3.1676 \(\int (A+B x) (d+e x)^4 (a^2+2 a b x+b^2 x^2)^2 \, dx\)

Optimal. Leaf size=204 \[ \frac{e^3 (a+b x)^9 (-5 a B e+A b e+4 b B d)}{9 b^6}+\frac{e^2 (a+b x)^8 (b d-a e) (-5 a B e+2 A b e+3 b B d)}{4 b^6}+\frac{2 e (a+b x)^7 (b d-a e)^2 (-5 a B e+3 A b e+2 b B d)}{7 b^6}+\frac{(a+b x)^6 (b d-a e)^3 (-5 a B e+4 A b e+b B d)}{6 b^6}+\frac{(a+b x)^5 (A b-a B) (b d-a e)^4}{5 b^6}+\frac{B e^4 (a+b x)^{10}}{10 b^6} \]

[Out]

((A*b - a*B)*(b*d - a*e)^4*(a + b*x)^5)/(5*b^6) + ((b*d - a*e)^3*(b*B*d + 4*A*b*e - 5*a*B*e)*(a + b*x)^6)/(6*b
^6) + (2*e*(b*d - a*e)^2*(2*b*B*d + 3*A*b*e - 5*a*B*e)*(a + b*x)^7)/(7*b^6) + (e^2*(b*d - a*e)*(3*b*B*d + 2*A*
b*e - 5*a*B*e)*(a + b*x)^8)/(4*b^6) + (e^3*(4*b*B*d + A*b*e - 5*a*B*e)*(a + b*x)^9)/(9*b^6) + (B*e^4*(a + b*x)
^10)/(10*b^6)

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Rubi [A]  time = 0.459401, antiderivative size = 204, 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} \[ \frac{e^3 (a+b x)^9 (-5 a B e+A b e+4 b B d)}{9 b^6}+\frac{e^2 (a+b x)^8 (b d-a e) (-5 a B e+2 A b e+3 b B d)}{4 b^6}+\frac{2 e (a+b x)^7 (b d-a e)^2 (-5 a B e+3 A b e+2 b B d)}{7 b^6}+\frac{(a+b x)^6 (b d-a e)^3 (-5 a B e+4 A b e+b B d)}{6 b^6}+\frac{(a+b x)^5 (A b-a B) (b d-a e)^4}{5 b^6}+\frac{B e^4 (a+b x)^{10}}{10 b^6} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x)*(d + e*x)^4*(a^2 + 2*a*b*x + b^2*x^2)^2,x]

[Out]

((A*b - a*B)*(b*d - a*e)^4*(a + b*x)^5)/(5*b^6) + ((b*d - a*e)^3*(b*B*d + 4*A*b*e - 5*a*B*e)*(a + b*x)^6)/(6*b
^6) + (2*e*(b*d - a*e)^2*(2*b*B*d + 3*A*b*e - 5*a*B*e)*(a + b*x)^7)/(7*b^6) + (e^2*(b*d - a*e)*(3*b*B*d + 2*A*
b*e - 5*a*B*e)*(a + b*x)^8)/(4*b^6) + (e^3*(4*b*B*d + A*b*e - 5*a*B*e)*(a + b*x)^9)/(9*b^6) + (B*e^4*(a + b*x)
^10)/(10*b^6)

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)^4 \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)^4 \, dx\\ &=\int \left (\frac{(A b-a B) (b d-a e)^4 (a+b x)^4}{b^5}+\frac{(b d-a e)^3 (b B d+4 A b e-5 a B e) (a+b x)^5}{b^5}+\frac{2 e (b d-a e)^2 (2 b B d+3 A b e-5 a B e) (a+b x)^6}{b^5}+\frac{2 e^2 (b d-a e) (3 b B d+2 A b e-5 a B e) (a+b x)^7}{b^5}+\frac{e^3 (4 b B d+A b e-5 a B e) (a+b x)^8}{b^5}+\frac{B e^4 (a+b x)^9}{b^5}\right ) \, dx\\ &=\frac{(A b-a B) (b d-a e)^4 (a+b x)^5}{5 b^6}+\frac{(b d-a e)^3 (b B d+4 A b e-5 a B e) (a+b x)^6}{6 b^6}+\frac{2 e (b d-a e)^2 (2 b B d+3 A b e-5 a B e) (a+b x)^7}{7 b^6}+\frac{e^2 (b d-a e) (3 b B d+2 A b e-5 a B e) (a+b x)^8}{4 b^6}+\frac{e^3 (4 b B d+A b e-5 a B e) (a+b x)^9}{9 b^6}+\frac{B e^4 (a+b x)^{10}}{10 b^6}\\ \end{align*}

