3.9.61 \(\int x^2 (a+b x)^n (c+d x)^2 \, dx\)
Optimal. Leaf size=157 \[ \frac {a^2 (b c-a d)^2 (a+b x)^{n+1}}{b^5 (n+1)}+\frac {\left (6 a^2 d^2-6 a b c d+b^2 c^2\right ) (a+b x)^{n+3}}{b^5 (n+3)}-\frac {2 a (b c-2 a d) (b c-a d) (a+b x)^{n+2}}{b^5 (n+2)}+\frac {2 d (b c-2 a d) (a+b x)^{n+4}}{b^5 (n+4)}+\frac {d^2 (a+b x)^{n+5}}{b^5 (n+5)} \]
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Rubi [A] time = 0.09, antiderivative size = 157, 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 =
{88} \begin {gather*} \frac {\left (6 a^2 d^2-6 a b c d+b^2 c^2\right ) (a+b x)^{n+3}}{b^5 (n+3)}+\frac {a^2 (b c-a d)^2 (a+b x)^{n+1}}{b^5 (n+1)}-\frac {2 a (b c-2 a d) (b c-a d) (a+b x)^{n+2}}{b^5 (n+2)}+\frac {2 d (b c-2 a d) (a+b x)^{n+4}}{b^5 (n+4)}+\frac {d^2 (a+b x)^{n+5}}{b^5 (n+5)} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[x^2*(a + b*x)^n*(c + d*x)^2,x]
[Out]
(a^2*(b*c - a*d)^2*(a + b*x)^(1 + n))/(b^5*(1 + n)) - (2*a*(b*c - 2*a*d)*(b*c - a*d)*(a + b*x)^(2 + n))/(b^5*(
2 + n)) + ((b^2*c^2 - 6*a*b*c*d + 6*a^2*d^2)*(a + b*x)^(3 + n))/(b^5*(3 + n)) + (2*d*(b*c - 2*a*d)*(a + b*x)^(
4 + n))/(b^5*(4 + n)) + (d^2*(a + b*x)^(5 + n))/(b^5*(5 + n))
Rule 88
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))
Rubi steps
\begin {align*} \int x^2 (a+b x)^n (c+d x)^2 \, dx &=\int \left (\frac {a^2 (-b c+a d)^2 (a+b x)^n}{b^4}+\frac {2 a (b c-2 a d) (-b c+a d) (a+b x)^{1+n}}{b^4}+\frac {\left (b^2 c^2-6 a b c d+6 a^2 d^2\right ) (a+b x)^{2+n}}{b^4}+\frac {2 d (b c-2 a d) (a+b x)^{3+n}}{b^4}+\frac {d^2 (a+b x)^{4+n}}{b^4}\right ) \, dx\\ &=\frac {a^2 (b c-a d)^2 (a+b x)^{1+n}}{b^5 (1+n)}-\frac {2 a (b c-2 a d) (b c-a d) (a+b x)^{2+n}}{b^5 (2+n)}+\frac {\left (b^2 c^2-6 a b c d+6 a^2 d^2\right ) (a+b x)^{3+n}}{b^5 (3+n)}+\frac {2 d (b c-2 a d) (a+b x)^{4+n}}{b^5 (4+n)}+\frac {d^2 (a+b x)^{5+n}}{b^5 (5+n)}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 136, normalized size = 0.87 \begin {gather*} \frac {(a+b x)^{n+1} \left (\frac {(a+b x)^2 \left (6 a^2 d^2-6 a b c d+b^2 c^2\right )}{n+3}+\frac {a^2 (b c-a d)^2}{n+1}+\frac {2 d (a+b x)^3 (b c-2 a d)}{n+4}-\frac {2 a (a+b x) (b c-2 a d) (b c-a d)}{n+2}+\frac {d^2 (a+b x)^4}{n+5}\right )}{b^5} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[x^2*(a + b*x)^n*(c + d*x)^2,x]
[Out]
((a + b*x)^(1 + n)*((a^2*(b*c - a*d)^2)/(1 + n) - (2*a*(b*c - 2*a*d)*(b*c - a*d)*(a + b*x))/(2 + n) + ((b^2*c^
2 - 6*a*b*c*d + 6*a^2*d^2)*(a + b*x)^2)/(3 + n) + (2*d*(b*c - 2*a*d)*(a + b*x)^3)/(4 + n) + (d^2*(a + b*x)^4)/
(5 + n)))/b^5
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IntegrateAlgebraic [F] time = 0.06, size = 0, normalized size = 0.00 \begin {gather*} \int x^2 (a+b x)^n (c+d x)^2 \, dx \end {gather*}
Verification is not applicable to the result.
