3.356 \(\int x (a+b x)^n (c+d x^2)^2 \, dx\)
Optimal. Leaf size=185 \[ -\frac {a \left (a^2 d+b^2 c\right )^2 (a+b x)^{n+1}}{b^6 (n+1)}+\frac {\left (a^2 d+b^2 c\right ) \left (5 a^2 d+b^2 c\right ) (a+b x)^{n+2}}{b^6 (n+2)}-\frac {2 a d \left (5 a^2 d+3 b^2 c\right ) (a+b x)^{n+3}}{b^6 (n+3)}+\frac {2 d \left (5 a^2 d+b^2 c\right ) (a+b x)^{n+4}}{b^6 (n+4)}-\frac {5 a d^2 (a+b x)^{n+5}}{b^6 (n+5)}+\frac {d^2 (a+b x)^{n+6}}{b^6 (n+6)} \]
[Out]
-a*(a^2*d+b^2*c)^2*(b*x+a)^(1+n)/b^6/(1+n)+(a^2*d+b^2*c)*(5*a^2*d+b^2*c)*(b*x+a)^(2+n)/b^6/(2+n)-2*a*d*(5*a^2*
d+3*b^2*c)*(b*x+a)^(3+n)/b^6/(3+n)+2*d*(5*a^2*d+b^2*c)*(b*x+a)^(4+n)/b^6/(4+n)-5*a*d^2*(b*x+a)^(5+n)/b^6/(5+n)
+d^2*(b*x+a)^(6+n)/b^6/(6+n)
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Rubi [A] time = 0.10, antiderivative size = 185, normalized size of antiderivative = 1.00,
number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used =
{772} \[ -\frac {a \left (a^2 d+b^2 c\right )^2 (a+b x)^{n+1}}{b^6 (n+1)}+\frac {\left (a^2 d+b^2 c\right ) \left (5 a^2 d+b^2 c\right ) (a+b x)^{n+2}}{b^6 (n+2)}-\frac {2 a d \left (5 a^2 d+3 b^2 c\right ) (a+b x)^{n+3}}{b^6 (n+3)}+\frac {2 d \left (5 a^2 d+b^2 c\right ) (a+b x)^{n+4}}{b^6 (n+4)}-\frac {5 a d^2 (a+b x)^{n+5}}{b^6 (n+5)}+\frac {d^2 (a+b x)^{n+6}}{b^6 (n+6)} \]
Antiderivative was successfully verified.
[In]
Int[x*(a + b*x)^n*(c + d*x^2)^2,x]
[Out]
-((a*(b^2*c + a^2*d)^2*(a + b*x)^(1 + n))/(b^6*(1 + n))) + ((b^2*c + a^2*d)*(b^2*c + 5*a^2*d)*(a + b*x)^(2 + n
))/(b^6*(2 + n)) - (2*a*d*(3*b^2*c + 5*a^2*d)*(a + b*x)^(3 + n))/(b^6*(3 + n)) + (2*d*(b^2*c + 5*a^2*d)*(a + b
*x)^(4 + n))/(b^6*(4 + n)) - (5*a*d^2*(a + b*x)^(5 + n))/(b^6*(5 + n)) + (d^2*(a + b*x)^(6 + n))/(b^6*(6 + n))
Rule 772
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegr
and[(d + e*x)^m*(f + g*x)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && IGtQ[p, 0]
Rubi steps
\begin {align*} \int x (a+b x)^n \left (c+d x^2\right )^2 \, dx &=\int \left (-\frac {a \left (b^2 c+a^2 d\right )^2 (a+b x)^n}{b^5}+\frac {\left (b^2 c+a^2 d\right ) \left (b^2 c+5 a^2 d\right ) (a+b x)^{1+n}}{b^5}-\frac {2 a d \left (3 b^2 c+5 a^2 d\right ) (a+b x)^{2+n}}{b^5}+\frac {2 d \left (b^2 c+5 a^2 d\right ) (a+b x)^{3+n}}{b^5}-\frac {5 a d^2 (a+b x)^{4+n}}{b^5}+\frac {d^2 (a+b x)^{5+n}}{b^5}\right ) \, dx\\ &=-\frac {a \left (b^2 c+a^2 d\right )^2 (a+b x)^{1+n}}{b^6 (1+n)}+\frac {\left (b^2 c+a^2 d\right ) \left (b^2 c+5 a^2 d\right ) (a+b x)^{2+n}}{b^6 (2+n)}-\frac {2 a d \left (3 b^2 c+5 a^2 d\right ) (a+b x)^{3+n}}{b^6 (3+n)}+\frac {2 d \left (b^2 c+5 a^2 d\right ) (a+b x)^{4+n}}{b^6 (4+n)}-\frac {5 a d^2 (a+b x)^{5+n}}{b^6 (5+n)}+\frac {d^2 (a+b x)^{6+n}}{b^6 (6+n)}\\ \end {align*}
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Mathematica [A] time = 0.50, size = 323, normalized size = 1.75 \[ \frac {(a+b x)^{n+1} \left (4 (n+1) (a+b x) \left ((n+5) \left (a^2 d+b^2 c\right ) \left (2 a^2 d-2 a b d (n+2) x+b^2 (n+3) \left (c (n+4)+d (n+2) x^2\right )\right )-a d (n+2) (a+b x) \left (2 a^2 d-2 a b d (n+3) x+b^2 (n+4) \left (c (n+5)+d (n+3) x^2\right )\right )\right )-a (n+6) \left (4 (n+4) \left (a^2 d+b^2 c\right ) \left (2 a^2 d-2 a b d (n+1) x+b^2 (n+2) \left (c (n+3)+d (n+1) x^2\right )\right )-4 a d (n+1) (a+b x) \left (2 a^2 d-2 a b d (n+2) x+b^2 (n+3) \left (c (n+4)+d (n+2) x^2\right )\right )+b^4 (n+1) (n+2) (n+3) (n+4) \left (c+d x^2\right )^2\right )+b^4 (n+1) (n+2) (n+3) (n+4) (n+5) (a+b x) \left (c+d x^2\right )^2\right )}{b^6 (n+1) (n+2) (n+3) (n+4) (n+5) (n+6)} \]
Antiderivative was successfully verified.
