3.1128 \(\int (A+B x) (d+e x)^4 (b x+c x^2)^3 \, dx\)

Optimal. Leaf size=412 \[ \frac{1}{10} c e^2 x^{10} \left (A c e (3 b e+4 c d)+3 B \left (b^2 e^2+4 b c d e+2 c^2 d^2\right )\right )+\frac{1}{9} e x^9 \left (3 A c e \left (b^2 e^2+4 b c d e+2 c^2 d^2\right )+B \left (12 b^2 c d e^2+b^3 e^3+18 b c^2 d^2 e+4 c^3 d^3\right )\right )+\frac{1}{8} x^8 \left (A e \left (12 b^2 c d e^2+b^3 e^3+18 b c^2 d^2 e+4 c^3 d^3\right )+B d \left (18 b^2 c d e^2+4 b^3 e^3+12 b c^2 d^2 e+c^3 d^3\right )\right )+\frac{1}{7} d x^7 \left (6 b^2 c d e (3 A e+2 B d)+2 b^3 e^2 (2 A e+3 B d)+3 b c^2 d^2 (4 A e+B d)+A c^3 d^3\right )+\frac{1}{6} b d^2 x^6 \left (2 b^2 e (3 A e+2 B d)+3 b c d (4 A e+B d)+3 A c^2 d^2\right )+\frac{1}{5} b^2 d^3 x^5 (4 A b e+3 A c d+b B d)+\frac{1}{4} A b^3 d^4 x^4+\frac{1}{11} c^2 e^3 x^{11} (A c e+3 b B e+4 B c d)+\frac{1}{12} B c^3 e^4 x^{12} \]

[Out]

(A*b^3*d^4*x^4)/4 + (b^2*d^3*(b*B*d + 3*A*c*d + 4*A*b*e)*x^5)/5 + (b*d^2*(3*A*c^2*d^2 + 2*b^2*e*(2*B*d + 3*A*e
) + 3*b*c*d*(B*d + 4*A*e))*x^6)/6 + (d*(A*c^3*d^3 + 2*b^3*e^2*(3*B*d + 2*A*e) + 6*b^2*c*d*e*(2*B*d + 3*A*e) +
3*b*c^2*d^2*(B*d + 4*A*e))*x^7)/7 + ((A*e*(4*c^3*d^3 + 18*b*c^2*d^2*e + 12*b^2*c*d*e^2 + b^3*e^3) + B*d*(c^3*d
^3 + 12*b*c^2*d^2*e + 18*b^2*c*d*e^2 + 4*b^3*e^3))*x^8)/8 + (e*(3*A*c*e*(2*c^2*d^2 + 4*b*c*d*e + b^2*e^2) + B*
(4*c^3*d^3 + 18*b*c^2*d^2*e + 12*b^2*c*d*e^2 + b^3*e^3))*x^9)/9 + (c*e^2*(A*c*e*(4*c*d + 3*b*e) + 3*B*(2*c^2*d
^2 + 4*b*c*d*e + b^2*e^2))*x^10)/10 + (c^2*e^3*(4*B*c*d + 3*b*B*e + A*c*e)*x^11)/11 + (B*c^3*e^4*x^12)/12

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Rubi [A]  time = 0.509999, antiderivative size = 412, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {771} \[ \frac{1}{10} c e^2 x^{10} \left (A c e (3 b e+4 c d)+3 B \left (b^2 e^2+4 b c d e+2 c^2 d^2\right )\right )+\frac{1}{9} e x^9 \left (3 A c e \left (b^2 e^2+4 b c d e+2 c^2 d^2\right )+B \left (12 b^2 c d e^2+b^3 e^3+18 b c^2 d^2 e+4 c^3 d^3\right )\right )+\frac{1}{8} x^8 \left (A e \left (12 b^2 c d e^2+b^3 e^3+18 b c^2 d^2 e+4 c^3 d^3\right )+B d \left (18 b^2 c d e^2+4 b^3 e^3+12 b c^2 d^2 e+c^3 d^3\right )\right )+\frac{1}{7} d x^7 \left (6 b^2 c d e (3 A e+2 B d)+2 b^3 e^2 (2 A e+3 B d)+3 b c^2 d^2 (4 A e+B d)+A c^3 d^3\right )+\frac{1}{6} b d^2 x^6 \left (2 b^2 e (3 A e+2 B d)+3 b c d (4 A e+B d)+3 A c^2 d^2\right )+\frac{1}{5} b^2 d^3 x^5 (4 A b e+3 A c d+b B d)+\frac{1}{4} A b^3 d^4 x^4+\frac{1}{11} c^2 e^3 x^{11} (A c e+3 b B e+4 B c d)+\frac{1}{12} B c^3 e^4 x^{12} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x)*(d + e*x)^4*(b*x + c*x^2)^3,x]

