Optimal. Leaf size=207 \[ \frac{e^2 x^2 \left (A c e (4 c d-b e)+B \left (b^2 e^2-4 b c d e+6 c^2 d^2\right )\right )}{2 c^3}+\frac{e x \left (A c e \left (b^2 e^2-4 b c d e+6 c^2 d^2\right )+B \left (4 b^2 c d e^2-b^3 e^3-6 b c^2 d^2 e+4 c^3 d^3\right )\right )}{c^4}+\frac{e^3 x^3 (A c e-b B e+4 B c d)}{3 c^2}+\frac{(b B-A c) (c d-b e)^4 \log (b+c x)}{b c^5}+\frac{A d^4 \log (x)}{b}+\frac{B e^4 x^4}{4 c} \]
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Rubi [A] time = 0.262597, antiderivative size = 207, 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} \[ \frac{e^2 x^2 \left (A c e (4 c d-b e)+B \left (b^2 e^2-4 b c d e+6 c^2 d^2\right )\right )}{2 c^3}+\frac{e x \left (A c e \left (b^2 e^2-4 b c d e+6 c^2 d^2\right )+B \left (4 b^2 c d e^2-b^3 e^3-6 b c^2 d^2 e+4 c^3 d^3\right )\right )}{c^4}+\frac{e^3 x^3 (A c e-b B e+4 B c d)}{3 c^2}+\frac{(b B-A c) (c d-b e)^4 \log (b+c x)}{b c^5}+\frac{A d^4 \log (x)}{b}+\frac{B e^4 x^4}{4 c} \]
Antiderivative was successfully verified.
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Rule 771
Rubi steps
\begin{align*} \int \frac{(A+B x) (d+e x)^4}{b x+c x^2} \, dx &=\int \left (\frac{e \left (A c e \left (6 c^2 d^2-4 b c d e+b^2 e^2\right )+B \left (4 c^3 d^3-6 b c^2 d^2 e+4 b^2 c d e^2-b^3 e^3\right )\right )}{c^4}+\frac{A d^4}{b x}+\frac{e^2 \left (A c e (4 c d-b e)+B \left (6 c^2 d^2-4 b c d e+b^2 e^2\right )\right ) x}{c^3}+\frac{e^3 (4 B c d-b B e+A c e) x^2}{c^2}+\frac{B e^4 x^3}{c}+\frac{(b B-A c) (-c d+b e)^4}{b c^4 (b+c x)}\right ) \, dx\\ &=\frac{e \left (A c e \left (6 c^2 d^2-4 b c d e+b^2 e^2\right )+B \left (4 c^3 d^3-6 b c^2 d^2 e+4 b^2 c d e^2-b^3 e^3\right )\right ) x}{c^4}+\frac{e^2 \left (A c e (4 c d-b e)+B \left (6 c^2 d^2-4 b c d e+b^2 e^2\right )\right ) x^2}{2 c^3}+\frac{e^3 (4 B c d-b B e+A c e) x^3}{3 c^2}+\frac{B e^4 x^4}{4 c}+\frac{A d^4 \log (x)}{b}+\frac{(b B-A c) (c d-b e)^4 \log (b+c x)}{b c^5}\\ \end{align*}
Mathematica [A] time = 0.149121, size = 187, normalized size = 0.9 \[ \frac{e x \left (2 A c e \left (6 b^2 e^2-3 b c e (8 d+e x)+2 c^2 \left (18 d^2+6 d e x+e^2 x^2\right )\right )+B \left (6 b^2 c e^2 (8 d+e x)-12 b^3 e^3-4 b c^2 e \left (18 d^2+6 d e x+e^2 x^2\right )+c^3 \left (36 d^2 e x+48 d^3+16 d e^2 x^2+3 e^3 x^3\right )\right )\right )}{12 c^4}+\frac{(b B-A c) (c d-b e)^4 \log (b+c x)}{b c^5}+\frac{A d^4 \log (x)}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 396, normalized size = 1.9 \begin{align*} -{\frac{{b}^{3}\ln \left ( cx+b \right ) A{e}^{4}}{{c}^{4}}}+{\frac{B{e}^{4}{x}^{4}}{4\,c}}+4\,{\frac{{b}^{2}\ln \left ( cx+b \right ) Ad{e}^{3}}{{c}^{3}}}-6\,{\frac{b\ln \left ( cx+b \right ) A{d}^{2}{e}^{2}}{{c}^{2}}}-4\,{\frac{{b}^{3}\ln \left ( cx+b \right ) Bd{e}^{3}}{{c}^{4}}}+6\,{\frac{{b}^{2}\ln \left ( cx+b \right ) B{d}^{2}{e}^{2}}{{c}^{3}}}-4\,{\frac{b\ln \left ( cx+b \right ) B{d}^{3}e}{{c}^{2}}}-2\,{\frac{{e}^{3}B{x}^{2}bd}{{c}^{2}}}-4\,{\frac{A{e}^{3}bdx}{{c}^{2}}}+4\,{\frac{B{e}^{3}{b}^{2}dx}{{c}^{3}}}-6\,{\frac{B{e}^{2}b{d}^{2}x}{{c}^{2}}}-{\frac{\ln \left ( cx+b \right ) A{d}^{4}}{b}}+{\frac{\ln \left ( cx+b \right ) B{d}^{4}}{c}}+{\frac{{e}^{4}A{x}^{3}}{3\,c}}+{\frac{A{d}^{4}\ln \left ( x \right ) }{b}}-{\frac{B{e}^{4}{b}^{3}x}{{c}^{4}}}+4\,{\frac{B{d}^{3}ex}{c}}-{\frac{B{e}^{4}{x}^{3}b}{3\,{c}^{2}}}+{\frac{4\,{e}^{3}B{x}^{3}d}{3\,c}}+{\frac{{b}^{4}\ln \left ( cx+b \right ) B{e}^{4}}{{c}^{5}}}-{\frac{{e}^{4}A{x}^{2}b}{2\,{c}^{2}}}+2\,{\frac{A{e}^{3}{x}^{2}d}{c}}+{\frac{B{e}^{4}{x}^{2}{b}^{2}}{2\,{c}^{3}}}+3\,{\frac{{e}^{2}B{x}^{2}{d}^{2}}{c}}+{\frac{{e}^{4}A{b}^{2}x}{{c}^{3}}}+6\,{\frac{A{d}^{2}{e}^{2}x}{c}}+4\,{\frac{\ln \left ( cx+b \right ) A{d}^{3}e}{c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.16961, size = 417, normalized size = 2.01 \begin{align*} \frac{A d^{4} \log \left (x\right )}{b} + \frac{3 \, B c^{3} e^{4} x^{4} + 4 \,{\left (4 \, B c^{3} d e^{3} -{\left (B b c^{2} - A c^{3}\right )} e^{4}\right )} x^{3} + 6 \,{\left (6 \, B c^{3} d^{2} e^{2} - 4 \,{\left (B b c^{2} - A c^{3}\right )} d e^{3} +{\left (B b^{2} c - A b c^{2}\right )} e^{4}\right )} x^{2} + 12 \,{\left (4 \, B c^{3} d^{3} e - 6 \,{\left (B b c^{2} - A c^{3}\right )} d^{2} e^{2} + 4 \,{\left (B b^{2} c - A b c^{2}\right )} d e^{3} -{\left (B b^{3} - A b^{2} c\right )} e^{4}\right )} x}{12 \, c^{4}} + \frac{{\left ({\left (B b c^{4} - A c^{5}\right )} d^{4} - 4 \,{\left (B b^{2} c^{3} - A b c^{4}\right )} d^{3} e + 6 \,{\left (B b^{3} c^{2} - A b^{2} c^{3}\right )} d^{2} e^{2} - 4 \,{\left (B b^{4} c - A b^{3} c^{2}\right )} d e^{3} +{\left (B b^{5} - A b^{4} c\right )} e^{4}\right )} \log \left (c x + b\right )}{b c^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.60668, size = 657, normalized size = 3.17 \begin{align*} \frac{3 \, B b c^{4} e^{4} x^{4} + 12 \, A c^{5} d^{4} \log \left (x\right ) + 4 \,{\left (4 \, B b c^{4} d e^{3} -{\left (B b^{2} c^{3} - A b c^{4}\right )} e^{4}\right )} x^{3} + 6 \,{\left (6 \, B b c^{4} d^{2} e^{2} - 4 \,{\left (B b^{2} c^{3} - A b c^{4}\right )} d e^{3} +{\left (B b^{3} c^{2} - A b^{2} c^{3}\right )} e^{4}\right )} x^{2} + 12 \,{\left (4 \, B b c^{4} d^{3} e - 6 \,{\left (B