[In]
Integrate[x*(a + b*x)^n*(c + d*x^2)^2,x]
[Out]
((a + b*x)^(1 + n)*(b^4*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(5 + n)*(a + b*x)*(c + d*x^2)^2 - a*(6 + n)*(b^4*(1 +
n)*(2 + n)*(3 + n)*(4 + n)*(c + d*x^2)^2 + 4*(b^2*c + a^2*d)*(4 + n)*(2*a^2*d - 2*a*b*d*(1 + n)*x + b^2*(2 + n
)*(c*(3 + n) + d*(1 + n)*x^2)) - 4*a*d*(1 + n)*(a + b*x)*(2*a^2*d - 2*a*b*d*(2 + n)*x + b^2*(3 + n)*(c*(4 + n)
+ d*(2 + n)*x^2))) + 4*(1 + n)*(a + b*x)*((b^2*c + a^2*d)*(5 + n)*(2*a^2*d - 2*a*b*d*(2 + n)*x + b^2*(3 + n)*
(c*(4 + n) + d*(2 + n)*x^2)) - a*d*(2 + n)*(a + b*x)*(2*a^2*d - 2*a*b*d*(3 + n)*x + b^2*(4 + n)*(c*(5 + n) + d
*(3 + n)*x^2)))))/(b^6*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(5 + n)*(6 + n))
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fricas [B] time = 0.96, size = 757, normalized size = 4.09 \[ -\frac {{\left (a^{2} b^{4} c^{2} n^{4} + 18 \, a^{2} b^{4} c^{2} n^{3} + 360 \, a^{2} b^{4} c^{2} + 360 \, a^{4} b^{2} c d + 120 \, a^{6} d^{2} - {\left (b^{6} d^{2} n^{5} + 15 \, b^{6} d^{2} n^{4} + 85 \, b^{6} d^{2} n^{3} + 225 \, b^{6} d^{2} n^{2} + 274 \, b^{6} d^{2} n + 120 \, b^{6} d^{2}\right )} x^{6} - {\left (a b^{5} d^{2} n^{5} + 10 \, a b^{5} d^{2} n^{4} + 35 \, a b^{5} d^{2} n^{3} + 50 \, a b^{5} d^{2} n^{2} + 24 \, a b^{5} d^{2} n\right )} x^{5} - {\left (2 \, b^{6} c d n^{5} + 360 \, b^{6} c d + {\left (34 \, b^{6} c d - 5 \, a^{2} b^{4} d^{2}\right )} n^{4} + 2 \, {\left (107 \, b^{6} c d - 15 \, a^{2} b^{4} d^{2}\right )} n^{3} + {\left (614 \, b^{6} c d - 55 \, a^{2} b^{4} d^{2}\right )} n^{2} + 6 \, {\left (132 \, b^{6} c d - 5 \, a^{2} b^{4} d^{2}\right )} n\right )} x^{4} - 2 \, {\left (a b^{5} c d n^{5} + 14 \, a b^{5} c d n^{4} + 5 \, {\left (13 \, a b^{5} c d + 2 \, a^{3} b^{3} d^{2}\right )} n^{3} + 2 \, {\left (56 \, a b^{5} c d + 15 \, a^{3} b^{3} d^{2}\right )} n^{2} + 20 \, {\left (3 \, a b^{5} c d + a^{3} b^{3} d^{2}\right )} n\right )} x^{3} + {\left (119 \, a^{2} b^{4} c^{2} + 12 \, a^{4} b^{2} c d\right )} n^{2} - {\left (b^{6} c^{2} n^{5} + 360 \, b^{6} c^{2} + {\left (19 \, b^{6} c^{2} - 6 \, a^{2} b^{4} c d\right )} n^{4} + {\left (137 \, b^{6} c^{2} - 72 \, a^{2} b^{4} c d\right )} n^{3} + {\left (461 \, b^{6} c^{2} - 246 \, a^{2} b^{4} c d - 60 \, a^{4} b^{2} d^{2}\right )} n^{2} + 6 \, {\left (117 \, b^{6} c^{2} - 30 \, a^{2} b^{4} c d - 10 \, a^{4} b^{2} d^{2}\right )} n\right )} x^{2} + 6 \, {\left (57 \, a^{2} b^{4} c^{2} + 22 \, a^{4} b^{2} c d\right )} n - {\left (a b^{5} c^{2} n^{5} + 18 \, a b^{5} c^{2} n^{4} + {\left (119 \, a b^{5} c^{2} + 12 \, a^{3} b^{3} c d\right )} n^{3} + 6 \, {\left (57 \, a b^{5} c^{2} + 22 \, a^{3} b^{3} c d\right )} n^{2} + 120 \, {\left (3 \, a b^{5} c^{2} + 3 \, a^{3} b^{3} c d + a^{5} b d^{2}\right )} n\right )} x\right )} {\left (b x + a\right )}^{n}}{b^{6} n^{6} + 21 \, b^{6} n^{5} + 175 \, b^{6} n^{4} + 735 \, b^{6} n^{3} + 1624 \, b^{6} n^{2} + 1764 \, b^{6} n + 720 \, b^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x*(b*x+a)^n*(d*x^2+c)^2,x, algorithm="fricas")
[Out]
-(a^2*b^4*c^2*n^4 + 18*a^2*b^4*c^2*n^3 + 360*a^2*b^4*c^2 + 360*a^4*b^2*c*d + 120*a^6*d^2 - (b^6*d^2*n^5 + 15*b
^6*d^2*n^4 + 85*b^6*d^2*n^3 + 225*b^6*d^2*n^2 + 274*b^6*d^2*n + 120*b^6*d^2)*x^6 - (a*b^5*d^2*n^5 + 10*a*b^5*d
^2*n^4 + 35*a*b^5*d^2*n^3 + 50*a*b^5*d^2*n^2 + 24*a*b^5*d^2*n)*x^5 - (2*b^6*c*d*n^5 + 360*b^6*c*d + (34*b^6*c*
d - 5*a^2*b^4*d^2)*n^4 + 2*(107*b^6*c*d - 15*a^2*b^4*d^2)*n^3 + (614*b^6*c*d - 55*a^2*b^4*d^2)*n^2 + 6*(132*b^
6*c*d - 5*a^2*b^4*d^2)*n)*x^4 - 2*(a*b^5*c*d*n^5 + 14*a*b^5*c*d*n^4 + 5*(13*a*b^5*c*d + 2*a^3*b^3*d^2)*n^3 + 2
*(56*a*b^5*c*d + 15*a^3*b^3*d^2)*n^2 + 20*(3*a*b^5*c*d + a^3*b^3*d^2)*n)*x^3 + (119*a^2*b^4*c^2 + 12*a^4*b^2*c
*d)*n^2 - (b^6*c^2*n^5 + 360*b^6*c^2 + (19*b^6*c^2 - 6*a^2*b^4*c*d)*n^4 + (137*b^6*c^2 - 72*a^2*b^4*c*d)*n^3 +
(461*b^6*c^2 - 246*a^2*b^4*c*d - 60*a^4*b^2*d^2)*n^2 + 6*(117*b^6*c^2 - 30*a^2*b^4*c*d - 10*a^4*b^2*d^2)*n)*x
^2 + 6*(57*a^2*b^4*c^2 + 22*a^4*b^2*c*d)*n - (a*b^5*c^2*n^5 + 18*a*b^5*c^2*n^4 + (119*a*b^5*c^2 + 12*a^3*b^3*c
*d)*n^3 + 6*(57*a*b^5*c^2 + 22*a^3*b^3*c*d)*n^2 + 120*(3*a*b^5*c^2 + 3*a^3*b^3*c*d + a^5*b*d^2)*n)*x)*(b*x + a
)^n/(b^6*n^6 + 21*b^6*n^5 + 175*b^6*n^4 + 735*b^6*n^3 + 1624*b^6*n^2 + 1764*b^6*n + 720*b^6)
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giac [B] time = 0.22, size = 1266, normalized size = 6.84 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x*(b*x+a)^n*(d*x^2+c)^2,x, algorithm="giac")