[Out]

(A*b^3*d^4*x^4)/4 + (b^2*d^3*(b*B*d + 3*A*c*d + 4*A*b*e)*x^5)/5 + (b*d^2*(3*A*c^2*d^2 + 2*b^2*e*(2*B*d + 3*A*e
) + 3*b*c*d*(B*d + 4*A*e))*x^6)/6 + (d*(A*c^3*d^3 + 2*b^3*e^2*(3*B*d + 2*A*e) + 6*b^2*c*d*e*(2*B*d + 3*A*e) +
3*b*c^2*d^2*(B*d + 4*A*e))*x^7)/7 + ((A*e*(4*c^3*d^3 + 18*b*c^2*d^2*e + 12*b^2*c*d*e^2 + b^3*e^3) + B*d*(c^3*d
^3 + 12*b*c^2*d^2*e + 18*b^2*c*d*e^2 + 4*b^3*e^3))*x^8)/8 + (e*(3*A*c*e*(2*c^2*d^2 + 4*b*c*d*e + b^2*e^2) + B*
(4*c^3*d^3 + 18*b*c^2*d^2*e + 12*b^2*c*d*e^2 + b^3*e^3))*x^9)/9 + (c*e^2*(A*c*e*(4*c*d + 3*b*e) + 3*B*(2*c^2*d
^2 + 4*b*c*d*e + b^2*e^2))*x^10)/10 + (c^2*e^3*(4*B*c*d + 3*b*B*e + A*c*e)*x^11)/11 + (B*c^3*e^4*x^12)/12

Rule 771

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(d + e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && N
eQ[b^2 - 4*a*c, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin{align*} \int (A+B x) (d+e x)^4 \left (b x+c x^2\right )^3 \, dx &=\int \left (A b^3 d^4 x^3+b^2 d^3 (b B d+3 A c d+4 A b e) x^4+b d^2 \left (3 A c^2 d^2+2 b^2 e (2 B d+3 A e)+3 b c d (B d+4 A e)\right ) x^5+d \left (A c^3 d^3+2 b^3 e^2 (3 B d+2 A e)+6 b^2 c d e (2 B d+3 A e)+3 b c^2 d^2 (B d+4 A e)\right ) x^6+\left (A e \left (4 c^3 d^3+18 b c^2 d^2 e+12 b^2 c d e^2+b^3 e^3\right )+B d \left (c^3 d^3+12 b c^2 d^2 e+18 b^2 c d e^2+4 b^3 e^3\right )\right ) x^7+e \left (3 A c e \left (2 c^2 d^2+4 b c d e+b^2 e^2\right )+B \left (4 c^3 d^3+18 b c^2 d^2 e+12 b^2 c d e^2+b^3 e^3\right )\right ) x^8+c e^2 \left (A c e (4 c d+3 b e)+3 B \left (2 c^2 d^2+4 b c d e+b^2 e^2\right )\right ) x^9+c^2 e^3 (4 B c d+3 b B e+A c e) x^{10}+B c^3 e^4 x^{11}\right ) \, dx\\ &=\frac{1}{4} A b^3 d^4 x^4+\frac{1}{5} b^2 d^3 (b B d+3 A c d+4 A b e) x^5+\frac{1}{6} b d^2 \left (3 A c^2 d^2+2 b^2 e (2 B d+3 A e)+3 b c d (B d+4 A e)\right ) x^6+\frac{1}{7} d \left (A c^3 d^3+2 b^3 e^2 (3 B d+2 A e)+6 b^2 c d e (2 B d+3 A e)+3 b c^2 d^2 (B d+4 A e)\right ) x^7+\frac{1}{8} \left (A e \left (4 c^3 d^3+18 b c^2 d^2 e+12 b^2 c d e^2+b^3 e^3\right )+B d \left (c^3 d^3+12 b c^2 d^2 e+18 b^2 c d e^2+4 b^3 e^3\right )\right ) x^8+\frac{1}{9} e \left (3 A c e \left (2 c^2 d^2+4 b c d e+b^2 e^2\right )+B \left (4 c^3 d^3+18 b c^2 d^2 e+12 b^2 c d e^2+b^3 e^3\right )\right ) x^9+\frac{1}{10} c e^2 \left (A c e (4 c d+3 b e)+3 B \left (2 c^2 d^2+4 b c d e+b^2 e^2\right )\right ) x^{10}+\frac{1}{11} c^2 e^3 (4 B c d+3 b B e+A c e) x^{11}+\frac{1}{12} B c^3 e^4 x^{12}\\ \end{align*}