[In]
IntegrateAlgebraic[x^2*(a + b*x)^n*(c + d*x)^2,x]
[Out]
Defer[IntegrateAlgebraic][x^2*(a + b*x)^n*(c + d*x)^2, x]
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fricas [B] time = 1.24, size = 584, normalized size = 3.72 \begin {gather*} \frac {{\left (2 \, a^{3} b^{2} c^{2} n^{2} + 40 \, a^{3} b^{2} c^{2} - 60 \, a^{4} b c d + 24 \, a^{5} d^{2} + {\left (b^{5} d^{2} n^{4} + 10 \, b^{5} d^{2} n^{3} + 35 \, b^{5} d^{2} n^{2} + 50 \, b^{5} d^{2} n + 24 \, b^{5} d^{2}\right )} x^{5} + {\left (60 \, b^{5} c d + {\left (2 \, b^{5} c d + a b^{4} d^{2}\right )} n^{4} + 2 \, {\left (11 \, b^{5} c d + 3 \, a b^{4} d^{2}\right )} n^{3} + {\left (82 \, b^{5} c d + 11 \, a b^{4} d^{2}\right )} n^{2} + 2 \, {\left (61 \, b^{5} c d + 3 \, a b^{4} d^{2}\right )} n\right )} x^{4} + {\left (40 \, b^{5} c^{2} + {\left (b^{5} c^{2} + 2 \, a b^{4} c d\right )} n^{4} + 4 \, {\left (3 \, b^{5} c^{2} + 4 \, a b^{4} c d - a^{2} b^{3} d^{2}\right )} n^{3} + {\left (49 \, b^{5} c^{2} + 34 \, a b^{4} c d - 12 \, a^{2} b^{3} d^{2}\right )} n^{2} + 2 \, {\left (39 \, b^{5} c^{2} + 10 \, a b^{4} c d - 4 \, a^{2} b^{3} d^{2}\right )} n\right )} x^{3} + {\left (a b^{4} c^{2} n^{4} + 2 \, {\left (5 \, a b^{4} c^{2} - 3 \, a^{2} b^{3} c d\right )} n^{3} + {\left (29 \, a b^{4} c^{2} - 36 \, a^{2} b^{3} c d + 12 \, a^{3} b^{2} d^{2}\right )} n^{2} + 2 \, {\left (10 \, a b^{4} c^{2} - 15 \, a^{2} b^{3} c d + 6 \, a^{3} b^{2} d^{2}\right )} n\right )} x^{2} + 6 \, {\left (3 \, a^{3} b^{2} c^{2} - 2 \, a^{4} b c d\right )} n - 2 \, {\left (a^{2} b^{3} c^{2} n^{3} + 3 \, {\left (3 \, a^{2} b^{3} c^{2} - 2 \, a^{3} b^{2} c d\right )} n^{2} + 2 \, {\left (10 \, a^{2} b^{3} c^{2} - 15 \, a^{3} b^{2} c d + 6 \, a^{4} b d^{2}\right )} n\right )} x\right )} {\left (b x + a\right )}^{n}}{b^{5} n^{5} + 15 \, b^{5} n^{4} + 85 \, b^{5} n^{3} + 225 \, b^{5} n^{2} + 274 \, b^{5} n + 120 \, b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^2*(b*x+a)^n*(d*x+c)^2,x, algorithm="fricas")
[Out]
(2*a^3*b^2*c^2*n^2 + 40*a^3*b^2*c^2 - 60*a^4*b*c*d + 24*a^5*d^2 + (b^5*d^2*n^4 + 10*b^5*d^2*n^3 + 35*b^5*d^2*n
^2 + 50*b^5*d^2*n + 24*b^5*d^2)*x^5 + (60*b^5*c*d + (2*b^5*c*d + a*b^4*d^2)*n^4 + 2*(11*b^5*c*d + 3*a*b^4*d^2)
*n^3 + (82*b^5*c*d + 11*a*b^4*d^2)*n^2 + 2*(61*b^5*c*d + 3*a*b^4*d^2)*n)*x^4 + (40*b^5*c^2 + (b^5*c^2 + 2*a*b^
4*c*d)*n^4 + 4*(3*b^5*c^2 + 4*a*b^4*c*d - a^2*b^3*d^2)*n^3 + (49*b^5*c^2 + 34*a*b^4*c*d - 12*a^2*b^3*d^2)*n^2
+ 2*(39*b^5*c^2 + 10*a*b^4*c*d - 4*a^2*b^3*d^2)*n)*x^3 + (a*b^4*c^2*n^4 + 2*(5*a*b^4*c^2 - 3*a^2*b^3*c*d)*n^3
+ (29*a*b^4*c^2 - 36*a^2*b^3*c*d + 12*a^3*b^2*d^2)*n^2 + 2*(10*a*b^4*c^2 - 15*a^2*b^3*c*d + 6*a^3*b^2*d^2)*n)*
x^2 + 6*(3*a^3*b^2*c^2 - 2*a^4*b*c*d)*n - 2*(a^2*b^3*c^2*n^3 + 3*(3*a^2*b^3*c^2 - 2*a^3*b^2*c*d)*n^2 + 2*(10*a
^2*b^3*c^2 - 15*a^3*b^2*c*d + 6*a^4*b*d^2)*n)*x)*(b*x + a)^n/(b^5*n^5 + 15*b^5*n^4 + 85*b^5*n^3 + 225*b^5*n^2
+ 274*b^5*n + 120*b^5)
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giac [B] time = 1.23, size = 1001, normalized size = 6.38 \begin {gather*} \frac {{\left (b x + a\right )}^{n} b^{5} d^{2} n^{4} x^{5} + 2 \, {\left (b x + a\right )}^{n} b^{5} c d n^{4} x^{4} + {\left (b x + a\right )}^{n} a b^{4} d^{2} n^{4} x^{4} + 10 \, {\left (b x + a\right )}^{n} b^{5} d^{2} n^{3} x^{5} + {\left (b x + a\right )}^{n} b^{5} c^{2} n^{4} x^{3} + 2 \, {\left (b x + a\right )}^{n} a b^{4} c d n^{4} x^{3} + 22 \, {\left (b x + a\right )}^{n} b^{5} c d n^{3} x^{4} + 6 \, {\left (b