Mathematica [B]  time = 0.174994, size = 512, normalized size = 2.51 \[ \frac{2}{7} b e x^7 \left (3 a^2 b e^2 (A e+4 B d)+2 a^3 B e^3+4 a b^2 d e (2 A e+3 B d)+b^3 d^2 (3 A e+2 B d)\right )+\frac{1}{6} x^6 \left (12 a^2 b^2 d e^2 (2 A e+3 B d)+4 a^3 b e^3 (A e+4 B d)+a^4 B e^4+8 a b^3 d^2 e (3 A e+2 B d)+b^4 d^3 (4 A e+B d)\right )+\frac{1}{5} x^5 \left (A \left (36 a^2 b^2 d^2 e^2+16 a^3 b d e^3+a^4 e^4+16 a b^3 d^3 e+b^4 d^4\right )+4 a B d \left (6 a^2 b d e^2+a^3 e^3+6 a b^2 d^2 e+b^3 d^3\right )\right )+\frac{1}{2} a d x^4 \left (2 A \left (6 a^2 b d e^2+a^3 e^3+6 a b^2 d^2 e+b^3 d^3\right )+a B d \left (3 a^2 e^2+8 a b d e+3 b^2 d^2\right )\right )+\frac{2}{3} a^2 d^2 x^3 \left (A \left (3 a^2 e^2+8 a b d e+3 b^2 d^2\right )+2 a B d (a e+b d)\right )+\frac{1}{4} b^2 e^2 x^8 \left (3 a^2 B e^2+2 a b e (A e+4 B d)+b^2 d (2 A e+3 B d)\right )+\frac{1}{2} a^3 d^3 x^2 (4 A (a e+b d)+a B d)+a^4 A d^4 x+\frac{1}{9} b^3 e^3 x^9 (4 a B e+A b e+4 b B d)+\frac{1}{10} b^4 B e^4 x^{10} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)*(d + e*x)^4*(a^2 + 2*a*b*x + b^2*x^2)^2,x]

[Out]

a^4*A*d^4*x + (a^3*d^3*(a*B*d + 4*A*(b*d + a*e))*x^2)/2 + (2*a^2*d^2*(2*a*B*d*(b*d + a*e) + A*(3*b^2*d^2 + 8*a
*b*d*e + 3*a^2*e^2))*x^3)/3 + (a*d*(a*B*d*(3*b^2*d^2 + 8*a*b*d*e + 3*a^2*e^2) + 2*A*(b^3*d^3 + 6*a*b^2*d^2*e +
6*a^2*b*d*e^2 + a^3*e^3))*x^4)/2 + ((4*a*B*d*(b^3*d^3 + 6*a*b^2*d^2*e + 6*a^2*b*d*e^2 + a^3*e^3) + A*(b^4*d^4
+ 16*a*b^3*d^3*e + 36*a^2*b^2*d^2*e^2 + 16*a^3*b*d*e^3 + a^4*e^4))*x^5)/5 + ((a^4*B*e^4 + 4*a^3*b*e^3*(4*B*d
+ A*e) + 12*a^2*b^2*d*e^2*(3*B*d + 2*A*e) + 8*a*b^3*d^2*e*(2*B*d + 3*A*e) + b^4*d^3*(B*d + 4*A*e))*x^6)/6 + (2
*b*e*(2*a^3*B*e^3 + 3*a^2*b*e^2*(4*B*d + A*e) + 4*a*b^2*d*e*(3*B*d + 2*A*e) + b^3*d^2*(2*B*d + 3*A*e))*x^7)/7
+ (b^2*e^2*(3*a^2*B*e^2 + 2*a*b*e*(4*B*d + A*e) + b^2*d*(3*B*d + 2*A*e))*x^8)/4 + (b^3*e^3*(4*b*B*d + A*b*e +
4*a*B*e)*x^9)/9 + (b^4*B*e^4*x^10)/10