Mathematica [A]  time = 0.16173, size = 412, normalized size = 1. \[ \frac{1}{10} c e^2 x^{10} \left (A c e (3 b e+4 c d)+3 B \left (b^2 e^2+4 b c d e+2 c^2 d^2\right )\right )+\frac{1}{9} e x^9 \left (3 A c e \left (b^2 e^2+4 b c d e+2 c^2 d^2\right )+B \left (12 b^2 c d e^2+b^3 e^3+18 b c^2 d^2 e+4 c^3 d^3\right )\right )+\frac{1}{8} x^8 \left (A e \left (12 b^2 c d e^2+b^3 e^3+18 b c^2 d^2 e+4 c^3 d^3\right )+B d \left (18 b^2 c d e^2+4 b^3 e^3+12 b c^2 d^2 e+c^3 d^3\right )\right )+\frac{1}{7} d x^7 \left (6 b^2 c d e (3 A e+2 B d)+2 b^3 e^2 (2 A e+3 B d)+3 b c^2 d^2 (4 A e+B d)+A c^3 d^3\right )+\frac{1}{6} b d^2 x^6 \left (2 b^2 e (3 A e+2 B d)+3 b c d (4 A e+B d)+3 A c^2 d^2\right )+\frac{1}{5} b^2 d^3 x^5 (4 A b e+3 A c d+b B d)+\frac{1}{4} A b^3 d^4 x^4+\frac{1}{11} c^2 e^3 x^{11} (A c e+3 b B e+4 B c d)+\frac{1}{12} B c^3 e^4 x^{12} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)*(d + e*x)^4*(b*x + c*x^2)^3,x]

[Out]

(A*b^3*d^4*x^4)/4 + (b^2*d^3*(b*B*d + 3*A*c*d + 4*A*b*e)*x^5)/5 + (b*d^2*(3*A*c^2*d^2 + 2*b^2*e*(2*B*d + 3*A*e
) + 3*b*c*d*(B*d + 4*A*e))*x^6)/6 + (d*(A*c^3*d^3 + 2*b^3*e^2*(3*B*d + 2*A*e) + 6*b^2*c*d*e*(2*B*d + 3*A*e) +
3*b*c^2*d^2*(B*d + 4*A*e))*x^7)/7 + ((A*e*(4*c^3*d^3 + 18*b*c^2*d^2*e + 12*b^2*c*d*e^2 + b^3*e^3) + B*d*(c^3*d
^3 + 12*b*c^2*d^2*e + 18*b^2*c*d*e^2 + 4*b^3*e^3))*x^8)/8 + (e*(3*A*c*e*(2*c^2*d^2 + 4*b*c*d*e + b^2*e^2) + B*
(4*c^3*d^3 + 18*b*c^2*d^2*e + 12*b^2*c*d*e^2 + b^3*e^3))*x^9)/9 + (c*e^2*(A*c*e*(4*c*d + 3*b*e) + 3*B*(2*c^2*d
^2 + 4*b*c*d*e + b^2*e^2))*x^10)/10 + (c^2*e^3*(4*B*c*d + 3*b*B*e + A*c*e)*x^11)/11 + (B*c^3*e^4*x^12)/12

