Optimal. Leaf size=118 \[ -\frac{(d+e x)^7 (-A c e-b B e+3 B c d)}{7 e^4}+\frac{(d+e x)^6 (B d (3 c d-2 b e)-A e (2 c d-b e))}{6 e^4}-\frac{d (d+e x)^5 (B d-A e) (c d-b e)}{5 e^4}+\frac{B c (d+e x)^8}{8 e^4} \]
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Rubi [A] time = 0.175077, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {771} \[ -\frac{(d+e x)^7 (-A c e-b B e+3 B c d)}{7 e^4}+\frac{(d+e x)^6 (B d (3 c d-2 b e)-A e (2 c d-b e))}{6 e^4}-\frac{d (d+e x)^5 (B d-A e) (c d-b e)}{5 e^4}+\frac{B c (d+e x)^8}{8 e^4} \]
Antiderivative was successfully verified.
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Rule 771
Rubi steps
\begin{align*} \int (A+B x) (d+e x)^4 \left (b x+c x^2\right ) \, dx &=\int \left (-\frac{d (B d-A e) (c d-b e) (d+e x)^4}{e^3}+\frac{(B d (3 c d-2 b e)-A e (2 c d-b e)) (d+e x)^5}{e^3}+\frac{(-3 B c d+b B e+A c e) (d+e x)^6}{e^3}+\frac{B c (d+e x)^7}{e^3}\right ) \, dx\\ &=-\frac{d (B d-A e) (c d-b e) (d+e x)^5}{5 e^4}+\frac{(B d (3 c d-2 b e)-A e (2 c d-b e)) (d+e x)^6}{6 e^4}-\frac{(3 B c d-b B e-A c e) (d+e x)^7}{7 e^4}+\frac{B c (d+e x)^8}{8 e^4}\\ \end{align*}
Mathematica [A] time = 0.0647882, size = 177, normalized size = 1.5 \[ \frac{1}{4} d^2 x^4 (2 A e (3 b e+2 c d)+B d (4 b e+c d))+\frac{1}{3} d^3 x^3 (4 A b e+A c d+b B d)+\frac{1}{7} e^3 x^7 (A c e+b B e+4 B c d)+\frac{1}{6} e^2 x^6 (A e (b e+4 c d)+2 B d (2 b e+3 c d))+\frac{2}{5} d e x^5 (A e (2 b e+3 c d)+B d (3 b e+2 c d))+\frac{1}{2} A b d^4 x^2+\frac{1}{8} B c e^4 x^8 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 200, normalized size = 1.7 \begin{align*}{\frac{B{e}^{4}c{x}^{8}}{8}}+{\frac{ \left ( \left ( A{e}^{4}+4\,Bd{e}^{3} \right ) c+B{e}^{4}b \right ){x}^{7}}{7}}+{\frac{ \left ( \left ( 4\,Ad{e}^{3}+6\,B{d}^{2}{e}^{2} \right ) c+ \left ( A{e}^{4}+4\,Bd{e}^{3} \right ) b \right ){x}^{6}}{6}}+{\frac{ \left ( \left ( 6\,A{d}^{2}{e}^{2}+4\,B{d}^{3}e \right ) c+ \left ( 4\,Ad{e}^{3}+6\,B{d}^{2}{e}^{2} \right ) b \right ){x}^{5}}{5}}+{\frac{ \left ( \left ( 4\,A{d}^{3}e+B{d}^{4} \right ) c+ \left ( 6\,A{d}^{2}{e}^{2}+4\,B{d}^{3}e \right ) b \right ){x}^{4}}{4}}+{\frac{ \left ( A{d}^{4}c+ \left ( 4\,A{d}^{3}e+B{d}^{4} \right ) b \right ){x}^{3}}{3}}+{\frac{A{d}^{4}b{x}^{2}}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.995372, size = 240, normalized size = 2.03 \begin{align*} \frac{1}{8} \, B c e^{4} x^{8} + \frac{1}{2} \, A b d^{4} x^{2} + \frac{1}{7} \,{\left (4 \, B c d e^{3} +{\left (B b + A c\right )} e^{4}\right )} x^{7} + \frac{1}{6} \,{\left (6 \, B c d^{2} e^{2} + A b e^{4} + 4 \,{\left (B b + A c\right )} d e^{3}\right )} x^{6} + \frac{2}{5} \,{\left (2 \, B c d^{3} e + 2 \, A b d e^{3} + 3 \,{\left (B b + A c\right )} d^{2} e^{2}\right )} x^{5} + \frac{1}{4} \,{\left (B c d^{4} + 6 \, A b d^{2} e^{2} + 4 \,{\left (B b + A c\right )} d^{3} e\right )} x^{4} + \frac{1}{3} \,{\left (4 \, A b d^{3} e +{\left (B b + A c\right )} d^{4}\right )} x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.24413, size = 512, normalized size = 4.34 \begin{align*} \frac{1}{8} x^{8} e^{4} c B + \frac{4}{7} x^{7} e^{3} d c B + \frac{1}{7} x^{7} e^{4} b B + \frac{1}{7} x^{7} e^{4} c A + x^{6} e^{2} d^{2} c B + \frac{2}{3} x^{6} e^{3} d b B + \frac{2}{3} x^{6} e^{3} d c A + \frac{1}{6} x^{6} e^{4} b A + \frac{4}{5} x^{5} e d^{3} c B + \frac{6}{5} x^{5} e^{2} d^{2} b B + \frac{6}{5} x^{5} e^{2} d^{2} c A + \frac{4}{5} x^{5} e^{3} d b A + \frac{1}{4} x^{4} d^{4} c B + x^{4} e d^{3} b B + x^{4} e d^{3} c A + \frac{3}{2} x^{4} e^{2} d^{2} b A + \frac{1}{3} x^{3} d^{4} b B + \frac{1}{3} x^{3} d^{4} c A + \frac{4}{3} x^{3} e d^{3} b A + \frac{1}{2} x^{2} d^{4} b A \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.107564, size = 230, normalized size = 1.95 \begin{align*} \frac{A b d^{4} x^{2}}{2} + \frac{B c e^{4} x^{8}}{8} + x^{7} \left (\frac{A c e^{4}}{7} + \frac{B b e^{4}}{7} + \frac{4 B c d e^{3}}{7}\right ) + x^{6} \left (\frac{A b e^{4}}{6} + \frac{2 A c d e^{3}}{3} + \frac{2 B b d e^{3}}{3} + B c d^{2} e^{2}\right ) + x^{5} \left (\frac{4 A b d e^{3}}{5} + \frac{6 A c d^{2} e^{2}}{5} + \frac{6 B b d^{2} e^{2}}{5} + \frac{4 B c d^{3} e}{5}\right ) + x^{4} \left (\frac{3 A b d^{2} e^{2}}{2} + A c d^{3} e + B b d^{3} e + \frac{B c d^{4}}{4}\right ) + x^{3} \left (\frac{4 A b d^{3} e}{3} + \frac{A c d^{4}}{3} + \frac{B b d^{4}}{3}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.27051, size = 284, normalized size = 2.41 \begin{align*} \frac{1}{8} \, B c x^{8} e^{4} + \frac{4}{7} \, B c d x^{7} e^{3} + B c d^{2} x^{6} e^{2} + \frac{4}{5} \, B c d^{3} x^{5} e + \frac{1}{4} \, B c d^{4} x^{4} + \frac{1}{7} \, B b x^{7} e^{4} + \frac{1}{7} \, A c x^{7} e^{4} + \frac{2}{3} \, B b d x^{6} e^{3} + \frac{2}{3} \, A c d x^{6} e^{3} + \frac{6}{5} \, B b d^{2} x^{5} e^{2} + \frac{6}{5} \, A c d^{2} x^{5} e^{2} + B b d^{3} x^{4} e + A c d^{3} x^{4} e + \frac{1}{3} \, B b d^{4} x^{3} + \frac{1}{3} \, A c d^{4} x^{3} + \frac{1}{6} \, A b x^{6} e^{4} + \frac{4}{5} \, A b d x^{5} e^{3} + \frac{3}{2} \, A b d^{2} x^{4} e^{2} + \frac{4}{3} \, A b d^{3} x^{3} e + \frac{1}{2} \, A b d^{4} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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