[Out]
((b*x + a)^n*b^6*d^2*n^5*x^6 + (b*x + a)^n*a*b^5*d^2*n^5*x^5 + 15*(b*x + a)^n*b^6*d^2*n^4*x^6 + 2*(b*x + a)^n*
b^6*c*d*n^5*x^4 + 10*(b*x + a)^n*a*b^5*d^2*n^4*x^5 + 85*(b*x + a)^n*b^6*d^2*n^3*x^6 + 2*(b*x + a)^n*a*b^5*c*d*
n^5*x^3 + 34*(b*x + a)^n*b^6*c*d*n^4*x^4 - 5*(b*x + a)^n*a^2*b^4*d^2*n^4*x^4 + 35*(b*x + a)^n*a*b^5*d^2*n^3*x^
5 + 225*(b*x + a)^n*b^6*d^2*n^2*x^6 + (b*x + a)^n*b^6*c^2*n^5*x^2 + 28*(b*x + a)^n*a*b^5*c*d*n^4*x^3 + 214*(b*
x + a)^n*b^6*c*d*n^3*x^4 - 30*(b*x + a)^n*a^2*b^4*d^2*n^3*x^4 + 50*(b*x + a)^n*a*b^5*d^2*n^2*x^5 + 274*(b*x +
a)^n*b^6*d^2*n*x^6 + (b*x + a)^n*a*b^5*c^2*n^5*x + 19*(b*x + a)^n*b^6*c^2*n^4*x^2 - 6*(b*x + a)^n*a^2*b^4*c*d*
n^4*x^2 + 130*(b*x + a)^n*a*b^5*c*d*n^3*x^3 + 20*(b*x + a)^n*a^3*b^3*d^2*n^3*x^3 + 614*(b*x + a)^n*b^6*c*d*n^2
*x^4 - 55*(b*x + a)^n*a^2*b^4*d^2*n^2*x^4 + 24*(b*x + a)^n*a*b^5*d^2*n*x^5 + 120*(b*x + a)^n*b^6*d^2*x^6 + 18*
(b*x + a)^n*a*b^5*c^2*n^4*x + 137*(b*x + a)^n*b^6*c^2*n^3*x^2 - 72*(b*x + a)^n*a^2*b^4*c*d*n^3*x^2 + 224*(b*x
+ a)^n*a*b^5*c*d*n^2*x^3 + 60*(b*x + a)^n*a^3*b^3*d^2*n^2*x^3 + 792*(b*x + a)^n*b^6*c*d*n*x^4 - 30*(b*x + a)^n
*a^2*b^4*d^2*n*x^4 - (b*x + a)^n*a^2*b^4*c^2*n^4 + 119*(b*x + a)^n*a*b^5*c^2*n^3*x + 12*(b*x + a)^n*a^3*b^3*c*
d*n^3*x + 461*(b*x + a)^n*b^6*c^2*n^2*x^2 - 246*(b*x + a)^n*a^2*b^4*c*d*n^2*x^2 - 60*(b*x + a)^n*a^4*b^2*d^2*n
^2*x^2 + 120*(b*x + a)^n*a*b^5*c*d*n*x^3 + 40*(b*x + a)^n*a^3*b^3*d^2*n*x^3 + 360*(b*x + a)^n*b^6*c*d*x^4 - 18
*(b*x + a)^n*a^2*b^4*c^2*n^3 + 342*(b*x + a)^n*a*b^5*c^2*n^2*x + 132*(b*x + a)^n*a^3*b^3*c*d*n^2*x + 702*(b*x
+ a)^n*b^6*c^2*n*x^2 - 180*(b*x + a)^n*a^2*b^4*c*d*n*x^2 - 60*(b*x + a)^n*a^4*b^2*d^2*n*x^2 - 119*(b*x + a)^n*
a^2*b^4*c^2*n^2 - 12*(b*x + a)^n*a^4*b^2*c*d*n^2 + 360*(b*x + a)^n*a*b^5*c^2*n*x + 360*(b*x + a)^n*a^3*b^3*c*d
*n*x + 120*(b*x + a)^n*a^5*b*d^2*n*x + 360*(b*x + a)^n*b^6*c^2*x^2 - 342*(b*x + a)^n*a^2*b^4*c^2*n - 132*(b*x
+ a)^n*a^4*b^2*c*d*n - 360*(b*x + a)^n*a^2*b^4*c^2 - 360*(b*x + a)^n*a^4*b^2*c*d - 120*(b*x + a)^n*a^6*d^2)/(b
^6*n^6 + 21*b^6*n^5 + 175*b^6*n^4 + 735*b^6*n^3 + 1624*b^6*n^2 + 1764*b^6*n + 720*b^6)
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maple [B] time = 0.01, size = 677, normalized size = 3.66 \[ -\frac {\left (-b^{5} d^{2} n^{5} x^{5}-15 b^{5} d^{2} n^{4} x^{5}+5 a \,b^{4} d^{2} n^{4} x^{4}-2 b^{5} c d \,n^{5} x^{3}-85 b^{5} d^{2} n^{3} x^{5}+50 a \,b^{4} d^{2} n^{3} x^{4}-34 b^{5} c d \,n^{4} x^{3}-225 b^{5} d^{2} n^{2} x^{5}-20 a^{2} b^{3} d^{2} n^{3} x^{3}+6 a \,b^{4} c d \,n^{4} x^{2}+175 a \,b^{4} d^{2} n^{2} x^{4}-b^{5} c^{2} n^{5} x -214 b^{5} c d \,n^{3} x^{3}-274 b^{5} d^{2} n \,x^{5}-120 a^{2} b^{3} d^{2} n^{2} x^{3}+84 a \,b^{4} c d \,n^{3} x^{2}+250 a \,b^{4} d^{2} n \,x^{4}-19 b^{5} c^{2} n^{4} x -614 b^{5} c d \,n^{2} x^{3}-120 d^{2} x^{5} b^{5}+60 a^{3} b^{2} d^{2} n^{2} x^{2}-12 a^{2} b^{3} c d \,n^{3} x -220 a^{2} b^{3} d^{2} n \,x^{3}+a \,b^{4} c^{2} n^{4}+390 a \,b^{4} c d \,n^{2} x^{2}+120 a \,d^{2} x^{4} b^{4}-137 b^{5} c^{2} n^{3} x -792 b^{5} c d n \,x^{3}+180 a^{3} b^{2} d^{2} n \,x^{2}-144 a^{2} b^{3} c d \,n^{2} x -120 a^{2} b^{3} d^{2} x^{3}+18 a \,b^{4} c^{2} n^{3}+672 a \,b^{4} c d n \,x^{2}-461 b^{5} c^{2} n^{2} x -360 b^{5} c d \,x^{3}-120 a^{4} b \,d^{2} n x +12 a^{3} b^{2} c d \,n^{2}+120 a^{3} b^{2} d^{2} x^{2}-492 a^{2} b^{3} c d n x +119 a \,b^{4} c^{2} n^{2}+360 a \,b^{4} c d \,x^{2}-702 b^{5} c^{2} n x -120 a^{4} b \,d^{2} x +132 a^{3} b^{2} c d n -360 a^{2} b^{3} c d x +342 a \,b^{4} c^{2} n -360 b^{5} c^{2} x +120 a^{5} d^{2}+360 a^{3} b^{2} c d +360 a \,b^{4} c^{2}\right ) \left (b x +a \right )^{n +1}}{\left (n^{6}+21 n^{5}+175 n^{4}+735 n^{3}+1624 n^{2}+1764 n +720\right ) b^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x*(b*x+a)^n*(d*x^2+c)^2,x)
[Out]
-(b*x+a)^(n+1)*(-b^5*d^2*n^5*x^5-15*b^5*d^2*n^4*x^5+5*a*b^4*d^2*n^4*x^4-2*b^5*c*d*n^5*x^3-85*b^5*d^2*n^3*x^5+5
0*a*b^4*d^2*n^3*x^4-34*b^5*c*d*n^4*x^3-225*b^5*d^2*n^2*x^5-20*a^2*b^3*d^2*n^3*x^3+6*a*b^4*c*d*n^4*x^2+175*a*b^
4*d^2*n^2*x^4-b^5*c^2*n^5*x-214*b^5*c*d*n^3*x^3-274*b^5*d^2*n*x^5-120*a^2*b^3*d^2*n^2*x^3+84*a*b^4*c*d*n^3*x^2
+250*a*b^4*d^2*n*x^4-19*b^5*c^2*n^4*x-614*b^5*c*d*n^2*x^3-120*b^5*d^2*x^5+60*a^3*b^2*d^2*n^2*x^2-12*a^2*b^3*c*
d*n^3*x-220*a^2*b^3*d^2*n*x^3+a*b^4*c^2*n^4+390*a*b^4*c*d*n^2*x^2+120*a*b^4*d^2*x^4-137*b^5*c^2*n^3*x-792*b^5*