x + a\right )}^{n} a b^{4} d^{2} n^{3} x^{4} + 35 \, {\left (b x + a\right )}^{n} b^{5} d^{2} n^{2} x^{5} + {\left (b x + a\right )}^{n} a b^{4} c^{2} n^{4} x^{2} + 12 \, {\left (b x + a\right )}^{n} b^{5} c^{2} n^{3} x^{3} + 16 \, {\left (b x + a\right )}^{n} a b^{4} c d n^{3} x^{3} - 4 \, {\left (b x + a\right )}^{n} a^{2} b^{3} d^{2} n^{3} x^{3} + 82 \, {\left (b x + a\right )}^{n} b^{5} c d n^{2} x^{4} + 11 \, {\left (b x + a\right )}^{n} a b^{4} d^{2} n^{2} x^{4} + 50 \, {\left (b x + a\right )}^{n} b^{5} d^{2} n x^{5} + 10 \, {\left (b x + a\right )}^{n} a b^{4} c^{2} n^{3} x^{2} - 6 \, {\left (b x + a\right )}^{n} a^{2} b^{3} c d n^{3} x^{2} + 49 \, {\left (b x + a\right )}^{n} b^{5} c^{2} n^{2} x^{3} + 34 \, {\left (b x + a\right )}^{n} a b^{4} c d n^{2} x^{3} - 12 \, {\left (b x + a\right )}^{n} a^{2} b^{3} d^{2} n^{2} x^{3} + 122 \, {\left (b x + a\right )}^{n} b^{5} c d n x^{4} + 6 \, {\left (b x + a\right )}^{n} a b^{4} d^{2} n x^{4} + 24 \, {\left (b x + a\right )}^{n} b^{5} d^{2} x^{5} - 2 \, {\left (b x + a\right )}^{n} a^{2} b^{3} c^{2} n^{3} x + 29 \, {\left (b x + a\right )}^{n} a b^{4} c^{2} n^{2} x^{2} - 36 \, {\left (b x + a\right )}^{n} a^{2} b^{3} c d n^{2} x^{2} + 12 \, {\left (b x + a\right )}^{n} a^{3} b^{2} d^{2} n^{2} x^{2} + 78 \, {\left (b x + a\right )}^{n} b^{5} c^{2} n x^{3} + 20 \, {\left (b x + a\right )}^{n} a b^{4} c d n x^{3} - 8 \, {\left (b x + a\right )}^{n} a^{2} b^{3} d^{2} n x^{3} + 60 \, {\left (b x + a\right )}^{n} b^{5} c d x^{4} - 18 \, {\left (b x + a\right )}^{n} a^{2} b^{3} c^{2} n^{2} x + 12 \, {\left (b x + a\right )}^{n} a^{3} b^{2} c d n^{2} x + 20 \, {\left (b x + a\right )}^{n} a b^{4} c^{2} n x^{2} - 30 \, {\left (b x + a\right )}^{n} a^{2} b^{3} c d n x^{2} + 12 \, {\left (b x + a\right )}^{n} a^{3} b^{2} d^{2} n x^{2} + 40 \, {\left (b x + a\right )}^{n} b^{5} c^{2} x^{3} + 2 \, {\left (b x + a\right )}^{n} a^{3} b^{2} c^{2} n^{2} - 40 \, {\left (b x + a\right )}^{n} a^{2} b^{3} c^{2} n x + 60 \, {\left (b x + a\right )}^{n} a^{3} b^{2} c d n x - 24 \, {\left (b x + a\right )}^{n} a^{4} b d^{2} n x + 18 \, {\left (b x + a\right )}^{n} a^{3} b^{2} c^{2} n - 12 \, {\left (b x + a\right )}^{n} a^{4} b c d n + 40 \, {\left (b x + a\right )}^{n} a^{3} b^{2} c^{2} - 60 \, {\left (b x + a\right )}^{n} a^{4} b c d + 24 \, {\left (b x + a\right )}^{n} a^{5} d^{2}}{b^{5} n^{5} + 15 \, b^{5} n^{4} + 85 \, b^{5} n^{3} + 225 \, b^{5} n^{2} + 274 \, b^{5} n + 120 \, b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^2*(b*x+a)^n*(d*x+c)^2,x, algorithm="giac")
[Out]
((b*x + a)^n*b^5*d^2*n^4*x^5 + 2*(b*x + a)^n*b^5*c*d*n^4*x^4 + (b*x + a)^n*a*b^4*d^2*n^4*x^4 + 10*(b*x + a)^n*
b^5*d^2*n^3*x^5 + (b*x + a)^n*b^5*c^2*n^4*x^3 + 2*(b*x + a)^n*a*b^4*c*d*n^4*x^3 + 22*(b*x + a)^n*b^5*c*d*n^3*x
^4 + 6*(b*x + a)^n*a*b^4*d^2*n^3*x^4 + 35*(b*x + a)^n*b^5*d^2*n^2*x^5 + (b*x + a)^n*a*b^4*c^2*n^4*x^2 + 12*(b*
x + a)^n*b^5*c^2*n^3*x^3 + 16*(b*x + a)^n*a*b^4*c*d*n^3*x^3 - 4*(b*x + a)^n*a^2*b^3*d^2*n^3*x^3 + 82*(b*x + a)
^n*b^5*c*d*n^2*x^4 + 11*(b*x + a)^n*a*b^4*d^2*n^2*x^4 + 50*(b*x + a)^n*b^5*d^2*n*x^5 + 10*(b*x + a)^n*a*b^4*c^
2*n^3*x^2 - 6*(b*x + a)^n*a^2*b^3*c*d*n^3*x^2 + 49*(b*x + a)^n*b^5*c^2*n^2*x^3 + 34*(b*x + a)^n*a*b^4*c*d*n^2*
x^3 - 12*(b*x + a)^n*a^2*b^3*d^2*n^2*x^3 + 122*(b*x + a)^n*b^5*c*d*n*x^4 + 6*(b*x + a)^n*a*b^4*d^2*n*x^4 + 24*
(b*x + a)^n*b^5*d^2*x^5 - 2*(b*x + a)^n*a^2*b^3*c^2*n^3*x + 29*(b*x + a)^n*a*b^4*c^2*n^2*x^2 - 36*(b*x + a)^n*
