Optimal. Leaf size=225 \[ \frac{1}{8} c x^8 \left (A c e (3 b e+2 c d)+B \left (3 b^2 e^2+6 b c d e+c^2 d^2\right )\right )+\frac{1}{7} x^7 \left (3 b^2 c e (A e+2 B d)+3 b c^2 d (2 A e+B d)+A c^3 d^2+b^3 B e^2\right )+\frac{1}{6} b x^6 \left (b^2 e (A e+2 B d)+3 b c d (2 A e+B d)+3 A c^2 d^2\right )+\frac{1}{5} b^2 d x^5 (2 A b e+3 A c d+b B d)+\frac{1}{4} A b^3 d^2 x^4+\frac{1}{9} c^2 e x^9 (A c e+3 b B e+2 B c d)+\frac{1}{10} B c^3 e^2 x^{10} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.298026, antiderivative size = 225, 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} \[ \frac{1}{8} c x^8 \left (A c e (3 b e+2 c d)+B \left (3 b^2 e^2+6 b c d e+c^2 d^2\right )\right )+\frac{1}{7} x^7 \left (3 b^2 c e (A e+2 B d)+3 b c^2 d (2 A e+B d)+A c^3 d^2+b^3 B e^2\right )+\frac{1}{6} b x^6 \left (b^2 e (A e+2 B d)+3 b c d (2 A e+B d)+3 A c^2 d^2\right )+\frac{1}{5} b^2 d x^5 (2 A b e+3 A c d+b B d)+\frac{1}{4} A b^3 d^2 x^4+\frac{1}{9} c^2 e x^9 (A c e+3 b B e+2 B c d)+\frac{1}{10} B c^3 e^2 x^{10} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 771
Rubi steps
\begin{align*} \int (A+B x) (d+e x)^2 \left (b x+c x^2\right )^3 \, dx &=\int \left (A b^3 d^2 x^3+b^2 d (b B d+3 A c d+2 A b e) x^4+b \left (3 A c^2 d^2+b^2 e (2 B d+A e)+3 b c d (B d+2 A e)\right ) x^5+\left (A c^3 d^2+b^3 B e^2+3 b^2 c e (2 B d+A e)+3 b c^2 d (B d+2 A e)\right ) x^6+c \left (A c e (2 c d+3 b e)+B \left (c^2 d^2+6 b c d e+3 b^2 e^2\right )\right ) x^7+c^2 e (2 B c d+3 b B e+A c e) x^8+B c^3 e^2 x^9\right ) \, dx\\ &=\frac{1}{4} A b^3 d^2 x^4+\frac{1}{5} b^2 d (b B d+3 A c d+2 A b e) x^5+\frac{1}{6} b \left (3 A c^2 d^2+b^2 e (2 B d+A e)+3 b c d (B d+2 A e)\right ) x^6+\frac{1}{7} \left (A c^3 d^2+b^3 B e^2+3 b^2 c e (2 B d+A e)+3 b c^2 d (B d+2 A e)\right ) x^7+\frac{1}{8} c \left (A c e (2 c d+3 b e)+B \left (c^2 d^2+6 b c d e+3 b^2 e^2\right )\right ) x^8+\frac{1}{9} c^2 e (2 B c d+3 b B e+A c e) x^9+\frac{1}{10} B c^3 e^2 x^{10}\\ \end{align*}
Mathematica [A] time = 0.0825138, size = 225, normalized size = 1. \[ \frac{1}{8} c x^8 \left (A c e (3 b e+2 c d)+B \left (3 b^2 e^2+6 b c d e+c^2 d^2\right )\right )+\frac{1}{7} x^7 \left (3 b^2 c e (A e+2 B d)+3 b c^2 d (2 A e+B d)+A c^3 d^2+b^3 B e^2\right )+\frac{1}{6} b x^6 \left (b^2 e (A e+2 B d)+3 b c d (2 A e+B d)+3 A c^2 d^2\right )+\frac{1}{5} b^2 d