x^2 + 24*(b*x + a)^n*a^3*b^4*c*d*n^4*x^2 + 121
9*(b*x + a)^n*b^7*c^2*n^3*x^3 - 664*(b*x + a)^n*a^2*b^5*c*d*n^3*x^3 - 120*(b*x + a)^n*a^4*b^3*d^2*n^3*x^3 + 10
80*(b*x + a)^n*a*b^6*c*d*n^2*x^4 + 330*(b*x + a)^n*a^3*b^4*d^2*n^2*x^4 + 4824*(b*x + a)^n*b^7*c*d*n*x^5 - 144*
(b*x + a)^n*a^2*b^5*d^2*n*x^5 - 44*(b*x + a)^n*a^2*b^5*c^2*n^4*x + 817*(b*x + a)^n*a*b^6*c^2*n^3*x^2 + 336*(b*
x + a)^n*a^3*b^4*c*d*n^3*x^2 + 3112*(b*x + a)^n*b^7*c^2*n^2*x^3 - 1216*(b*x + a)^n*a^2*b^5*c*d*n^2*x^3 - 360*(
b*x + a)^n*a^4*b^3*d^2*n^2*x^3 + 504*(b*x + a)^n*a*b^6*c*d*n*x^4 + 180*(b*x + a)^n*a^3*b^4*d^2*n*x^4 + 2016*(b
*x + a)^n*b^7*c*d*x^5 + 2*(b*x + a)^n*a^3*b^4*c^2*n^4 - 358*(b*x + a)^n*a^2*b^5*c^2*n^3*x - 48*(b*x + a)^n*a^4
*b^3*c*d*n^3*x + 1478*(b*x + a)^n*a*b^6*c^2*n^2*x^2 + 1320*(b*x + a)^n*a^3*b^4*c*d*n^2*x^2 + 360*(b*x + a)^n*a
^5*b^2*d^2*n^2*x^2 + 3796*(b*x + a)^n*b^7*c^2*n*x^3 - 672*(b*x + a)^n*a^2*b^5*c*d*n*x^3 - 240*(b*x + a)^n*a^4*
b^3*d^2*n*x^3 + 44*(b*x + a)^n*a^3*b^4*c^2*n^3 - 1276*(b*x + a)^n*a^2*b^5*c^2*n^2*x - 624*(b*x + a)^n*a^4*b^3*
c*d*n^2*x + 840*(b*x + a)^n*a*b^6*c^2*n*x^2 + 1008*(b*x + a)^n*a^3*b^4*c*d*n*x^2 + 360*(b*x + a)^n*a^5*b^2*d^2
*n*x^2 + 1680*(b*x + a)^n*b^7*c^2*x^3 + 358*(b*x + a)^n*a^3*b^4*c^2*n^2 + 48*(b*x + a)^n*a^5*b^2*c*d*n^2 - 168
0*(b*x + a)^n*a^2*b^5*c^2*n*x - 2016*(b*x + a)^n*a^4*b^3*c*d*n*x - 720*(b*x + a)^n*a^6*b*d^2*n*x + 1276*(b*x +
a)^n*a^3*b^4*c^2*n + 624*(b*x + a)^n*a^5*b^2*c*d*n + 1680*(b*x + a)^n*a^3*b^4*c^2 + 2016*(b*x + a)^n*a^5*b^2*
c*d + 720*(b*x + a)^n*a^7*d^2)/(b^7*n^7 + 28*b^7*n^6 + 322*b^7*n^5 + 1960*b^7*n^4 + 6769*b^7*n^3 + 13132*b^7*n
^2 + 13068*b^7*n + 5040*b^7)
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maple [B] time = 0.02, size = 1000, normalized size = 4.31 \begin {gather*} \frac {\left (b^{6} d^{2} n^{6} x^{6}+21 b^{6} d^{2} n^{5} x^{6}-6 a \,b^{5} d^{2} n^{5} x^{5}+2 b^{6} c d \,n^{6} x^{4}+175 b^{6} d^{2} n^{4} x^{6}-90 a \,b^{5} d^{2} n^{4} x^{5}+46 b^{6} c d \,n^{5} x^{4}+735 b^{6} d^{2} n^{3} x^{6}+30 a^{2} b^{4} d^{2} n^{4} x^{4}-8 a \,b^{5} c d \,n^{5} x^{3}-510 a \,b^{5} d^{2} n^{3} x^{5}+b^{6} c^{2} n^{6} x^{2}+414 b^{6} c d \,n^{4} x^{4}+1624 b^{6} d^{2} n^{2} x^{6}+300 a^{2} b^{4} d^{2} n^{3} x^{4}-152 a \,b^{5} c d \,n^{4} x^{3}-1350 a \,b^{5} d^{2} n^{2} x^{5}+25 b^{6} c^{2} n^{5} x^{2}+1850 b^{6} c d \,n^{3} x^{4}+1764 b^{6} d^{2} n \,x^{6}-120 a^{3} b^{3} d^{2} n^{3} x^{3}+24 a^{2} b^{4} c d \,n^{4} x^{2}+1050 a^{2} b^{4} d^{2} n^{2} x^{4}-2 a \,b^{5} c^{2} n^{5} x -1048 a \,b^{5} c d \,n^{3} x^{3}-1644 a \,b^{5} d^{2} n \,x^{5}+247 b^{6} c^{2} n^{4} x^{2}+4288 b^{6} c d \,n^{2} x^{4}+720 d^{2} x^{6} b^{6}-720 a^{3} b^{3} d^{2} n^{2} x^{3}+384 a^{2} b^{4} c d \,n^{3} x^{2}+1500 a^{2} b^{4} d^{2} n \,x^{4}-46 a \,b^{5} c^{2} n^{4} x -3208 a \,b^{5} c d \,n^{2} x^{3}-720 a \,d^{2} x^{5} b^{5}+1219 b^{6} c^{2} n^{3} x^{2}+4824 b^{6} c d n \,x^{4}+360 a^{4} b^{2} d^{2} n^{2} x^{2}-48 a^{3} b^{3} c d \,n^{3} x -1320 a^{3} b^{3} d^{2} n \,x^{3}+2 a^{2} b^{4} c^{2} n^{4}+1992 a^{2} b^{4} c d \,n^{2} x^{2}+720 a^{2} b^{4} d^{2} x^{4}-402 a \,b^{5} c^{2} n^{3} x -4320 a \,b^{5} c d n \,x^{3}+3112 b^{6} c^{2} n^{2} x^{2}+2016 b^{6} c d \,x^{4}+1080 a^{4} b^{2} d^{2} n \,x^{2}-672 a^{3} b^{3} c d \,n^{2} x -720 a^{3} b^{3} d^{2} x^{3}+44 a^{2} b^{4} c^{2} n^{3}+3648 a^{2} b^{4} c d n \,x^{2}-1634 a \,b^{5} c^{2} n^{2} x -2016 a \,b^{5} c d \,x^{3}+3796 b^{6} c^{2} n \,x^{2}-720 a^{5} b \,d^{2} n x +48 a^{4} b^{2} c d \,n^{2}+720 a^{4} b^{2} d^{2} x^{2}-2640 a^{3} b^{3} c d n x +358 a^{2} b^{4} c^{2} n^{2}+2016 a^{2} b^{4} c d \,x^{2}-2956 a \,b^{5} c^{2} n x +1680 b^{6} c^{2} x^{2}-720 a^{5} b \,d^{2} x +624 a^{4} b^{2} c d n -2016 a^{3} b^{3} c d x +1276 a^{2} b^{4} c^{2} n -1680 a \,b^{5} c^{2} x +720 a^{6} d^{2}+2016 a^{4} b^{2} c d +1680 a^{2} b^{4} c^{2}\right ) \left (b x +a \right )^{n +1}}{\left (n^{7}+28 n^{6}+322 n^{5}+1960 n^{4}+6769 n^{3}+13132 n^{2}+13068 n +5040\right ) b^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^2*(b*x+a)^n*(d*x^2+c)^2,x)
[Out]
(b*x+a)^(n+1)*(b^6*d^2*n^6*x^6+21*b^6*d^2*n^5*x^6-6*a*b^5*d^2*n^5*x^5+2*b^6*c*d*n^6*x^4+175*b^6*d^2*n^4*x^6-90
*a*b^5*d^2*n^4*x^5+46*b^6*c*d*n^5*x^4+735*b^6*d^2*n^3*x^6+30*a^2*b^4*d^2*n^4*x^4-8*a*b^5*c*d*n^5*x^3-510*a*b^5
*d^2*n^3*x^5+b^6*c^2*n^6*x^2+414*b^6*c*d*n^4*x^4+1624*b^6*d^2*n^2*x^6+300*a^2*b^4*d^2*n^3*x^4-152*a*b^5*c*d*n^
4*x^3-1350*a*b^5*d^2*n^2*x^5+25*b^6*c^2*n^5*x^2+1850*b^6*c*d*n^3*x^4+1764*b^6*d^2*n*x^6-120*a^3*b^3*d^2*n^3*x^
3+24*a^2*b^4*c*d*n^4*x^2+1050*a^2*b^4*d^2*n^2*x^4-2*a*b^5*c^2*n^5*x-1048*a*b^5*c*d*n^3*x^3-1644*a*b^5*d^2*n*x^
5+247*b^6*c^2*n^4*x^2+4288*b^6*c*d*n^2*x^4+720*b^6*d^2*x^6-720*a^3*b^3*d^2*n^2*x^3+384*a^2*b^4*c*d*n^3*x^2+150
0*a^2*b^4*d^2*n*x^4-46*a*b^5*c^2*n^4*x-3208*a*b^5*c*d*n^2*x^3-720*a*b^5*d^2*x^5+1219*b^6*c^2*n^3*x^2+4824*b^6*
c*d*n*x^4+360*a^4*b^2*d^2*n^2*x^2-48*a^3*b^3*c*d*n^3*x-1320*a^3*b^3*d^2*n*x^3+2*a^2*b^4*c^2*n^4+1992*a^2*b^4*c
