Optimal. Leaf size=198 \[ \frac{\log (x) \left (b^2 e (A e+2 B d)-3 b c d (2 A e+B d)+6 A c^2 d^2\right )}{b^5}-\frac{\log (b+c x) \left (b^2 e (A e+2 B d)-3 b c d (2 A e+B d)+6 A c^2 d^2\right )}{b^5}-\frac{d (2 A b e-3 A c d+b B d)}{b^4 x}-\frac{(c d-b e) (A b e-3 A c d+2 b B d)}{b^4 (b+c x)}-\frac{(b B-A c) (c d-b e)^2}{2 b^3 c (b+c x)^2}-\frac{A d^2}{2 b^3 x^2} \]
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Rubi [A] time = 0.244863, antiderivative size = 198, 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} \[ \frac{\log (x) \left (b^2 e (A e+2 B d)-3 b c d (2 A e+B d)+6 A c^2 d^2\right )}{b^5}-\frac{\log (b+c x) \left (b^2 e (A e+2 B d)-3 b c d (2 A e+B d)+6 A c^2 d^2\right )}{b^5}-\frac{d (2 A b e-3 A c d+b B d)}{b^4 x}-\frac{(c d-b e) (A b e-3 A c d+2 b B d)}{b^4 (b+c x)}-\frac{(b B-A c) (c d-b e)^2}{2 b^3 c (b+c x)^2}-\frac{A d^2}{2 b^3 x^2} \]
Antiderivative was successfully verified.
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Rule 771
Rubi steps
\begin{align*} \int \frac{(A+B x) (d+e x)^2}{\left (b x+c x^2\right )^3} \, dx &=\int \left (\frac{A d^2}{b^3 x^3}+\frac{d (b B d-3 A c d+2 A b e)}{b^4 x^2}+\frac{6 A c^2 d^2+b^2 e (2 B d+A e)-3 b c d (B d+2 A e)}{b^5 x}+\frac{(b B-A c) (-c d+b e)^2}{b^3 (b+c x)^3}-\frac{c (-c d+b e) (2 b B d-3 A c d+A b e)}{b^4 (b+c x)^2}+\frac{c \left (-6 A c^2 d^2-b^2 e (2 B d+A e)+3 b c d (B d+2 A e)\right )}{b^5 (b+c x)}\right ) \, dx\\ &=-\frac{A d^2}{2 b^3 x^2}-\frac{d (b B d-3 A c d+2 A b e)}{b^4 x}-\frac{(b B-A c) (c d-b e)^2}{2 b^3 c (b+c x)^2}-\frac{(c d-b e) (2 b B d-3 A c d+A b e)}{b^4 (b+c x)}+\frac{\left (6 A c^2 d^2+b^2 e (2 B d+A e)-3 b c d (B d+2 A e)\right ) \log (x)}{b^5}-\frac{\left (6 A c^2 d^2+b^2 e (2 B d+A e)-3 b c d (B d+2 A e)\right ) \log (b+c x)}{b^5}\\ \end{align*}
Mathematica [A] time = 0.308536, size = 190, normalized size = 0.96 \[ -\frac{-2 \log (x) \left (b^2 e (A e+2 B d)-3 b c d (2 A e+B d)+6 A c^2 d^2\right )+2 \log (b+c x) \left (b^2 e (A e+2 B d)-3 b c d (2 A e+B d)+6 A c^2 d^2\right )+\frac{b^2 (b B-A c) (c d-b e)^2}{c (b+c x)^2}+\frac{A b^2 d^2}{x^2}+\frac{2 b d (2 A b e-3 A c d+b B d)}{x}-\frac{2 b (b e-c d) (A b e-3 A c d+2 b B d)}{b+c x}}{2 b^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.014, size = 365, normalized size = 1.8 \begin{align*} -{\frac{A{d}^{2}}{2\,{b}^{3}{x}^{2}}}-{\frac{Acde}{{b}^{2} \left ( cx+b \right ) ^{2}}}+6\,{\frac{\ln \left ( cx+b \right ) Acde}{{b}^{4}}}-4\,{\frac{Acde}{{b}^{3} \left ( cx+b \right ) }}-6\,{\frac{Ac\ln \left ( x \right ) de}{{b}^{4}}}+{\frac{\ln \left ( x \right ) A{e}^{2}}{{b}^{3}}}-{\frac{B{d}^{2}}{{b}^{3}x}}+{\frac{A{e}^{2}}{2\,b \left ( cx+b \right ) ^{2}}}-{\frac{B{e}^{2}}{2\,c \left ( cx+b \right ) ^{2}}}-{\frac{\ln \left ( cx+b \right ) A{e}^{2}}{{b}^{3}}}+{\frac{A{e}^{2}}{{b}^{2} \left ( cx+b \right ) }}-{\frac{Bc{d}^{2}}{2\,{b}^{2} \left ( cx+b \right ) ^{2}}}+6\,{\frac{\ln \left ( x \right ) A{c}^{2}{d}^{2}}{{b}^{5}}}+2\,{\frac{\ln \left ( x \right ) Bde}{{b}^{3}}}-3\,{\frac{Bc\ln \left ( x \right ){d}^{2}}{{b}^{4}}}-2\,{\frac{Ade}{{b}^{3}x}}+3\,{\frac{Ac{d}^{2}}{{b}^{4}x}}-2\,{\frac{\ln \left ( cx+b \right ) Bde}{{b}^{3}}}+3\,{\frac{\ln \left ( cx+b \right ) Bc{d}^{2}}{{b}^{4}}}+3\,{\frac{A{c}^{2}{d}^{2}}{{b}^{4} \left ( cx+b \right ) }}+2\,{\frac{Bde}{{b}^{2} \left ( cx+b \right ) }}-2\,{\frac{Bc{d}^{2}}{{b}^{3} \left ( cx+b \right ) }}+{\frac{A{c}^{2}{d}^{2}}{2\,{b}^{3} \left ( cx+b \right ) ^{2}}}+{\frac{Bde}{b \left ( cx+b \right ) ^{2}}}-6\,{\frac{\ln \left ( cx+b \right ) A{c}^{2}{d}^{2}}{{b}^{5}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.21556, size = 396, normalized size = 2. \begin{align*} -\frac{A b^{3} c d^{2} - 2 \,{\left (A b^{2} c^{2} e^{2} - 3 \,{\left (B b c^{3} - 2 \, A c^{4}\right )} d^{2} + 2 \,{\left (B b^{2} c^{2} - 3 \, A b c^{3}\right )} d e\right )} x^{3} +{\left (9 \,{\left (B b^{2} c^{2} - 2 \, A b c^{3}\right )} d^{2} - 6 \,{\left (B b^{3} c - 3 \, A b^{2} c^{2}\right )} d e +{\left (B b^{4} - 3 \, A b^{3} c\right )} e^{2}\right )} x^{2} + 2 \,{\left (2 \, A b^{3} c d e +{\left (B b^{3} c - 2 \, A b^{2} c^{2}\right )} d^{2}\right )} x}{2 \,{\left (b^{4} c^{3} x^{4} + 2 \, b^{5} c^{2} x^{3} + b^{6} c x^{2}\right )}} - \frac{{\left (A b^{2} e^{2} - 3 \,{\left (B b c - 2 \, A c^{2}\right )} d^{2} + 2 \,{\left (B b^{2} - 3 \, A b c\right )} d e\right )} \log \left (c x + b\right )}{b^{5}} + \frac{{\left (A b^{2} e^{2} - 3 \,{\left (B b c - 2 \, A c^{2}\right )} d^{2} + 2 \,{\left (B b^{2} - 3 \, A b c\right )} d e\right )} \log \left (x\right )}{b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.89906, size = 1142, normalized size = 5.77 \begin{align*} -\frac{A b^{4} c d^{2} - 2 \,{\left (A b^{3} c^{2} e^{2} - 3 \,{\left (B b^{2} c^{3} - 2 \, A b c^{4}\right )} d^{2} + 2 \,{\left (B b^{3} c^{2} - 3 \, A b^{2} c^{3}\right )} d e\right )} x^{3} +{\left (9 \,{\left (B b^{3} c^{2} - 2 \, A b^{2} c^{3}\right )} d^{2} - 6 \,{\left (B b^{4} c - 3 \, A b^{3} c^{2}\right )} d e +{\left (B b^{5} - 3 \, A b^{4} c\right )} e^{2}\right )} x^{2} + 2 \,{\left (2 \, A b^{4} c d e +{\left (B b^{4} c - 2 \, A b^{3} c^{2}\right )} d^{2}\right )} x + 2 \,{\left ({\left (A b^{2} c^{3} e^{2} - 3 \,{\left (B b c^{4} - 2 \, A c^{5}\right )} d^{2} + 2 \,{\left (B b^{2} c^{3} - 3 \, A b c^{4}\right )} d e\right )} x^{4} + 2 \,{\left (A b^{3} c^{2} e^{2} - 3 \,{\left (B b^{2} c^{3} - 2 \, A b c^{4}\right )} d^{2} + 2 \,{\left (B b^{3} c^{2} - 3 \, A b^{2} c^{3}\right )} d e\right )} x^{3} +{\left (A b^{4} c e^{2} - 3 \,{\left (B b^{3} c^{2} - 2 \, A b^{2} c^{3}\right )} d^{2} + 2 \,{\left (B b^{4} c - 3 \, A b^{3} c^{2}\right )} d e\right )} x^{2}\right )} \log \left (c x + b\right ) - 2 \,{\left ({\left (A b^{2} c^{3} e^{2} - 3 \,{\left (B b c^{4} - 2 \, A c^{5}\right )} d^{2} + 2 \,{\left (B b^{2} c^{3} - 