3.1677 \(\int (A+B x) (d+e x)^3 (a^2+2 a b x+b^2 x^2)^2 \, dx\)

Optimal. Leaf size=159 \[ \frac{e^2 (a+b x)^8 (-4 a B e+A b e+3 b B d)}{8 b^5}+\frac{3 e (a+b x)^7 (b d-a e) (-2 a B e+A b e+b B d)}{7 b^5}+\frac{(a+b x)^6 (b d-a e)^2 (-4 a B e+3 A b e+b B d)}{6 b^5}+\frac{(a+b x)^5 (A b-a B) (b d-a e)^3}{5 b^5}+\frac{B e^3 (a+b x)^9}{9 b^5} \]

[Out]

((A*b - a*B)*(b*d - a*e)^3*(a + b*x)^5)/(5*b^5) + ((b*d - a*e)^2*(b*B*d + 3*A*b*e - 4*a*B*e)*(a + b*x)^6)/(6*b
^5) + (3*e*(b*d - a*e)*(b*B*d + A*b*e - 2*a*B*e)*(a + b*x)^7)/(7*b^5) + (e^2*(3*b*B*d + A*b*e - 4*a*B*e)*(a +
b*x)^8)/(8*b^5) + (B*e^3*(a + b*x)^9)/(9*b^5)

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Rubi [A]  time = 0.312856, antiderivative size = 159, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {27, 77} \[ \frac{e^2 (a+b x)^8 (-4 a B e+A b e+3 b B d)}{8 b^5}+\frac{3 e (a+b x)^7 (b d-a e) (-2 a B e+A b e+b B d)}{7 b^5}+\frac{(a+b x)^6 (b d-a e)^2 (-4 a B e+3 A b e+b B d)}{6 b^5}+\frac{(a+b x)^5 (A b-a B) (b d-a e)^3}{5 b^5}+\frac{B e^3 (a+b x)^9}{9 b^5} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x)*(d + e*x)^3*(a^2 + 2*a*b*x + b^2*x^2)^2,x]

[Out]

((A*b - a*B)*(b*d - a*e)^3*(a + b*x)^5)/(5*b^5) + ((b*d - a*e)^2*(b*B*d + 3*A*b*e - 4*a*B*e)*(a + b*x)^6)/(6*b
^5) + (3*e*(b*d - a*e)*(b*B*d + A*b*e - 2*a*B*e)*(a + b*x)^7)/(7*b^5) + (e^2*(3*b*B*d + A*b*e - 4*a*B*e)*(a +
b*x)^8)/(8*b^5) + (B*e^3*(a + b*x)^9)/(9*b^5)

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin{align*} \int (A+B x) (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx &=\int (a+b x)^4 (A+B x) (d+e x)^3 \, dx\\ &=\int \left (\frac{(A b-a B) (b d-a e)^3 (a+b x)^4}{b^4}+\frac{(b d-a e)^2 (b B d+3 A b e-4 a B e) (a+b x)^5}{b^4}+\frac{3 e (b d-a e) (b B d+A b e-2 a B e) (a+b x)^6}{b^4}+\frac{e^2 (3 b B d+A b e-4 a B e) (a+b x)^7}{b^4}+\frac{B e^3 (a+b x)^8}{b^4}\right ) \, dx\\ &=\frac{(A b-a B) (b d-a e)^3 (a+b x)^5}{5 b^5}+\frac{(b d-a e)^2 (b B d+3 A b e-4 a B e) (a+b x)^6}{6 b^5}+\frac{3 e (b d-a e) (b B d+A b e-2 a B e) (a+b x)^7}{7 b^5}+\frac{e^2 (3 b B d+A b e-4 a B e) (a+b x)^8}{8 b^5}+\frac{B e^3 (a+b x)^9}{9 b^5}\\ \end{align*}