*d*n^2*x^2+720*a^2*b^4*d^2*x^4-402*a*b^5*c^2*n^3*x-4320*a*b^5*c*d*n*x^3+3112*b^6*c^2*n^2*x^2+2016*b^6*c*d*x^4+
1080*a^4*b^2*d^2*n*x^2-672*a^3*b^3*c*d*n^2*x-720*a^3*b^3*d^2*x^3+44*a^2*b^4*c^2*n^3+3648*a^2*b^4*c*d*n*x^2-163
4*a*b^5*c^2*n^2*x-2016*a*b^5*c*d*x^3+3796*b^6*c^2*n*x^2-720*a^5*b*d^2*n*x+48*a^4*b^2*c*d*n^2+720*a^4*b^2*d^2*x
^2-2640*a^3*b^3*c*d*n*x+358*a^2*b^4*c^2*n^2+2016*a^2*b^4*c*d*x^2-2956*a*b^5*c^2*n*x+1680*b^6*c^2*x^2-720*a^5*b
*d^2*x+624*a^4*b^2*c*d*n-2016*a^3*b^3*c*d*x+1276*a^2*b^4*c^2*n-1680*a*b^5*c^2*x+720*a^6*d^2+2016*a^4*b^2*c*d+1
680*a^2*b^4*c^2)/b^7/(n^7+28*n^6+322*n^5+1960*n^4+6769*n^3+13132*n^2+13068*n+5040)
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maxima [A] time = 0.50, size = 447, normalized size = 1.93 \begin {gather*} \frac {{\left ({\left (n^{2} + 3 \, n + 2\right )} b^{3} x^{3} + {\left (n^{2} + n\right )} a b^{2} x^{2} - 2 \, a^{2} b n x + 2 \, a^{3}\right )} {\left (b x + a\right )}^{n} c^{2}}{{\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} b^{3}} + \frac {2 \, {\left ({\left (n^{4} + 10 \, n^{3} + 35 \, n^{2} + 50 \, n + 24\right )} b^{5} x^{5} + {\left (n^{4} + 6 \, n^{3} + 11 \, n^{2} + 6 \, n\right )} a b^{4} x^{4} - 4 \, {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a^{2} b^{3} x^{3} + 12 \, {\left (n^{2} + n\right )} a^{3} b^{2} x^{2} - 24 \, a^{4} b n x + 24 \, a^{5}\right )} {\left (b x + a\right )}^{n} c d}{{\left (n^{5} + 15 \, n^{4} + 85 \, n^{3} + 225 \, n^{2} + 274 \, n + 120\right )} b^{5}} + \frac {{\left ({\left (n^{6} + 21 \, n^{5} + 175 \, n^{4} + 735 \, n^{3} + 1624 \, n^{2} + 1764 \, n + 720\right )} b^{7} x^{7} + {\left (n^{6} + 15 \, n^{5} + 85 \, n^{4} + 225 \, n^{3} + 274 \, n^{2} + 120 \, n\right )} a b^{6} x^{6} - 6 \, {\left (n^{5} + 10 \, n^{4} + 35 \, n^{3} + 50 \, n^{2} + 24 \, n\right )} a^{2} b^{5} x^{5} + 30 \, {\left (n^{4} + 6 \, n^{3} + 11 \, n^{2} + 6 \, n\right )} a^{3} b^{4} x^{4} - 120 \, {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a^{4} b^{3} x^{3} + 360 \, {\left (n^{2} + n\right )} a^{5} b^{2} x^{2} - 720 \, a^{6} b n x + 720 \, a^{7}\right )} {\left (b x + a\right )}^{n} d^{2}}{{\left (n^{7} + 28 \, n^{6} + 322 \, n^{5} + 1960 \, n^{4} + 6769 \, n^{3} + 13132 \, n^{2} + 13068 \, n + 5040\right )} b^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^2*(b*x+a)^n*(d*x^2+c)^2,x, algorithm="maxima")
[Out]
((n^2 + 3*n + 2)*b^3*x^3 + (n^2 + n)*a*b^2*x^2 - 2*a^2*b*n*x + 2*a^3)*(b*x + a)^n*c^2/((n^3 + 6*n^2 + 11*n + 6
)*b^3) + 2*((n^4 + 10*n^3 + 35*n^2 + 50*n + 24)*b^5*x^5 + (n^4 + 6*n^3 + 11*n^2 + 6*n)*a*b^4*x^4 - 4*(n^3 + 3*
n^2 + 2*n)*a^2*b^3*x^3 + 12*(n^2 + n)*a^3*b^2*x^2 - 24*a^4*b*n*x + 24*a^5)*(b*x + a)^n*c*d/((n^5 + 15*n^4 + 85
*n^3 + 225*n^2 + 274*n + 120)*b^5) + ((n^6 + 21*n^5 + 175*n^4 + 735*n^3 + 1624*n^2 + 1764*n + 720)*b^7*x^7 + (
n^6 + 15*n^5 + 85*n^4 + 225*n^3 + 274*n^2 + 120*n)*a*b^6*x^6 - 6*(n^5 + 10*n^4 + 35*n^3 + 50*n^2 + 24*n)*a^2*b
^5*x^5 + 30*(n^4 + 6*n^3 + 11*n^2 + 6*n)*a^3*b^4*x^4 - 120*(n^3 + 3*n^2 + 2*n)*a^4*b^3*x^3 + 360*(n^2 + n)*a^5
*b^2*x^2 - 720*a^6*b*n*x + 720*a^7)*(b*x + a)^n*d^2/((n^7 + 28*n^6 + 322*n^5 + 1960*n^4 + 6769*n^3 + 13132*n^2
+ 13068*n + 5040)*b^7)
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mupad [B] time = 3.12, size = 932, normalized size = 4.02 \begin {gather*} \frac {2\,a^3\,{\left (a+b\,x\right )}^n\,\left (360\,a^4\,d^2+24\,a^2\,b^2\,c\,d\,n^2+312\,a^2\,b^2\,c\,d\,n+1008\,a^2\,b^2\,c\,d+b^4\,c^2\,n^4+22\,b^4\,c^2\,n^3+179\,b^4\,c^2\,n^2+638\,b^4\,c^2\,n+840\,b^4\,c^2\right )}{b^7\,\left (n^7+28\,n^6+322\,n^5+1960\,n^4+6769\,n^3+13132\,n^2+13068\,n+5040\right )}+\frac {d^2\,x^7\,{\left (a+b\,x\right )}^n\,\left (n^6+21\,n^5+175\,n^4+735\,n^3+1624\,n^2+1764\,n+720\right )}{n^7+28\,n^6+322\,n^5+1960\,n^4+6769\,n^3+13132\,n^2+13068\,n+5040}+\frac {x^3\,{\left (a+b\,x\right )}^n\,\left (n^2+3\,n+2\right )\,\left (-120\,a^4\,d^2\,n-8\,a^2\,b^2\,c\,d\,n^3-104\,a^2\,b^2\,c\,d\,n^2-336\,a^2\,b^2\,c\,d\,n+b^4\,c^2\,n^4+22\,b^4\,c^2\,n^3+179\,b^4\,c^2\,n^2+638\,b^4\,c^2\,n+840\,b^4\,c^2\right )}{b^4\,\left (n^7+28\,n^6+322\,n^5+1960\,n^4+6769\,n^3+13132\,n^2+13068\,n+5040\right )}-\frac {2\,a^2\,n\,x\,{\left (a+b\,x\right )}^n\,\left (360\,a^4\,d^2+24\,a^2\,b^2\,c\,d\,n^2+312\,a^2\,b^2\,c\,d\,n+1008\,a^2\,b^2\,c\,d+b^4\,c^2\,n^4+22\,b^4\,c^2\,n^3+179\,b^4\,c^2\,n^2+638\,b^4\,c^2\,n+840\,b^4\,c^2\right )}{b^6\,\left (n^7+28\,n^6+322\,n^5+1960\,n^4+6769\,n^3+13132\,n^2+13068\,n+5040\right )}+\frac {2\,d\,x^5\,{\left (a+b\,x\right )}^n\,\left (-3\,d\,a^2\,n+c\,b^2\,n^2+13\,c\,b^2\,n+42\,c\,b^2\right )\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}{b^2\,\left (n^7+28\,n^6+322\,n^5+1960\,n^4+6769\,n^3+13132\,n^2+13068\,n+5040\right )}+\frac {a\,d^2\,n\,x^6\,{\left (a+b\,x\right )}^n\,\left (n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120\right )}{b\,\left (n^7+28\,n^6+322\,n^5+1960\,n^4+6769\,n^3+13132\,n^2+13068\,n+5040\right )}+\frac {a\,n\,x^2\,\left (n+1\right )\,{\left (a+b\,x\right )}^n\,\left (360\,a^4\,d^2+24\,a^2\,b^2\,c\,d\,n^2+312\,a^2\,b^2\,c\,d\,n+1008\,a^2\,b^2\,c\,d+b^4\,c^2\,n^4+22\,b^4\,c^2\,n^3+179\,b^4\,c^2\,n^2+638\,b^4\,c^2\,n+840\,b^4\,c^2\right )}{b^5\,\left (n^7+28\,n^6+322\,n^5+1960\,n^4+6769\,n^3+13132\,n^2+13068\,n+5040\right )}+\frac {2\,a\,d\,n\,x^4\,{\left (a+b\,x\right )}^n\,\left (n^3+6\,n^2+11\,n+6\right )\,\left (15\,d\,a^2+c\,b^2\,n^2+13\,c\,b^2\,n+42\,c\,b^2\right )}{b^3\,\left (n^7+28\,n^6+322\,n^5+1960\,n^4+6769\,n^3+13132\,n^2+13068\,n+5040\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^2*(c + d*x^2)^2*(a + b*x)^n,x)
