Optimal. Leaf size=84 \[ \frac{x^2 (d+e x)}{2 c^2 \left (a^2-c^2 x^2\right )}+\frac{(3 a e+2 c d) \log (a-c x)}{4 c^5}+\frac{(2 c d-3 a e) \log (a+c x)}{4 c^5}+\frac{3 e x}{2 c^4} \]
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Rubi [A] time = 0.0734375, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {819, 774, 633, 31} \[ \frac{x^2 (d+e x)}{2 c^2 \left (a^2-c^2 x^2\right )}+\frac{(3 a e+2 c d) \log (a-c x)}{4 c^5}+\frac{(2 c d-3 a e) \log (a+c x)}{4 c^5}+\frac{3 e x}{2 c^4} \]
Antiderivative was successfully verified.
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Rule 819
Rule 774
Rule 633
Rule 31
Rubi steps
\begin{align*} \int \frac{x^3 (d+e x)}{\left (a^2-c^2 x^2\right )^2} \, dx &=\frac{x^2 (d+e x)}{2 c^2 \left (a^2-c^2 x^2\right )}-\frac{\int \frac{x \left (2 a^2 d+3 a^2 e x\right )}{a^2-c^2 x^2} \, dx}{2 a^2 c^2}\\ &=\frac{3 e x}{2 c^4}+\frac{x^2 (d+e x)}{2 c^2 \left (a^2-c^2 x^2\right )}+\frac{\int \frac{-3 a^4 e-2 a^2 c^2 d x}{a^2-c^2 x^2} \, dx}{2 a^2 c^4}\\ &=\frac{3 e x}{2 c^4}+\frac{x^2 (d+e x)}{2 c^2 \left (a^2-c^2 x^2\right )}-\frac{(2 c d-3 a e) \int \frac{1}{-a c-c^2 x} \, dx}{4 c^3}-\frac{(2 c d+3 a e) \int \frac{1}{a c-c^2 x} \, dx}{4 c^3}\\ &=\frac{3 e x}{2 c^4}+\frac{x^2 (d+e x)}{2 c^2 \left (a^2-c^2 x^2\right )}+\frac{(2 c d+3 a e) \log (a-c x)}{4 c^5}+\frac{(2 c d-3 a e) \log (a+c x)}{4 c^5}\\ \end{align*}
Mathematica [A] time = 0.0543913, size = 64, normalized size = 0.76 \[ \frac{\frac{a^2 c (d+e x)}{a^2-c^2 x^2}+c d \log \left (a^2-c^2 x^2\right )-3 a e \tanh ^{-1}\left (\frac{c x}{a}\right )+2 c e x}{2 c^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 126, normalized size = 1.5 \begin{align*}{\frac{ex}{{c}^{4}}}-{\frac{3\,\ln \left ( cx+a \right ) ae}{4\,{c}^{5}}}+{\frac{\ln \left ( cx+a \right ) d}{2\,{c}^{4}}}-{\frac{{a}^{2}e}{4\,{c}^{5} \left ( cx+a \right ) }}+{\frac{ad}{4\,{c}^{4} \left ( cx+a \right ) }}+{\frac{3\,\ln \left ( cx-a \right ) ae}{4\,{c}^{5}}}+{\frac{\ln \left ( cx-a \right ) d}{2\,{c}^{4}}}-{\frac{{a}^{2}e}{4\,{c}^{5} \left ( cx-a \right ) }}-{\frac{ad}{4\,{c}^{4} \left ( cx-a \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03764, size = 109, normalized size = 1.3 \begin{align*} -\frac{a^{2} e x + a^{2} d}{2 \,{\left (c^{6} x^{2} - a^{2} c^{4}\right )}} + \frac{e x}{c^{4}} + \frac{{\left (2 \, c d - 3 \, a e\right )} \log \left (c x + a\right )}{4 \, c^{5}} + \frac{{\left (2 \, c d + 3 \, a e\right )} \log \left (c x - a\right )}{4 \, c^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.53061, size = 263, normalized size = 3.13 \begin{align*} \frac{4 \, c^{3} e x^{3} - 6 \, a^{2} c e x - 2 \, a^{2} c d -{\left (2 \, a^{2} c d - 3 \, a^{3} e -{\left (2 \, c^{3} d - 3 \, a c^{2} e\right )} x^{2}\right )} \log \left (c x + a\right ) -{\left (2 \, a^{2} c d + 3 \, a^{3} e -{\left (2 \, c^{3} d + 3 \, a c^{2} e\right )} x^{2}\right )} \log \left (c x - a\right )}{4 \,{\left (c^{7} x^{2} - a^{2} c^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.15469, size = 109, normalized size = 1.3 \begin{align*} - \frac{a^{2} d + a^{2} e x}{- 2 a^{2} c^{4} + 2 c^{6} x^{2}} + \frac{e x}{c^{4}} - \frac{\left (3 a e - 2 c d\right ) \log{\left (x + \frac{2 d + \frac{3 a e - 2 c d}{c}}{3 e} \right )}}{4 c^{5}} + \frac{\left (3 a e + 2 c d\right ) \log{\left (x + \frac{2 d - \frac{3 a e + 2 c d}{c}}{3 e} \right )}}{4 c^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15325, size = 119, normalized size = 1.42 \begin{align*} \frac{x e}{c^{4}} + \frac{{\left (2 \, c d - 3 \, a e\right )} \log \left ({\left | c x + a \right |}\right )}{4 \, c^{5}} + \frac{{\left (2 \, c d + 3 \, a e\right )} \log \left ({\left | c x - a \right |}\right )}{4 \, c^{5}} - \frac{a^{2} x e + a^{2} d}{2 \,{\left (c x + a\right )}{\left (c x - a\right )} c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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