Mathematica [B]  time = 0.140617, size = 402, normalized size = 2.53 \[ \frac{1}{6} b x^6 \left (6 a^2 b e^2 (A e+3 B d)+4 a^3 B e^3+12 a b^2 d e (A e+B d)+b^3 d^2 (3 A e+B d)\right )+\frac{1}{5} x^5 \left (A b \left (18 a^2 b d e^2+4 a^3 e^3+12 a b^2 d^2 e+b^3 d^3\right )+a B \left (12 a^2 b d e^2+a^3 e^3+18 a b^2 d^2 e+4 b^3 d^3\right )\right )+\frac{1}{4} a x^4 \left (A \left (12 a^2 b d e^2+a^3 e^3+18 a b^2 d^2 e+4 b^3 d^3\right )+3 a B d \left (a^2 e^2+4 a b d e+2 b^2 d^2\right )\right )+\frac{1}{3} a^2 d x^3 \left (3 A \left (a^2 e^2+4 a b d e+2 b^2 d^2\right )+a B d (3 a e+4 b d)\right )+\frac{1}{7} b^2 e x^7 \left (6 a^2 B e^2+4 a b e (A e+3 B d)+3 b^2 d (A e+B d)\right )+\frac{1}{2} a^3 d^2 x^2 (3 a A e+a B d+4 A b d)+a^4 A d^3 x+\frac{1}{8} b^3 e^2 x^8 (4 a B e+A b e+3 b B d)+\frac{1}{9} b^4 B e^3 x^9 \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)*(d + e*x)^3*(a^2 + 2*a*b*x + b^2*x^2)^2,x]

[Out]

a^4*A*d^3*x + (a^3*d^2*(4*A*b*d + a*B*d + 3*a*A*e)*x^2)/2 + (a^2*d*(a*B*d*(4*b*d + 3*a*e) + 3*A*(2*b^2*d^2 + 4
*a*b*d*e + a^2*e^2))*x^3)/3 + (a*(3*a*B*d*(2*b^2*d^2 + 4*a*b*d*e + a^2*e^2) + A*(4*b^3*d^3 + 18*a*b^2*d^2*e +
12*a^2*b*d*e^2 + a^3*e^3))*x^4)/4 + ((a*B*(4*b^3*d^3 + 18*a*b^2*d^2*e + 12*a^2*b*d*e^2 + a^3*e^3) + A*b*(b^3*d
^3 + 12*a*b^2*d^2*e + 18*a^2*b*d*e^2 + 4*a^3*e^3))*x^5)/5 + (b*(4*a^3*B*e^3 + 12*a*b^2*d*e*(B*d + A*e) + 6*a^2
*b*e^2*(3*B*d + A*e) + b^3*d^2*(B*d + 3*A*e))*x^6)/6 + (b^2*e*(6*a^2*B*e^2 + 3*b^2*d*(B*d + A*e) + 4*a*b*e*(3*
B*d + A*e))*x^7)/7 + (b^3*e^2*(3*b*B*d + A*b*e + 4*a*B*e)*x^8)/8 + (b^4*B*e^3*x^9)/9