[Out]
(2*a^3*(a + b*x)^n*(360*a^4*d^2 + 840*b^4*c^2 + 638*b^4*c^2*n + 179*b^4*c^2*n^2 + 22*b^4*c^2*n^3 + b^4*c^2*n^4
+ 1008*a^2*b^2*c*d + 312*a^2*b^2*c*d*n + 24*a^2*b^2*c*d*n^2))/(b^7*(13068*n + 13132*n^2 + 6769*n^3 + 1960*n^4
+ 322*n^5 + 28*n^6 + n^7 + 5040)) + (d^2*x^7*(a + b*x)^n*(1764*n + 1624*n^2 + 735*n^3 + 175*n^4 + 21*n^5 + n^
6 + 720))/(13068*n + 13132*n^2 + 6769*n^3 + 1960*n^4 + 322*n^5 + 28*n^6 + n^7 + 5040) + (x^3*(a + b*x)^n*(3*n
+ n^2 + 2)*(840*b^4*c^2 - 120*a^4*d^2*n + 638*b^4*c^2*n + 179*b^4*c^2*n^2 + 22*b^4*c^2*n^3 + b^4*c^2*n^4 - 336
*a^2*b^2*c*d*n - 104*a^2*b^2*c*d*n^2 - 8*a^2*b^2*c*d*n^3))/(b^4*(13068*n + 13132*n^2 + 6769*n^3 + 1960*n^4 + 3
22*n^5 + 28*n^6 + n^7 + 5040)) - (2*a^2*n*x*(a + b*x)^n*(360*a^4*d^2 + 840*b^4*c^2 + 638*b^4*c^2*n + 179*b^4*c
^2*n^2 + 22*b^4*c^2*n^3 + b^4*c^2*n^4 + 1008*a^2*b^2*c*d + 312*a^2*b^2*c*d*n + 24*a^2*b^2*c*d*n^2))/(b^6*(1306
8*n + 13132*n^2 + 6769*n^3 + 1960*n^4 + 322*n^5 + 28*n^6 + n^7 + 5040)) + (2*d*x^5*(a + b*x)^n*(42*b^2*c + b^2
*c*n^2 - 3*a^2*d*n + 13*b^2*c*n)*(50*n + 35*n^2 + 10*n^3 + n^4 + 24))/(b^2*(13068*n + 13132*n^2 + 6769*n^3 + 1
960*n^4 + 322*n^5 + 28*n^6 + n^7 + 5040)) + (a*d^2*n*x^6*(a + b*x)^n*(274*n + 225*n^2 + 85*n^3 + 15*n^4 + n^5
+ 120))/(b*(13068*n + 13132*n^2 + 6769*n^3 + 1960*n^4 + 322*n^5 + 28*n^6 + n^7 + 5040)) + (a*n*x^2*(n + 1)*(a
+ b*x)^n*(360*a^4*d^2 + 840*b^4*c^2 + 638*b^4*c^2*n + 179*b^4*c^2*n^2 + 22*b^4*c^2*n^3 + b^4*c^2*n^4 + 1008*a^
2*b^2*c*d + 312*a^2*b^2*c*d*n + 24*a^2*b^2*c*d*n^2))/(b^5*(13068*n + 13132*n^2 + 6769*n^3 + 1960*n^4 + 322*n^5
+ 28*n^6 + n^7 + 5040)) + (2*a*d*n*x^4*(a + b*x)^n*(11*n + 6*n^2 + n^3 + 6)*(15*a^2*d + 42*b^2*c + b^2*c*n^2
+ 13*b^2*c*n))/(b^3*(13068*n + 13132*n^2 + 6769*n^3 + 1960*n^4 + 322*n^5 + 28*n^6 + n^7 + 5040))
________________________________________________________________________________________
sympy [A] time = 21.59, size = 14317, normalized size = 61.71
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x**2*(b*x+a)**n*(d*x**2+c)**2,x)
[Out]
Piecewise((a**n*(c**2*x**3/3 + 2*c*d*x**5/5 + d**2*x**7/7), Eq(b, 0)), (60*a**6*d**2*log(a/b + x)/(60*a**6*b**
7 + 360*a**5*b**8*x + 900*a**4*b**9*x**2 + 1200*a**3*b**10*x**3 + 900*a**2*b**11*x**4 + 360*a*b**12*x**5 + 60*
b**13*x**6) + 147*a**6*d**2/(60*a**6*b**7 + 360*a**5*b**8*x + 900*a**4*b**9*x**2 + 1200*a**3*b**10*x**3 + 900*
a**2*b**11*x**4 + 360*a*b**12*x**5 + 60*b**13*x**6) + 360*a**5*b*d**2*x*log(a/b + x)/(60*a**6*b**7 + 360*a**5*
b**8*x + 900*a**4*b**9*x**2 + 1200*a**3*b**10*x**3 + 900*a**2*b**11*x**4 + 360*a*b**12*x**5 + 60*b**13*x**6) +
822*a**5*b*d**2*x/(60*a**6*b**7 + 360*a**5*b**8*x + 900*a**4*b**9*x**2 + 1200*a**3*b**10*x**3 + 900*a**2*b**1
1*x**4 + 360*a*b**12*x**5 + 60*b**13*x**6) - 4*a**4*b**2*c*d/(60*a**6*b**7 + 360*a**5*b**8*x + 900*a**4*b**9*x
**2 + 1200*a**3*b**10*x**3 + 900*a**2*b**11*x**4 + 360*a*b**12*x**5 + 60*b**13*x**6) + 900*a**4*b**2*d**2*x**2
*log(a/b + x)/(60*a**6*b**7 + 360*a**5*b**8*x + 900*a**4*b**9*x**2 + 1200*a**3*b**10*x**3 + 900*a**2*b**11*x**
4 + 360*a*b**12*x**5 + 60*b**13*x**6) + 1875*a**4*b**2*d**2*x**2/(60*a**6*b**7 + 360*a**5*b**8*x + 900*a**4*b*
*9*x**2 + 1200*a**3*b**10*x**3 + 900*a**2*b**11*x**4 + 360*a*b**12*x**5 + 60*b**13*x**6) - 24*a**3*b**3*c*d*x/
(60*a**6*b**7 + 360*a**5*b**8*x + 900*a**4*b**9*x**2 + 1200*a**3*b**10*x**3 + 900*a**2*b**11*x**4 + 360*a*b**1
2*x**5 + 60*b**13*x**6) + 1200*a**3*b**3*d**2*x**3*log(a/b + x)/(60*a**6*b**7 + 360*a**5*b**8*x + 900*a**4*b**
9*x**2 + 1200*a**3*b**10*x**3 + 900*a**2*b**11*x**4 + 360*a*b**12*x**5 + 60*b**13*x**6) + 2200*a**3*b**3*d**2*
x**3/(60*a**6*b**7 + 360*a**5*b**8*x + 900*a**4*b**9*x**2 + 1200*a**3*b**10*x**3 + 900*a**2*b**11*x**4 + 360*a
*b**12*x**5 + 60*b**13*x**6) - a**2*b**4*c**2/(60*a**6*b**7 + 360*a**5*b**8*x + 900*a**4*b**9*x**2 + 1200*a**3
*b**10*x**3 + 900*a**2*b**11*x**4 + 360*a*b**12*x**5 + 60*b**13*x**6) - 60*a**2*b**4*c*d*x**2/(60*a**6*b**7 +
360*a**5*b**8*x + 900*a**4*b**9*x**2 + 1200*a**3*b**10*x**3 + 900*a**2*b**11*x**4 + 360*a*b**12*x**5 + 60*b**1
3*x**6) + 900*a**2*b**4*d**2*x**4*log(a/b + x)/(60*a**6*b**7 + 360*a**5*b**8*x + 900*a**4*b**9*x**2 + 1200*a**
3*b**10*x**3 + 900*a**2*b**11*x**4 + 360*a*b**12*x**5 + 60*b**13*x**6) + 1350*a**2*b**4*d**2*x**4/(60*a**6*b**
7 + 360*a**5*b**8*x + 900*a**4*b**9*x**2 + 1200*a**3*b**10*x**3 + 900*a**2*b**11*x**4 + 360*a*b**12*x**5 + 60*
