Optimal. Leaf size=150 \[ -\frac{b^2 (d+e x)^2 (-3 a B e-A b e+4 b B d)}{2 e^5}-\frac{(b d-a e)^3 (B d-A e)}{e^5 (d+e x)}+\frac{3 b x (b d-a e) (-a B e-A b e+2 b B d)}{e^4}-\frac{(b d-a e)^2 \log (d+e x) (-a B e-3 A b e+4 b B d)}{e^5}+\frac{b^3 B (d+e x)^3}{3 e^5} \]
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Rubi [A] time = 0.178001, antiderivative size = 150, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {77} \[ -\frac{b^2 (d+e x)^2 (-3 a B e-A b e+4 b B d)}{2 e^5}-\frac{(b d-a e)^3 (B d-A e)}{e^5 (d+e x)}+\frac{3 b x (b d-a e) (-a B e-A b e+2 b B d)}{e^4}-\frac{(b d-a e)^2 \log (d+e x) (-a B e-3 A b e+4 b B d)}{e^5}+\frac{b^3 B (d+e x)^3}{3 e^5} \]
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin{align*} \int \frac{(a+b x)^3 (A+B x)}{(d+e x)^2} \, dx &=\int \left (-\frac{3 b (b d-a e) (-2 b B d+A b e+a B e)}{e^4}+\frac{(-b d+a e)^3 (-B d+A e)}{e^4 (d+e x)^2}+\frac{(-b d+a e)^2 (-4 b B d+3 A b e+a B e)}{e^4 (d+e x)}+\frac{b^2 (-4 b B d+A b e+3 a B e) (d+e x)}{e^4}+\frac{b^3 B (d+e x)^2}{e^4}\right ) \, dx\\ &=\frac{3 b (b d-a e) (2 b B d-A b e-a B e) x}{e^4}-\frac{(b d-a e)^3 (B d-A e)}{e^5 (d+e x)}-\frac{b^2 (4 b B d-A b e-3 a B e) (d+e x)^2}{2 e^5}+\frac{b^3 B (d+e x)^3}{3 e^5}-\frac{(b d-a e)^2 (4 b B d-3 A b e-a B e) \log (d+e x)}{e^5}\\ \end{align*}
Mathematica [A] time = 0.107922, size = 244, normalized size = 1.63 \[ \frac{18 a^2 b e^2 \left (A d e+B \left (-d^2+d e x+e^2 x^2\right )\right )+6 a^3 e^3 (B d-A e)+9 a b^2 e \left (2 A e \left (-d^2+d e x+e^2 x^2\right )+B \left (-4 d^2 e x+2 d^3-3 d e^2 x^2+e^3 x^3\right )\right )-6 (d+e x) (b d-a e)^2 \log (d+e x) (-a B e-3 A b e+4 b B d)+b^3 \left (3 A e \left (-4 d^2 e x+2 d^3-3 d e^2 x^2+e^3 x^3\right )+2 B \left (6 d^2 e^2 x^2+9 d^3 e x-3 d^4-2 d e^3 x^3+e^4 x^4\right )\right )}{6 e^5 (d+e x)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.009, size = 376, normalized size = 2.5 \begin{align*}{\frac{{b}^{3}B{x}^{3}}{3\,{e}^{2}}}+{\frac{{b}^{3}A{x}^{2}}{2\,{e}^{2}}}+{\frac{3\,{b}^{2}B{x}^{2}a}{2\,{e}^{2}}}-{\frac{{b}^{3}B{x}^{2}d}{{e}^{3}}}+3\,{\frac{a{b}^{2}Ax}{{e}^{2}}}-2\,{\frac{{b}^{3}Adx}{{e}^{3}}}+3\,{\frac{{a}^{2}bBx}{{e}^{2}}}-6\,{\frac{a{b}^{2}Bdx}{{e}^{3}}}+3\,{\frac{{b}^{3}B{d}^{2}x}{{e}^{4}}}-{\frac{{a}^{3}A}{e \left ( ex+d \right ) }}+3\,{\frac{Ad{a}^{2}b}{{e}^{2} \left ( ex+d \right ) }}-3\,{\frac{a{b}^{2}A{d}^{2}}{{e}^{3} \left ( ex+d \right ) }}+{\frac{{b}^{3}A{d}^{3}}{{e}^{4} \left ( ex+d \right ) }}+{\frac{Bd{a}^{3}}{{e}^{2} \left ( ex+d \right ) }}-3\,{\frac{{a}^{2}bB{d}^{2}}{{e}^{3} \left ( ex+d \right ) }}+3\,{\frac{a{b}^{2}B{d}^{3}}{{e}^{4} \left ( ex+d \right ) }}-{\frac{{b}^{3}B{d}^{4}}{{e}^{5} \left ( ex+d \right ) }}+3\,{\frac{\ln \left ( ex+d \right ) A{a}^{2}b}{{e}^{2}}}-6\,{\frac{\ln \left ( ex+d \right ) Aa{b}^{2}d}{{e}^{3}}}+3\,{\frac{\ln \left ( ex+d \right ) A{b}^{3}{d}^{2}}{{e}^{4}}}+{\frac{\ln \left ( ex+d \right ) B{a}^{3}}{{e}^{2}}}-6\,{\frac{\ln \left ( ex+d \right ) B{a}^{2}bd}{{e}^{3}}}+9\,{\frac{\ln \left ( ex+d \right ) Ba{b}^{2}{d}^{2}}{{e}^{4}}}-4\,{\frac{\ln \left ( ex+d \right ){b}^{3}B{d}^{3}}{{e}^{5}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.64134, size = 360, normalized size = 2.4 \begin{align*} -\frac{B b^{3} d^{4} + A a^{3} e^{4} -{\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 3 \,{\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} -{\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3}}{e^{6} x + d e^{5}} + \frac{2 \, B b^{3} e^{2} x^{3} - 3 \,{\left (2 \, B b^{3} d e -{\left (3 \, B a b^{2} + A b^{3}\right )} e^{2}\right )} x^{2} + 