Optimal. Leaf size=108 \[ \frac{d+e x}{2 a^2 x^2 \left (a^2-c^2 x^2\right )}+\frac{2 c^2 d \log (x)}{a^6}-\frac{c (3 a e+4 c d) \log (a-c x)}{4 a^6}-\frac{c (4 c d-3 a e) \log (a+c x)}{4 a^6}-\frac{d}{a^4 x^2}-\frac{3 e}{2 a^4 x} \]
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Rubi [A] time = 0.108293, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {823, 801} \[ \frac{d+e x}{2 a^2 x^2 \left (a^2-c^2 x^2\right )}+\frac{2 c^2 d \log (x)}{a^6}-\frac{c (3 a e+4 c d) \log (a-c x)}{4 a^6}-\frac{c (4 c d-3 a e) \log (a+c x)}{4 a^6}-\frac{d}{a^4 x^2}-\frac{3 e}{2 a^4 x} \]
Antiderivative was successfully verified.
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Rule 823
Rule 801
Rubi steps
\begin{align*} \int \frac{d+e x}{x^3 \left (a^2-c^2 x^2\right )^2} \, dx &=\frac{d+e x}{2 a^2 x^2 \left (a^2-c^2 x^2\right )}+\frac{\int \frac{4 a^2 c^2 d+3 a^2 c^2 e x}{x^3 \left (a^2-c^2 x^2\right )} \, dx}{2 a^4 c^2}\\ &=\frac{d+e x}{2 a^2 x^2 \left (a^2-c^2 x^2\right )}+\frac{\int \left (\frac{4 c^2 d}{x^3}+\frac{3 c^2 e}{x^2}+\frac{4 c^4 d}{a^2 x}+\frac{c^4 (4 c d+3 a e)}{2 a^2 (a-c x)}+\frac{c^4 (-4 c d+3 a e)}{2 a^2 (a+c x)}\right ) \, dx}{2 a^4 c^2}\\ &=-\frac{d}{a^4 x^2}-\frac{3 e}{2 a^4 x}+\frac{d+e x}{2 a^2 x^2 \left (a^2-c^2 x^2\right )}+\frac{2 c^2 d \log (x)}{a^6}-\frac{c (4 c d+3 a e) \log (a-c x)}{4 a^6}-\frac{c (4 c d-3 a e) \log (a+c x)}{4 a^6}\\ \end{align*}
Mathematica [A] time = 0.0722124, size = 91, normalized size = 0.84 \[ \frac{\frac{a^2 c^2 (d+e x)}{a^2-c^2 x^2}-2 c^2 d \log \left (a^2-c^2 x^2\right )-\frac{a^2 d}{x^2}-\frac{2 a^2 e}{x}+3 a c e \tanh ^{-1}\left (\frac{c x}{a}\right )+4 c^2 d \log (x)}{2 a^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.016, size = 155, normalized size = 1.4 \begin{align*} -{\frac{e}{{a}^{4}x}}-{\frac{d}{2\,{a}^{4}{x}^{2}}}+2\,{\frac{d{c}^{2}\ln \left ( x \right ) }{{a}^{6}}}+{\frac{3\,c\ln \left ( cx+a \right ) e}{4\,{a}^{5}}}-{\frac{{c}^{2}\ln \left ( cx+a \right ) d}{{a}^{6}}}-{\frac{ec}{4\,{a}^{4} \left ( cx+a \right ) }}+{\frac{d{c}^{2}}{4\,{a}^{5} \left ( cx+a \right ) }}-{\frac{3\,c\ln \left ( cx-a \right ) e}{4\,{a}^{5}}}-{\frac{{c}^{2}\ln \left ( cx-a \right ) d}{{a}^{6}}}-{\frac{ec}{4\,{a}^{4} \left ( cx-a \right ) }}-{\frac{d{c}^{2}}{4\,{a}^{5} \left ( cx-a \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.07508, size = 155, normalized size = 1.44 \begin{align*} -\frac{3 \, c^{2} e x^{3} + 2 \, c^{2} d x^{2} - 2 \, a^{2} e x - a^{2} d}{2 \,{\left (a^{4} c^{2} x^{4} - a^{6} x^{2}\right )}} + \frac{2 \, c^{2} d \log \left (x\right )}{a^{6}} - \frac{{\left (4 \, c^{2} d - 3 \, a c e\right )} \log \left (c x + a\right )}{4 \, a^{6}} - \frac{{\left (4 \, c^{2} d + 3 \, a c e\right )} \log \left (c x - a\right )}{4 \, a^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.6168, size = 374, normalized size = 3.46 \begin{align*} -\frac{6 \, a^{2} c^{2} e x^{3} + 4 \, a^{2} c^{2} d x^{2} - 4 \, a^{4} e x - 2 \, a^{4} d +{\left ({\left (4 \, c^{4} d - 3 \, a c^{3} e\right )} x^{4} -{\left (4 \, a^{2} c^{2} d - 3 \, a^{3} c e\right )} x^{2}\right )} \log \left (c x + a\right ) +{\left ({\left (4 \, c^{4} d + 3 \, a c^{3} e\right )} x^{4} -{\left (4 \, a^{2} c^{2} d + 3 \, a^{3} c e\right )} x^{2}\right )} \log \left (c x - a\right ) - 8 \,{\left (c^{4} d x^{4} - a^{2} c^{2} d x^{2}\right )} \log \left (x\right )}{4 \,{\left (a^{6} c^{2} x^{4} - a^{8} x^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 2.29428, size = 311, normalized size = 2.88 \begin{align*} - \frac{- a^{2} d - 2 a^{2} e x + 2 c^{2} d x^{2} + 3 c^{2} e x^{3}}{- 2 a^{6} x^{2} + 2 a^{4} c^{2} x^{4}} + \frac{2 c^{2} d \log{\left (x \right )}}{a^{6}} + \frac{c \left (3 a e - 4 c d\right ) \log{\left (x + \frac{- 24 a^{2} c^{2} d e^{2} + 3 a^{2} c e^{2} \left (3 a e - 4 c d\right ) - 128 c^{4} d^{3} - 16 c^{3} d^{2} \left (3 a e - 4 c d\right ) + 4 c^{2} d \left (3 a e - 4 c d\right )^{2}}{9 a^{2} c^{2} e^{3} - 144 c^{4} d^{2} e} \right )}}{4 a^{6}} - \frac{c \left (3 a e + 4 c d\right ) \log{\left (x + \frac{- 24 a^{2} c^{2} d e^{2} - 3 a^{2} c e^{2} \left (3 a e + 4 c d\right ) - 128 c^{4} d^{3} + 16 c^{3} d^{2} \left (3 a e + 4 c d\right ) + 4 c^{2} d \left (3 a e + 4 c d\right )^{2}}{9 a^{2} c^{2} e^{3} - 144 c^{4} d^{2} e} \right )}}{4 a^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14705, size = 188, normalized size = 1.74 \begin{align*} \frac{2 \, c^{2} d \log \left ({\left | x \right |}\right )}{a^{6}} - \frac{{\left (4 \, c^{3} d - 3 \, a c^{2} e\right )} \log \left ({\left | c x + a \right |}\right )}{4 \, a^{6} c} - \frac{{\left (4 \, c^{3} d + 3 \, a c^{2} e\right )} \log \left ({\left | c x - a \right |}\right )}{4 \, a^{6} c} - \frac{3 \, a^{2} c^{2} x^{3} e + 2 \, a^{2} c^{2} d x^{2} - 2 \, a^{4} x e - a^{4} d}{2 \,{\left (c x + a\right )}{\left (c x - a\right )} a^{6} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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