c*d*n*x^3+180*a^3*b^2*d^2*n*x^2-144*a^2*b^3*c*d*n^2*x-120*a^2*b^3*d^2*x^3+18*a*b^4*c^2*n^3+672*a*b^4*c*d*n*x^2
-461*b^5*c^2*n^2*x-360*b^5*c*d*x^3-120*a^4*b*d^2*n*x+12*a^3*b^2*c*d*n^2+120*a^3*b^2*d^2*x^2-492*a^2*b^3*c*d*n*
x+119*a*b^4*c^2*n^2+360*a*b^4*c*d*x^2-702*b^5*c^2*n*x-120*a^4*b*d^2*x+132*a^3*b^2*c*d*n-360*a^2*b^3*c*d*x+342*
a*b^4*c^2*n-360*b^5*c^2*x+120*a^5*d^2+360*a^3*b^2*c*d+360*a*b^4*c^2)/b^6/(n^6+21*n^5+175*n^4+735*n^3+1624*n^2+
1764*n+720)
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maxima [A] time = 0.50, size = 335, normalized size = 1.81 \[ \frac {{\left (b^{2} {\left (n + 1\right )} x^{2} + a b n x - a^{2}\right )} {\left (b x + a\right )}^{n} c^{2}}{{\left (n^{2} + 3 \, n + 2\right )} b^{2}} + \frac {2 \, {\left ({\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} b^{4} x^{4} + {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a b^{3} x^{3} - 3 \, {\left (n^{2} + n\right )} a^{2} b^{2} x^{2} + 6 \, a^{3} b n x - 6 \, a^{4}\right )} {\left (b x + a\right )}^{n} c d}{{\left (n^{4} + 10 \, n^{3} + 35 \, n^{2} + 50 \, n + 24\right )} b^{4}} + \frac {{\left ({\left (n^{5} + 15 \, n^{4} + 85 \, n^{3} + 225 \, n^{2} + 274 \, n + 120\right )} b^{6} x^{6} + {\left (n^{5} + 10 \, n^{4} + 35 \, n^{3} + 50 \, n^{2} + 24 \, n\right )} a b^{5} x^{5} - 5 \, {\left (n^{4} + 6 \, n^{3} + 11 \, n^{2} + 6 \, n\right )} a^{2} b^{4} x^{4} + 20 \, {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a^{3} b^{3} x^{3} - 60 \, {\left (n^{2} + n\right )} a^{4} b^{2} x^{2} + 120 \, a^{5} b n x - 120 \, a^{6}\right )} {\left (b x + a\right )}^{n} d^{2}}{{\left (n^{6} + 21 \, n^{5} + 175 \, n^{4} + 735 \, n^{3} + 1624 \, n^{2} + 1764 \, n + 720\right )} b^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x*(b*x+a)^n*(d*x^2+c)^2,x, algorithm="maxima")
[Out]
(b^2*(n + 1)*x^2 + a*b*n*x - a^2)*(b*x + a)^n*c^2/((n^2 + 3*n + 2)*b^2) + 2*((n^3 + 6*n^2 + 11*n + 6)*b^4*x^4
+ (n^3 + 3*n^2 + 2*n)*a*b^3*x^3 - 3*(n^2 + n)*a^2*b^2*x^2 + 6*a^3*b*n*x - 6*a^4)*(b*x + a)^n*c*d/((n^4 + 10*n^
3 + 35*n^2 + 50*n + 24)*b^4) + ((n^5 + 15*n^4 + 85*n^3 + 225*n^2 + 274*n + 120)*b^6*x^6 + (n^5 + 10*n^4 + 35*n
^3 + 50*n^2 + 24*n)*a*b^5*x^5 - 5*(n^4 + 6*n^3 + 11*n^2 + 6*n)*a^2*b^4*x^4 + 20*(n^3 + 3*n^2 + 2*n)*a^3*b^3*x^
3 - 60*(n^2 + n)*a^4*b^2*x^2 + 120*a^5*b*n*x - 120*a^6)*(b*x + a)^n*d^2/((n^6 + 21*n^5 + 175*n^4 + 735*n^3 + 1
624*n^2 + 1764*n + 720)*b^6)
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mupad [B] time = 3.05, size = 723, normalized size = 3.91 \[ \frac {d^2\,x^6\,{\left (a+b\,x\right )}^n\,\left (n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120\right )}{n^6+21\,n^5+175\,n^4+735\,n^3+1624\,n^2+1764\,n+720}-\frac {a^2\,{\left (a+b\,x\right )}^n\,\left (120\,a^4\,d^2+12\,a^2\,b^2\,c\,d\,n^2+132\,a^2\,b^2\,c\,d\,n+360\,a^2\,b^2\,c\,d+b^4\,c^2\,n^4+18\,b^4\,c^2\,n^3+119\,b^4\,c^2\,n^2+342\,b^4\,c^2\,n+360\,b^4\,c^2\right )}{b^6\,\left (n^6+21\,n^5+175\,n^4+735\,n^3+1624\,n^2+1764\,n+720\right )}+\frac {x^2\,\left (n+1\right )\,{\left (a+b\,x\right )}^n\,\left (-60\,a^4\,d^2\,n-6\,a^2\,b^2\,c\,d\,n^3-66\,a^2\,b^2\,c\,d\,n^2-180\,a^2\,b^2\,c\,d\,n+b^4\,c^2\,n^4+18\,b^4\,c^2\,n^3+119\,b^4\,c^2\,n^2+342\,b^4\,c^2\,n+360\,b^4\,c^2\right )}{b^4\,\left (n^6+21\,n^5+175\,n^4+735\,n^3+1624\,n^2+1764\,n+720\right )}+\frac {d\,x^4\,{\left (a+b\,x\right )}^n\,\left (-5\,d\,a^2\,n+2\,c\,b^2\,n^2+22\,c\,b^2\,n+60\,c\,b^2\right )\,\left (n^3+6\,n^2+11\,n+6\right )}{b^2\,\left (n^6+21\,n^5+175\,n^4+735\,n^3+1624\,n^2+1764\,n+720\right )}+\frac {a\,n\,x\,{\left (a+b\,x\right )}^n\,\left (120\,a^4\,d^2+12\,a^2\,b^2\,c\,d\,n^2+132\,a^2\,b^2\,c\,d\,n+360\,a^2\,b^2\,c\,d+b^4\,c^2\,n^4+18\,b^4\,c^2\,n^3+119\,b^4\,c^2\,n^2+342\,b^4\,c^2\,n+360\,b^4\,c^2\right )}{b^5\,\left (n^6+21\,n^5+175\,n^4+735\,n^3+1624\,n^2+1764\,n+720\right )}+\frac {a\,d^2\,n\,x^5\,{\left (a+b\,x\right )}^n\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}{b\,\left (n^6+21\,n^5+175\,n^4+735\,n^3+1624\,n^2+1764\,n+720\right )}+\frac {2\,a\,d\,n\,x^3\,{\left (a+b\,x\right )}^n\,\left (n^2+3\,n+2\right )\,\left (10\,d\,a^2+c\,b^2\,n^2+11\,c\,b^2\,n+30\,c\,b^2\right )}{b^3\,\left (n^6+21\,n^5+175\,n^4+735\,n^3+1624\,n^2+1764\,n+720\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x*(c + d*x^2)^2*(a + b*x)^n,x)
[Out]
(d^2*x^6*(a + b*x)^n*(274*n + 225*n^2 + 85*n^3 + 15*n^4 + n^5 + 120))/(1764*n + 1624*n^2 + 735*n^3 + 175*n^4 +
21*n^5 + n^6 + 720) - (a^2*(a + b*x)^n*(120*a^4*d^2 + 360*b^4*c^2 + 342*b^4*c^2*n + 119*b^4*c^2*n^2 + 18*b^4*