6 \,{\left (3 \, B b^{3} d^{2} - 2 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d e + 3 \,{\left (B a^{2} b + A a b^{2}\right )} e^{2}\right )} x}{6 \, e^{4}} - \frac{{\left (4 \, B b^{3} d^{3} - 3 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e + 6 \,{\left (B a^{2} b + A a b^{2}\right )} d e^{2} -{\left (B a^{3} + 3 \, A a^{2} b\right )} e^{3}\right )} \log \left (e x + d\right )}{e^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.75716, size = 819, normalized size = 5.46 \begin{align*} \frac{2 \, B b^{3} e^{4} x^{4} - 6 \, B b^{3} d^{4} - 6 \, A a^{3} e^{4} + 6 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e - 18 \,{\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} + 6 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3} -{\left (4 \, B b^{3} d e^{3} - 3 \,{\left (3 \, B a b^{2} + A b^{3}\right )} e^{4}\right )} x^{3} + 3 \,{\left (4 \, B b^{3} d^{2} e^{2} - 3 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d e^{3} + 6 \,{\left (B a^{2} b + A a b^{2}\right )} e^{4}\right )} x^{2} + 6 \,{\left (3 \, B b^{3} d^{3} e - 2 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2} + 3 \,{\left (B a^{2} b + A a b^{2}\right )} d e^{3}\right )} x - 6 \,{\left (4 \, B b^{3} d^{4} - 3 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 6 \,{\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} -{\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3} +{\left (4 \, B b^{3} d^{3} e - 3 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2} + 6 \,{\left (B a^{2} b + A a b^{2}\right )} d e^{3} -{\left (B a^{3} + 3 \, A a^{2} b\right )} e^{4}\right )} x\right )} \log \left (e x + d\right )}{6 \,{\left (e^{6} x + d e^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.11098, size = 250, normalized size = 1.67 \begin{align*} \frac{B b^{3} x^{3}}{3 e^{2}} + \frac{- A a^{3} e^{4} + 3 A a^{2} b d e^{3} - 3 A a b^{2} d^{2} e^{2} + A b^{3} d^{3} e + B a^{3} d e^{3} - 3 B a^{2} b d^{2} e^{2} + 3 B a b^{2} d^{3} e - B b^{3} d^{4}}{d e^{5} + e^{6} x} + \frac{x^{2} \left (A b^{3} e + 3 B a b^{2} e - 2 B b^{3} d\right )}{2 e^{3}} + \frac{x \left (3 A a b^{2} e^{2} - 2 A b^{3} d e + 3 B a^{2} b e^{2} - 6 B a b^{2} d e + 3 B b^{3} d^{2}\right )}{e^{4}} + \frac{\left (a e - b d\right )^{2} \left (3 A b e + B a e - 4 B b d\right ) \log{\left (d + e x \right )}}{e^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.23718, size = 489, normalized size = 3.26 \begin{align*} \frac{1}{6} \,{\left (2 \, B b^{3} - \frac{3 \,{\left (4 \, B b^{3} d e - 3 \, B a b^{2} e^{2} - A b^{3} e^{2}\right )} e^{\left (-1\right )}}{x e + d} + \frac{18 \,{\left (2 \, B b^{3} d^{2} e^{2} - 3 \, B a b^{2} d e^{3} - A b^{3} d e^{3} + B a^{2} b e^{4} + A a b^{2} e^{4}\right )} e^{\left (-2\right )}}{{\left (x e + d\right )}^{2}}\right )}{\left (x e + d\right )}^{3} e^{\left (-5\right )} +{\left (4 \, B b^{3} d^{3} - 9 \, B a b^{2} d^{2} e - 3 \, A b^{3} d^{2} e + 6 \, B a^{2} b d e^{2} + 6 \, A a b^{2} d e^{2} - B a^{3} e^{3} - 3 \, A a^{2} b e^{3}\right )} e^{\left (-5\right )} \log \left (\frac{{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) -{\left (\frac{B b^{3} d^{4} e^{3}}{x e + d} - \frac{3 \, B a b^{2} d^{3} e^{4}}{x e + d} - \frac{A b^{3} d^{3} e^{4}}{x e + d} + \frac{3 \, B a^{2} b d^{2} e^{5}}{x e + d} + \frac{3 \, A a b^{2} d^{2} e^{5}}{x e + d} - \frac{B a^{3} d e^{6}}{x e + d} - \frac{3 \, A a^{2} b d e^{6}}{x e + d} + \frac{A a^{3} e^{7}}{x e + d}\right )} e^{\left (-8\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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