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Maple [B]  time = 0., size = 434, normalized size = 2.7 \begin{align*}{\frac{{b}^{4}B{e}^{3}{x}^{9}}{9}}+{\frac{ \left ( \left ( A{e}^{3}+3\,Bd{e}^{2} \right ){b}^{4}+4\,B{e}^{3}a{b}^{3} \right ){x}^{8}}{8}}+{\frac{ \left ( \left ( 3\,Ad{e}^{2}+3\,B{d}^{2}e \right ){b}^{4}+4\, \left ( A{e}^{3}+3\,Bd{e}^{2} \right ) a{b}^{3}+6\,B{e}^{3}{a}^{2}{b}^{2} \right ){x}^{7}}{7}}+{\frac{ \left ( \left ( 3\,A{d}^{2}e+B{d}^{3} \right ){b}^{4}+4\, \left ( 3\,Ad{e}^{2}+3\,B{d}^{2}e \right ) a{b}^{3}+6\, \left ( A{e}^{3}+3\,Bd{e}^{2} \right ){a}^{2}{b}^{2}+4\,B{e}^{3}{a}^{3}b \right ){x}^{6}}{6}}+{\frac{ \left ( A{d}^{3}{b}^{4}+4\, \left ( 3\,A{d}^{2}e+B{d}^{3} \right ) a{b}^{3}+6\, \left ( 3\,Ad{e}^{2}+3\,B{d}^{2}e \right ){a}^{2}{b}^{2}+4\, \left ( A{e}^{3}+3\,Bd{e}^{2} \right ){a}^{3}b+B{e}^{3}{a}^{4} \right ){x}^{5}}{5}}+{\frac{ \left ( 4\,A{d}^{3}a{b}^{3}+6\, \left ( 3\,A{d}^{2}e+B{d}^{3} \right ){a}^{2}{b}^{2}+4\, \left ( 3\,Ad{e}^{2}+3\,B{d}^{2}e \right ){a}^{3}b+ \left ( A{e}^{3}+3\,Bd{e}^{2} \right ){a}^{4} \right ){x}^{4}}{4}}+{\frac{ \left ( 6\,A{d}^{3}{a}^{2}{b}^{2}+4\, \left ( 3\,A{d}^{2}e+B{d}^{3} \right ){a}^{3}b+ \left ( 3\,Ad{e}^{2}+3\,B{d}^{2}e \right ){a}^{4} \right ){x}^{3}}{3}}+{\frac{ \left ( 4\,A{d}^{3}{a}^{3}b+ \left ( 3\,A{d}^{2}e+B{d}^{3} \right ){a}^{4} \right ){x}^{2}}{2}}+A{d}^{3}{a}^{4}x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(e*x+d)^3*(b^2*x^2+2*a*b*x+a^2)^2,x)

[Out]

1/9*b^4*B*e^3*x^9+1/8*((A*e^3+3*B*d*e^2)*b^4+4*B*e^3*a*b^3)*x^8+1/7*((3*A*d*e^2+3*B*d^2*e)*b^4+4*(A*e^3+3*B*d*
e^2)*a*b^3+6*B*e^3*a^2*b^2)*x^7+1/6*((3*A*d^2*e+B*d^3)*b^4+4*(3*A*d*e^2+3*B*d^2*e)*a*b^3+6*(A*e^3+3*B*d*e^2)*a
^2*b^2+4*B*e^3*a^3*b)*x^6+1/5*(A*d^3*b^4+4*(3*A*d^2*e+B*d^3)*a*b^3+6*(3*A*d*e^2+3*B*d^2*e)*a^2*b^2+4*(A*e^3+3*
B*d*e^2)*a^3*b+B*e^3*a^4)*x^5+1/4*(4*A*d^3*a*b^3+6*(3*A*d^2*e+B*d^3)*a^2*b^2+4*(3*A*d*e^2+3*B*d^2*e)*a^3*b+(A*
e^3+3*B*d*e^2)*a^4)*x^4+1/3*(6*A*d^3*a^2*b^2+4*(3*A*d^2*e+B*d^3)*a^3*b+(3*A*d*e^2+3*B*d^2*e)*a^4)*x^3+1/2*(4*A
*d^3*a^3*b+(3*A*d^2*e+B*d^3)*a^4)*x^2+A*d^3*a^4*x

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Maxima [B]  time = 1.02244, size = 599, normalized size = 3.77 \begin{align*} \frac{1}{9} \, B b^{4} e^{3} x^{9} + A a^{4} d^{3} x + \frac{1}{8} \,{\left (3 \, B b^{4} d e^{2} +{\left (4 \, B a b^{3} + A b^{4}\right )} e^{3}\right )} x^{8} + \frac{1}{7} \,{\left (3 \, B b^{4} d^{2} e + 3 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d e^{2} + 2 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{3}\right )} x^{7} + \frac{1}{6} \,{\left (B b^{4} d^{3} + 3 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e + 6 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{2} + 2 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{3}\right )} x^{6} + \frac{1}{5} \,{\left ({\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} + 6 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e + 6 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{2} +{\left (B a^{4} + 4 \, A a^{3} b\right )} e^{3}\right )} x^{5} + \frac{1}{4} \,{\left (A a^{4} e^{3} + 2 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} + 6 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e + 3 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{2}\right )} x^{4} + \frac{1}{3} \,{\left (3 \, A a^{4} d e^{2} + 2 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{3} + 3 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} d^{2} e\right )} x^{3} + \frac{1}{2} \,{\left (3 \, A a^{4} d^{2} e +{\left (B a^{4} + 4 \, A a^{3} b\right )} d^{3}\right )} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^3*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="maxima")