c^2*n^3 + b^4*c^2*n^4 + 360*a^2*b^2*c*d + 132*a^2*b^2*c*d*n + 12*a^2*b^2*c*d*n^2))/(b^6*(1764*n + 1624*n^2 + 7
35*n^3 + 175*n^4 + 21*n^5 + n^6 + 720)) + (x^2*(n + 1)*(a + b*x)^n*(360*b^4*c^2 - 60*a^4*d^2*n + 342*b^4*c^2*n
+ 119*b^4*c^2*n^2 + 18*b^4*c^2*n^3 + b^4*c^2*n^4 - 180*a^2*b^2*c*d*n - 66*a^2*b^2*c*d*n^2 - 6*a^2*b^2*c*d*n^3
))/(b^4*(1764*n + 1624*n^2 + 735*n^3 + 175*n^4 + 21*n^5 + n^6 + 720)) + (d*x^4*(a + b*x)^n*(60*b^2*c + 2*b^2*c
*n^2 - 5*a^2*d*n + 22*b^2*c*n)*(11*n + 6*n^2 + n^3 + 6))/(b^2*(1764*n + 1624*n^2 + 735*n^3 + 175*n^4 + 21*n^5
+ n^6 + 720)) + (a*n*x*(a + b*x)^n*(120*a^4*d^2 + 360*b^4*c^2 + 342*b^4*c^2*n + 119*b^4*c^2*n^2 + 18*b^4*c^2*n
^3 + b^4*c^2*n^4 + 360*a^2*b^2*c*d + 132*a^2*b^2*c*d*n + 12*a^2*b^2*c*d*n^2))/(b^5*(1764*n + 1624*n^2 + 735*n^
3 + 175*n^4 + 21*n^5 + n^6 + 720)) + (a*d^2*n*x^5*(a + b*x)^n*(50*n + 35*n^2 + 10*n^3 + n^4 + 24))/(b*(1764*n
+ 1624*n^2 + 735*n^3 + 175*n^4 + 21*n^5 + n^6 + 720)) + (2*a*d*n*x^3*(a + b*x)^n*(3*n + n^2 + 2)*(10*a^2*d + 3
0*b^2*c + b^2*c*n^2 + 11*b^2*c*n))/(b^3*(1764*n + 1624*n^2 + 735*n^3 + 175*n^4 + 21*n^5 + n^6 + 720))
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sympy [A] time = 13.80, size = 8940, normalized size = 48.32 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x*(b*x+a)**n*(d*x**2+c)**2,x)
[Out]
Piecewise((a**n*(c**2*x**2/2 + c*d*x**4/2 + d**2*x**6/6), Eq(b, 0)), (60*a**5*d**2*log(a/b + x)/(60*a**5*b**6
+ 300*a**4*b**7*x + 600*a**3*b**8*x**2 + 600*a**2*b**9*x**3 + 300*a*b**10*x**4 + 60*b**11*x**5) + 137*a**5*d**
2/(60*a**5*b**6 + 300*a**4*b**7*x + 600*a**3*b**8*x**2 + 600*a**2*b**9*x**3 + 300*a*b**10*x**4 + 60*b**11*x**5
) + 300*a**4*b*d**2*x*log(a/b + x)/(60*a**5*b**6 + 300*a**4*b**7*x + 600*a**3*b**8*x**2 + 600*a**2*b**9*x**3 +
300*a*b**10*x**4 + 60*b**11*x**5) + 625*a**4*b*d**2*x/(60*a**5*b**6 + 300*a**4*b**7*x + 600*a**3*b**8*x**2 +
600*a**2*b**9*x**3 + 300*a*b**10*x**4 + 60*b**11*x**5) - 6*a**3*b**2*c*d/(60*a**5*b**6 + 300*a**4*b**7*x + 600
*a**3*b**8*x**2 + 600*a**2*b**9*x**3 + 300*a*b**10*x**4 + 60*b**11*x**5) + 600*a**3*b**2*d**2*x**2*log(a/b + x
)/(60*a**5*b**6 + 300*a**4*b**7*x + 600*a**3*b**8*x**2 + 600*a**2*b**9*x**3 + 300*a*b**10*x**4 + 60*b**11*x**5
) + 1100*a**3*b**2*d**2*x**2/(60*a**5*b**6 + 300*a**4*b**7*x + 600*a**3*b**8*x**2 + 600*a**2*b**9*x**3 + 300*a
*b**10*x**4 + 60*b**11*x**5) - 30*a**2*b**3*c*d*x/(60*a**5*b**6 + 300*a**4*b**7*x + 600*a**3*b**8*x**2 + 600*a
**2*b**9*x**3 + 300*a*b**10*x**4 + 60*b**11*x**5) + 600*a**2*b**3*d**2*x**3*log(a/b + x)/(60*a**5*b**6 + 300*a
**4*b**7*x + 600*a**3*b**8*x**2 + 600*a**2*b**9*x**3 + 300*a*b**10*x**4 + 60*b**11*x**5) + 900*a**2*b**3*d**2*
x**3/(60*a**5*b**6 + 300*a**4*b**7*x + 600*a**3*b**8*x**2 + 600*a**2*b**9*x**3 + 300*a*b**10*x**4 + 60*b**11*x
**5) - 3*a*b**4*c**2/(60*a**5*b**6 + 300*a**4*b**7*x + 600*a**3*b**8*x**2 + 600*a**2*b**9*x**3 + 300*a*b**10*x
**4 + 60*b**11*x**5) - 60*a*b**4*c*d*x**2/(60*a**5*b**6 + 300*a**4*b**7*x + 600*a**3*b**8*x**2 + 600*a**2*b**9
*x**3 + 300*a*b**10*x**4 + 60*b**11*x**5) + 300*a*b**4*d**2*x**4*log(a/b + x)/(60*a**5*b**6 + 300*a**4*b**7*x
+ 600*a**3*b**8*x**2 + 600*a**2*b**9*x**3 + 300*a*b**10*x**4 + 60*b**11*x**5) + 300*a*b**4*d**2*x**4/(60*a**5*
b**6 + 300*a**4*b**7*x + 600*a**3*b**8*x**2 + 600*a**2*b**9*x**3 + 300*a*b**10*x**4 + 60*b**11*x**5) - 15*b**5
*c**2*x/(60*a**5*b**6 + 300*a**4*b**7*x + 600*a**3*b**8*x**2 + 600*a**2*b**9*x**3 + 300*a*b**10*x**4 + 60*b**1
1*x**5) - 60*b**5*c*d*x**3/(60*a**5*b**6 + 300*a**4*b**7*x + 600*a**3*b**8*x**2 + 600*a**2*b**9*x**3 + 300*a*b
**10*x**4 + 60*b**11*x**5) + 60*b**5*d**2*x**5*log(a/b + x)/(60*a**5*b**6 + 300*a**4*b**7*x + 600*a**3*b**8*x*
*2 + 600*a**2*b**9*x**3 + 300*a*b**10*x**4 + 60*b**11*x**5), Eq(n, -6)), (-60*a**5*d**2*log(a/b + x)/(12*a**4*
b**6 + 48*a**3*b**7*x + 72*a**2*b**8*x**2 + 48*a*b**9*x**3 + 12*b**10*x**4) - 125*a**5*d**2/(12*a**4*b**6 + 48
*a**3*b**7*x + 72*a**2*b**8*x**2 + 48*a*b**9*x**3 + 12*b**10*x**4) - 240*a**4*b*d**2*x*log(a/b + x)/(12*a**4*b
**6 + 48*a**3*b**7*x + 72*a**2*b**8*x**2 + 48*a*b**9*x**3 + 12*b**10*x**4) - 440*a**4*b*d**2*x/(12*a**4*b**6 +
48*a**3*b**7*x + 72*a**2*b**8*x**2 + 48*a*b**9*x**3 + 12*b**10*x**4) - 6*a**3*b**2*c*d/(12*a**4*b**6 + 48*a**
3*b**7*x + 72*a**2*b**8*x**2 + 48*a*b**9*x**3 + 12*b**10*x**4) - 360*a**3*b**2*d**2*x**2*log(a/b + x)/(12*a**4
