Optimal. Leaf size=118 \[ -\frac{(d+e x)^6 (-A c e-b B e+3 B c d)}{6 e^4}+\frac{(d+e x)^5 (B d (3 c d-2 b e)-A e (2 c d-b e))}{5 e^4}-\frac{d (d+e x)^4 (B d-A e) (c d-b e)}{4 e^4}+\frac{B c (d+e x)^7}{7 e^4} \]
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Rubi [A] time = 0.136868, 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} \[ -\frac{(d+e x)^6 (-A c e-b B e+3 B c d)}{6 e^4}+\frac{(d+e x)^5 (B d (3 c d-2 b e)-A e (2 c d-b e))}{5 e^4}-\frac{d (d+e x)^4 (B d-A e) (c d-b e)}{4 e^4}+\frac{B c (d+e x)^7}{7 e^4} \]
Antiderivative was successfully verified.
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Rule 771
Rubi steps
\begin{align*} \int (A+B x) (d+e x)^3 \left (b x+c x^2\right ) \, dx &=\int \left (-\frac{d (B d-A e) (c d-b e) (d+e x)^3}{e^3}+\frac{(B d (3 c d-2 b e)-A e (2 c d-b e)) (d+e x)^4}{e^3}+\frac{(-3 B c d+b B e+A c e) (d+e x)^5}{e^3}+\frac{B c (d+e x)^6}{e^3}\right ) \, dx\\ &=-\frac{d (B d-A e) (c d-b e) (d+e x)^4}{4 e^4}+\frac{(B d (3 c d-2 b e)-A e (2 c d-b e)) (d+e x)^5}{5 e^4}-\frac{(3 B c d-b B e-A c e) (d+e x)^6}{6 e^4}+\frac{B c (d+e x)^7}{7 e^4}\\ \end{align*}
Mathematica [A] time = 0.0448671, size = 135, normalized size = 1.14 \[ \frac{1}{3} d^2 x^3 (3 A b e+A c d+b B d)+\frac{1}{6} e^2 x^6 (A c e+b B e+3 B c d)+\frac{1}{5} e x^5 (A e (b e+3 c d)+3 B d (b e+c d))+\frac{1}{4} d x^4 (3 A e (b e+c d)+B d (3 b e+c d))+\frac{1}{2} A b d^3 x^2+\frac{1}{7} B c e^3 x^7 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 152, normalized size = 1.3 \begin{align*}{\frac{B{e}^{3}c{x}^{7}}{7}}+{\frac{ \left ( \left ( A{e}^{3}+3\,Bd{e}^{2} \right ) c+B{e}^{3}b \right ){x}^{6}}{6}}+{\frac{ \left ( \left ( 3\,Ad{e}^{2}+3\,B{d}^{2}e \right ) c+ \left ( A{e}^{3}+3\,Bd{e}^{2} \right ) b \right ){x}^{5}}{5}}+{\frac{ \left ( \left ( 3\,A{d}^{2}e+B{d}^{3} \right ) c+ \left ( 3\,Ad{e}^{2}+3\,B{d}^{2}e \right ) b \right ){x}^{4}}{4}}+{\frac{ \left ( A{d}^{3}c+ \left ( 3\,A{d}^{2}e+B{d}^{3} \right ) b \right ){x}^{3}}{3}}+{\frac{A{d}^{3}b{x}^{2}}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03196, size = 185, normalized size = 1.57 \begin{align*} \frac{1}{7} \, B c e^{3} x^{7} + \frac{1}{2} \, A b d^{3} x^{2} + \frac{1}{6} \,{\left (3 \, B c d e^{2} +{\left (B b + A c\right )} e^{3}\right )} x^{6} + \frac{1}{5} \,{\left (3 \, B c d^{2} e + A b e^{3} + 3 \,{\left (B b + A c\right )} d e^{2}\right )} x^{5} + \frac{1}{4} \,{\left (B c d^{3} + 3 \, A b d e^{2} + 3 \,{\left (B b + A c\right )} d^{2} e\right )} x^{4} + \frac{1}{3} \,{\left (3 \, A b d^{2} e +{\left (B b + A c\right )} d^{3}\right )} x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.25495, size = 404, normalized size = 3.42 \begin{align*} \frac{1}{7} x^{7} e^{3} c B + \frac{1}{2} x^{6} e^{2} d c B + \frac{1}{6} x^{6} e^{3} b B + \frac{1}{6} x^{6} e^{3} c A + \frac{3}{5} x^{5} e d^{2} c B + \frac{3}{5} x^{5} e^{2} d b B + \frac{3}{5} x^{5} e^{2} d c A + \frac{1}{5} x^{5} e^{3} b A + \frac{1}{4} x^{4} d^{3} c B + \frac{3}{4} x^{4} e d^{2} b B + \frac{3}{4} x^{4} e d^{2} c A + \frac{3}{4} x^{4} e^{2} d b A + \frac{1}{3} x^{3} d^{3} b B + \frac{1}{3} x^{3} d^{3} c A + x^{3} e d^{2} b A + \frac{1}{2} x^{2} d^{3} b A \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.090861, size = 177, normalized size = 1.5 \begin{align*} \frac{A b d^{3} x^{2}}{2} + \frac{B c e^{3} x^{7}}{7} + x^{6} \left (\frac{A c e^{3}}{6} + \frac{B b e^{3}}{6} + \frac{B c d e^{2}}{2}\right ) + x^{5} \left (\frac{A b e^{3}}{5} + \frac{3 A c d e^{2}}{5} + \frac{3 B b d e^{2}}{5} + \frac{3 B c d^{2} e}{5}\right ) + x^{4} \left (\frac{3 A b d e^{2}}{4} + \frac{3 A c d^{2} e}{4} + \frac{3 B b d^{2} e}{4} + \frac{B c d^{3}}{4}\right ) + x^{3} \left (A b d^{2} e + \frac{A c d^{3}}{3} + \frac{B b d^{3}}{3}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.24203, size = 221, normalized size = 1.87 \begin{align*} \frac{1}{7} \, B c x^{7} e^{3} + \frac{1}{2} \, B c d x^{6} e^{2} + \frac{3}{5} \, B c d^{2} x^{5} e + \frac{1}{4} \, B c d^{3} x^{4} + \frac{1}{6} \, B b x^{6} e^{3} + \frac{1}{6} \, A c x^{6} e^{3} + \frac{3}{5} \, B b d x^{5} e^{2} + \frac{3}{5} \, A c d x^{5} e^{2} + \frac{3}{4} \, B b d^{2} x^{4} e + \frac{3}{4} \, A c d^{2} x^{4} e + \frac{1}{3} \, B b d^{3} x^{3} + \frac{1}{3} \, A c d^{3} x^{3} + \frac{1}{5} \, A b x^{5} e^{3} + \frac{3}{4} \, A b d x^{4} e^{2} + A b d^{2} x^{3} e + \frac{1}{2} \, A b d^{3} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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