[Out]

1/9*B*b^4*e^3*x^9 + A*a^4*d^3*x + 1/8*(3*B*b^4*d*e^2 + (4*B*a*b^3 + A*b^4)*e^3)*x^8 + 1/7*(3*B*b^4*d^2*e + 3*(
4*B*a*b^3 + A*b^4)*d*e^2 + 2*(3*B*a^2*b^2 + 2*A*a*b^3)*e^3)*x^7 + 1/6*(B*b^4*d^3 + 3*(4*B*a*b^3 + A*b^4)*d^2*e
+ 6*(3*B*a^2*b^2 + 2*A*a*b^3)*d*e^2 + 2*(2*B*a^3*b + 3*A*a^2*b^2)*e^3)*x^6 + 1/5*((4*B*a*b^3 + A*b^4)*d^3 + 6
*(3*B*a^2*b^2 + 2*A*a*b^3)*d^2*e + 6*(2*B*a^3*b + 3*A*a^2*b^2)*d*e^2 + (B*a^4 + 4*A*a^3*b)*e^3)*x^5 + 1/4*(A*a
^4*e^3 + 2*(3*B*a^2*b^2 + 2*A*a*b^3)*d^3 + 6*(2*B*a^3*b + 3*A*a^2*b^2)*d^2*e + 3*(B*a^4 + 4*A*a^3*b)*d*e^2)*x^
4 + 1/3*(3*A*a^4*d*e^2 + 2*(2*B*a^3*b + 3*A*a^2*b^2)*d^3 + 3*(B*a^4 + 4*A*a^3*b)*d^2*e)*x^3 + 1/2*(3*A*a^4*d^2
*e + (B*a^4 + 4*A*a^3*b)*d^3)*x^2

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Fricas [B]  time = 1.30712, size = 1175, normalized size = 7.39 \begin{align*} \frac{1}{9} x^{9} e^{3} b^{4} B + \frac{3}{8} x^{8} e^{2} d b^{4} B + \frac{1}{2} x^{8} e^{3} b^{3} a B + \frac{1}{8} x^{8} e^{3} b^{4} A + \frac{3}{7} x^{7} e d^{2} b^{4} B + \frac{12}{7} x^{7} e^{2} d b^{3} a B + \frac{6}{7} x^{7} e^{3} b^{2} a^{2} B + \frac{3}{7} x^{7} e^{2} d b^{4} A + \frac{4}{7} x^{7} e^{3} b^{3} a A + \frac{1}{6} x^{6} d^{3} b^{4} B + 2 x^{6} e d^{2} b^{3} a B + 3 x^{6} e^{2} d b^{2} a^{2} B + \frac{2}{3} x^{6} e^{3} b a^{3} B + \frac{1}{2} x^{6} e d^{2} b^{4} A + 2 x^{6} e^{2} d b^{3} a A + x^{6} e^{3} b^{2} a^{2} A + \frac{4}{5} x^{5} d^{3} b^{3} a B + \frac{18}{5} x^{5} e d^{2} b^{2} a^{2} B + \frac{12}{5} x^{5} e^{2} d b a^{3} B + \frac{1}{5} x^{5} e^{3} a^{4} B + \frac{1}{5} x^{5} d^{3} b^{4} A + \frac{12}{5} x^{5} e d^{2} b^{3} a A + \frac{18}{5} x^{5} e^{2} d b^{2} a^{2} A + \frac{4}{5} x^{5} e^{3} b a^{3} A + \frac{3}{2} x^{4} d^{3} b^{2} a^{2} B + 3 x^{4} e d^{2} b a^{3} B + \frac{3}{4} x^{4} e^{2} d a^{4} B + x^{4} d^{3} b^{3} a A + \frac{9}{2} x^{4} e d^{2} b^{2} a^{2} A + 3 x^{4} e^{2} d b a^{3} A + \frac{1}{4} x^{4} e^{3} a^{4} A + \frac{4}{3} x^{3} d^{3} b a^{3} B + x^{3} e d^{2} a^{4} B + 2 x^{3} d^{3} b^{2} a^{2} A + 4 x^{3} e d^{2} b a^{3} A + x^{3} e^{2} d a^{4} A + \frac{1}{2} x^{2} d^{3} a^{4} B + 2 x^{2} d^{3} b a^{3} A + \frac{3}{2} x^{2} e d^{2} a^{4} A + x d^{3} a^{4} A \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^3*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="fricas")