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Maple [A]  time = 0.001, size = 444, normalized size = 1.1 \begin{align*}{\frac{B{c}^{3}{e}^{4}{x}^{12}}{12}}+{\frac{ \left ( \left ( A{e}^{4}+4\,Bd{e}^{3} \right ){c}^{3}+3\,B{e}^{4}b{c}^{2} \right ){x}^{11}}{11}}+{\frac{ \left ( \left ( 4\,Ad{e}^{3}+6\,B{d}^{2}{e}^{2} \right ){c}^{3}+3\, \left ( A{e}^{4}+4\,Bd{e}^{3} \right ) b{c}^{2}+3\,B{e}^{4}{b}^{2}c \right ){x}^{10}}{10}}+{\frac{ \left ( \left ( 6\,A{d}^{2}{e}^{2}+4\,B{d}^{3}e \right ){c}^{3}+3\, \left ( 4\,Ad{e}^{3}+6\,B{d}^{2}{e}^{2} \right ) b{c}^{2}+3\, \left ( A{e}^{4}+4\,Bd{e}^{3} \right ){b}^{2}c+B{e}^{4}{b}^{3} \right ){x}^{9}}{9}}+{\frac{ \left ( \left ( 4\,A{d}^{3}e+B{d}^{4} \right ){c}^{3}+3\, \left ( 6\,A{d}^{2}{e}^{2}+4\,B{d}^{3}e \right ) b{c}^{2}+3\, \left ( 4\,Ad{e}^{3}+6\,B{d}^{2}{e}^{2} \right ){b}^{2}c+ \left ( A{e}^{4}+4\,Bd{e}^{3} \right ){b}^{3} \right ){x}^{8}}{8}}+{\frac{ \left ( A{d}^{4}{c}^{3}+3\, \left ( 4\,A{d}^{3}e+B{d}^{4} \right ) b{c}^{2}+3\, \left ( 6\,A{d}^{2}{e}^{2}+4\,B{d}^{3}e \right ){b}^{2}c+ \left ( 4\,Ad{e}^{3}+6\,B{d}^{2}{e}^{2} \right ){b}^{3} \right ){x}^{7}}{7}}+{\frac{ \left ( 3\,A{d}^{4}b{c}^{2}+3\, \left ( 4\,A{d}^{3}e+B{d}^{4} \right ){b}^{2}c+ \left ( 6\,A{d}^{2}{e}^{2}+4\,B{d}^{3}e \right ){b}^{3} \right ){x}^{6}}{6}}+{\frac{ \left ( 3\,A{d}^{4}{b}^{2}c+ \left ( 4\,A{d}^{3}e+B{d}^{4} \right ){b}^{3} \right ){x}^{5}}{5}}+{\frac{A{b}^{3}{d}^{4}{x}^{4}}{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(e*x+d)^4*(c*x^2+b*x)^3,x)

[Out]

1/12*B*c^3*e^4*x^12+1/11*((A*e^4+4*B*d*e^3)*c^3+3*B*e^4*b*c^2)*x^11+1/10*((4*A*d*e^3+6*B*d^2*e^2)*c^3+3*(A*e^4
+4*B*d*e^3)*b*c^2+3*B*e^4*b^2*c)*x^10+1/9*((6*A*d^2*e^2+4*B*d^3*e)*c^3+3*(4*A*d*e^3+6*B*d^2*e^2)*b*c^2+3*(A*e^
4+4*B*d*e^3)*b^2*c+B*e^4*b^3)*x^9+1/8*((4*A*d^3*e+B*d^4)*c^3+3*(6*A*d^2*e^2+4*B*d^3*e)*b*c^2+3*(4*A*d*e^3+6*B*
d^2*e^2)*b^2*c+(A*e^4+4*B*d*e^3)*b^3)*x^8+1/7*(A*d^4*c^3+3*(4*A*d^3*e+B*d^4)*b*c^2+3*(6*A*d^2*e^2+4*B*d^3*e)*b
^2*c+(4*A*d*e^3+6*B*d^2*e^2)*b^3)*x^7+1/6*(3*A*d^4*b*c^2+3*(4*A*d^3*e+B*d^4)*b^2*c+(6*A*d^2*e^2+4*B*d^3*e)*b^3
)*x^6+1/5*(3*A*d^4*b^2*c+(4*A*d^3*e+B*d^4)*b^3)*x^5+1/4*A*b^3*d^4*x^4