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Maple [B]  time = 0., size = 563, normalized size = 2.8 \begin{align*}{\frac{B{e}^{4}{b}^{4}{x}^{10}}{10}}+{\frac{ \left ( \left ( A{e}^{4}+4\,Bd{e}^{3} \right ){b}^{4}+4\,B{e}^{4}a{b}^{3} \right ){x}^{9}}{9}}+{\frac{ \left ( \left ( 4\,Ad{e}^{3}+6\,B{d}^{2}{e}^{2} \right ){b}^{4}+4\, \left ( A{e}^{4}+4\,Bd{e}^{3} \right ) a{b}^{3}+6\,B{e}^{4}{a}^{2}{b}^{2} \right ){x}^{8}}{8}}+{\frac{ \left ( \left ( 6\,A{d}^{2}{e}^{2}+4\,B{d}^{3}e \right ){b}^{4}+4\, \left ( 4\,Ad{e}^{3}+6\,B{d}^{2}{e}^{2} \right ) a{b}^{3}+6\, \left ( A{e}^{4}+4\,Bd{e}^{3} \right ){a}^{2}{b}^{2}+4\,B{e}^{4}{a}^{3}b \right ){x}^{7}}{7}}+{\frac{ \left ( \left ( 4\,A{d}^{3}e+B{d}^{4} \right ){b}^{4}+4\, \left ( 6\,A{d}^{2}{e}^{2}+4\,B{d}^{3}e \right ) a{b}^{3}+6\, \left ( 4\,Ad{e}^{3}+6\,B{d}^{2}{e}^{2} \right ){a}^{2}{b}^{2}+4\, \left ( A{e}^{4}+4\,Bd{e}^{3} \right ){a}^{3}b+B{e}^{4}{a}^{4} \right ){x}^{6}}{6}}+{\frac{ \left ( A{d}^{4}{b}^{4}+4\, \left ( 4\,A{d}^{3}e+B{d}^{4} \right ) a{b}^{3}+6\, \left ( 6\,A{d}^{2}{e}^{2}+4\,B{d}^{3}e \right ){a}^{2}{b}^{2}+4\, \left ( 4\,Ad{e}^{3}+6\,B{d}^{2}{e}^{2} \right ){a}^{3}b+ \left ( A{e}^{4}+4\,Bd{e}^{3} \right ){a}^{4} \right ){x}^{5}}{5}}+{\frac{ \left ( 4\,A{d}^{4}a{b}^{3}+6\, \left ( 4\,A{d}^{3}e+B{d}^{4} \right ){a}^{2}{b}^{2}+4\, \left ( 6\,A{d}^{2}{e}^{2}+4\,B{d}^{3}e \right ){a}^{3}b+ \left ( 4\,Ad{e}^{3}+6\,B{d}^{2}{e}^{2} \right ){a}^{4} \right ){x}^{4}}{4}}+{\frac{ \left ( 6\,A{d}^{4}{a}^{2}{b}^{2}+4\, \left ( 4\,A{d}^{3}e+B{d}^{4} \right ){a}^{3}b+ \left ( 6\,A{d}^{2}{e}^{2}+4\,B{d}^{3}e \right ){a}^{4} \right ){x}^{3}}{3}}+{\frac{ \left ( 4\,A{d}^{4}{a}^{3}b+ \left ( 4\,A{d}^{3}e+B{d}^{4} \right ){a}^{4} \right ){x}^{2}}{2}}+A{d}^{4}{a}^{4}x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(e*x+d)^4*(b^2*x^2+2*a*b*x+a^2)^2,x)

[Out]

1/10*B*e^4*b^4*x^10+1/9*((A*e^4+4*B*d*e^3)*b^4+4*B*e^4*a*b^3)*x^9+1/8*((4*A*d*e^3+6*B*d^2*e^2)*b^4+4*(A*e^4+4*
B*d*e^3)*a*b^3+6*B*e^4*a^2*b^2)*x^8+1/7*((6*A*d^2*e^2+4*B*d^3*e)*b^4+4*(4*A*d*e^3+6*B*d^2*e^2)*a*b^3+6*(A*e^4+
4*B*d*e^3)*a^2*b^2+4*B*e^4*a^3*b)*x^7+1/6*((4*A*d^3*e+B*d^4)*b^4+4*(6*A*d^2*e^2+4*B*d^3*e)*a*b^3+6*(4*A*d*e^3+
6*B*d^2*e^2)*a^2*b^2+4*(A*e^4+4*B*d*e^3)*a^3*b+B*e^4*a^4)*x^6+1/5*(A*d^4*b^4+4*(4*A*d^3*e+B*d^4)*a*b^3+6*(6*A*
d^2*e^2+4*B*d^3*e)*a^2*b^2+4*(4*A*d*e^3+6*B*d^2*e^2)*a^3*b+(A*e^4+4*B*d*e^3)*a^4)*x^5+1/4*(4*A*d^4*a*b^3+6*(4*
A*d^3*e+B*d^4)*a^2*b^2+4*(6*A*d^2*e^2+4*B*d^3*e)*a^3*b+(4*A*d*e^3+6*B*d^2*e^2)*a^4)*x^4+1/3*(6*A*d^4*a^2*b^2+4
*(4*A*d^3*e+B*d^4)*a^3*b+(6*A*d^2*e^2+4*B*d^3*e)*a^4)*x^3+1/2*(4*A*d^4*a^3*b+(4*A*d^3*e+B*d^4)*a^4)*x^2+A*d^4*
a^4*x