[Out]

1/9*x^9*e^3*b^4*B + 3/8*x^8*e^2*d*b^4*B + 1/2*x^8*e^3*b^3*a*B + 1/8*x^8*e^3*b^4*A + 3/7*x^7*e*d^2*b^4*B + 12/7
*x^7*e^2*d*b^3*a*B + 6/7*x^7*e^3*b^2*a^2*B + 3/7*x^7*e^2*d*b^4*A + 4/7*x^7*e^3*b^3*a*A + 1/6*x^6*d^3*b^4*B + 2
*x^6*e*d^2*b^3*a*B + 3*x^6*e^2*d*b^2*a^2*B + 2/3*x^6*e^3*b*a^3*B + 1/2*x^6*e*d^2*b^4*A + 2*x^6*e^2*d*b^3*a*A +
x^6*e^3*b^2*a^2*A + 4/5*x^5*d^3*b^3*a*B + 18/5*x^5*e*d^2*b^2*a^2*B + 12/5*x^5*e^2*d*b*a^3*B + 1/5*x^5*e^3*a^4
*B + 1/5*x^5*d^3*b^4*A + 12/5*x^5*e*d^2*b^3*a*A + 18/5*x^5*e^2*d*b^2*a^2*A + 4/5*x^5*e^3*b*a^3*A + 3/2*x^4*d^3
*b^2*a^2*B + 3*x^4*e*d^2*b*a^3*B + 3/4*x^4*e^2*d*a^4*B + x^4*d^3*b^3*a*A + 9/2*x^4*e*d^2*b^2*a^2*A + 3*x^4*e^2
*d*b*a^3*A + 1/4*x^4*e^3*a^4*A + 4/3*x^3*d^3*b*a^3*B + x^3*e*d^2*a^4*B + 2*x^3*d^3*b^2*a^2*A + 4*x^3*e*d^2*b*a
^3*A + x^3*e^2*d*a^4*A + 1/2*x^2*d^3*a^4*B + 2*x^2*d^3*b*a^3*A + 3/2*x^2*e*d^2*a^4*A + x*d^3*a^4*A