a^2*b^3*c*d*n^2*x^2 + 12*(b*x + a)^n*a^3*b^2*d^2*n^2*x^2 + 78*(b*x + a)^n*b^5*c^2*n*x^3 + 20*(b*x + a)^n*a*b^4
*c*d*n*x^3 - 8*(b*x + a)^n*a^2*b^3*d^2*n*x^3 + 60*(b*x + a)^n*b^5*c*d*x^4 - 18*(b*x + a)^n*a^2*b^3*c^2*n^2*x +
12*(b*x + a)^n*a^3*b^2*c*d*n^2*x + 20*(b*x + a)^n*a*b^4*c^2*n*x^2 - 30*(b*x + a)^n*a^2*b^3*c*d*n*x^2 + 12*(b*
x + a)^n*a^3*b^2*d^2*n*x^2 + 40*(b*x + a)^n*b^5*c^2*x^3 + 2*(b*x + a)^n*a^3*b^2*c^2*n^2 - 40*(b*x + a)^n*a^2*b
^3*c^2*n*x + 60*(b*x + a)^n*a^3*b^2*c*d*n*x - 24*(b*x + a)^n*a^4*b*d^2*n*x + 18*(b*x + a)^n*a^3*b^2*c^2*n - 12
*(b*x + a)^n*a^4*b*c*d*n + 40*(b*x + a)^n*a^3*b^2*c^2 - 60*(b*x + a)^n*a^4*b*c*d + 24*(b*x + a)^n*a^5*d^2)/(b^
5*n^5 + 15*b^5*n^4 + 85*b^5*n^3 + 225*b^5*n^2 + 274*b^5*n + 120*b^5)
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maple [B] time = 0.01, size = 547, normalized size = 3.48 \begin {gather*} \frac {\left (b^{4} d^{2} n^{4} x^{4}+2 b^{4} c d \,n^{4} x^{3}+10 b^{4} d^{2} n^{3} x^{4}-4 a \,b^{3} d^{2} n^{3} x^{3}+b^{4} c^{2} n^{4} x^{2}+22 b^{4} c d \,n^{3} x^{3}+35 b^{4} d^{2} n^{2} x^{4}-6 a \,b^{3} c d \,n^{3} x^{2}-24 a \,b^{3} d^{2} n^{2} x^{3}+12 b^{4} c^{2} n^{3} x^{2}+82 b^{4} c d \,n^{2} x^{3}+50 b^{4} d^{2} n \,x^{4}+12 a^{2} b^{2} d^{2} n^{2} x^{2}-2 a \,b^{3} c^{2} n^{3} x -48 a \,b^{3} c d \,n^{2} x^{2}-44 a \,b^{3} d^{2} n \,x^{3}+49 b^{4} c^{2} n^{2} x^{2}+122 b^{4} c d n \,x^{3}+24 d^{2} x^{4} b^{4}+12 a^{2} b^{2} c d \,n^{2} x +36 a^{2} b^{2} d^{2} n \,x^{2}-20 a \,b^{3} c^{2} n^{2} x -102 a \,b^{3} c d n \,x^{2}-24 a \,b^{3} d^{2} x^{3}+78 b^{4} c^{2} n \,x^{2}+60 b^{4} c d \,x^{3}-24 a^{3} b \,d^{2} n x +2 a^{2} b^{2} c^{2} n^{2}+72 a^{2} b^{2} c d n x +24 a^{2} b^{2} d^{2} x^{2}-58 a \,b^{3} c^{2} n x -60 a \,b^{3} c d \,x^{2}+40 b^{4} c^{2} x^{2}-12 a^{3} b c d n -24 a^{3} b \,d^{2} x +18 a^{2} b^{2} c^{2} n +60 a^{2} b^{2} c d x -40 a \,b^{3} c^{2} x +24 a^{4} d^{2}-60 a^{3} b c d +40 a^{2} b^{2} c^{2}\right ) \left (b x +a \right )^{n +1}}{\left (n^{5}+15 n^{4}+85 n^{3}+225 n^{2}+274 n +120\right ) b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^2*(b*x+a)^n*(d*x+c)^2,x)
[Out]
(b*x+a)^(n+1)*(b^4*d^2*n^4*x^4+2*b^4*c*d*n^4*x^3+10*b^4*d^2*n^3*x^4-4*a*b^3*d^2*n^3*x^3+b^4*c^2*n^4*x^2+22*b^4
*c*d*n^3*x^3+35*b^4*d^2*n^2*x^4-6*a*b^3*c*d*n^3*x^2-24*a*b^3*d^2*n^2*x^3+12*b^4*c^2*n^3*x^2+82*b^4*c*d*n^2*x^3
+50*b^4*d^2*n*x^4+12*a^2*b^2*d^2*n^2*x^2-2*a*b^3*c^2*n^3*x-48*a*b^3*c*d*n^2*x^2-44*a*b^3*d^2*n*x^3+49*b^4*c^2*
n^2*x^2+122*b^4*c*d*n*x^3+24*b^4*d^2*x^4+12*a^2*b^2*c*d*n^2*x+36*a^2*b^2*d^2*n*x^2-20*a*b^3*c^2*n^2*x-102*a*b^
3*c*d*n*x^2-24*a*b^3*d^2*x^3+78*b^4*c^2*n*x^2+60*b^4*c*d*x^3-24*a^3*b*d^2*n*x+2*a^2*b^2*c^2*n^2+72*a^2*b^2*c*d
*n*x+24*a^2*b^2*d^2*x^2-58*a*b^3*c^2*n*x-60*a*b^3*c*d*x^2+40*b^4*c^2*x^2-12*a^3*b*c*d*n-24*a^3*b*d^2*x+18*a^2*
b^2*c^2*n+60*a^2*b^2*c*d*x-40*a*b^3*c^2*x+24*a^4*d^2-60*a^3*b*c*d+40*a^2*b^2*c^2)/b^5/(n^5+15*n^4+85*n^3+225*n
^2+274*n+120)
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maxima [B] time = 0.56, size = 318, normalized size = 2.03 \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^{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^{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} d^{2}}{{\left (n^{5} + 15 \, n^{4} + 85 \, n^{3} + 225 \, n^{2} + 274 \, n + 120\right )} b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^2*(b*x+a)^n*(d*x+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^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^4 + 10*n^3 + 35*n^2 + 50*n + 2