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Maxima [A]  time = 1.02075, size = 578, normalized size = 1.4 \begin{align*} \frac{1}{12} \, B c^{3} e^{4} x^{12} + \frac{1}{4} \, A b^{3} d^{4} x^{4} + \frac{1}{11} \,{\left (4 \, B c^{3} d e^{3} +{\left (3 \, B b c^{2} + A c^{3}\right )} e^{4}\right )} x^{11} + \frac{1}{10} \,{\left (6 \, B c^{3} d^{2} e^{2} + 4 \,{\left (3 \, B b c^{2} + A c^{3}\right )} d e^{3} + 3 \,{\left (B b^{2} c + A b c^{2}\right )} e^{4}\right )} x^{10} + \frac{1}{9} \,{\left (4 \, B c^{3} d^{3} e + 6 \,{\left (3 \, B b c^{2} + A c^{3}\right )} d^{2} e^{2} + 12 \,{\left (B b^{2} c + A b c^{2}\right )} d e^{3} +{\left (B b^{3} + 3 \, A b^{2} c\right )} e^{4}\right )} x^{9} + \frac{1}{8} \,{\left (B c^{3} d^{4} + A b^{3} e^{4} + 4 \,{\left (3 \, B b c^{2} + A c^{3}\right )} d^{3} e + 18 \,{\left (B b^{2} c + A b c^{2}\right )} d^{2} e^{2} + 4 \,{\left (B b^{3} + 3 \, A b^{2} c\right )} d e^{3}\right )} x^{8} + \frac{1}{7} \,{\left (4 \, A b^{3} d e^{3} +{\left (3 \, B b c^{2} + A c^{3}\right )} d^{4} + 12 \,{\left (B b^{2} c + A b c^{2}\right )} d^{3} e + 6 \,{\left (B b^{3} + 3 \, A b^{2} c\right )} d^{2} e^{2}\right )} x^{7} + \frac{1}{6} \,{\left (6 \, A b^{3} d^{2} e^{2} + 3 \,{\left (B b^{2} c + A b c^{2}\right )} d^{4} + 4 \,{\left (B b^{3} + 3 \, A b^{2} c\right )} d^{3} e\right )} x^{6} + \frac{1}{5} \,{\left (4 \, A b^{3} d^{3} e +{\left (B b^{3} + 3 \, A b^{2} c\right )} d^{4}\right )} x^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^4*(c*x^2+b*x)^3,x, algorithm="maxima")

[Out]

1/12*B*c^3*e^4*x^12 + 1/4*A*b^3*d^4*x^4 + 1/11*(4*B*c^3*d*e^3 + (3*B*b*c^2 + A*c^3)*e^4)*x^11 + 1/10*(6*B*c^3*
d^2*e^2 + 4*(3*B*b*c^2 + A*c^3)*d*e^3 + 3*(B*b^2*c + A*b*c^2)*e^4)*x^10 + 1/9*(4*B*c^3*d^3*e + 6*(3*B*b*c^2 +
A*c^3)*d^2*e^2 + 12*(B*b^2*c + A*b*c^2)*d*e^3 + (B*b^3 + 3*A*b^2*c)*e^4)*x^9 + 1/8*(B*c^3*d^4 + A*b^3*e^4 + 4*
(3*B*b*c^2 + A*c^3)*d^3*e + 18*(B*b^2*c + A*b*c^2)*d^2*e^2 + 4*(B*b^3 + 3*A*b^2*c)*d*e^3)*x^8 + 1/7*(4*A*b^3*d
*e^3 + (3*B*b*c^2 + A*c^3)*d^4 + 12*(B*b^2*c + A*b*c^2)*d^3*e + 6*(B*b^3 + 3*A*b^2*c)*d^2*e^2)*x^7 + 1/6*(6*A*
b^3*d^2*e^2 + 3*(B*b^2*c + A*b*c^2)*d^4 + 4*(B*b^3 + 3*A*b^2*c)*d^3*e)*x^6 + 1/5*(4*A*b^3*d^3*e + (B*b^3 + 3*A
*b^2*c)*d^4)*x^5

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Fricas [A]  time = 1.48102, size = 1233, normalized size = 2.99 \begin{align*} \frac{1}{12} x^{12} e^{4} c^{3} B + \frac{4}{11} x^{11} e^{3} d c^{3} B + \frac{3}{11} x^{11} e^{4} c^{2} b B + \frac{1}{11} x^{11} e^{4} c^{3} A + \frac{3}{5} x^{10} e^{2} d^{2} c^{3} B + \frac{6}{5} x^{10} e^{3} d c^{2} b B + \frac{3}{10} x^{10} e^{4} c b^{2} B + \frac{2}{5} x^{10} e^{3} d c^{3} A + \frac{3}{10} x^{10} e^{4} c^{2} b A + \frac{4}{9} x^{9} e d^{3} c^{3} B + 2 x^{9} e^{2} d^{2} c^{2} b B + \frac{4}{3} x^{9} e^{3} d c b^{2} B + \frac{1}{9} x^{9} e^{4} b^{3} B + \frac{2}{3} x^{9} e^{2} d^{2} c^{3} A + \frac{4}{3} x^{9} e^{3} d c^{2} b A + \frac{1}{3} x^{9} e^{4} c b^{2} A + \frac{1}{8} x^{8} d^{4} c^{3} B + \frac{3}{2} x^{8} e d^{3} c^{2} b B + \frac{9}{4} x^{8} e^{2} d^{2} c b^{2} B + \frac{1}{2} x^{8} e^{3} d b^{3} B + \frac{1}{2} x^{8} e d^{3} c^{3} A + \frac{9}{4} x^{8} e^{2} d^{2} c^{2} b A + \frac{3}{2} x^{8} e^{3} d c b^{2} A + \frac{1}{8} x^{8} e^{4} b^{3} A + \frac{3}{7} x^{7} d^{4} c^{2} b B + \frac{12}{7} x^{7} e d^{3} c b^{2} B + \frac{6}{7} x^{7} e^{2} d^{2} b^{3} B + \frac{1}{7} x^{7} d^{4} c^{3} A + \frac{12}{7} x^{7} e d^{3} c^{2} b A + \frac{18}{7} x^{7} e^{2} d^{2} c b^{2} A + \frac{4}{7} x^{7} e^{3} d b^{3} A + \frac{1}{2} x^{6} d^{4} c b^{2} B + \frac{2}{3} x^{6} e d^{3} b^{3} B + \frac{1}{2} x^{6} d^{4} c^{2} b A + 2 x^{6} e d^{3} c b^{2} A + x^{6} e^{2} d^{2} b^{3} A + \frac{1}{5} x^{5} d^{4} b^{3} B + \frac{3}{5} x^{5} d^{4} c b^{2} A + \frac{4}{5} x^{5} e d^{3} b^{3} A + \frac{1}{4} x^{4} d^{4} b^{3} A \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^4*(c*x^2+b*x)^3,x, algorithm="fricas")