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Sympy [B]  time = 0.156383, size = 546, normalized size = 3.43 \begin{align*} A a^{4} d^{3} x + \frac{B b^{4} e^{3} x^{9}}{9} + x^{8} \left (\frac{A b^{4} e^{3}}{8} + \frac{B a b^{3} e^{3}}{2} + \frac{3 B b^{4} d e^{2}}{8}\right ) + x^{7} \left (\frac{4 A a b^{3} e^{3}}{7} + \frac{3 A b^{4} d e^{2}}{7} + \frac{6 B a^{2} b^{2} e^{3}}{7} + \frac{12 B a b^{3} d e^{2}}{7} + \frac{3 B b^{4} d^{2} e}{7}\right ) + x^{6} \left (A a^{2} b^{2} e^{3} + 2 A a b^{3} d e^{2} + \frac{A b^{4} d^{2} e}{2} + \frac{2 B a^{3} b e^{3}}{3} + 3 B a^{2} b^{2} d e^{2} + 2 B a b^{3} d^{2} e + \frac{B b^{4} d^{3}}{6}\right ) + x^{5} \left (\frac{4 A a^{3} b e^{3}}{5} + \frac{18 A a^{2} b^{2} d e^{2}}{5} + \frac{12 A a b^{3} d^{2} e}{5} + \frac{A b^{4} d^{3}}{5} + \frac{B a^{4} e^{3}}{5} + \frac{12 B a^{3} b d e^{2}}{5} + \frac{18 B a^{2} b^{2} d^{2} e}{5} + \frac{4 B a b^{3} d^{3}}{5}\right ) + x^{4} \left (\frac{A a^{4} e^{3}}{4} + 3 A a^{3} b d e^{2} + \frac{9 A a^{2} b^{2} d^{2} e}{2} + A a b^{3} d^{3} + \frac{3 B a^{4} d e^{2}}{4} + 3 B a^{3} b d^{2} e + \frac{3 B a^{2} b^{2} d^{3}}{2}\right ) + x^{3} \left (A a^{4} d e^{2} + 4 A a^{3} b d^{2} e + 2 A a^{2} b^{2} d^{3} + B a^{4} d^{2} e + \frac{4 B a^{3} b d^{3}}{3}\right ) + x^{2} \left (\frac{3 A a^{4} d^{2} e}{2} + 2 A a^{3} b d^{3} + \frac{B a^{4} d^{3}}{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)**3*(b**2*x**2+2*a*b*x+a**2)**2,x)

[Out]

A*a**4*d**3*x + B*b**4*e**3*x**9/9 + x**8*(A*b**4*e**3/8 + B*a*b**3*e**3/2 + 3*B*b**4*d*e**2/8) + x**7*(4*A*a*
b**3*e**3/7 + 3*A*b**4*d*e**2/7 + 6*B*a**2*b**2*e**3/7 + 12*B*a*b**3*d*e**2/7 + 3*B*b**4*d**2*e/7) + x**6*(A*a
**2*b**2*e**3 + 2*A*a*b**3*d*e**2 + A*b**4*d**2*e/2 + 2*B*a**3*b*e**3/3 + 3*B*a**2*b**2*d*e**2 + 2*B*a*b**3*d*
*2*e + B*b**4*d**3/6) + x**5*(4*A*a**3*b*e**3/5 + 18*A*a**2*b**2*d*e**2/5 + 12*A*a*b**3*d**2*e/5 + A*b**4*d**3
/5 + B*a**4*e**3/5 + 12*B*a**3*b*d*e**2/5 + 18*B*a**2*b**2*d**2*e/5 + 4*B*a*b**3*d**3/5) + x**4*(A*a**4*e**3/4
+ 3*A*a**3*b*d*e**2 + 9*A*a**2*b**2*d**2*e/2 + A*a*b**3*d**3 + 3*B*a**4*d*e**2/4 + 3*B*a**3*b*d**2*e + 3*B*a*
*2*b**2*d**3/2) + x**3*(A*a**4*d*e**2 + 4*A*a**3*b*d**2*e + 2*A*a**2*b**2*d**3 + B*a**4*d**2*e + 4*B*a**3*b*d*
*3/3) + x**2*(3*A*a**4*d**2*e/2 + 2*A*a**3*b*d**3 + B*a**4*d**3/2)