b^{2} c^{3} - A b c^{4}\right )} d^{2} e^{2} + 4 \,{\left (B b^{3} c^{2} - A b^{2} c^{3}\right )} d e^{3} -{\left (B b^{4} c - A b^{3} c^{2}\right )} e^{4}\right )} x + 12 \,{\left ({\left (B b c^{4} - A c^{5}\right )} d^{4} - 4 \,{\left (B b^{2} c^{3} - A b c^{4}\right )} d^{3} e + 6 \,{\left (B b^{3} c^{2} - A b^{2} c^{3}\right )} d^{2} e^{2} - 4 \,{\left (B b^{4} c - A b^{3} c^{2}\right )} d e^{3} +{\left (B b^{5} - A b^{4} c\right )} e^{4}\right )} \log \left (c x + b\right )}{12 \, b c^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 8.65103, size = 391, normalized size = 1.89 \begin{align*} \frac{A d^{4} \log{\left (x \right )}}{b} + \frac{B e^{4} x^{4}}{4 c} - \frac{x^{3} \left (- A c e^{4} + B b e^{4} - 4 B c d e^{3}\right )}{3 c^{2}} + \frac{x^{2} \left (- A b c e^{4} + 4 A c^{2} d e^{3} + B b^{2} e^{4} - 4 B b c d e^{3} + 6 B c^{2} d^{2} e^{2}\right )}{2 c^{3}} - \frac{x \left (- A b^{2} c e^{4} + 4 A b c^{2} d e^{3} - 6 A c^{3} d^{2} e^{2} + B b^{3} e^{4} - 4 B b^{2} c d e^{3} + 6 B b c^{2} d^{2} e^{2} - 4 B c^{3} d^{3} e\right )}{c^{4}} + \frac{\left (- A c + B b\right ) \left (b e - c d\right )^{4} \log{\left (x + \frac{- A b c^{4} d^{4} + \frac{b \left (- A c + B b\right ) \left (b e - c d\right )^{4}}{c}}{- A b^{4} c e^{4} + 4 A b^{3} c^{2} d e^{3} - 6 A b^{2} c^{3} d^{2} e^{2} + 4 A b c^{4} d^{3} e - 2 A c^{5} d^{4} + B b^{5} e^{4} - 4 B b^{4} c d e^{3} + 6 B b^{3} c^{2} d^{2} e^{2} - 4 B b^{2} c^{3} d^{3} e + B b c^{4} d^{4}} \right )}}{b c^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19658, size = 446, normalized size = 2.15 \begin{align*} \frac{A d^{4} \log \left ({\left | x \right |}\right )}{b} + \frac{3 \, B c^{3} x^{4} e^{4} + 16 \, B c^{3} d x^{3} e^{3} + 36 \, B c^{3} d^{2} x^{2} e^{2} + 48 \, B c^{3} d^{3} x e - 4 \, B b c^{2} x^{3} e^{4} + 4 \, A c^{3} x^{3} e^{4} - 24 \, B b c^{2} d x^{2} e^{3} + 24 \, A c^{3} d x^{2} e^{3} - 72 \, B b c^{2} d^{2} x e^{2} + 72 \, A c^{3} d^{2} x e^{2} + 6 \, B b^{2} c x^{2} e^{4} - 6 \, A b c^{2} x^{2} e^{4} + 48 \, B b^{2} c d x e^{3} - 48 \, A b c^{2} d x e^{3} - 12 \, B b^{3} x e^{4} + 12 \, A b^{2} c x e^{4}}{12 \, c^{4}} + \frac{{\left (B b c^{4} d^{4} - A c^{5} d^{4} - 4 \, B b^{2} c^{3} d^{3} e + 4 \, A b c^{4} d^{3} e + 6 \, B b^{3} c^{2} d^{2} e^{2} - 6 \, A b^{2} c^{3} d^{2} e^{2} - 4 \, B b^{4} c d e^{3} + 4 \, A b^{3} c^{2} d e^{3} + B b^{5} e^{4} - A b^{4} c e^{4}\right )} \log \left ({\left | c x + b \right |}\right )}{b c^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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