b**13*x**6) - 6*a*b**5*c**2*x/(60*a**6*b**7 + 360*a**5*b**8*x + 900*a**4*b**9*x**2 + 1200*a**3*b**10*x**3 + 90
0*a**2*b**11*x**4 + 360*a*b**12*x**5 + 60*b**13*x**6) - 80*a*b**5*c*d*x**3/(60*a**6*b**7 + 360*a**5*b**8*x + 9
00*a**4*b**9*x**2 + 1200*a**3*b**10*x**3 + 900*a**2*b**11*x**4 + 360*a*b**12*x**5 + 60*b**13*x**6) + 360*a*b**
5*d**2*x**5*log(a/b + x)/(60*a**6*b**7 + 360*a**5*b**8*x + 900*a**4*b**9*x**2 + 1200*a**3*b**10*x**3 + 900*a**
2*b**11*x**4 + 360*a*b**12*x**5 + 60*b**13*x**6) + 360*a*b**5*d**2*x**5/(60*a**6*b**7 + 360*a**5*b**8*x + 900*
a**4*b**9*x**2 + 1200*a**3*b**10*x**3 + 900*a**2*b**11*x**4 + 360*a*b**12*x**5 + 60*b**13*x**6) - 15*b**6*c**2
*x**2/(60*a**6*b**7 + 360*a**5*b**8*x + 900*a**4*b**9*x**2 + 1200*a**3*b**10*x**3 + 900*a**2*b**11*x**4 + 360*
a*b**12*x**5 + 60*b**13*x**6) - 60*b**6*c*d*x**4/(60*a**6*b**7 + 360*a**5*b**8*x + 900*a**4*b**9*x**2 + 1200*a
**3*b**10*x**3 + 900*a**2*b**11*x**4 + 360*a*b**12*x**5 + 60*b**13*x**6) + 60*b**6*d**2*x**6*log(a/b + x)/(60*
a**6*b**7 + 360*a**5*b**8*x + 900*a**4*b**9*x**2 + 1200*a**3*b**10*x**3 + 900*a**2*b**11*x**4 + 360*a*b**12*x*
*5 + 60*b**13*x**6), Eq(n, -7)), (-180*a**6*d**2*log(a/b + x)/(30*a**5*b**7 + 150*a**4*b**8*x + 300*a**3*b**9*
x**2 + 300*a**2*b**10*x**3 + 150*a*b**11*x**4 + 30*b**12*x**5) - 411*a**6*d**2/(30*a**5*b**7 + 150*a**4*b**8*x
+ 300*a**3*b**9*x**2 + 300*a**2*b**10*x**3 + 150*a*b**11*x**4 + 30*b**12*x**5) - 900*a**5*b*d**2*x*log(a/b +
x)/(30*a**5*b**7 + 150*a**4*b**8*x + 300*a**3*b**9*x**2 + 300*a**2*b**10*x**3 + 150*a*b**11*x**4 + 30*b**12*x*
*5) - 1875*a**5*b*d**2*x/(30*a**5*b**7 + 150*a**4*b**8*x + 300*a**3*b**9*x**2 + 300*a**2*b**10*x**3 + 150*a*b*
*11*x**4 + 30*b**12*x**5) - 12*a**4*b**2*c*d/(30*a**5*b**7 + 150*a**4*b**8*x + 300*a**3*b**9*x**2 + 300*a**2*b
**10*x**3 + 150*a*b**11*x**4 + 30*b**12*x**5) - 1800*a**4*b**2*d**2*x**2*log(a/b + x)/(30*a**5*b**7 + 150*a**4
*b**8*x + 300*a**3*b**9*x**2 + 300*a**2*b**10*x**3 + 150*a*b**11*x**4 + 30*b**12*x**5) - 3300*a**4*b**2*d**2*x
**2/(30*a**5*b**7 + 150*a**4*b**8*x + 300*a**3*b**9*x**2 + 300*a**2*b**10*x**3 + 150*a*b**11*x**4 + 30*b**12*x
**5) - 60*a**3*b**3*c*d*x/(30*a**5*b**7 + 150*a**4*b**8*x + 300*a**3*b**9*x**2 + 300*a**2*b**10*x**3 + 150*a*b
**11*x**4 + 30*b**12*x**5) - 1800*a**3*b**3*d**2*x**3*log(a/b + x)/(30*a**5*b**7 + 150*a**4*b**8*x + 300*a**3*
b**9*x**2 + 300*a**2*b**10*x**3 + 150*a*b**11*x**4 + 30*b**12*x**5) - 2700*a**3*b**3*d**2*x**3/(30*a**5*b**7 +
150*a**4*b**8*x + 300*a**3*b**9*x**2 + 300*a**2*b**10*x**3 + 150*a*b**11*x**4 + 30*b**12*x**5) - a**2*b**4*c*
*2/(30*a**5*b**7 + 150*a**4*b**8*x + 300*a**3*b**9*x**2 + 300*a**2*b**10*x**3 + 150*a*b**11*x**4 + 30*b**12*x*
*5) - 120*a**2*b**4*c*d*x**2/(30*a**5*b**7 + 150*a**4*b**8*x + 300*a**3*b**9*x**2 + 300*a**2*b**10*x**3 + 150*
a*b**11*x**4 + 30*b**12*x**5) - 900*a**2*b**4*d**2*x**4*log(a/b + x)/(30*a**5*b**7 + 150*a**4*b**8*x + 300*a**
3*b**9*x**2 + 300*a**2*b**10*x**3 + 150*a*b**11*x**4 + 30*b**12*x**5) - 900*a**2*b**4*d**2*x**4/(30*a**5*b**7
+ 150*a**4*b**8*x + 300*a**3*b**9*x**2 + 300*a**2*b**10*x**3 + 150*a*b**11*x**4 + 30*b**12*x**5) - 5*a*b**5*c*
*2*x/(30*a**5*b**7 + 150*a**4*b**8*x + 300*a**3*b**9*x**2 + 300*a**2*b**10*x**3 + 150*a*b**11*x**4 + 30*b**12*
x**5) - 120*a*b**5*c*d*x**3/(30*a**5*b**7 + 150*a**4*b**8*x + 300*a**3*b**9*x**2 + 300*a**2*b**10*x**3 + 150*a
*b**11*x**4 + 30*b**12*x**5) - 180*a*b**5*d**2*x**5*log(a/b + x)/(30*a**5*b**7 + 150*a**4*b**8*x + 300*a**3*b*
*9*x**2 + 300*a**2*b**10*x**3 + 150*a*b**11*x**4 + 30*b**12*x**5) - 10*b**6*c**2*x**2/(30*a**5*b**7 + 150*a**4
*b**8*x + 300*a**3*b**9*x**2 + 300*a**2*b**10*x**3 + 150*a*b**11*x**4 + 30*b**12*x**5) - 60*b**6*c*d*x**4/(30*
a**5*b**7 + 150*a**4*b**8*x + 300*a**3*b**9*x**2 + 300*a**2*b**10*x**3 + 150*a*b**11*x**4 + 30*b**12*x**5) + 3
0*b**6*d**2*x**6/(30*a**5*b**7 + 150*a**4*b**8*x + 300*a**3*b**9*x**2 + 300*a**2*b**10*x**3 + 150*a*b**11*x**4
+ 30*b**12*x**5), Eq(n, -6)), (180*a**6*d**2*log(a/b + x)/(12*a**4*b**7 + 48*a**3*b**8*x + 72*a**2*b**9*x**2
+ 48*a*b**10*x**3 + 12*b**11*x**4) + 375*a**6*d**2/(12*a**4*b**7 + 48*a**3*b**8*x + 72*a**2*b**9*x**2 + 48*a*b
**10*x**3 + 12*b**11*x**4) + 720*a**5*b*d**2*x*log(a/b + x)/(12*a**4*b**7 + 48*a**3*b**8*x + 72*a**2*b**9*x**2
+ 48*a*b**10*x**3 + 12*b**11*x**4) + 1320*a**5*b*d**2*x/(12*a**4*b**7 + 48*a**3*b**8*x + 72*a**2*b**9*x**2 +
48*a*b**10*x**3 + 12*b**11*x**4) + 24*a**4*b**2*c*d*log(a/b + x)/(12*a**4*b**7 + 48*a**3*b**8*x + 72*a**2*b**9
*x**2 + 48*a*b**10*x**3 + 12*b**11*x**4) + 50*a**4*b**2*c*d/(12*a**4*b**7 + 48*a**3*b**8*x + 72*a**2*b**9*x**2
+ 48*a*b**10*x**3 + 12*b**11*x**4) + 1080*a**4*b**2*d**2*x**2*log(a/b + x)/(12*a**4*b**7 + 48*a**3*b**8*x + 7
2*a**2*b**9*x**2 + 48*a*b**10*x**3 + 12*b**11*x**4) + 1620*a**4*b**2*d**2*x**2/(12*a**4*b**7 + 48*a**3*b**8*x
+ 72*a**2*b**9*x**2 + 48*a*b**10*x**3 + 12*b**11*x**4) + 96*a**3*b**3*c*d*x*log(a/b + x)/(12*a**4*b**7 + 48*a*