3 \, A b c^{4}\right )} d e\right )} x^{4} + 2 \,{\left (A b^{3} c^{2} e^{2} - 3 \,{\left (B b^{2} c^{3} - 2 \, A b c^{4}\right )} d^{2} + 2 \,{\left (B b^{3} c^{2} - 3 \, A b^{2} c^{3}\right )} d e\right )} x^{3} +{\left (A b^{4} c e^{2} - 3 \,{\left (B b^{3} c^{2} - 2 \, A b^{2} c^{3}\right )} d^{2} + 2 \,{\left (B b^{4} c - 3 \, A b^{3} c^{2}\right )} d e\right )} x^{2}\right )} \log \left (x\right )}{2 \,{\left (b^{5} c^{3} x^{4} + 2 \, b^{6} c^{2} x^{3} + b^{7} c x^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 20.631, size = 660, normalized size = 3.33 \begin{align*} \frac{- A b^{3} c d^{2} + x^{3} \left (2 A b^{2} c^{2} e^{2} - 12 A b c^{3} d e + 12 A c^{4} d^{2} + 4 B b^{2} c^{2} d e - 6 B b c^{3} d^{2}\right ) + x^{2} \left (3 A b^{3} c e^{2} - 18 A b^{2} c^{2} d e + 18 A b c^{3} d^{2} - B b^{4} e^{2} + 6 B b^{3} c d e - 9 B b^{2} c^{2} d^{2}\right ) + x \left (- 4 A b^{3} c d e + 4 A b^{2} c^{2} d^{2} - 2 B b^{3} c d^{2}\right )}{2 b^{6} c x^{2} + 4 b^{5} c^{2} x^{3} + 2 b^{4} c^{3} x^{4}} + \frac{\left (A b^{2} e^{2} - 6 A b c d e + 6 A c^{2} d^{2} + 2 B b^{2} d e - 3 B b c d^{2}\right ) \log{\left (x + \frac{A b^{3} e^{2} - 6 A b^{2} c d e + 6 A b c^{2} d^{2} + 2 B b^{3} d e - 3 B b^{2} c d^{2} - b \left (A b^{2} e^{2} - 6 A b c d e + 6 A c^{2} d^{2} + 2 B b^{2} d e - 3 B b c d^{2}\right )}{2 A b^{2} c e^{2} - 12 A b c^{2} d e + 12 A c^{3} d^{2} + 4 B b^{2} c d e - 6 B b c^{2} d^{2}} \right )}}{b^{5}} - \frac{\left (A b^{2} e^{2} - 6 A b c d e + 6 A c^{2} d^{2} + 2 B b^{2} d e - 3 B b c d^{2}\right ) \log{\left (x + \frac{A b^{3} e^{2} - 6 A b^{2} c d e + 6 A b c^{2} d^{2} + 2 B b^{3} d e - 3 B b^{2} c d^{2} + b \left (A b^{2} e^{2} - 6 A b c d e + 6 A c^{2} d^{2} + 2 B b^{2} d e - 3 B b c d^{2}\right )}{2 A b^{2} c e^{2} - 12 A b c^{2} d e + 12 A c^{3} d^{2} + 4 B b^{2} c d e - 6 B b c^{2} d^{2}} \right )}}{b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.38448, size = 437, normalized size = 2.21 \begin{align*} -\frac{{\left (3 \, B b c d^{2} - 6 \, A c^{2} d^{2} - 2 \, B b^{2} d e + 6 \, A b c d e - A b^{2} e^{2}\right )} \log \left ({\left | x \right |}\right )}{b^{5}} + \frac{{\left (3 \, B b c^{2} d^{2} - 6 \, A c^{3} d^{2} - 2 \, B b^{2} c d e + 6 \, A b c^{2} d e - A b^{2} c e^{2}\right )} \log \left ({\left | c x + b \right |}\right )}{b^{5} c} - \frac{6 \, B b c^{3} d^{2} x^{3} - 12 \, A c^{4} d^{2} x^{3} - 4 \, B b^{2} c^{2} d x^{3} e + 12 \, A b c^{3} d x^{3} e + 9 \, B b^{2} c^{2} d^{2} x^{2} - 18 \, A b c^{3} d^{2} x^{2} - 2 \, A b^{2} c^{2} x^{3} e^{2} - 6 \, B b^{3} c d x^{2} e + 18 \, A b^{2} c^{2} d x^{2} e + 2 \, B b^{3} c d^{2} x - 4 \, A b^{2} c^{2} d^{2} x + B b^{4} x^{2} e^{2} - 3 \, A b^{3} c x^{2} e^{2} + 4 \, A b^{3} c d x e + A b^{3} c d^{2}}{2 \,{\left (c x^{2} + b x\right )}^{2} b^{4} c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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