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Maxima [B]  time = 1.06737, size = 759, normalized size = 3.72 \begin{align*} \frac{1}{10} \, B b^{4} e^{4} x^{10} + A a^{4} d^{4} x + \frac{1}{9} \,{\left (4 \, B b^{4} d e^{3} +{\left (4 \, B a b^{3} + A b^{4}\right )} e^{4}\right )} x^{9} + \frac{1}{4} \,{\left (3 \, B b^{4} d^{2} e^{2} + 2 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d e^{3} +{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{4}\right )} x^{8} + \frac{2}{7} \,{\left (2 \, B b^{4} d^{3} e + 3 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e^{2} + 4 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{3} +{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{4}\right )} x^{7} + \frac{1}{6} \,{\left (B b^{4} d^{4} + 4 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e + 12 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{2} + 8 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{3} +{\left (B a^{4} + 4 \, A a^{3} b\right )} e^{4}\right )} x^{6} + \frac{1}{5} \,{\left (A a^{4} e^{4} +{\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} + 8 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e + 12 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{2} + 4 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{3}\right )} x^{5} + \frac{1}{2} \,{\left (2 \, A a^{4} d e^{3} +{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{4} + 4 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{3} e + 3 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} d^{2} e^{2}\right )} x^{4} + \frac{2}{3} \,{\left (3 \, A a^{4} d^{2} e^{2} +{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{4} + 2 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} d^{3} e\right )} x^{3} + \frac{1}{2} \,{\left (4 \, A a^{4} d^{3} e +{\left (B a^{4} + 4 \, A a^{3} b\right )} d^{4}\right )} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^4*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="maxima")

[Out]

1/10*B*b^4*e^4*x^10 + A*a^4*d^4*x + 1/9*(4*B*b^4*d*e^3 + (4*B*a*b^3 + A*b^4)*e^4)*x^9 + 1/4*(3*B*b^4*d^2*e^2 +
2*(4*B*a*b^3 + A*b^4)*d*e^3 + (3*B*a^2*b^2 + 2*A*a*b^3)*e^4)*x^8 + 2/7*(2*B*b^4*d^3*e + 3*(4*B*a*b^3 + A*b^4)
*d^2*e^2 + 4*(3*B*a^2*b^2 + 2*A*a*b^3)*d*e^3 + (2*B*a^3*b + 3*A*a^2*b^2)*e^4)*x^7 + 1/6*(B*b^4*d^4 + 4*(4*B*a*
b^3 + A*b^4)*d^3*e + 12*(3*B*a^2*b^2 + 2*A*a*b^3)*d^2*e^2 + 8*(2*B*a^3*b + 3*A*a^2*b^2)*d*e^3 + (B*a^4 + 4*A*a
^3*b)*e^4)*x^6 + 1/5*(A*a^4*e^4 + (4*B*a*b^3 + A*b^4)*d^4 + 8*(3*B*a^2*b^2 + 2*A*a*b^3)*d^3*e + 12*(2*B*a^3*b
+ 3*A*a^2*b^2)*d^2*e^2 + 4*(B*a^4 + 4*A*a^3*b)*d*e^3)*x^5 + 1/2*(2*A*a^4*d*e^3 + (3*B*a^2*b^2 + 2*A*a*b^3)*d^4
+ 4*(2*B*a^3*b + 3*A*a^2*b^2)*d^3*e + 3*(B*a^4 + 4*A*a^3*b)*d^2*e^2)*x^4 + 2/3*(3*A*a^4*d^2*e^2 + (2*B*a^3*b
+ 3*A*a^2*b^2)*d^4 + 2*(B*a^4 + 4*A*a^3*b)*d^3*e)*x^3 + 1/2*(4*A*a^4*d^3*e + (B*a^4 + 4*A*a^3*b)*d^4)*x^2