[Out]

1/12*x^12*e^4*c^3*B + 4/11*x^11*e^3*d*c^3*B + 3/11*x^11*e^4*c^2*b*B + 1/11*x^11*e^4*c^3*A + 3/5*x^10*e^2*d^2*c
^3*B + 6/5*x^10*e^3*d*c^2*b*B + 3/10*x^10*e^4*c*b^2*B + 2/5*x^10*e^3*d*c^3*A + 3/10*x^10*e^4*c^2*b*A + 4/9*x^9
*e*d^3*c^3*B + 2*x^9*e^2*d^2*c^2*b*B + 4/3*x^9*e^3*d*c*b^2*B + 1/9*x^9*e^4*b^3*B + 2/3*x^9*e^2*d^2*c^3*A + 4/3
*x^9*e^3*d*c^2*b*A + 1/3*x^9*e^4*c*b^2*A + 1/8*x^8*d^4*c^3*B + 3/2*x^8*e*d^3*c^2*b*B + 9/4*x^8*e^2*d^2*c*b^2*B
+ 1/2*x^8*e^3*d*b^3*B + 1/2*x^8*e*d^3*c^3*A + 9/4*x^8*e^2*d^2*c^2*b*A + 3/2*x^8*e^3*d*c*b^2*A + 1/8*x^8*e^4*b
^3*A + 3/7*x^7*d^4*c^2*b*B + 12/7*x^7*e*d^3*c*b^2*B + 6/7*x^7*e^2*d^2*b^3*B + 1/7*x^7*d^4*c^3*A + 12/7*x^7*e*d
^3*c^2*b*A + 18/7*x^7*e^2*d^2*c*b^2*A + 4/7*x^7*e^3*d*b^3*A + 1/2*x^6*d^4*c*b^2*B + 2/3*x^6*e*d^3*b^3*B + 1/2*
x^6*d^4*c^2*b*A + 2*x^6*e*d^3*c*b^2*A + x^6*e^2*d^2*b^3*A + 1/5*x^5*d^4*b^3*B + 3/5*x^5*d^4*c*b^2*A + 4/5*x^5*
e*d^3*b^3*A + 1/4*x^4*d^4*b^3*A