*3*b**8*x + 72*a**2*b**9*x**2 + 48*a*b**10*x**3 + 12*b**11*x**4) + 176*a**3*b**3*c*d*x/(12*a**4*b**7 + 48*a**3
*b**8*x + 72*a**2*b**9*x**2 + 48*a*b**10*x**3 + 12*b**11*x**4) + 720*a**3*b**3*d**2*x**3*log(a/b + x)/(12*a**4
*b**7 + 48*a**3*b**8*x + 72*a**2*b**9*x**2 + 48*a*b**10*x**3 + 12*b**11*x**4) + 720*a**3*b**3*d**2*x**3/(12*a*
*4*b**7 + 48*a**3*b**8*x + 72*a**2*b**9*x**2 + 48*a*b**10*x**3 + 12*b**11*x**4) - a**2*b**4*c**2/(12*a**4*b**7
+ 48*a**3*b**8*x + 72*a**2*b**9*x**2 + 48*a*b**10*x**3 + 12*b**11*x**4) + 144*a**2*b**4*c*d*x**2*log(a/b + x)
/(12*a**4*b**7 + 48*a**3*b**8*x + 72*a**2*b**9*x**2 + 48*a*b**10*x**3 + 12*b**11*x**4) + 216*a**2*b**4*c*d*x**
2/(12*a**4*b**7 + 48*a**3*b**8*x + 72*a**2*b**9*x**2 + 48*a*b**10*x**3 + 12*b**11*x**4) + 180*a**2*b**4*d**2*x
**4*log(a/b + x)/(12*a**4*b**7 + 48*a**3*b**8*x + 72*a**2*b**9*x**2 + 48*a*b**10*x**3 + 12*b**11*x**4) - 4*a*b
**5*c**2*x/(12*a**4*b**7 + 48*a**3*b**8*x + 72*a**2*b**9*x**2 + 48*a*b**10*x**3 + 12*b**11*x**4) + 96*a*b**5*c
*d*x**3*log(a/b + x)/(12*a**4*b**7 + 48*a**3*b**8*x + 72*a**2*b**9*x**2 + 48*a*b**10*x**3 + 12*b**11*x**4) + 9
6*a*b**5*c*d*x**3/(12*a**4*b**7 + 48*a**3*b**8*x + 72*a**2*b**9*x**2 + 48*a*b**10*x**3 + 12*b**11*x**4) - 36*a
*b**5*d**2*x**5/(12*a**4*b**7 + 48*a**3*b**8*x + 72*a**2*b**9*x**2 + 48*a*b**10*x**3 + 12*b**11*x**4) - 6*b**6
*c**2*x**2/(12*a**4*b**7 + 48*a**3*b**8*x + 72*a**2*b**9*x**2 + 48*a*b**10*x**3 + 12*b**11*x**4) + 24*b**6*c*d
*x**4*log(a/b + x)/(12*a**4*b**7 + 48*a**3*b**8*x + 72*a**2*b**9*x**2 + 48*a*b**10*x**3 + 12*b**11*x**4) + 6*b
**6*d**2*x**6/(12*a**4*b**7 + 48*a**3*b**8*x + 72*a**2*b**9*x**2 + 48*a*b**10*x**3 + 12*b**11*x**4), Eq(n, -5)
), (-60*a**6*d**2*log(a/b + x)/(3*a**3*b**7 + 9*a**2*b**8*x + 9*a*b**9*x**2 + 3*b**10*x**3) - 110*a**6*d**2/(3
*a**3*b**7 + 9*a**2*b**8*x + 9*a*b**9*x**2 + 3*b**10*x**3) - 180*a**5*b*d**2*x*log(a/b + x)/(3*a**3*b**7 + 9*a
**2*b**8*x + 9*a*b**9*x**2 + 3*b**10*x**3) - 270*a**5*b*d**2*x/(3*a**3*b**7 + 9*a**2*b**8*x + 9*a*b**9*x**2 +
3*b**10*x**3) - 24*a**4*b**2*c*d*log(a/b + x)/(3*a**3*b**7 + 9*a**2*b**8*x + 9*a*b**9*x**2 + 3*b**10*x**3) - 4
4*a**4*b**2*c*d/(3*a**3*b**7 + 9*a**2*b**8*x + 9*a*b**9*x**2 + 3*b**10*x**3) - 180*a**4*b**2*d**2*x**2*log(a/b
+ x)/(3*a**3*b**7 + 9*a**2*b**8*x + 9*a*b**9*x**2 + 3*b**10*x**3) - 180*a**4*b**2*d**2*x**2/(3*a**3*b**7 + 9*
a**2*b**8*x + 9*a*b**9*x**2 + 3*b**10*x**3) - 72*a**3*b**3*c*d*x*log(a/b + x)/(3*a**3*b**7 + 9*a**2*b**8*x + 9
*a*b**9*x**2 + 3*b**10*x**3) - 108*a**3*b**3*c*d*x/(3*a**3*b**7 + 9*a**2*b**8*x + 9*a*b**9*x**2 + 3*b**10*x**3
) - 60*a**3*b**3*d**2*x**3*log(a/b + x)/(3*a**3*b**7 + 9*a**2*b**8*x + 9*a*b**9*x**2 + 3*b**10*x**3) - a**2*b*
*4*c**2/(3*a**3*b**7 + 9*a**2*b**8*x + 9*a*b**9*x**2 + 3*b**10*x**3) - 72*a**2*b**4*c*d*x**2*log(a/b + x)/(3*a
**3*b**7 + 9*a**2*b**8*x + 9*a*b**9*x**2 + 3*b**10*x**3) - 72*a**2*b**4*c*d*x**2/(3*a**3*b**7 + 9*a**2*b**8*x
+ 9*a*b**9*x**2 + 3*b**10*x**3) + 15*a**2*b**4*d**2*x**4/(3*a**3*b**7 + 9*a**2*b**8*x + 9*a*b**9*x**2 + 3*b**1
0*x**3) - 3*a*b**5*c**2*x/(3*a**3*b**7 + 9*a**2*b**8*x + 9*a*b**9*x**2 + 3*b**10*x**3) - 24*a*b**5*c*d*x**3*lo
g(a/b + x)/(3*a**3*b**7 + 9*a**2*b**8*x + 9*a*b**9*x**2 + 3*b**10*x**3) - 3*a*b**5*d**2*x**5/(3*a**3*b**7 + 9*
a**2*b**8*x + 9*a*b**9*x**2 + 3*b**10*x**3) - 3*b**6*c**2*x**2/(3*a**3*b**7 + 9*a**2*b**8*x + 9*a*b**9*x**2 +
3*b**10*x**3) + 6*b**6*c*d*x**4/(3*a**3*b**7 + 9*a**2*b**8*x + 9*a*b**9*x**2 + 3*b**10*x**3) + b**6*d**2*x**6/
(3*a**3*b**7 + 9*a**2*b**8*x + 9*a*b**9*x**2 + 3*b**10*x**3), Eq(n, -4)), (60*a**6*d**2*log(a/b + x)/(4*a**2*b
**7 + 8*a*b**8*x + 4*b**9*x**2) + 90*a**6*d**2/(4*a**2*b**7 + 8*a*b**8*x + 4*b**9*x**2) + 120*a**5*b*d**2*x*lo
g(a/b + x)/(4*a**2*b**7 + 8*a*b**8*x + 4*b**9*x**2) + 120*a**5*b*d**2*x/(4*a**2*b**7 + 8*a*b**8*x + 4*b**9*x**
2) + 48*a**4*b**2*c*d*log(a/b + x)/(4*a**2*b**7 + 8*a*b**8*x + 4*b**9*x**2) + 72*a**4*b**2*c*d/(4*a**2*b**7 +
8*a*b**8*x + 4*b**9*x**2) + 60*a**4*b**2*d**2*x**2*log(a/b + x)/(4*a**2*b**7 + 8*a*b**8*x + 4*b**9*x**2) + 96*
a**3*b**3*c*d*x*log(a/b + x)/(4*a**2*b**7 + 8*a*b**8*x + 4*b**9*x**2) + 96*a**3*b**3*c*d*x/(4*a**2*b**7 + 8*a*
b**8*x + 4*b**9*x**2) - 20*a**3*b**3*d**2*x**3/(4*a**2*b**7 + 8*a*b**8*x + 4*b**9*x**2) + 4*a**2*b**4*c**2*log
(a/b + x)/(4*a**2*b**7 + 8*a*b**8*x + 4*b**9*x**2) + 6*a**2*b**4*c**2/(4*a**2*b**7 + 8*a*b**8*x + 4*b**9*x**2)
+ 48*a**2*b**4*c*d*x**2*log(a/b + x)/(4*a**2*b**7 + 8*a*b**8*x + 4*b**9*x**2) + 5*a**2*b**4*d**2*x**4/(4*a**2
*b**7 + 8*a*b**8*x + 4*b**9*x**2) + 8*a*b**5*c**2*x*log(a/b + x)/(4*a**2*b**7 + 8*a*b**8*x + 4*b**9*x**2) + 8*
a*b**5*c**2*x/(4*a**2*b**7 + 8*a*b**8*x + 4*b**9*x**2) - 16*a*b**5*c*d*x**3/(4*a**2*b**7 + 8*a*b**8*x + 4*b**9
*x**2) - 2*a*b**5*d**2*x**5/(4*a**2*b**7 + 8*a*b**8*x + 4*b**9*x**2) + 4*b**6*c**2*x**2*log(a/b + x)/(4*a**2*b
**7 + 8*a*b**8*x + 4*b**9*x**2) + 4*b**6*c*d*x**4/(4*a**2*b**7 + 8*a*b**8*x + 4*b**9*x**2) + b**6*d**2*x**6/(4
*a**2*b**7 + 8*a*b**8*x + 4*b**9*x**2), Eq(n, -3)), (-180*a**6*d**2*log(a/b + x)/(30*a*b**7 + 30*b**8*x) - 180