4)*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*d^2/((n^5 + 15*n^4 + 85*n^3 + 225*n^2 + 274*n + 120)*b^5)
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mupad [B] time = 1.37, size = 464, normalized size = 2.96 \begin {gather*} {\left (a+b\,x\right )}^n\,\left (\frac {2\,a^3\,\left (12\,a^2\,d^2-6\,a\,b\,c\,d\,n-30\,a\,b\,c\,d+b^2\,c^2\,n^2+9\,b^2\,c^2\,n+20\,b^2\,c^2\right )}{b^5\,\left (n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120\right )}+\frac {d^2\,x^5\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}{n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120}+\frac {x^3\,\left (n^2+3\,n+2\right )\,\left (-4\,a^2\,d^2\,n+2\,a\,b\,c\,d\,n^2+10\,a\,b\,c\,d\,n+b^2\,c^2\,n^2+9\,b^2\,c^2\,n+20\,b^2\,c^2\right )}{b^2\,\left (n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120\right )}+\frac {d\,x^4\,\left (10\,b\,c+a\,d\,n+2\,b\,c\,n\right )\,\left (n^3+6\,n^2+11\,n+6\right )}{b\,\left (n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120\right )}-\frac {2\,a^2\,n\,x\,\left (12\,a^2\,d^2-6\,a\,b\,c\,d\,n-30\,a\,b\,c\,d+b^2\,c^2\,n^2+9\,b^2\,c^2\,n+20\,b^2\,c^2\right )}{b^4\,\left (n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120\right )}+\frac {a\,n\,x^2\,\left (n+1\right )\,\left (12\,a^2\,d^2-6\,a\,b\,c\,d\,n-30\,a\,b\,c\,d+b^2\,c^2\,n^2+9\,b^2\,c^2\,n+20\,b^2\,c^2\right )}{b^3\,\left (n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120\right )}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^2*(a + b*x)^n*(c + d*x)^2,x)
[Out]
(a + b*x)^n*((2*a^3*(12*a^2*d^2 + 20*b^2*c^2 + 9*b^2*c^2*n + b^2*c^2*n^2 - 30*a*b*c*d - 6*a*b*c*d*n))/(b^5*(27
4*n + 225*n^2 + 85*n^3 + 15*n^4 + n^5 + 120)) + (d^2*x^5*(50*n + 35*n^2 + 10*n^3 + n^4 + 24))/(274*n + 225*n^2
+ 85*n^3 + 15*n^4 + n^5 + 120) + (x^3*(3*n + n^2 + 2)*(20*b^2*c^2 - 4*a^2*d^2*n + 9*b^2*c^2*n + b^2*c^2*n^2 +
10*a*b*c*d*n + 2*a*b*c*d*n^2))/(b^2*(274*n + 225*n^2 + 85*n^3 + 15*n^4 + n^5 + 120)) + (d*x^4*(10*b*c + a*d*n
+ 2*b*c*n)*(11*n + 6*n^2 + n^3 + 6))/(b*(274*n + 225*n^2 + 85*n^3 + 15*n^4 + n^5 + 120)) - (2*a^2*n*x*(12*a^2
*d^2 + 20*b^2*c^2 + 9*b^2*c^2*n + b^2*c^2*n^2 - 30*a*b*c*d - 6*a*b*c*d*n))/(b^4*(274*n + 225*n^2 + 85*n^3 + 15
*n^4 + n^5 + 120)) + (a*n*x^2*(n + 1)*(12*a^2*d^2 + 20*b^2*c^2 + 9*b^2*c^2*n + b^2*c^2*n^2 - 30*a*b*c*d - 6*a*
b*c*d*n))/(b^3*(274*n + 225*n^2 + 85*n^3 + 15*n^4 + n^5 + 120)))
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sympy [A] time = 6.92, size = 6418, normalized size = 40.88
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+c)**2,x)
[Out]
Piecewise((a**n*(c**2*x**3/3 + c*d*x**4/2 + d**2*x**5/5), Eq(b, 0)), (12*a**4*d**2*log(a/b + x)/(12*a**4*b**5
+ 48*a**3*b**6*x + 72*a**2*b**7*x**2 + 48*a*b**8*x**3 + 12*b**9*x**4) + 25*a**4*d**2/(12*a**4*b**5 + 48*a**3*b
**6*x + 72*a**2*b**7*x**2 + 48*a*b**8*x**3 + 12*b**9*x**4) - 6*a**3*b*c*d/(12*a**4*b**5 + 48*a**3*b**6*x + 72*
a**2*b**7*x**2 + 48*a*b**8*x**3 + 12*b**9*x**4) + 48*a**3*b*d**2*x*log(a/b + x)/(12*a**4*b**5 + 48*a**3*b**6*x
+ 72*a**2*b**7*x**2 + 48*a*b**8*x**3 + 12*b**9*x**4) + 88*a**3*b*d**2*x/(12*a**4*b**5 + 48*a**3*b**6*x + 72*a
**2*b**7*x**2 + 48*a*b**8*x**3 + 12*b**9*x**4) - a**2*b**2*c**2/(12*a**4*b**5 + 48*a**3*b**6*x + 72*a**2*b**7*
x**2 + 48*a*b**8*x**3 + 12*b**9*x**4) - 24*a**2*b**2*c*d*x/(12*a**4*b**5 + 48*a**3*b**6*x + 72*a**2*b**7*x**2
+ 48*a*b**8*x**3 + 12*b**9*x**4) + 72*a**2*b**2*d**2*x**2*log(a/b + x)/(12*a**4*b**5 + 48*a**3*b**6*x + 72*a**
