Optimal. Leaf size=141 \[ \frac{e^2 x (-3 a B e+A b e+3 b B d)}{b^4}-\frac{(b d-a e)^2 (-4 a B e+3 A b e+b B d)}{b^5 (a+b x)}-\frac{(A b-a B) (b d-a e)^3}{2 b^5 (a+b x)^2}+\frac{3 e (b d-a e) \log (a+b x) (-2 a B e+A b e+b B d)}{b^5}+\frac{B e^3 x^2}{2 b^3} \]
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Rubi [A] time = 0.156016, antiderivative size = 141, 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} \[ \frac{e^2 x (-3 a B e+A b e+3 b B d)}{b^4}-\frac{(b d-a e)^2 (-4 a B e+3 A b e+b B d)}{b^5 (a+b x)}-\frac{(A b-a B) (b d-a e)^3}{2 b^5 (a+b x)^2}+\frac{3 e (b d-a e) \log (a+b x) (-2 a B e+A b e+b B d)}{b^5}+\frac{B e^3 x^2}{2 b^3} \]
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin{align*} \int \frac{(A+B x) (d+e x)^3}{(a+b x)^3} \, dx &=\int \left (\frac{e^2 (3 b B d+A b e-3 a B e)}{b^4}+\frac{B e^3 x}{b^3}+\frac{(A b-a B) (b d-a e)^3}{b^4 (a+b x)^3}+\frac{(b d-a e)^2 (b B d+3 A b e-4 a B e)}{b^4 (a+b x)^2}+\frac{3 e (b d-a e) (b B d+A b e-2 a B e)}{b^4 (a+b x)}\right ) \, dx\\ &=\frac{e^2 (3 b B d+A b e-3 a B e) x}{b^4}+\frac{B e^3 x^2}{2 b^3}-\frac{(A b-a B) (b d-a e)^3}{2 b^5 (a+b x)^2}-\frac{(b d-a e)^2 (b B d+3 A b e-4 a B e)}{b^5 (a+b x)}+\frac{3 e (b d-a e) (b B d+A b e-2 a B e) \log (a+b x)}{b^5}\\ \end{align*}
Mathematica [A] time = 0.130818, size = 245, normalized size = 1.74 \[ \frac{-A b \left (a^2 b e^2 (4 e x-9 d)+5 a^3 e^3+a b^2 e \left (3 d^2-12 d e x-4 e^2 x^2\right )+b^3 \left (6 d^2 e x+d^3-2 e^3 x^3\right )\right )+B \left (a^2 b^2 e \left (9 d^2-12 d e x-11 e^2 x^2\right )+a^3 b e^2 (2 e x-15 d)+7 a^4 e^3-a b^3 \left (-12 d^2 e x+d^3-12 d e^2 x^2+4 e^3 x^3\right )+b^4 x \left (-2 d^3+6 d e^2 x^2+e^3 x^3\right )\right )+6 e (a+b x)^2 (b d-a e) \log (a+b x) (-2 a B e+A b e+b B d)}{2 b^5 (a+b x)^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.01, size = 404, normalized size = 2.9 \begin{align*}{\frac{B{e}^{3}{x}^{2}}{2\,{b}^{3}}}+{\frac{{e}^{3}Ax}{{b}^{3}}}-3\,{\frac{Ba{e}^{3}x}{{b}^{4}}}+3\,{\frac{{e}^{2}Bdx}{{b}^{3}}}-3\,{\frac{{e}^{3}\ln \left ( bx+a \right ) Aa}{{b}^{4}}}+3\,{\frac{{e}^{2}\ln \left ( bx+a \right ) Ad}{{b}^{3}}}+6\,{\frac{{e}^{3}\ln \left ( bx+a \right ) B{a}^{2}}{{b}^{5}}}-9\,{\frac{{e}^{2}\ln \left ( bx+a \right ) Bad}{{b}^{4}}}+3\,{\frac{e\ln \left ( bx+a \right ) B{d}^{2}}{{b}^{3}}}-3\,{\frac{{a}^{2}A{e}^{3}}{{b}^{4} \left ( bx+a \right ) }}+6\,{\frac{Aad{e}^{2}}{{b}^{3} \left ( bx+a \right ) }}-3\,{\frac{A{d}^{2}e}{{b}^{2} \left ( bx+a \right ) }}+4\,{\frac{B{a}^{3}{e}^{3}}{{b}^{5} \left ( bx+a \right ) }}-9\,{\frac{B{a}^{2}d{e}^{2}}{{b}^{4} \left ( bx+a \right ) }}+6\,{\frac{Ba{d}^{2}e}{{b}^{3} \left ( bx+a \right ) }}-{\frac{B{d}^{3}}{{b}^{2} \left ( bx+a \right ) }}+{\frac{{a}^{3}A{e}^{3}}{2\,{b}^{4} \left ( bx+a \right ) ^{2}}}-{\frac{3\,{a}^{2}Ad{e}^{2}}{2\,{b}^{3} \left ( bx+a \right ) ^{2}}}+{\frac{3\,Aa{d}^{2}e}{2\,{b}^{2} \left ( bx+a \right ) ^{2}}}-{\frac{A{d}^{3}}{2\,b \left ( bx+a \right ) ^{2}}}-{\frac{B{a}^{4}{e}^{3}}{2\,{b}^{5} \left ( bx+a \right ) ^{2}}}+{\frac{3\,B{a}^{3}d{e}^{2}}{2\,{b}^{4} \left ( bx+a \right ) ^{2}}}-{\frac{3\,B{a}^{2}{d}^{2}e}{2\,{b}^{3} \left ( bx+a \right ) ^{2}}}+{\frac{Ba{d}^{3}}{2\,{b}^{2} \left ( bx+a \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.18452, size = 381, normalized size = 2.7 \begin{align*} -\frac{{\left (B a b^{3} + A b^{4}\right )} d^{3} - 3 \,{\left (3 \, B a^{2} b^{2} - A a b^{3}\right )} d^{2} e + 3 \,{\left (5 \, B a^{3} b - 3 \, A a^{2} b^{2}\right )} d e^{2} -{\left (7 \, B a^{4} - 5 \, A a^{3} b\right )} e^{3} + 2 \,{\left (B b^{4} d^{3} - 3 \,{\left (2 \, B a b^{3} - A b^{4}\right )} d^{2} e + 3 \,{\left (3 \, B a^{2} b^{2} - 2 \, A a b^{3}\right )} d e^{2} -{\left (4 \, B a^{3} b - 3 \, A a^{2} b^{2}\right )} e^{3}\right )} x}{2 \,{\left (b^{7} x^{2} + 2 \, a b^{6} x + a^{2} b^{5}\right )}} + \frac{B b e^{3} x^{2} + 2 \,{\left (3 \, B b d e^{2} -{\left (3 \, B a - A b\right )} e^{3}\right )} x}{2 \, b^{4}} + \frac{3 \,{\left (B b^{2} d^{2} e -{\left (3 \, B a b - A b^{2}\right )} d e^{2} +{\left (2 \, B a^{2} - A a b\right )} e^{3}\right )} \log \left (b x + a\right )}{b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.51626, size = 875, normalized size = 6.21 \begin{align*} \frac{B b^{4} e^{3} x^{4} -{\left (B a b^{3} + A b^{4}\right )} d^{3} + 3 \,{\left (3 \, B a^{2} b^{2} - A a b^{3}\right )} d^{2} e - 3 \,{\left (5 \, B a^{3} b - 3 \, A a^{2} b^{2}\right )} d e^{2} +{\left (7 \, B a^{4} - 5 \, A a^{3} b\right )} e^{3} + 2 \,{\left (3 \, B b^{4} d e^{2} -{\left (2 \, B a b^{3} - A b^{4}\right )} e^{3}\right )} x^{3} +{\left (12 \, B a b^{3} d e^{2} -{\left (11 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} e^{3}\right )} x^{2} - 2 \,{\left (B b^{4} d^{3} - 