*b**6 + 48*a**3*b**7*x + 72*a**2*b**8*x**2 + 48*a*b**9*x**3 + 12*b**10*x**4) - 540*a**3*b**2*d**2*x**2/(12*a**
4*b**6 + 48*a**3*b**7*x + 72*a**2*b**8*x**2 + 48*a*b**9*x**3 + 12*b**10*x**4) - 24*a**2*b**3*c*d*x/(12*a**4*b*
*6 + 48*a**3*b**7*x + 72*a**2*b**8*x**2 + 48*a*b**9*x**3 + 12*b**10*x**4) - 240*a**2*b**3*d**2*x**3*log(a/b +
x)/(12*a**4*b**6 + 48*a**3*b**7*x + 72*a**2*b**8*x**2 + 48*a*b**9*x**3 + 12*b**10*x**4) - 240*a**2*b**3*d**2*x
**3/(12*a**4*b**6 + 48*a**3*b**7*x + 72*a**2*b**8*x**2 + 48*a*b**9*x**3 + 12*b**10*x**4) - a*b**4*c**2/(12*a**
4*b**6 + 48*a**3*b**7*x + 72*a**2*b**8*x**2 + 48*a*b**9*x**3 + 12*b**10*x**4) - 36*a*b**4*c*d*x**2/(12*a**4*b*
*6 + 48*a**3*b**7*x + 72*a**2*b**8*x**2 + 48*a*b**9*x**3 + 12*b**10*x**4) - 60*a*b**4*d**2*x**4*log(a/b + x)/(
12*a**4*b**6 + 48*a**3*b**7*x + 72*a**2*b**8*x**2 + 48*a*b**9*x**3 + 12*b**10*x**4) - 4*b**5*c**2*x/(12*a**4*b
**6 + 48*a**3*b**7*x + 72*a**2*b**8*x**2 + 48*a*b**9*x**3 + 12*b**10*x**4) - 24*b**5*c*d*x**3/(12*a**4*b**6 +
48*a**3*b**7*x + 72*a**2*b**8*x**2 + 48*a*b**9*x**3 + 12*b**10*x**4) + 12*b**5*d**2*x**5/(12*a**4*b**6 + 48*a*
*3*b**7*x + 72*a**2*b**8*x**2 + 48*a*b**9*x**3 + 12*b**10*x**4), Eq(n, -5)), (60*a**5*d**2*log(a/b + x)/(6*a**
3*b**6 + 18*a**2*b**7*x + 18*a*b**8*x**2 + 6*b**9*x**3) + 110*a**5*d**2/(6*a**3*b**6 + 18*a**2*b**7*x + 18*a*b
**8*x**2 + 6*b**9*x**3) + 180*a**4*b*d**2*x*log(a/b + x)/(6*a**3*b**6 + 18*a**2*b**7*x + 18*a*b**8*x**2 + 6*b*
*9*x**3) + 270*a**4*b*d**2*x/(6*a**3*b**6 + 18*a**2*b**7*x + 18*a*b**8*x**2 + 6*b**9*x**3) + 12*a**3*b**2*c*d*
log(a/b + x)/(6*a**3*b**6 + 18*a**2*b**7*x + 18*a*b**8*x**2 + 6*b**9*x**3) + 22*a**3*b**2*c*d/(6*a**3*b**6 + 1
8*a**2*b**7*x + 18*a*b**8*x**2 + 6*b**9*x**3) + 180*a**3*b**2*d**2*x**2*log(a/b + x)/(6*a**3*b**6 + 18*a**2*b*
*7*x + 18*a*b**8*x**2 + 6*b**9*x**3) + 180*a**3*b**2*d**2*x**2/(6*a**3*b**6 + 18*a**2*b**7*x + 18*a*b**8*x**2
+ 6*b**9*x**3) + 36*a**2*b**3*c*d*x*log(a/b + x)/(6*a**3*b**6 + 18*a**2*b**7*x + 18*a*b**8*x**2 + 6*b**9*x**3)
+ 54*a**2*b**3*c*d*x/(6*a**3*b**6 + 18*a**2*b**7*x + 18*a*b**8*x**2 + 6*b**9*x**3) + 60*a**2*b**3*d**2*x**3*l
og(a/b + x)/(6*a**3*b**6 + 18*a**2*b**7*x + 18*a*b**8*x**2 + 6*b**9*x**3) - a*b**4*c**2/(6*a**3*b**6 + 18*a**2
*b**7*x + 18*a*b**8*x**2 + 6*b**9*x**3) + 36*a*b**4*c*d*x**2*log(a/b + x)/(6*a**3*b**6 + 18*a**2*b**7*x + 18*a
*b**8*x**2 + 6*b**9*x**3) + 36*a*b**4*c*d*x**2/(6*a**3*b**6 + 18*a**2*b**7*x + 18*a*b**8*x**2 + 6*b**9*x**3) -
15*a*b**4*d**2*x**4/(6*a**3*b**6 + 18*a**2*b**7*x + 18*a*b**8*x**2 + 6*b**9*x**3) - 3*b**5*c**2*x/(6*a**3*b**
6 + 18*a**2*b**7*x + 18*a*b**8*x**2 + 6*b**9*x**3) + 12*b**5*c*d*x**3*log(a/b + x)/(6*a**3*b**6 + 18*a**2*b**7
*x + 18*a*b**8*x**2 + 6*b**9*x**3) + 3*b**5*d**2*x**5/(6*a**3*b**6 + 18*a**2*b**7*x + 18*a*b**8*x**2 + 6*b**9*
x**3), Eq(n, -4)), (-60*a**5*d**2*log(a/b + x)/(6*a**2*b**6 + 12*a*b**7*x + 6*b**8*x**2) - 90*a**5*d**2/(6*a**
2*b**6 + 12*a*b**7*x + 6*b**8*x**2) - 120*a**4*b*d**2*x*log(a/b + x)/(6*a**2*b**6 + 12*a*b**7*x + 6*b**8*x**2)
- 120*a**4*b*d**2*x/(6*a**2*b**6 + 12*a*b**7*x + 6*b**8*x**2) - 36*a**3*b**2*c*d*log(a/b + x)/(6*a**2*b**6 +
12*a*b**7*x + 6*b**8*x**2) - 54*a**3*b**2*c*d/(6*a**2*b**6 + 12*a*b**7*x + 6*b**8*x**2) - 60*a**3*b**2*d**2*x*
*2*log(a/b + x)/(6*a**2*b**6 + 12*a*b**7*x + 6*b**8*x**2) - 72*a**2*b**3*c*d*x*log(a/b + x)/(6*a**2*b**6 + 12*
a*b**7*x + 6*b**8*x**2) - 72*a**2*b**3*c*d*x/(6*a**2*b**6 + 12*a*b**7*x + 6*b**8*x**2) + 20*a**2*b**3*d**2*x**
3/(6*a**2*b**6 + 12*a*b**7*x + 6*b**8*x**2) - 3*a*b**4*c**2/(6*a**2*b**6 + 12*a*b**7*x + 6*b**8*x**2) - 36*a*b
**4*c*d*x**2*log(a/b + x)/(6*a**2*b**6 + 12*a*b**7*x + 6*b**8*x**2) - 5*a*b**4*d**2*x**4/(6*a**2*b**6 + 12*a*b
**7*x + 6*b**8*x**2) - 6*b**5*c**2*x/(6*a**2*b**6 + 12*a*b**7*x + 6*b**8*x**2) + 12*b**5*c*d*x**3/(6*a**2*b**6
+ 12*a*b**7*x + 6*b**8*x**2) + 2*b**5*d**2*x**5/(6*a**2*b**6 + 12*a*b**7*x + 6*b**8*x**2), Eq(n, -3)), (60*a*
*5*d**2*log(a/b + x)/(12*a*b**6 + 12*b**7*x) + 60*a**5*d**2/(12*a*b**6 + 12*b**7*x) + 60*a**4*b*d**2*x*log(a/b
+ x)/(12*a*b**6 + 12*b**7*x) + 72*a**3*b**2*c*d*log(a/b + x)/(12*a*b**6 + 12*b**7*x) + 72*a**3*b**2*c*d/(12*a
*b**6 + 12*b**7*x) - 30*a**3*b**2*d**2*x**2/(12*a*b**6 + 12*b**7*x) + 72*a**2*b**3*c*d*x*log(a/b + x)/(12*a*b*
*6 + 12*b**7*x) + 10*a**2*b**3*d**2*x**3/(12*a*b**6 + 12*b**7*x) + 12*a*b**4*c**2*log(a/b + x)/(12*a*b**6 + 12
*b**7*x) + 12*a*b**4*c**2/(12*a*b**6 + 12*b**7*x) - 36*a*b**4*c*d*x**2/(12*a*b**6 + 12*b**7*x) - 5*a*b**4*d**2