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Sympy [A]  time = 0.247579, size = 564, normalized size = 1.37 \begin{align*} \frac{A b^{3} d^{4} x^{4}}{4} + \frac{B c^{3} e^{4} x^{12}}{12} + x^{11} \left (\frac{A c^{3} e^{4}}{11} + \frac{3 B b c^{2} e^{4}}{11} + \frac{4 B c^{3} d e^{3}}{11}\right ) + x^{10} \left (\frac{3 A b c^{2} e^{4}}{10} + \frac{2 A c^{3} d e^{3}}{5} + \frac{3 B b^{2} c e^{4}}{10} + \frac{6 B b c^{2} d e^{3}}{5} + \frac{3 B c^{3} d^{2} e^{2}}{5}\right ) + x^{9} \left (\frac{A b^{2} c e^{4}}{3} + \frac{4 A b c^{2} d e^{3}}{3} + \frac{2 A c^{3} d^{2} e^{2}}{3} + \frac{B b^{3} e^{4}}{9} + \frac{4 B b^{2} c d e^{3}}{3} + 2 B b c^{2} d^{2} e^{2} + \frac{4 B c^{3} d^{3} e}{9}\right ) + x^{8} \left (\frac{A b^{3} e^{4}}{8} + \frac{3 A b^{2} c d e^{3}}{2} + \frac{9 A b c^{2} d^{2} e^{2}}{4} + \frac{A c^{3} d^{3} e}{2} + \frac{B b^{3} d e^{3}}{2} + \frac{9 B b^{2} c d^{2} e^{2}}{4} + \frac{3 B b c^{2} d^{3} e}{2} + \frac{B c^{3} d^{4}}{8}\right ) + x^{7} \left (\frac{4 A b^{3} d e^{3}}{7} + \frac{18 A b^{2} c d^{2} e^{2}}{7} + \frac{12 A b c^{2} d^{3} e}{7} + \frac{A c^{3} d^{4}}{7} + \frac{6 B b^{3} d^{2} e^{2}}{7} + \frac{12 B b^{2} c d^{3} e}{7} + \frac{3 B b c^{2} d^{4}}{7}\right ) + x^{6} \left (A b^{3} d^{2} e^{2} + 2 A b^{2} c d^{3} e + \frac{A b c^{2} d^{4}}{2} + \frac{2 B b^{3} d^{3} e}{3} + \frac{B b^{2} c d^{4}}{2}\right ) + x^{5} \left (\frac{4 A b^{3} d^{3} e}{5} + \frac{3 A b^{2} c d^{4}}{5} + \frac{B b^{3} d^{4}}{5}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)**4*(c*x**2+b*x)**3,x)

[Out]

A*b**3*d**4*x**4/4 + B*c**3*e**4*x**12/12 + x**11*(A*c**3*e**4/11 + 3*B*b*c**2*e**4/11 + 4*B*c**3*d*e**3/11) +
x**10*(3*A*b*c**2*e**4/10 + 2*A*c**3*d*e**3/5 + 3*B*b**2*c*e**4/10 + 6*B*b*c**2*d*e**3/5 + 3*B*c**3*d**2*e**2
/5) + x**9*(A*b**2*c*e**4/3 + 4*A*b*c**2*d*e**3/3 + 2*A*c**3*d**2*e**2/3 + B*b**3*e**4/9 + 4*B*b**2*c*d*e**3/3
+ 2*B*b*c**2*d**2*e**2 + 4*B*c**3*d**3*e/9) + x**8*(A*b**3*e**4/8 + 3*A*b**2*c*d*e**3/2 + 9*A*b*c**2*d**2*e**
2/4 + A*c**3*d**3*e/2 + B*b**3*d*e**3/2 + 9*B*b**2*c*d**2*e**2/4 + 3*B*b*c**2*d**3*e/2 + B*c**3*d**4/8) + x**7
*(4*A*b**3*d*e**3/7 + 18*A*b**2*c*d**2*e**2/7 + 12*A*b*c**2*d**3*e/7 + A*c**3*d**4/7 + 6*B*b**3*d**2*e**2/7 +
12*B*b**2*c*d**3*e/7 + 3*B*b*c**2*d**4/7) + x**6*(A*b**3*d**2*e**2 + 2*A*b**2*c*d**3*e + A*b*c**2*d**4/2 + 2*B
*b**3*d**3*e/3 + B*b**2*c*d**4/2) + x**5*(4*A*b**3*d**3*e/5 + 3*A*b**2*c*d**4/5 + B*b**3*d**4/5)