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Fricas [B]  time = 1.35711, size = 1531, normalized size = 7.5 \begin{align*} \frac{1}{10} x^{10} e^{4} b^{4} B + \frac{4}{9} x^{9} e^{3} d b^{4} B + \frac{4}{9} x^{9} e^{4} b^{3} a B + \frac{1}{9} x^{9} e^{4} b^{4} A + \frac{3}{4} x^{8} e^{2} d^{2} b^{4} B + 2 x^{8} e^{3} d b^{3} a B + \frac{3}{4} x^{8} e^{4} b^{2} a^{2} B + \frac{1}{2} x^{8} e^{3} d b^{4} A + \frac{1}{2} x^{8} e^{4} b^{3} a A + \frac{4}{7} x^{7} e d^{3} b^{4} B + \frac{24}{7} x^{7} e^{2} d^{2} b^{3} a B + \frac{24}{7} x^{7} e^{3} d b^{2} a^{2} B + \frac{4}{7} x^{7} e^{4} b a^{3} B + \frac{6}{7} x^{7} e^{2} d^{2} b^{4} A + \frac{16}{7} x^{7} e^{3} d b^{3} a A + \frac{6}{7} x^{7} e^{4} b^{2} a^{2} A + \frac{1}{6} x^{6} d^{4} b^{4} B + \frac{8}{3} x^{6} e d^{3} b^{3} a B + 6 x^{6} e^{2} d^{2} b^{2} a^{2} B + \frac{8}{3} x^{6} e^{3} d b a^{3} B + \frac{1}{6} x^{6} e^{4} a^{4} B + \frac{2}{3} x^{6} e d^{3} b^{4} A + 4 x^{6} e^{2} d^{2} b^{3} a A + 4 x^{6} e^{3} d b^{2} a^{2} A + \frac{2}{3} x^{6} e^{4} b a^{3} A + \frac{4}{5} x^{5} d^{4} b^{3} a B + \frac{24}{5} x^{5} e d^{3} b^{2} a^{2} B + \frac{24}{5} x^{5} e^{2} d^{2} b a^{3} B + \frac{4}{5} x^{5} e^{3} d a^{4} B + \frac{1}{5} x^{5} d^{4} b^{4} A + \frac{16}{5} x^{5} e d^{3} b^{3} a A + \frac{36}{5} x^{5} e^{2} d^{2} b^{2} a^{2} A + \frac{16}{5} x^{5} e^{3} d b a^{3} A + \frac{1}{5} x^{5} e^{4} a^{4} A + \frac{3}{2} x^{4} d^{4} b^{2} a^{2} B + 4 x^{4} e d^{3} b a^{3} B + \frac{3}{2} x^{4} e^{2} d^{2} a^{4} B + x^{4} d^{4} b^{3} a A + 6 x^{4} e d^{3} b^{2} a^{2} A + 6 x^{4} e^{2} d^{2} b a^{3} A + x^{4} e^{3} d a^{4} A + \frac{4}{3} x^{3} d^{4} b a^{3} B + \frac{4}{3} x^{3} e d^{3} a^{4} B + 2 x^{3} d^{4} b^{2} a^{2} A + \frac{16}{3} x^{3} e d^{3} b a^{3} A + 2 x^{3} e^{2} d^{2} a^{4} A + \frac{1}{2} x^{2} d^{4} a^{4} B + 2 x^{2} d^{4} b a^{3} A + 2 x^{2} e d^{3} a^{4} A + x d^{4} a^{4} A \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^4*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="fricas")

[Out]