*a**6*d**2/(30*a*b**7 + 30*b**8*x) - 180*a**5*b*d**2*x*log(a/b + x)/(30*a*b**7 + 30*b**8*x) - 240*a**4*b**2*c*
d*log(a/b + x)/(30*a*b**7 + 30*b**8*x) - 240*a**4*b**2*c*d/(30*a*b**7 + 30*b**8*x) + 90*a**4*b**2*d**2*x**2/(3
0*a*b**7 + 30*b**8*x) - 240*a**3*b**3*c*d*x*log(a/b + x)/(30*a*b**7 + 30*b**8*x) - 30*a**3*b**3*d**2*x**3/(30*
a*b**7 + 30*b**8*x) - 60*a**2*b**4*c**2*log(a/b + x)/(30*a*b**7 + 30*b**8*x) - 60*a**2*b**4*c**2/(30*a*b**7 +
30*b**8*x) + 120*a**2*b**4*c*d*x**2/(30*a*b**7 + 30*b**8*x) + 15*a**2*b**4*d**2*x**4/(30*a*b**7 + 30*b**8*x) -
60*a*b**5*c**2*x*log(a/b + x)/(30*a*b**7 + 30*b**8*x) - 40*a*b**5*c*d*x**3/(30*a*b**7 + 30*b**8*x) - 9*a*b**5
*d**2*x**5/(30*a*b**7 + 30*b**8*x) + 30*b**6*c**2*x**2/(30*a*b**7 + 30*b**8*x) + 20*b**6*c*d*x**4/(30*a*b**7 +
30*b**8*x) + 6*b**6*d**2*x**6/(30*a*b**7 + 30*b**8*x), Eq(n, -2)), (a**6*d**2*log(a/b + x)/b**7 - a**5*d**2*x
/b**6 + 2*a**4*c*d*log(a/b + x)/b**5 + a**4*d**2*x**2/(2*b**5) - 2*a**3*c*d*x/b**4 - a**3*d**2*x**3/(3*b**4) +
a**2*c**2*log(a/b + x)/b**3 + a**2*c*d*x**2/b**3 + a**2*d**2*x**4/(4*b**3) - a*c**2*x/b**2 - 2*a*c*d*x**3/(3*
b**2) - a*d**2*x**5/(5*b**2) + c**2*x**2/(2*b) + c*d*x**4/(2*b) + d**2*x**6/(6*b), Eq(n, -1)), (720*a**7*d**2*
(a + b*x)**n/(b**7*n**7 + 28*b**7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**3 + 13132*b**7*n**2 + 1
3068*b**7*n + 5040*b**7) - 720*a**6*b*d**2*n*x*(a + b*x)**n/(b**7*n**7 + 28*b**7*n**6 + 322*b**7*n**5 + 1960*b
**7*n**4 + 6769*b**7*n**3 + 13132*b**7*n**2 + 13068*b**7*n + 5040*b**7) + 48*a**5*b**2*c*d*n**2*(a + b*x)**n/(
b**7*n**7 + 28*b**7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**3 + 13132*b**7*n**2 + 13068*b**7*n +
5040*b**7) + 624*a**5*b**2*c*d*n*(a + b*x)**n/(b**7*n**7 + 28*b**7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 + 676
9*b**7*n**3 + 13132*b**7*n**2 + 13068*b**7*n + 5040*b**7) + 2016*a**5*b**2*c*d*(a + b*x)**n/(b**7*n**7 + 28*b*
*7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**3 + 13132*b**7*n**2 + 13068*b**7*n + 5040*b**7) + 360*
a**5*b**2*d**2*n**2*x**2*(a + b*x)**n/(b**7*n**7 + 28*b**7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n
**3 + 13132*b**7*n**2 + 13068*b**7*n + 5040*b**7) + 360*a**5*b**2*d**2*n*x**2*(a + b*x)**n/(b**7*n**7 + 28*b**
7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**3 + 13132*b**7*n**2 + 13068*b**7*n + 5040*b**7) - 48*a*
*4*b**3*c*d*n**3*x*(a + b*x)**n/(b**7*n**7 + 28*b**7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**3 +
13132*b**7*n**2 + 13068*b**7*n + 5040*b**7) - 624*a**4*b**3*c*d*n**2*x*(a + b*x)**n/(b**7*n**7 + 28*b**7*n**6
+ 322*b**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**3 + 13132*b**7*n**2 + 13068*b**7*n + 5040*b**7) - 2016*a**4*b*
*3*c*d*n*x*(a + b*x)**n/(b**7*n**7 + 28*b**7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**3 + 13132*b*
*7*n**2 + 13068*b**7*n + 5040*b**7) - 120*a**4*b**3*d**2*n**3*x**3*(a + b*x)**n/(b**7*n**7 + 28*b**7*n**6 + 32
2*b**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**3 + 13132*b**7*n**2 + 13068*b**7*n + 5040*b**7) - 360*a**4*b**3*d*
*2*n**2*x**3*(a + b*x)**n/(b**7*n**7 + 28*b**7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**3 + 13132*
b**7*n**2 + 13068*b**7*n + 5040*b**7) - 240*a**4*b**3*d**2*n*x**3*(a + b*x)**n/(b**7*n**7 + 28*b**7*n**6 + 322
*b**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**3 + 13132*b**7*n**2 + 13068*b**7*n + 5040*b**7) + 2*a**3*b**4*c**2*
n**4*(a + b*x)**n/(b**7*n**7 + 28*b**7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**3 + 13132*b**7*n**
2 + 13068*b**7*n + 5040*b**7) + 44*a**3*b**4*c**2*n**3*(a + b*x)**n/(b**7*n**7 + 28*b**7*n**6 + 322*b**7*n**5
+ 1960*b**7*n**4 + 6769*b**7*n**3 + 13132*b**7*n**2 + 13068*b**7*n + 5040*b**7) + 358*a**3*b**4*c**2*n**2*(a +
b*x)**n/(b**7*n**7 + 28*b**7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**3 + 13132*b**7*n**2 + 13068
*b**7*n + 5040*b**7) + 1276*a**3*b**4*c**2*n*(a + b*x)**n/(b**7*n**7 + 28*b**7*n**6 + 322*b**7*n**5 + 1960*b**
7*n**4 + 6769*b**7*n**3 + 13132*b**7*n**2 + 13068*b**7*n + 5040*b**7) + 1680*a**3*b**4*c**2*(a + b*x)**n/(b**7
*n**7 + 28*b**7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**3 + 13132*b**7*n**2 + 13068*b**7*n + 5040
*b**7) + 24*a**3*b**4*c*d*n**4*x**2*(a + b*x)**n/(b**7*n**7 + 28*b**7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 +
6769*b**7*n**3 + 13132*b**7*n**2 + 13068*b**7*n + 5040*b**7) + 336*a**3*b**4*c*d*n**3*x**2*(a + b*x)**n/(b**7*
n**7 + 28*b**7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**3 + 13132*b**7*n**2 + 13068*b**7*n + 5040*
b**7) + 1320*a**3*b**4*c*d*n**2*x**2*(a + b*x)**n/(b**7*n**7 + 28*b**7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 +
6769*b**7*n**3 + 13132*b**7*n**2 + 13068*b**7*n + 5040*b**7) + 1008*a**3*b**4*c*d*n*x**2*(a + b*x)**n/(b**7*n