3 \,{\left (2 \, B a b^{3} - A b^{4}\right )} d^{2} e + 6 \,{\left (B a^{2} b^{2} - A a b^{3}\right )} d e^{2} -{\left (B a^{3} b - 2 \, A a^{2} b^{2}\right )} e^{3}\right )} x + 6 \,{\left (B a^{2} b^{2} d^{2} e -{\left (3 \, B a^{3} b - A a^{2} b^{2}\right )} d e^{2} +{\left (2 \, B a^{4} - A a^{3} b\right )} e^{3} +{\left (B b^{4} d^{2} e -{\left (3 \, B a b^{3} - A b^{4}\right )} d e^{2} +{\left (2 \, B a^{2} b^{2} - A a b^{3}\right )} e^{3}\right )} x^{2} + 2 \,{\left (B a b^{3} d^{2} e -{\left (3 \, B a^{2} b^{2} - A a b^{3}\right )} d e^{2} +{\left (2 \, B a^{3} b - A a^{2} b^{2}\right )} e^{3}\right )} x\right )} \log \left (b x + a\right )}{2 \,{\left (b^{7} x^{2} + 2 \, a b^{6} x + a^{2} b^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 5.27665, size = 296, normalized size = 2.1 \begin{align*} \frac{B e^{3} x^{2}}{2 b^{3}} + \frac{- 5 A a^{3} b e^{3} + 9 A a^{2} b^{2} d e^{2} - 3 A a b^{3} d^{2} e - A b^{4} d^{3} + 7 B a^{4} e^{3} - 15 B a^{3} b d e^{2} + 9 B a^{2} b^{2} d^{2} e - B a b^{3} d^{3} + x \left (- 6 A a^{2} b^{2} e^{3} + 12 A a b^{3} d e^{2} - 6 A b^{4} d^{2} e + 8 B a^{3} b e^{3} - 18 B a^{2} b^{2} d e^{2} + 12 B a b^{3} d^{2} e - 2 B b^{4} d^{3}\right )}{2 a^{2} b^{5} + 4 a b^{6} x + 2 b^{7} x^{2}} - \frac{x \left (- A b e^{3} + 3 B a e^{3} - 3 B b d e^{2}\right )}{b^{4}} + \frac{3 e \left (a e - b d\right ) \left (- A b e + 2 B a e - B b d\right ) \log{\left (a + b x \right )}}{b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 3.56607, size = 367, normalized size = 2.6 \begin{align*} \frac{3 \,{\left (B b^{2} d^{2} e - 3 \, B a b d e^{2} + A b^{2} d e^{2} + 2 \, B a^{2} e^{3} - A a b e^{3}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{5}} + \frac{B b^{3} x^{2} e^{3} + 6 \, B b^{3} d x e^{2} - 6 \, B a b^{2} x e^{3} + 2 \, A b^{3} x e^{3}}{2 \, b^{6}} - \frac{B a b^{3} d^{3} + A b^{4} d^{3} - 9 \, B a^{2} b^{2} d^{2} e + 3 \, A a b^{3} d^{2} e + 15 \, B a^{3} b d e^{2} - 9 \, A a^{2} b^{2} d e^{2} - 7 \, B a^{4} e^{3} + 5 \, A a^{3} b e^{3} + 2 \,{\left (B b^{4} d^{3} - 6 \, B a b^{3} d^{2} e + 3 \, A b^{4} d^{2} e + 9 \, B a^{2} b^{2} d e^{2} - 6 \, A a b^{3} d e^{2} - 4 \, B a^{3} b e^{3} + 3 \, A a^{2} b^{2} e^{3}\right )} x}{2 \,{\left (b x + a\right )}^{2} b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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