2*b**7*x**2 + 48*a*b**8*x**3 + 12*b**9*x**4) + 108*a**2*b**2*d**2*x**2/(12*a**4*b**5 + 48*a**3*b**6*x + 72*a**
2*b**7*x**2 + 48*a*b**8*x**3 + 12*b**9*x**4) - 4*a*b**3*c**2*x/(12*a**4*b**5 + 48*a**3*b**6*x + 72*a**2*b**7*x
**2 + 48*a*b**8*x**3 + 12*b**9*x**4) - 36*a*b**3*c*d*x**2/(12*a**4*b**5 + 48*a**3*b**6*x + 72*a**2*b**7*x**2 +
48*a*b**8*x**3 + 12*b**9*x**4) + 48*a*b**3*d**2*x**3*log(a/b + x)/(12*a**4*b**5 + 48*a**3*b**6*x + 72*a**2*b*
*7*x**2 + 48*a*b**8*x**3 + 12*b**9*x**4) + 48*a*b**3*d**2*x**3/(12*a**4*b**5 + 48*a**3*b**6*x + 72*a**2*b**7*x
**2 + 48*a*b**8*x**3 + 12*b**9*x**4) - 6*b**4*c**2*x**2/(12*a**4*b**5 + 48*a**3*b**6*x + 72*a**2*b**7*x**2 + 4
8*a*b**8*x**3 + 12*b**9*x**4) - 24*b**4*c*d*x**3/(12*a**4*b**5 + 48*a**3*b**6*x + 72*a**2*b**7*x**2 + 48*a*b**
8*x**3 + 12*b**9*x**4) + 12*b**4*d**2*x**4*log(a/b + x)/(12*a**4*b**5 + 48*a**3*b**6*x + 72*a**2*b**7*x**2 + 4
8*a*b**8*x**3 + 12*b**9*x**4), Eq(n, -5)), (-12*a**4*d**2*log(a/b + x)/(3*a**3*b**5 + 9*a**2*b**6*x + 9*a*b**7
*x**2 + 3*b**8*x**3) - 22*a**4*d**2/(3*a**3*b**5 + 9*a**2*b**6*x + 9*a*b**7*x**2 + 3*b**8*x**3) + 6*a**3*b*c*d
*log(a/b + x)/(3*a**3*b**5 + 9*a**2*b**6*x + 9*a*b**7*x**2 + 3*b**8*x**3) + 11*a**3*b*c*d/(3*a**3*b**5 + 9*a**
2*b**6*x + 9*a*b**7*x**2 + 3*b**8*x**3) - 36*a**3*b*d**2*x*log(a/b + x)/(3*a**3*b**5 + 9*a**2*b**6*x + 9*a*b**
7*x**2 + 3*b**8*x**3) - 54*a**3*b*d**2*x/(3*a**3*b**5 + 9*a**2*b**6*x + 9*a*b**7*x**2 + 3*b**8*x**3) - a**2*b*
*2*c**2/(3*a**3*b**5 + 9*a**2*b**6*x + 9*a*b**7*x**2 + 3*b**8*x**3) + 18*a**2*b**2*c*d*x*log(a/b + x)/(3*a**3*
b**5 + 9*a**2*b**6*x + 9*a*b**7*x**2 + 3*b**8*x**3) + 27*a**2*b**2*c*d*x/(3*a**3*b**5 + 9*a**2*b**6*x + 9*a*b*
*7*x**2 + 3*b**8*x**3) - 36*a**2*b**2*d**2*x**2*log(a/b + x)/(3*a**3*b**5 + 9*a**2*b**6*x + 9*a*b**7*x**2 + 3*
b**8*x**3) - 36*a**2*b**2*d**2*x**2/(3*a**3*b**5 + 9*a**2*b**6*x + 9*a*b**7*x**2 + 3*b**8*x**3) - 3*a*b**3*c**
2*x/(3*a**3*b**5 + 9*a**2*b**6*x + 9*a*b**7*x**2 + 3*b**8*x**3) + 18*a*b**3*c*d*x**2*log(a/b + x)/(3*a**3*b**5
+ 9*a**2*b**6*x + 9*a*b**7*x**2 + 3*b**8*x**3) + 18*a*b**3*c*d*x**2/(3*a**3*b**5 + 9*a**2*b**6*x + 9*a*b**7*x
**2 + 3*b**8*x**3) - 12*a*b**3*d**2*x**3*log(a/b + x)/(3*a**3*b**5 + 9*a**2*b**6*x + 9*a*b**7*x**2 + 3*b**8*x*
*3) - 3*b**4*c**2*x**2/(3*a**3*b**5 + 9*a**2*b**6*x + 9*a*b**7*x**2 + 3*b**8*x**3) + 6*b**4*c*d*x**3*log(a/b +
x)/(3*a**3*b**5 + 9*a**2*b**6*x + 9*a*b**7*x**2 + 3*b**8*x**3) + 3*b**4*d**2*x**4/(3*a**3*b**5 + 9*a**2*b**6*
x + 9*a*b**7*x**2 + 3*b**8*x**3), Eq(n, -4)), (12*a**4*d**2*log(a/b + x)/(2*a**2*b**5 + 4*a*b**6*x + 2*b**7*x*
*2) + 18*a**4*d**2/(2*a**2*b**5 + 4*a*b**6*x + 2*b**7*x**2) - 12*a**3*b*c*d*log(a/b + x)/(2*a**2*b**5 + 4*a*b*
*6*x + 2*b**7*x**2) - 18*a**3*b*c*d/(2*a**2*b**5 + 4*a*b**6*x + 2*b**7*x**2) + 24*a**3*b*d**2*x*log(a/b + x)/(
2*a**2*b**5 + 4*a*b**6*x + 2*b**7*x**2) + 24*a**3*b*d**2*x/(2*a**2*b**5 + 4*a*b**6*x + 2*b**7*x**2) + 2*a**2*b
**2*c**2*log(a/b + x)/(2*a**2*b**5 + 4*a*b**6*x + 2*b**7*x**2) + 3*a**2*b**2*c**2/(2*a**2*b**5 + 4*a*b**6*x +
2*b**7*x**2) - 24*a**2*b**2*c*d*x*log(a/b + x)/(2*a**2*b**5 + 4*a*b**6*x + 2*b**7*x**2) - 24*a**2*b**2*c*d*x/(
2*a**2*b**5 + 4*a*b**6*x + 2*b**7*x**2) + 12*a**2*b**2*d**2*x**2*log(a/b + x)/(2*a**2*b**5 + 4*a*b**6*x + 2*b*
*7*x**2) + 4*a*b**3*c**2*x*log(a/b + x)/(2*a**2*b**5 + 4*a*b**6*x + 2*b**7*x**2) + 4*a*b**3*c**2*x/(2*a**2*b**
5 + 4*a*b**6*x + 2*b**7*x**2) - 12*a*b**3*c*d*x**2*log(a/b + x)/(2*a**2*b**5 + 4*a*b**6*x + 2*b**7*x**2) - 4*a