x^5 (2 A b e+3 A c d+b B d)+\frac{1}{4} A b^3 d^2 x^4+\frac{1}{9} c^2 e x^9 (A c e+3 b B e+2 B c d)+\frac{1}{10} B c^3 e^2 x^{10} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.001, size = 240, normalized size = 1.1 \begin{align*}{\frac{B{c}^{3}{e}^{2}{x}^{10}}{10}}+{\frac{ \left ( \left ( A{e}^{2}+2\,Bde \right ){c}^{3}+3\,B{e}^{2}b{c}^{2} \right ){x}^{9}}{9}}+{\frac{ \left ( \left ( 2\,Ade+B{d}^{2} \right ){c}^{3}+3\, \left ( A{e}^{2}+2\,Bde \right ) b{c}^{2}+3\,B{e}^{2}{b}^{2}c \right ){x}^{8}}{8}}+{\frac{ \left ( A{c}^{3}{d}^{2}+3\, \left ( 2\,Ade+B{d}^{2} \right ) b{c}^{2}+3\, \left ( A{e}^{2}+2\,Bde \right ){b}^{2}c+{b}^{3}B{e}^{2} \right ){x}^{7}}{7}}+{\frac{ \left ( 3\,A{d}^{2}b{c}^{2}+3\, \left ( 2\,Ade+B{d}^{2} \right ){b}^{2}c+ \left ( A{e}^{2}+2\,Bde \right ){b}^{3} \right ){x}^{6}}{6}}+{\frac{ \left ( 3\,A{d}^{2}{b}^{2}c+ \left ( 2\,Ade+B{d}^{2} \right ){b}^{3} \right ){x}^{5}}{5}}+{\frac{A{b}^{3}{d}^{2}{x}^{4}}{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.989138, size = 327, normalized size = 1.45 \begin{align*} \frac{1}{10} \, B c^{3} e^{2} x^{10} + \frac{1}{4} \, A b^{3} d^{2} x^{4} + \frac{1}{9} \,{\left (2 \, B c^{3} d e +{\left (3 \, B b c^{2} + A c^{3}\right )} e^{2}\right )} x^{9} + \frac{1}{8} \,{\left (B c^{3} d^{2} + 2 \,{\left (3 \, B b c^{2} + A c^{3}\right )} d e + 3 \,{\left (B b^{2} c + A b c^{2}\right )} e^{2}\right )} x^{8} + \frac{1}{7} \,{\left ({\left (3 \, B b c^{2} + A c^{3}\right )} d^{2} + 6 \,{\left (B b^{2} c + A b c^{2}\right )} d e +{\left (B b^{3} + 3 \, A b^{2} c\right )} e^{2}\right )} x^{7} + \frac{1}{6} \,{\left (A b^{3} e^{2} + 3 \,{\left (B b^{2} c + A b c^{2}\right )} d^{2} + 2 \,{\left (B b^{3} + 3 \, A b^{2} c\right )} d e\right )} x^{6} + \frac{1}{5} \,{\left (2 \, A b^{3} d e +{\left (B b^{3} + 3 \, A b^{2} c\right )} d^{2}\right )} x^{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.31952, size = 676, normalized size = 3. \begin{align*} \frac{1}{10} x^{10} e^{2} c^{3} B + \frac{2}{9} x^{9} e d c^{3} B + \frac{1}{3} x^{9} e^{2} c^{2} b B + \frac{1}{9} x^{9} e^{2} c^{3} A + \frac{1}{8} x^{8} d^{2} c^{3} B + \frac{3}{4} x^{8} e d c^{2} b B + \frac{3}{8} x^{8} e^{2} c b^{2} B + \frac{1}{4} x^{8} e d c^{3} A + \frac{3}{8} x^{8} e^{2} c^{2} b A + \frac{3}{7} x^{7} d^{2} c^{2} b B + \frac{6}{7} x^{7} e d c b^{2} B + \frac{1}{7} x^{7} e^{2} b^{3} B + \frac{1}{7} x^{7} d^{2} c^{3} A + \frac{6}{7} x^{7} e d c^{2} b A + \frac{3}{7} x^{7} e^{2} c b^{2} A + \frac{1}{2} x^{6} d^{2} c b^{2} B + \frac{1}{3} x^{6} e d b^{3} B + \frac{1}{2} x^{6} d^{2} c^{2} b A + x^{6} e d c b^{2} A + \frac{1}{6} x^{6} e^{2} b^{3} A + \frac{1}{5} x^{5} d^{2} b^{3} B + \frac{3}{5} x^{5} d^{2} c b^{2} A + \frac{2}{5} x^{5} e d b^{3} A + \frac{1}{4} x^{4} d^{2} b^{3} A \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.110345, size = 303, normalized size = 1.35 \begin{align*} \frac{A b^{3} d^{2} x^{4}}{4} + \frac{B c^{3} e^{2} x^{10}}{10} + x^{9} \left (\frac{A c^{3} e^{2}}{9} + \frac{B b c^{2} e^{2}}{3} + \frac{2 B c^{3} d e}{9}\right ) + x^{8} \left (\frac{3 A b c^{2} e^{2}}{8} + \frac{A c^{3} d e}{4} + \frac{3 B b^{2} c e^{2}}{8} + \frac{3 B b c^{2} d e}{4} + \frac{B c^{3} d^{2}}{8}\right ) + x^{7} \left (\frac{3 A b^{2} c e^{2}}{7} + \frac{6 A b c^{2} d e}{7} + \frac{A c^{3} d^{2}}{7} + \frac{B b^{3} e^{2}}{7} + \frac{6 B b^{2} c d e}{7} + \frac{3 B b c^{2} d^{2}}{7}\right ) + x^{6} \left (\frac{A b^{3} e^{2}}{6} + A b^{2} c d e + \frac{A b c^{2} d^{2}}{2} + \frac{B b^{3} d e}{3} + \frac{B b^{2} c d^{2}}{2}\right ) + x^{5} \left (\frac{2 A b^{3} d e}{5} + \frac{3 A b^{2} c d^{2}}{5} + \frac{B b^{3} d^{2}}{5}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.16377, size = 394, normalized size = 1.75 \begin{align*} \frac{1}{10} \, B c^{3} x^{10} e^{2} + \frac{2}{9} \, B c^{3} d x^{9} e + \frac{1}{8} \, B c^{3} d^{2} x^{8} + \frac{1}{3} \, B b c^{2} x^{9} e^{2} + \frac{1}{9} \, A c^{3} x^{9} e^{2} + \frac{3}{4} \, B b c^{2} d x^{8} e + \frac{1}{4} \, A c^{3} d x^{8} e + \frac{3}{7} \, B b c^{2} d^{2} x^{7} + \frac{1}{7} \, A c^{3} d^{2} x^{7} + \frac{3}{8} \, B b^{2} c x^{8} e^{2} + \frac{3}{8} \, A b c^{2} x^{8} e^{2} + \frac{6}{7} \, B b^{2} c d x^{7} e + \frac{6}{7} \, A b c^{2} d x^{7} e + \frac{1}{2} \, B b^{2} c d^{2} x^{6} + \frac{1}{2} \, A b c^{2} d^{2} x^{6} + \frac{1}{7} \, B b^{3} x^{7} e^{2} + \frac{3}{7} \, A b^{2} c x^{7} e^{2} + \frac{1}{3} \, B b^{3} d x^{6} e + A b^{2} c d x^{6} e + \frac{1}{5} \, B b^{3} d^{2} x^{5} + \frac{3}{5} \, A b^{2} c d^{2} x^{5} + \frac{1}{6} \, A b^{3} x^{6} e^{2} + \frac{2}{5} \, A b^{3} d x^{5} e + \frac{1}{4} \, A b^{3} d^{2} x^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]