Optimal. Leaf size=94 \[ \frac{x^3 (d+e x)}{2 c^2 \left (a^2-c^2 x^2\right )}+\frac{a (4 a e+3 c d) \log (a-c x)}{4 c^6}-\frac{a (3 c d-4 a e) \log (a+c x)}{4 c^6}+\frac{3 d x}{2 c^4}+\frac{e x^2}{c^4} \]
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Rubi [A] time = 0.0988786, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {819, 801, 633, 31} \[ \frac{x^3 (d+e x)}{2 c^2 \left (a^2-c^2 x^2\right )}+\frac{a (4 a e+3 c d) \log (a-c x)}{4 c^6}-\frac{a (3 c d-4 a e) \log (a+c x)}{4 c^6}+\frac{3 d x}{2 c^4}+\frac{e x^2}{c^4} \]
Antiderivative was successfully verified.
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Rule 819
Rule 801
Rule 633
Rule 31
Rubi steps
\begin{align*} \int \frac{x^4 (d+e x)}{\left (a^2-c^2 x^2\right )^2} \, dx &=\frac{x^3 (d+e x)}{2 c^2 \left (a^2-c^2 x^2\right )}-\frac{\int \frac{x^2 \left (3 a^2 d+4 a^2 e x\right )}{a^2-c^2 x^2} \, dx}{2 a^2 c^2}\\ &=\frac{x^3 (d+e x)}{2 c^2 \left (a^2-c^2 x^2\right )}-\frac{\int \left (-\frac{3 a^2 d}{c^2}-\frac{4 a^2 e x}{c^2}+\frac{3 a^4 d+4 a^4 e x}{c^2 \left (a^2-c^2 x^2\right )}\right ) \, dx}{2 a^2 c^2}\\ &=\frac{3 d x}{2 c^4}+\frac{e x^2}{c^4}+\frac{x^3 (d+e x)}{2 c^2 \left (a^2-c^2 x^2\right )}-\frac{\int \frac{3 a^4 d+4 a^4 e x}{a^2-c^2 x^2} \, dx}{2 a^2 c^4}\\ &=\frac{3 d x}{2 c^4}+\frac{e x^2}{c^4}+\frac{x^3 (d+e x)}{2 c^2 \left (a^2-c^2 x^2\right )}+\frac{(a (3 c d-4 a e)) \int \frac{1}{-a c-c^2 x} \, dx}{4 c^4}-\frac{(a (3 c d+4 a e)) \int \frac{1}{a c-c^2 x} \, dx}{4 c^4}\\ &=\frac{3 d x}{2 c^4}+\frac{e x^2}{c^4}+\frac{x^3 (d+e x)}{2 c^2 \left (a^2-c^2 x^2\right )}+\frac{a (3 c d+4 a e) \log (a-c x)}{4 c^6}-\frac{a (3 c d-4 a e) \log (a+c x)}{4 c^6}\\ \end{align*}
Mathematica [A] time = 0.0478109, size = 84, normalized size = 0.89 \[ \frac{\frac{a^2 c^2 d x+a^4 e}{a^2-c^2 x^2}+2 a^2 e \log \left (a^2-c^2 x^2\right )-3 a c d \tanh ^{-1}\left (\frac{c x}{a}\right )+2 c^2 d x+c^2 e x^2}{2 c^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.014, size = 143, normalized size = 1.5 \begin{align*}{\frac{e{x}^{2}}{2\,{c}^{4}}}+{\frac{dx}{{c}^{4}}}+{\frac{{a}^{2}\ln \left ( cx+a \right ) e}{{c}^{6}}}-{\frac{3\,a\ln \left ( cx+a \right ) d}{4\,{c}^{5}}}+{\frac{{a}^{3}e}{4\,{c}^{6} \left ( cx+a \right ) }}-{\frac{{a}^{2}d}{4\,{c}^{5} \left ( cx+a \right ) }}+{\frac{{a}^{2}\ln \left ( cx-a \right ) e}{{c}^{6}}}+{\frac{3\,a\ln \left ( cx-a \right ) d}{4\,{c}^{5}}}-{\frac{{a}^{3}e}{4\,{c}^{6} \left ( cx-a \right ) }}-{\frac{{a}^{2}d}{4\,{c}^{5} \left ( cx-a \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.07973, size = 134, normalized size = 1.43 \begin{align*} -\frac{a^{2} c^{2} d x + a^{4} e}{2 \,{\left (c^{8} x^{2} - a^{2} c^{6}\right )}} + \frac{e x^{2} + 2 \, d x}{2 \, c^{4}} - \frac{{\left (3 \, a c d - 4 \, a^{2} e\right )} \log \left (c x + a\right )}{4 \, c^{6}} + \frac{{\left (3 \, a c d + 4 \, a^{2} e\right )} \log \left (c x - a\right )}{4 \, c^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.53232, size = 317, normalized size = 3.37 \begin{align*} \frac{2 \, c^{4} e x^{4} + 4 \, c^{4} d x^{3} - 2 \, a^{2} c^{2} e x^{2} - 6 \, a^{2} c^{2} d x - 2 \, a^{4} e +{\left (3 \, a^{3} c d - 4 \, a^{4} e -{\left (3 \, a c^{3} d - 4 \, a^{2} c^{2} e\right )} x^{2}\right )} \log \left (c x + a\right ) -{\left (3 \, a^{3} c d + 4 \, a^{4} e -{\left (3 \, a c^{3} d + 4 \, a^{2} c^{2} e\right )} x^{2}\right )} \log \left (c x - a\right )}{4 \,{\left (c^{8} x^{2} - a^{2} c^{6}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.21857, size = 139, normalized size = 1.48 \begin{align*} \frac{a \left (4 a e - 3 c d\right ) \log{\left (x + \frac{4 a^{2} e - a \left (4 a e - 3 c d\right )}{3 c^{2} d} \right )}}{4 c^{6}} + \frac{a \left (4 a e + 3 c d\right ) \log{\left (x + \frac{4 a^{2} e - a \left (4 a e + 3 c d\right )}{3 c^{2} d} \right )}}{4 c^{6}} - \frac{a^{4} e + a^{2} c^{2} d x}{- 2 a^{2} c^{6} + 2 c^{8} x^{2}} + \frac{d x}{c^{4}} + \frac{e x^{2}}{2 c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13846, size = 151, normalized size = 1.61 \begin{align*} -\frac{{\left (3 \, a c d - 4 \, a^{2} e\right )} \log \left ({\left | c x + a \right |}\right )}{4 \, c^{6}} + \frac{{\left (3 \, a c d + 4 \, a^{2} e\right )} \log \left ({\left | c x - a \right |}\right )}{4 \, c^{6}} + \frac{c^{4} x^{2} e + 2 \, c^{4} d x}{2 \, c^{8}} - \frac{a^{2} c^{2} d x + a^{4} e}{2 \,{\left (c x + a\right )}{\left (c x - a\right )} c^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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