*b**3*d**2*x**3/(2*a**2*b**5 + 4*a*b**6*x + 2*b**7*x**2) + 2*b**4*c**2*x**2*log(a/b + x)/(2*a**2*b**5 + 4*a*b*
*6*x + 2*b**7*x**2) + 4*b**4*c*d*x**3/(2*a**2*b**5 + 4*a*b**6*x + 2*b**7*x**2) + b**4*d**2*x**4/(2*a**2*b**5 +
4*a*b**6*x + 2*b**7*x**2), Eq(n, -3)), (-12*a**4*d**2*log(a/b + x)/(3*a*b**5 + 3*b**6*x) - 12*a**4*d**2/(3*a*
b**5 + 3*b**6*x) + 18*a**3*b*c*d*log(a/b + x)/(3*a*b**5 + 3*b**6*x) + 18*a**3*b*c*d/(3*a*b**5 + 3*b**6*x) - 12
*a**3*b*d**2*x*log(a/b + x)/(3*a*b**5 + 3*b**6*x) - 6*a**2*b**2*c**2*log(a/b + x)/(3*a*b**5 + 3*b**6*x) - 6*a*
*2*b**2*c**2/(3*a*b**5 + 3*b**6*x) + 18*a**2*b**2*c*d*x*log(a/b + x)/(3*a*b**5 + 3*b**6*x) + 6*a**2*b**2*d**2*
x**2/(3*a*b**5 + 3*b**6*x) - 6*a*b**3*c**2*x*log(a/b + x)/(3*a*b**5 + 3*b**6*x) - 9*a*b**3*c*d*x**2/(3*a*b**5
+ 3*b**6*x) - 2*a*b**3*d**2*x**3/(3*a*b**5 + 3*b**6*x) + 3*b**4*c**2*x**2/(3*a*b**5 + 3*b**6*x) + 3*b**4*c*d*x
**3/(3*a*b**5 + 3*b**6*x) + b**4*d**2*x**4/(3*a*b**5 + 3*b**6*x), Eq(n, -2)), (a**4*d**2*log(a/b + x)/b**5 - 2
*a**3*c*d*log(a/b + x)/b**4 - a**3*d**2*x/b**4 + a**2*c**2*log(a/b + x)/b**3 + 2*a**2*c*d*x/b**3 + a**2*d**2*x
**2/(2*b**3) - a*c**2*x/b**2 - a*c*d*x**2/b**2 - a*d**2*x**3/(3*b**2) + c**2*x**2/(2*b) + 2*c*d*x**3/(3*b) + d
**2*x**4/(4*b), Eq(n, -1)), (24*a**5*d**2*(a + b*x)**n/(b**5*n**5 + 15*b**5*n**4 + 85*b**5*n**3 + 225*b**5*n**
2 + 274*b**5*n + 120*b**5) - 12*a**4*b*c*d*n*(a + b*x)**n/(b**5*n**5 + 15*b**5*n**4 + 85*b**5*n**3 + 225*b**5*
n**2 + 274*b**5*n + 120*b**5) - 60*a**4*b*c*d*(a + b*x)**n/(b**5*n**5 + 15*b**5*n**4 + 85*b**5*n**3 + 225*b**5
*n**2 + 274*b**5*n + 120*b**5) - 24*a**4*b*d**2*n*x*(a + b*x)**n/(b**5*n**5 + 15*b**5*n**4 + 85*b**5*n**3 + 22
5*b**5*n**2 + 274*b**5*n + 120*b**5) + 2*a**3*b**2*c**2*n**2*(a + b*x)**n/(b**5*n**5 + 15*b**5*n**4 + 85*b**5*
n**3 + 225*b**5*n**2 + 274*b**5*n + 120*b**5) + 18*a**3*b**2*c**2*n*(a + b*x)**n/(b**5*n**5 + 15*b**5*n**4 + 8
5*b**5*n**3 + 225*b**5*n**2 + 274*b**5*n + 120*b**5) + 40*a**3*b**2*c**2*(a + b*x)**n/(b**5*n**5 + 15*b**5*n**
4 + 85*b**5*n**3 + 225*b**5*n**2 + 274*b**5*n + 120*b**5) + 12*a**3*b**2*c*d*n**2*x*(a + b*x)**n/(b**5*n**5 +
15*b**5*n**4 + 85*b**5*n**3 + 225*b**5*n**2 + 274*b**5*n + 120*b**5) + 60*a**3*b**2*c*d*n*x*(a + b*x)**n/(b**5
*n**5 + 15*b**5*n**4 + 85*b**5*n**3 + 225*b**5*n**2 + 274*b**5*n + 120*b**5) + 12*a**3*b**2*d**2*n**2*x**2*(a
+ b*x)**n/(b**5*n**5 + 15*b**5*n**4 + 85*b**5*n**3 + 225*b**5*n**2 + 274*b**5*n + 120*b**5) + 12*a**3*b**2*d**
2*n*x**2*(a + b*x)**n/(b**5*n**5 + 15*b**5*n**4 + 85*b**5*n**3 + 225*b**5*n**2 + 274*b**5*n + 120*b**5) - 2*a*
*2*b**3*c**2*n**3*x*(a + b*x)**n/(b**5*n**5 + 15*b**5*n**4 + 85*b**5*n**3 + 225*b**5*n**2 + 274*b**5*n + 120*b
**5) - 18*a**2*b**3*c**2*n**2*x*(a + b*x)**n/(b**5*n**5 + 15*b**5*n**4 + 85*b**5*n**3 + 225*b**5*n**2 + 274*b*
*5*n + 120*b**5) - 40*a**2*b**3*c**2*n*x*(a + b*x)**n/(b**5*n**5 + 15*b**5*n**4 + 85*b**5*n**3 + 225*b**5*n**2
+ 274*b**5*n + 120*b**5) - 6*a**2*b**3*c*d*n**3*x**2*(a + b*x)**n/(b**5*n**5 + 15*b**5*n**4 + 85*b**5*n**3 +
225*b**5*n**2 + 274*b**5*n + 120*b**5) - 36*a**2*b**3*c*d*n**2*x**2*(a + b*x)**n/(b**5*n**5 + 15*b**5*n**4 + 8
5*b**5*n**3 + 225*b**5*n**2 + 274*b**5*n + 120*b**5) - 30*a**2*b**3*c*d*n*x**2*(a + b*x)**n/(b**5*n**5 + 15*b*
*5*n**4 + 85*b**5*n**3 + 225*b**5*n**2 + 274*b**5*n + 120*b**5) - 4*a**2*b**3*d**2*n**3*x**3*(a + b*x)**n/(b**
5*n**5 + 15*b**5*n**4 + 85*b**5*n**3 + 225*b**5*n**2 + 274*b**5*n + 120*b**5) - 12*a**2*b**3*d**2*n**2*x**3*(a
