Optimal. Leaf size=103 \[ -\frac{d^2 x \left (a^2 d^2+6 b^2 c^2\right )}{b^4}-\frac{(a d+b c)^4 \log (a-b x)}{2 a b^5}+\frac{(b c-a d)^4 \log (a+b x)}{2 a b^5}-\frac{2 c d^3 x^2}{b^2}-\frac{d^4 x^3}{3 b^2} \]
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Rubi [A] time = 0.101209, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043, Rules used = {72} \[ -\frac{d^2 x \left (a^2 d^2+6 b^2 c^2\right )}{b^4}-\frac{(a d+b c)^4 \log (a-b x)}{2 a b^5}+\frac{(b c-a d)^4 \log (a+b x)}{2 a b^5}-\frac{2 c d^3 x^2}{b^2}-\frac{d^4 x^3}{3 b^2} \]
Antiderivative was successfully verified.
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Rule 72
Rubi steps
\begin{align*} \int \frac{(c+d x)^4}{(a-b x) (a+b x)} \, dx &=\int \left (\frac{-6 b^2 c^2 d^2-a^2 d^4}{b^4}-\frac{4 c d^3 x}{b^2}-\frac{d^4 x^2}{b^2}+\frac{(b c+a d)^4}{2 a b^4 (a-b x)}+\frac{(-b c+a d)^4}{2 a b^4 (a+b x)}\right ) \, dx\\ &=-\frac{d^2 \left (6 b^2 c^2+a^2 d^2\right ) x}{b^4}-\frac{2 c d^3 x^2}{b^2}-\frac{d^4 x^3}{3 b^2}-\frac{(b c+a d)^4 \log (a-b x)}{2 a b^5}+\frac{(b c-a d)^4 \log (a+b x)}{2 a b^5}\\ \end{align*}
Mathematica [A] time = 0.0485051, size = 86, normalized size = 0.83 \[ \frac{-2 a b d^2 x \left (3 a^2 d^2+b^2 \left (18 c^2+6 c d x+d^2 x^2\right )\right )+3 (b c-a d)^4 \log (a+b x)-3 (a d+b c)^4 \log (a-b x)}{6 a b^5} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.007, size = 229, normalized size = 2.2 \begin{align*} -{\frac{{d}^{4}{x}^{3}}{3\,{b}^{2}}}-2\,{\frac{c{d}^{3}{x}^{2}}{{b}^{2}}}-{\frac{{d}^{4}{a}^{2}x}{{b}^{4}}}-6\,{\frac{{d}^{2}{c}^{2}x}{{b}^{2}}}+{\frac{{a}^{3}\ln \left ( bx+a \right ){d}^{4}}{2\,{b}^{5}}}-2\,{\frac{{a}^{2}\ln \left ( bx+a \right ) c{d}^{3}}{{b}^{4}}}+3\,{\frac{a\ln \left ( bx+a \right ){c}^{2}{d}^{2}}{{b}^{3}}}-2\,{\frac{\ln \left ( bx+a \right ){c}^{3}d}{{b}^{2}}}+{\frac{\ln \left ( bx+a \right ){c}^{4}}{2\,ab}}-{\frac{{a}^{3}\ln \left ( bx-a \right ){d}^{4}}{2\,{b}^{5}}}-2\,{\frac{{a}^{2}\ln \left ( bx-a \right ) c{d}^{3}}{{b}^{4}}}-3\,{\frac{a\ln \left ( bx-a \right ){c}^{2}{d}^{2}}{{b}^{3}}}-2\,{\frac{\ln \left ( bx-a \right ){c}^{3}d}{{b}^{2}}}-{\frac{\ln \left ( bx-a \right ){c}^{4}}{2\,ab}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.06548, size = 242, normalized size = 2.35 \begin{align*} -\frac{b^{2} d^{4} x^{3} + 6 \, b^{2} c d^{3} x^{2} + 3 \,{\left (6 \, b^{2} c^{2} d^{2} + a^{2} d^{4}\right )} x}{3 \, b^{4}} + \frac{{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \log \left (b x + a\right )}{2 \, a b^{5}} - \frac{{\left (b^{4} c^{4} + 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} + 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \log \left (b x - a\right )}{2 \, a b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.38803, size = 360, normalized size = 3.5 \begin{align*} -\frac{2 \, a b^{3} d^{4} x^{3} + 12 \, a b^{3} c d^{3} x^{2} + 6 \,{\left (6 \, a b^{3} c^{2} d^{2} + a^{3} b d^{4}\right )} x - 3 \,{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \log \left (b x + a\right ) + 3 \,{\left (b^{4} c^{4} + 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} + 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \log \left (b x - a\right )}{6 \, a b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.35187, size = 214, normalized size = 2.08 \begin{align*} - \frac{2 c d^{3} x^{2}}{b^{2}} - \frac{d^{4} x^{3}}{3 b^{2}} - \frac{x \left (a^{2} d^{4} + 6 b^{2} c^{2} d^{2}\right )}{b^{4}} + \frac{\left (a d - b c\right )^{4} \log{\left (x + \frac{4 a^{4} c d^{3} + 4 a^{2} b^{2} c^{3} d + \frac{a \left (a d - b c\right )^{4}}{b}}{a^{4} d^{4} + 6 a^{2} b^{2} c^{2} d^{2} + b^{4} c^{4}} \right )}}{2 a b^{5}} - \frac{\left (a d + b c\right )^{4} \log{\left (x + \frac{4 a^{4} c d^{3} + 4 a^{2} b^{2} c^{3} d - \frac{a \left (a d + b c\right )^{4}}{b}}{a^{4} d^{4} + 6 a^{2} b^{2} c^{2} d^{2} + b^{4} c^{4}} \right )}}{2 a b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 3.28404, size = 247, normalized size = 2.4 \begin{align*} \frac{{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \log \left ({\left | b x + a \right |}\right )}{2 \, a b^{5}} - \frac{{\left (b^{4} c^{4} + 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} + 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \log \left ({\left | b x - a \right |}\right )}{2 \, a b^{5}} - \frac{b^{4} d^{4} x^{3} + 6 \, b^{4} c d^{3} x^{2} + 18 \, b^{4} c^{2} d^{2} x + 3 \, a^{2} b^{2} d^{4} x}{3 \, b^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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