1/10*x^10*e^4*b^4*B + 4/9*x^9*e^3*d*b^4*B + 4/9*x^9*e^4*b^3*a*B + 1/9*x^9*e^4*b^4*A + 3/4*x^8*e^2*d^2*b^4*B +
2*x^8*e^3*d*b^3*a*B + 3/4*x^8*e^4*b^2*a^2*B + 1/2*x^8*e^3*d*b^4*A + 1/2*x^8*e^4*b^3*a*A + 4/7*x^7*e*d^3*b^4*B
+ 24/7*x^7*e^2*d^2*b^3*a*B + 24/7*x^7*e^3*d*b^2*a^2*B + 4/7*x^7*e^4*b*a^3*B + 6/7*x^7*e^2*d^2*b^4*A + 16/7*x^7
*e^3*d*b^3*a*A + 6/7*x^7*e^4*b^2*a^2*A + 1/6*x^6*d^4*b^4*B + 8/3*x^6*e*d^3*b^3*a*B + 6*x^6*e^2*d^2*b^2*a^2*B +
8/3*x^6*e^3*d*b*a^3*B + 1/6*x^6*e^4*a^4*B + 2/3*x^6*e*d^3*b^4*A + 4*x^6*e^2*d^2*b^3*a*A + 4*x^6*e^3*d*b^2*a^2
*A + 2/3*x^6*e^4*b*a^3*A + 4/5*x^5*d^4*b^3*a*B + 24/5*x^5*e*d^3*b^2*a^2*B + 24/5*x^5*e^2*d^2*b*a^3*B + 4/5*x^5
*e^3*d*a^4*B + 1/5*x^5*d^4*b^4*A + 16/5*x^5*e*d^3*b^3*a*A + 36/5*x^5*e^2*d^2*b^2*a^2*A + 16/5*x^5*e^3*d*b*a^3*
A + 1/5*x^5*e^4*a^4*A + 3/2*x^4*d^4*b^2*a^2*B + 4*x^4*e*d^3*b*a^3*B + 3/2*x^4*e^2*d^2*a^4*B + x^4*d^4*b^3*a*A
+ 6*x^4*e*d^3*b^2*a^2*A + 6*x^4*e^2*d^2*b*a^3*A + x^4*e^3*d*a^4*A + 4/3*x^3*d^4*b*a^3*B + 4/3*x^3*e*d^3*a^4*B
+ 2*x^3*d^4*b^2*a^2*A + 16/3*x^3*e*d^3*b*a^3*A + 2*x^3*e^2*d^2*a^4*A + 1/2*x^2*d^4*a^4*B + 2*x^2*d^4*b*a^3*A +
2*x^2*e*d^3*a^4*A + x*d^4*a^4*A

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Sympy [B]  time = 0.151795, size = 717, normalized size = 3.51 \begin{align*} A a^{4} d^{4} x + \frac{B b^{4} e^{4} x^{10}}{10} + x^{9} \left (\frac{A b^{4} e^{4}}{9} + \frac{4 B a b^{3} e^{4}}{9} + \frac{4 B b^{4} d e^{3}}{9}\right ) + x^{8} \left (\frac{A a b^{3} e^{4}}{2} + \frac{A b^{4} d e^{3}}{2} + \frac{3 B a^{2} b^{2} e^{4}}{4} + 2 B a b^{3} d e^{3} + \frac{3 B b^{4} d^{2} e^{2}}{4}\right ) + x^{7} \left (\frac{6 A a^{2} b^{2} e^{4}}{7} + \frac{16 A a b^{3} d e^{3}}{7} + \frac{6 A b^{4} d^{2} e^{2}}{7} + \frac{4 B a^{3} b e^{4}}{7} + \frac{24 B a^{2} b^{2} d e^{3}}{7} + \frac{24 B a b^{3} d^{2} e^{2}}{7} + \frac{4 B b^{4} d^{3} e}{7}\right ) + x^{6} \left (\frac{2 A a^{3} b e^{4}}{3} + 4 A a^{2} b^{2} d e^{3} + 4 A a b^{3} d^{2} e^{2} + \frac{2 A b^{4} d^{3} e}{3} + \frac{B a^{4} e^{4}}{6} + \frac{8 B a^{3} b d e^{3}}{3} + 6 B a^{2} b^{2} d^{2} e^{2} + \frac{8 B a b^{3} d^{3} e}{3} + \frac{B b^{4} d^{4}}{6}\right ) + x^{5} \left (\frac{A a^{4} e^{4}}{5} + \frac{16 A a^{3} b d e^{3}}{5} + \frac{36 A a^{2} b^{2} d^{2} e^{2}}{5} + \frac{16 A a b^{3} d^{3} e}{5} + \frac{A b^{4} d^{4}}{5} + \frac{4 B a^{4} d e^{3}}{5} + \frac{24 B a^{3} b d^{2} e^{2}}{5} + \frac{24 B a^{2} b^{2} d^{3} e}{5} + \frac{4 B a b^{3} d^{4}}{5}\right ) + x^{4} \left (A a^{4} d e^{3} + 6 A a^{3} b d^{2} e^{2} + 6 A a^{2} b^{2} d^{3} e + A a b^{3} d^{4} + \frac{3 B a^{4} d^{2} e^{2}}{2} + 4 B a^{3} b d^{3} e + \frac{3 B a^{2} b^{2} d^{4}}{2}\right ) + x^{3} \left (2 A a^{4} d^{2} e^{2} + \frac{16 A a^{3} b d^{3} e}{3} + 2 A a^{2} b^{2} d^{4} + \frac{4 B a^{4} d^{3} e}{3} + \frac{4 B a^{3} b d^{4}}{3}\right ) + x^{2} \left (2 A a^{4} d^{3} e + 2 A a^{3} b d^{4} + \frac{B a^{4} d^{4}}{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)**4*(b**2*x**2+2*a*b*x+a**2)**2,x)