**7 + 28*b**7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**3 + 13132*b**7*n**2 + 13068*b**7*n + 5040*b
**7) + 30*a**3*b**4*d**2*n**4*x**4*(a + b*x)**n/(b**7*n**7 + 28*b**7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 + 6
769*b**7*n**3 + 13132*b**7*n**2 + 13068*b**7*n + 5040*b**7) + 180*a**3*b**4*d**2*n**3*x**4*(a + b*x)**n/(b**7*
n**7 + 28*b**7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**3 + 13132*b**7*n**2 + 13068*b**7*n + 5040*
b**7) + 330*a**3*b**4*d**2*n**2*x**4*(a + b*x)**n/(b**7*n**7 + 28*b**7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 +
6769*b**7*n**3 + 13132*b**7*n**2 + 13068*b**7*n + 5040*b**7) + 180*a**3*b**4*d**2*n*x**4*(a + b*x)**n/(b**7*n
**7 + 28*b**7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**3 + 13132*b**7*n**2 + 13068*b**7*n + 5040*b
**7) - 2*a**2*b**5*c**2*n**5*x*(a + b*x)**n/(b**7*n**7 + 28*b**7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 + 6769*
b**7*n**3 + 13132*b**7*n**2 + 13068*b**7*n + 5040*b**7) - 44*a**2*b**5*c**2*n**4*x*(a + b*x)**n/(b**7*n**7 + 2
8*b**7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**3 + 13132*b**7*n**2 + 13068*b**7*n + 5040*b**7) -
358*a**2*b**5*c**2*n**3*x*(a + b*x)**n/(b**7*n**7 + 28*b**7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 + 6769*b**7*
n**3 + 13132*b**7*n**2 + 13068*b**7*n + 5040*b**7) - 1276*a**2*b**5*c**2*n**2*x*(a + b*x)**n/(b**7*n**7 + 28*b
**7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**3 + 13132*b**7*n**2 + 13068*b**7*n + 5040*b**7) - 168
0*a**2*b**5*c**2*n*x*(a + b*x)**n/(b**7*n**7 + 28*b**7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**3
+ 13132*b**7*n**2 + 13068*b**7*n + 5040*b**7) - 8*a**2*b**5*c*d*n**5*x**3*(a + b*x)**n/(b**7*n**7 + 28*b**7*n*
*6 + 322*b**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**3 + 13132*b**7*n**2 + 13068*b**7*n + 5040*b**7) - 128*a**2*
b**5*c*d*n**4*x**3*(a + b*x)**n/(b**7*n**7 + 28*b**7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**3 +
13132*b**7*n**2 + 13068*b**7*n + 5040*b**7) - 664*a**2*b**5*c*d*n**3*x**3*(a + b*x)**n/(b**7*n**7 + 28*b**7*n*
*6 + 322*b**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**3 + 13132*b**7*n**2 + 13068*b**7*n + 5040*b**7) - 1216*a**2
*b**5*c*d*n**2*x**3*(a + b*x)**n/(b**7*n**7 + 28*b**7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**3 +
13132*b**7*n**2 + 13068*b**7*n + 5040*b**7) - 672*a**2*b**5*c*d*n*x**3*(a + b*x)**n/(b**7*n**7 + 28*b**7*n**6
+ 322*b**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**3 + 13132*b**7*n**2 + 13068*b**7*n + 5040*b**7) - 6*a**2*b**5
*d**2*n**5*x**5*(a + b*x)**n/(b**7*n**7 + 28*b**7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**3 + 131
32*b**7*n**2 + 13068*b**7*n + 5040*b**7) - 60*a**2*b**5*d**2*n**4*x**5*(a + b*x)**n/(b**7*n**7 + 28*b**7*n**6
+ 322*b**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**3 + 13132*b**7*n**2 + 13068*b**7*n + 5040*b**7) - 210*a**2*b**
5*d**2*n**3*x**5*(a + b*x)**n/(b**7*n**7 + 28*b**7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**3 + 13
132*b**7*n**2 + 13068*b**7*n + 5040*b**7) - 300*a**2*b**5*d**2*n**2*x**5*(a + b*x)**n/(b**7*n**7 + 28*b**7*n**
6 + 322*b**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**3 + 13132*b**7*n**2 + 13068*b**7*n + 5040*b**7) - 144*a**2*b
**5*d**2*n*x**5*(a + b*x)**n/(b**7*n**7 + 28*b**7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**3 + 131
32*b**7*n**2 + 13068*b**7*n + 5040*b**7) + a*b**6*c**2*n**6*x**2*(a + b*x)**n/(b**7*n**7 + 28*b**7*n**6 + 322*
b**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**3 + 13132*b**7*n**2 + 13068*b**7*n + 5040*b**7) + 23*a*b**6*c**2*n**
5*x**2*(a + b*x)**n/(b**7*n**7 + 28*b**7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**3 + 13132*b**7*n
**2 + 13068*b**7*n + 5040*b**7) + 201*a*b**6*c**2*n**4*x**2*(a + b*x)**n/(b**7*n**7 + 28*b**7*n**6 + 322*b**7*
n**5 + 1960*b**7*n**4 + 6769*b**7*n**3 + 13132*b**7*n**2 + 13068*b**7*n + 5040*b**7) + 817*a*b**6*c**2*n**3*x*
*2*(a + b*x)**n/(b**7*n**7 + 28*b**7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**3 + 13132*b**7*n**2
+ 13068*b**7*n + 5040*b**7) + 1478*a*b**6*c**2*n**2*x**2*(a + b*x)**n/(b**7*n**7 + 28*b**7*n**6 + 322*b**7*n**
5 + 1960*b**7*n**4 + 6769*b**7*n**3 + 13132*b**7*n**2 + 13068*b**7*n + 5040*b**7) + 840*a*b**6*c**2*n*x**2*(a
+ b*x)**n/(b**7*n**7 + 28*b**7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**3 + 13132*b**7*n**2 + 1306
8*b**7*n + 5040*b**7) + 2*a*b**6*c*d*n**6*x**4*(a + b*x)**n/(b**7*n**7 + 28*b**7*n**6 + 322*b**7*n**5 + 1960*b
**7*n**4 + 6769*b**7*n**3 + 13132*b**7*n**2 + 13068*b**7*n + 5040*b**7) + 38*a*b**6*c*d*n**5*x**4*(a + b*x)**n
/(b**7*n**7 + 28*b**7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**3 + 13132*b**7*n**2 + 13068*b**7*n
+ 5040*b**7) + 262*a*b**6*c*d*n**4*x**4*(a + b*x)**n/(b**7*n**7 + 28*b**7*n**6 + 322*b**7*n**5 + 1960*b**7*n**
4 + 6769*b**7*n**3 + 13132*b**7*n**2 + 13068*b**7*n + 5040*b**7) + 802*a*b**6*c*d*n**3*x**4*(a + b*x)**n/(b**7
*n**7 + 28*b**7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**3 + 13132*b**7*n**2 + 13068*b**7*n + 5040