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Giac [B]  time = 1.15265, size = 707, normalized size = 4.45 \begin{align*} \frac{1}{9} \, B b^{4} x^{9} e^{3} + \frac{3}{8} \, B b^{4} d x^{8} e^{2} + \frac{3}{7} \, B b^{4} d^{2} x^{7} e + \frac{1}{6} \, B b^{4} d^{3} x^{6} + \frac{1}{2} \, B a b^{3} x^{8} e^{3} + \frac{1}{8} \, A b^{4} x^{8} e^{3} + \frac{12}{7} \, B a b^{3} d x^{7} e^{2} + \frac{3}{7} \, A b^{4} d x^{7} e^{2} + 2 \, B a b^{3} d^{2} x^{6} e + \frac{1}{2} \, A b^{4} d^{2} x^{6} e + \frac{4}{5} \, B a b^{3} d^{3} x^{5} + \frac{1}{5} \, A b^{4} d^{3} x^{5} + \frac{6}{7} \, B a^{2} b^{2} x^{7} e^{3} + \frac{4}{7} \, A a b^{3} x^{7} e^{3} + 3 \, B a^{2} b^{2} d x^{6} e^{2} + 2 \, A a b^{3} d x^{6} e^{2} + \frac{18}{5} \, B a^{2} b^{2} d^{2} x^{5} e + \frac{12}{5} \, A a b^{3} d^{2} x^{5} e + \frac{3}{2} \, B a^{2} b^{2} d^{3} x^{4} + A a b^{3} d^{3} x^{4} + \frac{2}{3} \, B a^{3} b x^{6} e^{3} + A a^{2} b^{2} x^{6} e^{3} + \frac{12}{5} \, B a^{3} b d x^{5} e^{2} + \frac{18}{5} \, A a^{2} b^{2} d x^{5} e^{2} + 3 \, B a^{3} b d^{2} x^{4} e + \frac{9}{2} \, A a^{2} b^{2} d^{2} x^{4} e + \frac{4}{3} \, B a^{3} b d^{3} x^{3} + 2 \, A a^{2} b^{2} d^{3} x^{3} + \frac{1}{5} \, B a^{4} x^{5} e^{3} + \frac{4}{5} \, A a^{3} b x^{5} e^{3} + \frac{3}{4} \, B a^{4} d x^{4} e^{2} + 3 \, A a^{3} b d x^{4} e^{2} + B a^{4} d^{2} x^{3} e + 4 \, A a^{3} b d^{2} x^{3} e + \frac{1}{2} \, B a^{4} d^{3} x^{2} + 2 \, A a^{3} b d^{3} x^{2} + \frac{1}{4} \, A a^{4} x^{4} e^{3} + A a^{4} d x^{3} e^{2} + \frac{3}{2} \, A a^{4} d^{2} x^{2} e + A a^{4} d^{3} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^3*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="giac")

[Out]

1/9*B*b^4*x^9*e^3 + 3/8*B*b^4*d*x^8*e^2 + 3/7*B*b^4*d^2*x^7*e + 1/6*B*b^4*d^3*x^6 + 1/2*B*a*b^3*x^8*e^3 + 1/8*
A*b^4*x^8*e^3 + 12/7*B*a*b^3*d*x^7*e^2 + 3/7*A*b^4*d*x^7*e^2 + 2*B*a*b^3*d^2*x^6*e + 1/2*A*b^4*d^2*x^6*e + 4/5
*B*a*b^3*d^3*x^5 + 1/5*A*b^4*d^3*x^5 + 6/7*B*a^2*b^2*x^7*e^3 + 4/7*A*a*b^3*x^7*e^3 + 3*B*a^2*b^2*d*x^6*e^2 + 2
*A*a*b^3*d*x^6*e^2 + 18/5*B*a^2*b^2*d^2*x^5*e + 12/5*A*a*b^3*d^2*x^5*e + 3/2*B*a^2*b^2*d^3*x^4 + A*a*b^3*d^3*x
^4 + 2/3*B*a^3*b*x^6*e^3 + A*a^2*b^2*x^6*e^3 + 12/5*B*a^3*b*d*x^5*e^2 + 18/5*A*a^2*b^2*d*x^5*e^2 + 3*B*a^3*b*d
^2*x^4*e + 9/2*A*a^2*b^2*d^2*x^4*e + 4/3*B*a^3*b*d^3*x^3 + 2*A*a^2*b^2*d^3*x^3 + 1/5*B*a^4*x^5*e^3 + 4/5*A*a^3
*b*x^5*e^3 + 3/4*B*a^4*d*x^4*e^2 + 3*A*a^3*b*d*x^4*e^2 + B*a^4*d^2*x^3*e + 4*A*a^3*b*d^2*x^3*e + 1/2*B*a^4*d^3
*x^2 + 2*A*a^3*b*d^3*x^2 + 1/4*A*a^4*x^4*e^3 + A*a^4*d*x^3*e^2 + 3/2*A*a^4*d^2*x^2*e + A*a^4*d^3*x