[Out]

A*a**4*d**4*x + B*b**4*e**4*x**10/10 + x**9*(A*b**4*e**4/9 + 4*B*a*b**3*e**4/9 + 4*B*b**4*d*e**3/9) + x**8*(A*
a*b**3*e**4/2 + A*b**4*d*e**3/2 + 3*B*a**2*b**2*e**4/4 + 2*B*a*b**3*d*e**3 + 3*B*b**4*d**2*e**2/4) + x**7*(6*A
*a**2*b**2*e**4/7 + 16*A*a*b**3*d*e**3/7 + 6*A*b**4*d**2*e**2/7 + 4*B*a**3*b*e**4/7 + 24*B*a**2*b**2*d*e**3/7
+ 24*B*a*b**3*d**2*e**2/7 + 4*B*b**4*d**3*e/7) + x**6*(2*A*a**3*b*e**4/3 + 4*A*a**2*b**2*d*e**3 + 4*A*a*b**3*d
**2*e**2 + 2*A*b**4*d**3*e/3 + B*a**4*e**4/6 + 8*B*a**3*b*d*e**3/3 + 6*B*a**2*b**2*d**2*e**2 + 8*B*a*b**3*d**3
*e/3 + B*b**4*d**4/6) + x**5*(A*a**4*e**4/5 + 16*A*a**3*b*d*e**3/5 + 36*A*a**2*b**2*d**2*e**2/5 + 16*A*a*b**3*
d**3*e/5 + A*b**4*d**4/5 + 4*B*a**4*d*e**3/5 + 24*B*a**3*b*d**2*e**2/5 + 24*B*a**2*b**2*d**3*e/5 + 4*B*a*b**3*
d**4/5) + x**4*(A*a**4*d*e**3 + 6*A*a**3*b*d**2*e**2 + 6*A*a**2*b**2*d**3*e + A*a*b**3*d**4 + 3*B*a**4*d**2*e*
*2/2 + 4*B*a**3*b*d**3*e + 3*B*a**2*b**2*d**4/2) + x**3*(2*A*a**4*d**2*e**2 + 16*A*a**3*b*d**3*e/3 + 2*A*a**2*
b**2*d**4 + 4*B*a**4*d**3*e/3 + 4*B*a**3*b*d**4/3) + x**2*(2*A*a**4*d**3*e + 2*A*a**3*b*d**4 + B*a**4*d**4/2)

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Giac [B]  time = 1.16584, size = 913, normalized size = 4.48 \begin{align*} \frac{1}{10} \, B b^{4} x^{10} e^{4} + \frac{4}{9} \, B b^{4} d x^{9} e^{3} + \frac{3}{4} \, B b^{4} d^{2} x^{8} e^{2} + \frac{4}{7} \, B b^{4} d^{3} x^{7} e + \frac{1}{6} \, B b^{4} d^{4} x^{6} + \frac{4}{9} \, B a b^{3} x^{9} e^{4} + \frac{1}{9} \, A b^{4} x^{9} e^{4} + 2 \, B a b^{3} d x^{8} e^{3} + \frac{1}{2} \, A b^{4} d x^{8} e^{3} + \frac{24}{7} \, B a b^{3} d^{2} x^{7} e^{2} + \frac{6}{7} \, A b^{4} d^{2} x^{7} e^{2} + \frac{8}{3} \, B a b^{3} d^{3} x^{6} e + \frac{2}{3} \, A b^{4} d^{3} x^{6} e + \frac{4}{5} \, B a b^{3} d^{4} x^{5} + \frac{1}{5} \, A b^{4} d^{4} x^{5} + \frac{3}{4} \, B a^{2} b^{2} x^{8} e^{4} + \frac{1}{2} \, A a b^{3} x^{8} e^{4} + \frac{24}{7} \, B a^{2} b^{2} d x^{7} e^{3} + \frac{16}{7} \, A a b^{3} d x^{7} e^{3} + 6 \, B a^{2} b^{2} d^{2} x^{6} e^{2} + 4 \, A a b^{3} d^{2} x^{6} e^{2} + \frac{24}{5} \, B a^{2} b^{2} d^{3} x^{5} e + \frac{16}{5} \, A a b^{3} d^{3} x^{5} e + \frac{3}{2} \, B a^{2} b^{2} d^{4} x^{4} + A a b^{3} d^{4} x^{4} + \frac{4}{7} \, B a^{3} b x^{7} e^{4} + \frac{6}{7} \, A a^{2} b^{2} x^{7} e^{4} + \frac{8}{3} \, B a^{3} b d x^{6} e^{3} + 4 \, A a^{2} b^{2} d x^{6} e^{3} + \frac{24}{5} \, B a^{3} b d^{2} x^{5} e^{2} + \frac{36}{5} \, A a^{2} b^{2} d^{2} x^{5} e^{2} + 4 \, B a^{3} b d^{3} x^{4} e + 6 \, A a^{2} b^{2} d^{3} x^{4} e + \frac{4}{3} \, B a^{3} b d^{4} x^{3} + 2 \, A a^{2} b^{2} d^{4} x^{3} + \frac{1}{6} \, B a^{4} x^{6} e^{4} + \frac{2}{3} \, A a^{3} b x^{6} e^{4} + \frac{4}{5} \, B a^{4} d x^{5} e^{3} + \frac{16}{5} \, A a^{3} b d x^{5} e^{3} + \frac{3}{2} \, B a^{4} d^{2} x^{4} e^{2} + 6 \, A a^{3} b d^{2} x^{4} e^{2} + \frac{4}{3} \, B a^{4} d^{3} x^{3} e + \frac{16}{3} \, A a^{3} b d^{3} x^{3} e + \frac{1}{2} \, B a^{4} d^{4} x^{2} + 2 \, A a^{3} b d^{4} x^{2} + \frac{1}{5} \, A a^{4} x^{5} e^{4} + A a^{4} d x^{4} e^{3} + 2 \, A a^{4} d^{2} x^{3} e^{2} + 2 \, A a^{4} d^{3} x^{2} e + A a^{4} d^{4} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^4*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="giac")