*b**7) + 1080*a*b**6*c*d*n**2*x**4*(a + b*x)**n/(b**7*n**7 + 28*b**7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 + 6
769*b**7*n**3 + 13132*b**7*n**2 + 13068*b**7*n + 5040*b**7) + 504*a*b**6*c*d*n*x**4*(a + b*x)**n/(b**7*n**7 +
28*b**7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**3 + 13132*b**7*n**2 + 13068*b**7*n + 5040*b**7) +
a*b**6*d**2*n**6*x**6*(a + b*x)**n/(b**7*n**7 + 28*b**7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**
3 + 13132*b**7*n**2 + 13068*b**7*n + 5040*b**7) + 15*a*b**6*d**2*n**5*x**6*(a + b*x)**n/(b**7*n**7 + 28*b**7*n
**6 + 322*b**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**3 + 13132*b**7*n**2 + 13068*b**7*n + 5040*b**7) + 85*a*b**
6*d**2*n**4*x**6*(a + b*x)**n/(b**7*n**7 + 28*b**7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**3 + 13
132*b**7*n**2 + 13068*b**7*n + 5040*b**7) + 225*a*b**6*d**2*n**3*x**6*(a + b*x)**n/(b**7*n**7 + 28*b**7*n**6 +
322*b**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**3 + 13132*b**7*n**2 + 13068*b**7*n + 5040*b**7) + 274*a*b**6*d*
*2*n**2*x**6*(a + b*x)**n/(b**7*n**7 + 28*b**7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**3 + 13132*
b**7*n**2 + 13068*b**7*n + 5040*b**7) + 120*a*b**6*d**2*n*x**6*(a + b*x)**n/(b**7*n**7 + 28*b**7*n**6 + 322*b*
*7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**3 + 13132*b**7*n**2 + 13068*b**7*n + 5040*b**7) + b**7*c**2*n**6*x**3*
(a + b*x)**n/(b**7*n**7 + 28*b**7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**3 + 13132*b**7*n**2 + 1
3068*b**7*n + 5040*b**7) + 25*b**7*c**2*n**5*x**3*(a + b*x)**n/(b**7*n**7 + 28*b**7*n**6 + 322*b**7*n**5 + 196
0*b**7*n**4 + 6769*b**7*n**3 + 13132*b**7*n**2 + 13068*b**7*n + 5040*b**7) + 247*b**7*c**2*n**4*x**3*(a + b*x)
**n/(b**7*n**7 + 28*b**7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**3 + 13132*b**7*n**2 + 13068*b**7
*n + 5040*b**7) + 1219*b**7*c**2*n**3*x**3*(a + b*x)**n/(b**7*n**7 + 28*b**7*n**6 + 322*b**7*n**5 + 1960*b**7*
n**4 + 6769*b**7*n**3 + 13132*b**7*n**2 + 13068*b**7*n + 5040*b**7) + 3112*b**7*c**2*n**2*x**3*(a + b*x)**n/(b
**7*n**7 + 28*b**7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**3 + 13132*b**7*n**2 + 13068*b**7*n + 5
040*b**7) + 3796*b**7*c**2*n*x**3*(a + b*x)**n/(b**7*n**7 + 28*b**7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 + 67
69*b**7*n**3 + 13132*b**7*n**2 + 13068*b**7*n + 5040*b**7) + 1680*b**7*c**2*x**3*(a + b*x)**n/(b**7*n**7 + 28*
b**7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**3 + 13132*b**7*n**2 + 13068*b**7*n + 5040*b**7) + 2*
b**7*c*d*n**6*x**5*(a + b*x)**n/(b**7*n**7 + 28*b**7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**3 +
13132*b**7*n**2 + 13068*b**7*n + 5040*b**7) + 46*b**7*c*d*n**5*x**5*(a + b*x)**n/(b**7*n**7 + 28*b**7*n**6 + 3
22*b**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**3 + 13132*b**7*n**2 + 13068*b**7*n + 5040*b**7) + 414*b**7*c*d*n*
*4*x**5*(a + b*x)**n/(b**7*n**7 + 28*b**7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**3 + 13132*b**7*
n**2 + 13068*b**7*n + 5040*b**7) + 1850*b**7*c*d*n**3*x**5*(a + b*x)**n/(b**7*n**7 + 28*b**7*n**6 + 322*b**7*n
**5 + 1960*b**7*n**4 + 6769*b**7*n**3 + 13132*b**7*n**2 + 13068*b**7*n + 5040*b**7) + 4288*b**7*c*d*n**2*x**5*
(a + b*x)**n/(b**7*n**7 + 28*b**7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**3 + 13132*b**7*n**2 + 1
3068*b**7*n + 5040*b**7) + 4824*b**7*c*d*n*x**5*(a + b*x)**n/(b**7*n**7 + 28*b**7*n**6 + 322*b**7*n**5 + 1960*
b**7*n**4 + 6769*b**7*n**3 + 13132*b**7*n**2 + 13068*b**7*n + 5040*b**7) + 2016*b**7*c*d*x**5*(a + b*x)**n/(b*
*7*n**7 + 28*b**7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**3 + 13132*b**7*n**2 + 13068*b**7*n + 50
40*b**7) + b**7*d**2*n**6*x**7*(a + b*x)**n/(b**7*n**7 + 28*b**7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 + 6769*
b**7*n**3 + 13132*b**7*n**2 + 13068*b**7*n + 5040*b**7) + 21*b**7*d**2*n**5*x**7*(a + b*x)**n/(b**7*n**7 + 28*
b**7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**3 + 13132*b**7*n**2 + 13068*b**7*n + 5040*b**7) + 17
5*b**7*d**2*n**4*x**7*(a + b*x)**n/(b**7*n**7 + 28*b**7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**3
+ 13132*b**7*n**2 + 13068*b**7*n + 5040*b**7) + 735*b**7*d**2*n**3*x**7*(a + b*x)**n/(b**7*n**7 + 28*b**7*n**
6 + 322*b**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**3 + 13132*b**7*n**2 + 13068*b**7*n + 5040*b**7) + 1624*b**7*
d**2*n**2*x**7*(a + b*x)**n/(b**7*n**7 + 28*b**7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**3 + 1313
2*b**7*n**2 + 13068*b**7*n + 5040*b**7) + 1764*b**7*d**2*n*x**7*(a + b*x)**n/(b**7*n**7 + 28*b**7*n**6 + 322*b
**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**3 + 13132*b**7*n**2 + 13068*b**7*n + 5040*b**7) + 720*b**7*d**2*x**7*
(a + b*x)**n/(b**7*n**7 + 28*b**7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**3 + 13132*b**7*n**2 + 1
3068*b**7*n + 5040*b**7), True))
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