+ b*x)**n/(b**5*n**5 + 15*b**5*n**4 + 85*b**5*n**3 + 225*b**5*n**2 + 274*b**5*n + 120*b**5) - 8*a**2*b**3*d**
2*n*x**3*(a + b*x)**n/(b**5*n**5 + 15*b**5*n**4 + 85*b**5*n**3 + 225*b**5*n**2 + 274*b**5*n + 120*b**5) + a*b*
*4*c**2*n**4*x**2*(a + b*x)**n/(b**5*n**5 + 15*b**5*n**4 + 85*b**5*n**3 + 225*b**5*n**2 + 274*b**5*n + 120*b**
5) + 10*a*b**4*c**2*n**3*x**2*(a + b*x)**n/(b**5*n**5 + 15*b**5*n**4 + 85*b**5*n**3 + 225*b**5*n**2 + 274*b**5
*n + 120*b**5) + 29*a*b**4*c**2*n**2*x**2*(a + b*x)**n/(b**5*n**5 + 15*b**5*n**4 + 85*b**5*n**3 + 225*b**5*n**
2 + 274*b**5*n + 120*b**5) + 20*a*b**4*c**2*n*x**2*(a + b*x)**n/(b**5*n**5 + 15*b**5*n**4 + 85*b**5*n**3 + 225
*b**5*n**2 + 274*b**5*n + 120*b**5) + 2*a*b**4*c*d*n**4*x**3*(a + b*x)**n/(b**5*n**5 + 15*b**5*n**4 + 85*b**5*
n**3 + 225*b**5*n**2 + 274*b**5*n + 120*b**5) + 16*a*b**4*c*d*n**3*x**3*(a + b*x)**n/(b**5*n**5 + 15*b**5*n**4
+ 85*b**5*n**3 + 225*b**5*n**2 + 274*b**5*n + 120*b**5) + 34*a*b**4*c*d*n**2*x**3*(a + b*x)**n/(b**5*n**5 + 1
5*b**5*n**4 + 85*b**5*n**3 + 225*b**5*n**2 + 274*b**5*n + 120*b**5) + 20*a*b**4*c*d*n*x**3*(a + b*x)**n/(b**5*
n**5 + 15*b**5*n**4 + 85*b**5*n**3 + 225*b**5*n**2 + 274*b**5*n + 120*b**5) + a*b**4*d**2*n**4*x**4*(a + b*x)*
*n/(b**5*n**5 + 15*b**5*n**4 + 85*b**5*n**3 + 225*b**5*n**2 + 274*b**5*n + 120*b**5) + 6*a*b**4*d**2*n**3*x**4
*(a + b*x)**n/(b**5*n**5 + 15*b**5*n**4 + 85*b**5*n**3 + 225*b**5*n**2 + 274*b**5*n + 120*b**5) + 11*a*b**4*d*
*2*n**2*x**4*(a + b*x)**n/(b**5*n**5 + 15*b**5*n**4 + 85*b**5*n**3 + 225*b**5*n**2 + 274*b**5*n + 120*b**5) +
6*a*b**4*d**2*n*x**4*(a + b*x)**n/(b**5*n**5 + 15*b**5*n**4 + 85*b**5*n**3 + 225*b**5*n**2 + 274*b**5*n + 120*
b**5) + b**5*c**2*n**4*x**3*(a + b*x)**n/(b**5*n**5 + 15*b**5*n**4 + 85*b**5*n**3 + 225*b**5*n**2 + 274*b**5*n
+ 120*b**5) + 12*b**5*c**2*n**3*x**3*(a + b*x)**n/(b**5*n**5 + 15*b**5*n**4 + 85*b**5*n**3 + 225*b**5*n**2 +
274*b**5*n + 120*b**5) + 49*b**5*c**2*n**2*x**3*(a + b*x)**n/(b**5*n**5 + 15*b**5*n**4 + 85*b**5*n**3 + 225*b*
*5*n**2 + 274*b**5*n + 120*b**5) + 78*b**5*c**2*n*x**3*(a + b*x)**n/(b**5*n**5 + 15*b**5*n**4 + 85*b**5*n**3 +
225*b**5*n**2 + 274*b**5*n + 120*b**5) + 40*b**5*c**2*x**3*(a + b*x)**n/(b**5*n**5 + 15*b**5*n**4 + 85*b**5*n
**3 + 225*b**5*n**2 + 274*b**5*n + 120*b**5) + 2*b**5*c*d*n**4*x**4*(a + b*x)**n/(b**5*n**5 + 15*b**5*n**4 + 8
5*b**5*n**3 + 225*b**5*n**2 + 274*b**5*n + 120*b**5) + 22*b**5*c*d*n**3*x**4*(a + b*x)**n/(b**5*n**5 + 15*b**5
*n**4 + 85*b**5*n**3 + 225*b**5*n**2 + 274*b**5*n + 120*b**5) + 82*b**5*c*d*n**2*x**4*(a + b*x)**n/(b**5*n**5
+ 15*b**5*n**4 + 85*b**5*n**3 + 225*b**5*n**2 + 274*b**5*n + 120*b**5) + 122*b**5*c*d*n*x**4*(a + b*x)**n/(b**
5*n**5 + 15*b**5*n**4 + 85*b**5*n**3 + 225*b**5*n**2 + 274*b**5*n + 120*b**5) + 60*b**5*c*d*x**4*(a + b*x)**n/
(b**5*n**5 + 15*b**5*n**4 + 85*b**5*n**3 + 225*b**5*n**2 + 274*b**5*n + 120*b**5) + b**5*d**2*n**4*x**5*(a + b
*x)**n/(b**5*n**5 + 15*b**5*n**4 + 85*b**5*n**3 + 225*b**5*n**2 + 274*b**5*n + 120*b**5) + 10*b**5*d**2*n**3*x
**5*(a + b*x)**n/(b**5*n**5 + 15*b**5*n**4 + 85*b**5*n**3 + 225*b**5*n**2 + 274*b**5*n + 120*b**5) + 35*b**5*d
**2*n**2*x**5*(a + b*x)**n/(b**5*n**5 + 15*b**5*n**4 + 85*b**5*n**3 + 225*b**5*n**2 + 274*b**5*n + 120*b**5) +
50*b**5*d**2*n*x**5*(a + b*x)**n/(b**5*n**5 + 15*b**5*n**4 + 85*b**5*n**3 + 225*b**5*n**2 + 274*b**5*n + 120*
b**5) + 24*b**5*d**2*x**5*(a + b*x)**n/(b**5*n**5 + 15*b**5*n**4 + 85*b**5*n**3 + 225*b**5*n**2 + 274*b**5*n +
120*b**5), True))
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