[Out]

1/10*B*b^4*x^10*e^4 + 4/9*B*b^4*d*x^9*e^3 + 3/4*B*b^4*d^2*x^8*e^2 + 4/7*B*b^4*d^3*x^7*e + 1/6*B*b^4*d^4*x^6 +
4/9*B*a*b^3*x^9*e^4 + 1/9*A*b^4*x^9*e^4 + 2*B*a*b^3*d*x^8*e^3 + 1/2*A*b^4*d*x^8*e^3 + 24/7*B*a*b^3*d^2*x^7*e^2
+ 6/7*A*b^4*d^2*x^7*e^2 + 8/3*B*a*b^3*d^3*x^6*e + 2/3*A*b^4*d^3*x^6*e + 4/5*B*a*b^3*d^4*x^5 + 1/5*A*b^4*d^4*x
^5 + 3/4*B*a^2*b^2*x^8*e^4 + 1/2*A*a*b^3*x^8*e^4 + 24/7*B*a^2*b^2*d*x^7*e^3 + 16/7*A*a*b^3*d*x^7*e^3 + 6*B*a^2
*b^2*d^2*x^6*e^2 + 4*A*a*b^3*d^2*x^6*e^2 + 24/5*B*a^2*b^2*d^3*x^5*e + 16/5*A*a*b^3*d^3*x^5*e + 3/2*B*a^2*b^2*d
^4*x^4 + A*a*b^3*d^4*x^4 + 4/7*B*a^3*b*x^7*e^4 + 6/7*A*a^2*b^2*x^7*e^4 + 8/3*B*a^3*b*d*x^6*e^3 + 4*A*a^2*b^2*d
*x^6*e^3 + 24/5*B*a^3*b*d^2*x^5*e^2 + 36/5*A*a^2*b^2*d^2*x^5*e^2 + 4*B*a^3*b*d^3*x^4*e + 6*A*a^2*b^2*d^3*x^4*e
+ 4/3*B*a^3*b*d^4*x^3 + 2*A*a^2*b^2*d^4*x^3 + 1/6*B*a^4*x^6*e^4 + 2/3*A*a^3*b*x^6*e^4 + 4/5*B*a^4*d*x^5*e^3 +
16/5*A*a^3*b*d*x^5*e^3 + 3/2*B*a^4*d^2*x^4*e^2 + 6*A*a^3*b*d^2*x^4*e^2 + 4/3*B*a^4*d^3*x^3*e + 16/3*A*a^3*b*d
^3*x^3*e + 1/2*B*a^4*d^4*x^2 + 2*A*a^3*b*d^4*x^2 + 1/5*A*a^4*x^5*e^4 + A*a^4*d*x^4*e^3 + 2*A*a^4*d^2*x^3*e^2 +
2*A*a^4*d^3*x^2*e + A*a^4*d^4*x