Optimal. Leaf size=95 \[ -\frac{a^3 (a e+c d) \log (a-c x)}{2 c^6}+\frac{a^3 (c d-a e) \log (a+c x)}{2 c^6}-\frac{a^2 d x}{c^4}-\frac{a^2 e x^2}{2 c^4}-\frac{d x^3}{3 c^2}-\frac{e x^4}{4 c^2} \]
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Rubi [A] time = 0.0700957, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {801, 633, 31} \[ -\frac{a^3 (a e+c d) \log (a-c x)}{2 c^6}+\frac{a^3 (c d-a e) \log (a+c x)}{2 c^6}-\frac{a^2 d x}{c^4}-\frac{a^2 e x^2}{2 c^4}-\frac{d x^3}{3 c^2}-\frac{e x^4}{4 c^2} \]
Antiderivative was successfully verified.
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Rule 801
Rule 633
Rule 31
Rubi steps
\begin{align*} \int \frac{x^4 (d+e x)}{a^2-c^2 x^2} \, dx &=\int \left (-\frac{a^2 d}{c^4}-\frac{a^2 e x}{c^4}-\frac{d x^2}{c^2}-\frac{e x^3}{c^2}+\frac{a^4 d+a^4 e x}{c^4 \left (a^2-c^2 x^2\right )}\right ) \, dx\\ &=-\frac{a^2 d x}{c^4}-\frac{a^2 e x^2}{2 c^4}-\frac{d x^3}{3 c^2}-\frac{e x^4}{4 c^2}+\frac{\int \frac{a^4 d+a^4 e x}{a^2-c^2 x^2} \, dx}{c^4}\\ &=-\frac{a^2 d x}{c^4}-\frac{a^2 e x^2}{2 c^4}-\frac{d x^3}{3 c^2}-\frac{e x^4}{4 c^2}-\frac{\left (a^3 (c d-a e)\right ) \int \frac{1}{-a c-c^2 x} \, dx}{2 c^4}+\frac{\left (a^3 (c d+a e)\right ) \int \frac{1}{a c-c^2 x} \, dx}{2 c^4}\\ &=-\frac{a^2 d x}{c^4}-\frac{a^2 e x^2}{2 c^4}-\frac{d x^3}{3 c^2}-\frac{e x^4}{4 c^2}-\frac{a^3 (c d+a e) \log (a-c x)}{2 c^6}+\frac{a^3 (c d-a e) \log (a+c x)}{2 c^6}\\ \end{align*}
Mathematica [A] time = 0.0158961, size = 86, normalized size = 0.91 \[ -\frac{a^2 d x}{c^4}+\frac{a^3 d \tanh ^{-1}\left (\frac{c x}{a}\right )}{c^5}-\frac{a^2 e x^2}{2 c^4}-\frac{a^4 e \log \left (a^2-c^2 x^2\right )}{2 c^6}-\frac{d x^3}{3 c^2}-\frac{e x^4}{4 c^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 106, normalized size = 1.1 \begin{align*} -{\frac{e{x}^{4}}{4\,{c}^{2}}}-{\frac{d{x}^{3}}{3\,{c}^{2}}}-{\frac{{a}^{2}e{x}^{2}}{2\,{c}^{4}}}-{\frac{{a}^{2}dx}{{c}^{4}}}-{\frac{{a}^{4}\ln \left ( cx+a \right ) e}{2\,{c}^{6}}}+{\frac{{a}^{3}\ln \left ( cx+a \right ) d}{2\,{c}^{5}}}-{\frac{{a}^{4}\ln \left ( cx-a \right ) e}{2\,{c}^{6}}}-{\frac{{a}^{3}\ln \left ( cx-a \right ) d}{2\,{c}^{5}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0776, size = 122, normalized size = 1.28 \begin{align*} -\frac{3 \, c^{2} e x^{4} + 4 \, c^{2} d x^{3} + 6 \, a^{2} e x^{2} + 12 \, a^{2} d x}{12 \, c^{4}} + \frac{{\left (a^{3} c d - a^{4} e\right )} \log \left (c x + a\right )}{2 \, c^{6}} - \frac{{\left (a^{3} c d + a^{4} e\right )} \log \left (c x - a\right )}{2 \, c^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.49947, size = 194, normalized size = 2.04 \begin{align*} -\frac{3 \, c^{4} e x^{4} + 4 \, c^{4} d x^{3} + 6 \, a^{2} c^{2} e x^{2} + 12 \, a^{2} c^{2} d x - 6 \,{\left (a^{3} c d - a^{4} e\right )} \log \left (c x + a\right ) + 6 \,{\left (a^{3} c d + a^{4} e\right )} \log \left (c x - a\right )}{12 \, c^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.580899, size = 129, normalized size = 1.36 \begin{align*} - \frac{a^{3} \left (a e - c d\right ) \log{\left (x + \frac{a^{4} e - a^{3} \left (a e - c d\right )}{a^{2} c^{2} d} \right )}}{2 c^{6}} - \frac{a^{3} \left (a e + c d\right ) \log{\left (x + \frac{a^{4} e - a^{3} \left (a e + c d\right )}{a^{2} c^{2} d} \right )}}{2 c^{6}} - \frac{a^{2} d x}{c^{4}} - \frac{a^{2} e x^{2}}{2 c^{4}} - \frac{d x^{3}}{3 c^{2}} - \frac{e x^{4}}{4 c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11857, size = 138, normalized size = 1.45 \begin{align*} \frac{{\left (a^{3} c d - a^{4} e\right )} \log \left ({\left | c x + a \right |}\right )}{2 \, c^{6}} - \frac{{\left (a^{3} c d + a^{4} e\right )} \log \left ({\left | c x - a \right |}\right )}{2 \, c^{6}} - \frac{3 \, c^{6} x^{4} e + 4 \, c^{6} d x^{3} + 6 \, a^{2} c^{4} x^{2} e + 12 \, a^{2} c^{4} d x}{12 \, c^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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