*x**4/(12*a*b**6 + 12*b**7*x) + 12*b**5*c**2*x*log(a/b + x)/(12*a*b**6 + 12*b**7*x) + 12*b**5*c*d*x**3/(12*a*b
**6 + 12*b**7*x) + 3*b**5*d**2*x**5/(12*a*b**6 + 12*b**7*x), Eq(n, -2)), (-a**5*d**2*log(a/b + x)/b**6 + a**4*
d**2*x/b**5 - 2*a**3*c*d*log(a/b + x)/b**4 - a**3*d**2*x**2/(2*b**4) + 2*a**2*c*d*x/b**3 + a**2*d**2*x**3/(3*b
**3) - a*c**2*log(a/b + x)/b**2 - a*c*d*x**2/b**2 - a*d**2*x**4/(4*b**2) + c**2*x/b + 2*c*d*x**3/(3*b) + d**2*
x**5/(5*b), Eq(n, -1)), (-120*a**6*d**2*(a + b*x)**n/(b**6*n**6 + 21*b**6*n**5 + 175*b**6*n**4 + 735*b**6*n**3
+ 1624*b**6*n**2 + 1764*b**6*n + 720*b**6) + 120*a**5*b*d**2*n*x*(a + b*x)**n/(b**6*n**6 + 21*b**6*n**5 + 175
*b**6*n**4 + 735*b**6*n**3 + 1624*b**6*n**2 + 1764*b**6*n + 720*b**6) - 12*a**4*b**2*c*d*n**2*(a + b*x)**n/(b*
*6*n**6 + 21*b**6*n**5 + 175*b**6*n**4 + 735*b**6*n**3 + 1624*b**6*n**2 + 1764*b**6*n + 720*b**6) - 132*a**4*b
**2*c*d*n*(a + b*x)**n/(b**6*n**6 + 21*b**6*n**5 + 175*b**6*n**4 + 735*b**6*n**3 + 1624*b**6*n**2 + 1764*b**6*
n + 720*b**6) - 360*a**4*b**2*c*d*(a + b*x)**n/(b**6*n**6 + 21*b**6*n**5 + 175*b**6*n**4 + 735*b**6*n**3 + 162
4*b**6*n**2 + 1764*b**6*n + 720*b**6) - 60*a**4*b**2*d**2*n**2*x**2*(a + b*x)**n/(b**6*n**6 + 21*b**6*n**5 + 1
75*b**6*n**4 + 735*b**6*n**3 + 1624*b**6*n**2 + 1764*b**6*n + 720*b**6) - 60*a**4*b**2*d**2*n*x**2*(a + b*x)**
n/(b**6*n**6 + 21*b**6*n**5 + 175*b**6*n**4 + 735*b**6*n**3 + 1624*b**6*n**2 + 1764*b**6*n + 720*b**6) + 12*a*
*3*b**3*c*d*n**3*x*(a + b*x)**n/(b**6*n**6 + 21*b**6*n**5 + 175*b**6*n**4 + 735*b**6*n**3 + 1624*b**6*n**2 + 1
764*b**6*n + 720*b**6) + 132*a**3*b**3*c*d*n**2*x*(a + b*x)**n/(b**6*n**6 + 21*b**6*n**5 + 175*b**6*n**4 + 735
*b**6*n**3 + 1624*b**6*n**2 + 1764*b**6*n + 720*b**6) + 360*a**3*b**3*c*d*n*x*(a + b*x)**n/(b**6*n**6 + 21*b**
6*n**5 + 175*b**6*n**4 + 735*b**6*n**3 + 1624*b**6*n**2 + 1764*b**6*n + 720*b**6) + 20*a**3*b**3*d**2*n**3*x**
3*(a + b*x)**n/(b**6*n**6 + 21*b**6*n**5 + 175*b**6*n**4 + 735*b**6*n**3 + 1624*b**6*n**2 + 1764*b**6*n + 720*
b**6) + 60*a**3*b**3*d**2*n**2*x**3*(a + b*x)**n/(b**6*n**6 + 21*b**6*n**5 + 175*b**6*n**4 + 735*b**6*n**3 + 1
624*b**6*n**2 + 1764*b**6*n + 720*b**6) + 40*a**3*b**3*d**2*n*x**3*(a + b*x)**n/(b**6*n**6 + 21*b**6*n**5 + 17
5*b**6*n**4 + 735*b**6*n**3 + 1624*b**6*n**2 + 1764*b**6*n + 720*b**6) - a**2*b**4*c**2*n**4*(a + b*x)**n/(b**
6*n**6 + 21*b**6*n**5 + 175*b**6*n**4 + 735*b**6*n**3 + 1624*b**6*n**2 + 1764*b**6*n + 720*b**6) - 18*a**2*b**
4*c**2*n**3*(a + b*x)**n/(b**6*n**6 + 21*b**6*n**5 + 175*b**6*n**4 + 735*b**6*n**3 + 1624*b**6*n**2 + 1764*b**
6*n + 720*b**6) - 119*a**2*b**4*c**2*n**2*(a + b*x)**n/(b**6*n**6 + 21*b**6*n**5 + 175*b**6*n**4 + 735*b**6*n*
*3 + 1624*b**6*n**2 + 1764*b**6*n + 720*b**6) - 342*a**2*b**4*c**2*n*(a + b*x)**n/(b**6*n**6 + 21*b**6*n**5 +
175*b**6*n**4 + 735*b**6*n**3 + 1624*b**6*n**2 + 1764*b**6*n + 720*b**6) - 360*a**2*b**4*c**2*(a + b*x)**n/(b*
*6*n**6 + 21*b**6*n**5 + 175*b**6*n**4 + 735*b**6*n**3 + 1624*b**6*n**2 + 1764*b**6*n + 720*b**6) - 6*a**2*b**
4*c*d*n**4*x**2*(a + b*x)**n/(b**6*n**6 + 21*b**6*n**5 + 175*b**6*n**4 + 735*b**6*n**3 + 1624*b**6*n**2 + 1764
*b**6*n + 720*b**6) - 72*a**2*b**4*c*d*n**3*x**2*(a + b*x)**n/(b**6*n**6 + 21*b**6*n**5 + 175*b**6*n**4 + 735*
b**6*n**3 + 1624*b**6*n**2 + 1764*b**6*n + 720*b**6) - 246*a**2*b**4*c*d*n**2*x**2*(a + b*x)**n/(b**6*n**6 + 2
1*b**6*n**5 + 175*b**6*n**4 + 735*b**6*n**3 + 1624*b**6*n**2 + 1764*b**6*n + 720*b**6) - 180*a**2*b**4*c*d*n*x
**2*(a + b*x)**n/(b**6*n**6 + 21*b**6*n**5 + 175*b**6*n**4 + 735*b**6*n**3 + 1624*b**6*n**2 + 1764*b**6*n + 72
0*b**6) - 5*a**2*b**4*d**2*n**4*x**4*(a + b*x)**n/(b**6*n**6 + 21*b**6*n**5 + 175*b**6*n**4 + 735*b**6*n**3 +
1624*b**6*n**2 + 1764*b**6*n + 720*b**6) - 30*a**2*b**4*d**2*n**3*x**4*(a + b*x)**n/(b**6*n**6 + 21*b**6*n**5
+ 175*b**6*n**4 + 735*b**6*n**3 + 1624*b**6*n**2 + 1764*b**6*n + 720*b**6) - 55*a**2*b**4*d**2*n**2*x**4*(a +
b*x)**n/(b**6*n**6 + 21*b**6*n**5 + 175*b**6*n**4 + 735*b**6*n**3 + 1624*b**6*n**2 + 1764*b**6*n + 720*b**6) -
30*a**2*b**4*d**2*n*x**4*(a + b*x)**n/(b**6*n**6 + 21*b**6*n**5 + 175*b**6*n**4 + 735*b**6*n**3 + 1624*b**6*n
**2 + 1764*b**6*n + 720*b**6) + a*b**5*c**2*n**5*x*(a + b*x)**n/(b**6*n**6 + 21*b**6*n**5 + 175*b**6*n**4 + 73
5*b**6*n**3 + 1624*b**6*n**2 + 1764*b**6*n + 720*b**6) + 18*a*b**5*c**2*n**4*x*(a + b*x)**n/(b**6*n**6 + 21*b*
*6*n**5 + 175*b**6*n**4 + 735*b**6*n**3 + 1624*b**6*n**2 + 1764*b**6*n + 720*b**6) + 119*a*b**5*c**2*n**3*x*(a