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Giac [A]  time = 1.23865, size = 707, normalized size = 1.72 \begin{align*} \frac{1}{12} \, B c^{3} x^{12} e^{4} + \frac{4}{11} \, B c^{3} d x^{11} e^{3} + \frac{3}{5} \, B c^{3} d^{2} x^{10} e^{2} + \frac{4}{9} \, B c^{3} d^{3} x^{9} e + \frac{1}{8} \, B c^{3} d^{4} x^{8} + \frac{3}{11} \, B b c^{2} x^{11} e^{4} + \frac{1}{11} \, A c^{3} x^{11} e^{4} + \frac{6}{5} \, B b c^{2} d x^{10} e^{3} + \frac{2}{5} \, A c^{3} d x^{10} e^{3} + 2 \, B b c^{2} d^{2} x^{9} e^{2} + \frac{2}{3} \, A c^{3} d^{2} x^{9} e^{2} + \frac{3}{2} \, B b c^{2} d^{3} x^{8} e + \frac{1}{2} \, A c^{3} d^{3} x^{8} e + \frac{3}{7} \, B b c^{2} d^{4} x^{7} + \frac{1}{7} \, A c^{3} d^{4} x^{7} + \frac{3}{10} \, B b^{2} c x^{10} e^{4} + \frac{3}{10} \, A b c^{2} x^{10} e^{4} + \frac{4}{3} \, B b^{2} c d x^{9} e^{3} + \frac{4}{3} \, A b c^{2} d x^{9} e^{3} + \frac{9}{4} \, B b^{2} c d^{2} x^{8} e^{2} + \frac{9}{4} \, A b c^{2} d^{2} x^{8} e^{2} + \frac{12}{7} \, B b^{2} c d^{3} x^{7} e + \frac{12}{7} \, A b c^{2} d^{3} x^{7} e + \frac{1}{2} \, B b^{2} c d^{4} x^{6} + \frac{1}{2} \, A b c^{2} d^{4} x^{6} + \frac{1}{9} \, B b^{3} x^{9} e^{4} + \frac{1}{3} \, A b^{2} c x^{9} e^{4} + \frac{1}{2} \, B b^{3} d x^{8} e^{3} + \frac{3}{2} \, A b^{2} c d x^{8} e^{3} + \frac{6}{7} \, B b^{3} d^{2} x^{7} e^{2} + \frac{18}{7} \, A b^{2} c d^{2} x^{7} e^{2} + \frac{2}{3} \, B b^{3} d^{3} x^{6} e + 2 \, A b^{2} c d^{3} x^{6} e + \frac{1}{5} \, B b^{3} d^{4} x^{5} + \frac{3}{5} \, A b^{2} c d^{4} x^{5} + \frac{1}{8} \, A b^{3} x^{8} e^{4} + \frac{4}{7} \, A b^{3} d x^{7} e^{3} + A b^{3} d^{2} x^{6} e^{2} + \frac{4}{5} \, A b^{3} d^{3} x^{5} e + \frac{1}{4} \, A b^{3} d^{4} x^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^4*(c*x^2+b*x)^3,x, algorithm="giac")

[Out]

1/12*B*c^3*x^12*e^4 + 4/11*B*c^3*d*x^11*e^3 + 3/5*B*c^3*d^2*x^10*e^2 + 4/9*B*c^3*d^3*x^9*e + 1/8*B*c^3*d^4*x^8
+ 3/11*B*b*c^2*x^11*e^4 + 1/11*A*c^3*x^11*e^4 + 6/5*B*b*c^2*d*x^10*e^3 + 2/5*A*c^3*d*x^10*e^3 + 2*B*b*c^2*d^2
*x^9*e^2 + 2/3*A*c^3*d^2*x^9*e^2 + 3/2*B*b*c^2*d^3*x^8*e + 1/2*A*c^3*d^3*x^8*e + 3/7*B*b*c^2*d^4*x^7 + 1/7*A*c
^3*d^4*x^7 + 3/10*B*b^2*c*x^10*e^4 + 3/10*A*b*c^2*x^10*e^4 + 4/3*B*b^2*c*d*x^9*e^3 + 4/3*A*b*c^2*d*x^9*e^3 + 9
/4*B*b^2*c*d^2*x^8*e^2 + 9/4*A*b*c^2*d^2*x^8*e^2 + 12/7*B*b^2*c*d^3*x^7*e + 12/7*A*b*c^2*d^3*x^7*e + 1/2*B*b^2
*c*d^4*x^6 + 1/2*A*b*c^2*d^4*x^6 + 1/9*B*b^3*x^9*e^4 + 1/3*A*b^2*c*x^9*e^4 + 1/2*B*b^3*d*x^8*e^3 + 3/2*A*b^2*c
*d*x^8*e^3 + 6/7*B*b^3*d^2*x^7*e^2 + 18/7*A*b^2*c*d^2*x^7*e^2 + 2/3*B*b^3*d^3*x^6*e + 2*A*b^2*c*d^3*x^6*e + 1/
5*B*b^3*d^4*x^5 + 3/5*A*b^2*c*d^4*x^5 + 1/8*A*b^3*x^8*e^4 + 4/7*A*b^3*d*x^7*e^3 + A*b^3*d^2*x^6*e^2 + 4/5*A*b^
3*d^3*x^5*e + 1/4*A*b^3*d^4*x^4