+ b*x)**n/(b**6*n**6 + 21*b**6*n**5 + 175*b**6*n**4 + 735*b**6*n**3 + 1624*b**6*n**2 + 1764*b**6*n + 720*b**6
) + 342*a*b**5*c**2*n**2*x*(a + b*x)**n/(b**6*n**6 + 21*b**6*n**5 + 175*b**6*n**4 + 735*b**6*n**3 + 1624*b**6*
n**2 + 1764*b**6*n + 720*b**6) + 360*a*b**5*c**2*n*x*(a + b*x)**n/(b**6*n**6 + 21*b**6*n**5 + 175*b**6*n**4 +
735*b**6*n**3 + 1624*b**6*n**2 + 1764*b**6*n + 720*b**6) + 2*a*b**5*c*d*n**5*x**3*(a + b*x)**n/(b**6*n**6 + 21
*b**6*n**5 + 175*b**6*n**4 + 735*b**6*n**3 + 1624*b**6*n**2 + 1764*b**6*n + 720*b**6) + 28*a*b**5*c*d*n**4*x**
3*(a + b*x)**n/(b**6*n**6 + 21*b**6*n**5 + 175*b**6*n**4 + 735*b**6*n**3 + 1624*b**6*n**2 + 1764*b**6*n + 720*
b**6) + 130*a*b**5*c*d*n**3*x**3*(a + b*x)**n/(b**6*n**6 + 21*b**6*n**5 + 175*b**6*n**4 + 735*b**6*n**3 + 1624
*b**6*n**2 + 1764*b**6*n + 720*b**6) + 224*a*b**5*c*d*n**2*x**3*(a + b*x)**n/(b**6*n**6 + 21*b**6*n**5 + 175*b
**6*n**4 + 735*b**6*n**3 + 1624*b**6*n**2 + 1764*b**6*n + 720*b**6) + 120*a*b**5*c*d*n*x**3*(a + b*x)**n/(b**6
*n**6 + 21*b**6*n**5 + 175*b**6*n**4 + 735*b**6*n**3 + 1624*b**6*n**2 + 1764*b**6*n + 720*b**6) + a*b**5*d**2*
n**5*x**5*(a + b*x)**n/(b**6*n**6 + 21*b**6*n**5 + 175*b**6*n**4 + 735*b**6*n**3 + 1624*b**6*n**2 + 1764*b**6*
n + 720*b**6) + 10*a*b**5*d**2*n**4*x**5*(a + b*x)**n/(b**6*n**6 + 21*b**6*n**5 + 175*b**6*n**4 + 735*b**6*n**
3 + 1624*b**6*n**2 + 1764*b**6*n + 720*b**6) + 35*a*b**5*d**2*n**3*x**5*(a + b*x)**n/(b**6*n**6 + 21*b**6*n**5
+ 175*b**6*n**4 + 735*b**6*n**3 + 1624*b**6*n**2 + 1764*b**6*n + 720*b**6) + 50*a*b**5*d**2*n**2*x**5*(a + b*
x)**n/(b**6*n**6 + 21*b**6*n**5 + 175*b**6*n**4 + 735*b**6*n**3 + 1624*b**6*n**2 + 1764*b**6*n + 720*b**6) + 2
4*a*b**5*d**2*n*x**5*(a + b*x)**n/(b**6*n**6 + 21*b**6*n**5 + 175*b**6*n**4 + 735*b**6*n**3 + 1624*b**6*n**2 +
1764*b**6*n + 720*b**6) + b**6*c**2*n**5*x**2*(a + b*x)**n/(b**6*n**6 + 21*b**6*n**5 + 175*b**6*n**4 + 735*b*
*6*n**3 + 1624*b**6*n**2 + 1764*b**6*n + 720*b**6) + 19*b**6*c**2*n**4*x**2*(a + b*x)**n/(b**6*n**6 + 21*b**6*
n**5 + 175*b**6*n**4 + 735*b**6*n**3 + 1624*b**6*n**2 + 1764*b**6*n + 720*b**6) + 137*b**6*c**2*n**3*x**2*(a +
b*x)**n/(b**6*n**6 + 21*b**6*n**5 + 175*b**6*n**4 + 735*b**6*n**3 + 1624*b**6*n**2 + 1764*b**6*n + 720*b**6)
+ 461*b**6*c**2*n**2*x**2*(a + b*x)**n/(b**6*n**6 + 21*b**6*n**5 + 175*b**6*n**4 + 735*b**6*n**3 + 1624*b**6*n
**2 + 1764*b**6*n + 720*b**6) + 702*b**6*c**2*n*x**2*(a + b*x)**n/(b**6*n**6 + 21*b**6*n**5 + 175*b**6*n**4 +
735*b**6*n**3 + 1624*b**6*n**2 + 1764*b**6*n + 720*b**6) + 360*b**6*c**2*x**2*(a + b*x)**n/(b**6*n**6 + 21*b**
6*n**5 + 175*b**6*n**4 + 735*b**6*n**3 + 1624*b**6*n**2 + 1764*b**6*n + 720*b**6) + 2*b**6*c*d*n**5*x**4*(a +
b*x)**n/(b**6*n**6 + 21*b**6*n**5 + 175*b**6*n**4 + 735*b**6*n**3 + 1624*b**6*n**2 + 1764*b**6*n + 720*b**6) +
34*b**6*c*d*n**4*x**4*(a + b*x)**n/(b**6*n**6 + 21*b**6*n**5 + 175*b**6*n**4 + 735*b**6*n**3 + 1624*b**6*n**2
+ 1764*b**6*n + 720*b**6) + 214*b**6*c*d*n**3*x**4*(a + b*x)**n/(b**6*n**6 + 21*b**6*n**5 + 175*b**6*n**4 + 7
35*b**6*n**3 + 1624*b**6*n**2 + 1764*b**6*n + 720*b**6) + 614*b**6*c*d*n**2*x**4*(a + b*x)**n/(b**6*n**6 + 21*
b**6*n**5 + 175*b**6*n**4 + 735*b**6*n**3 + 1624*b**6*n**2 + 1764*b**6*n + 720*b**6) + 792*b**6*c*d*n*x**4*(a
+ b*x)**n/(b**6*n**6 + 21*b**6*n**5 + 175*b**6*n**4 + 735*b**6*n**3 + 1624*b**6*n**2 + 1764*b**6*n + 720*b**6)
+ 360*b**6*c*d*x**4*(a + b*x)**n/(b**6*n**6 + 21*b**6*n**5 + 175*b**6*n**4 + 735*b**6*n**3 + 1624*b**6*n**2 +
1764*b**6*n + 720*b**6) + b**6*d**2*n**5*x**6*(a + b*x)**n/(b**6*n**6 + 21*b**6*n**5 + 175*b**6*n**4 + 735*b*
*6*n**3 + 1624*b**6*n**2 + 1764*b**6*n + 720*b**6) + 15*b**6*d**2*n**4*x**6*(a + b*x)**n/(b**6*n**6 + 21*b**6*
n**5 + 175*b**6*n**4 + 735*b**6*n**3 + 1624*b**6*n**2 + 1764*b**6*n + 720*b**6) + 85*b**6*d**2*n**3*x**6*(a +
b*x)**n/(b**6*n**6 + 21*b**6*n**5 + 175*b**6*n**4 + 735*b**6*n**3 + 1624*b**6*n**2 + 1764*b**6*n + 720*b**6) +
225*b**6*d**2*n**2*x**6*(a + b*x)**n/(b**6*n**6 + 21*b**6*n**5 + 175*b**6*n**4 + 735*b**6*n**3 + 1624*b**6*n*
*2 + 1764*b**6*n + 720*b**6) + 274*b**6*d**2*n*x**6*(a + b*x)**n/(b**6*n**6 + 21*b**6*n**5 + 175*b**6*n**4 + 7
35*b**6*n**3 + 1624*b**6*n**2 + 1764*b**6*n + 720*b**6) + 120*b**6*d**2*x**6*(a + b*x)**n/(b**6*n**6 + 21*b**6
*n**5 + 175*b**6*n**4 + 735*b**6*n**3 + 1624*b**6*n**2 + 1764*b**6*n + 720*b**6), True))
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