Optimal. Leaf size=75 \[ \frac {c^2 d \log (x)}{a^4}-\frac {c (a e+c d) \log (a-c x)}{2 a^4}-\frac {c (c d-a e) \log (a+c x)}{2 a^4}-\frac {d}{2 a^2 x^2}-\frac {e}{a^2 x} \]
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Rubi [A] time = 0.07, antiderivative size = 75, normalized size of antiderivative = 1.00, 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 = {801} \[ \frac {c^2 d \log (x)}{a^4}-\frac {c (a e+c d) \log (a-c x)}{2 a^4}-\frac {c (c d-a e) \log (a+c x)}{2 a^4}-\frac {d}{2 a^2 x^2}-\frac {e}{a^2 x} \]
Antiderivative was successfully verified.
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Rule 801
Rubi steps
\begin {align*} \int \frac {d+e x}{x^3 \left (a^2-c^2 x^2\right )} \, dx &=\int \left (\frac {d}{a^2 x^3}+\frac {e}{a^2 x^2}+\frac {c^2 d}{a^4 x}+\frac {c^2 (c d+a e)}{2 a^4 (a-c x)}+\frac {c^2 (-c d+a e)}{2 a^4 (a+c x)}\right ) \, dx\\ &=-\frac {d}{2 a^2 x^2}-\frac {e}{a^2 x}+\frac {c^2 d \log (x)}{a^4}-\frac {c (c d+a e) \log (a-c x)}{2 a^4}-\frac {c (c d-a e) \log (a+c x)}{2 a^4}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 68, normalized size = 0.91 \[ \frac {c^2 d \log (x)}{a^4}+\frac {c e \tanh ^{-1}\left (\frac {c x}{a}\right )}{a^3}-\frac {d}{2 a^2 x^2}-\frac {e}{a^2 x}-\frac {c^2 d \log \left (a^2-c^2 x^2\right )}{2 a^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.79, size = 78, normalized size = 1.04 \[ \frac {2 \, c^{2} d x^{2} \log \relax (x) - 2 \, a^{2} e x - {\left (c^{2} d - a c e\right )} x^{2} \log \left (c x + a\right ) - {\left (c^{2} d + a c e\right )} x^{2} \log \left (c x - a\right ) - a^{2} d}{2 \, a^{4} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 93, normalized size = 1.24 \[ \frac {c^{2} d \log \left ({\left | x \right |}\right )}{a^{4}} - \frac {{\left (c^{3} d - a c^{2} e\right )} \log \left ({\left | c x + a \right |}\right )}{2 \, a^{4} c} - \frac {{\left (c^{3} d + a c^{2} e\right )} \log \left ({\left | c x - a \right |}\right )}{2 \, a^{4} c} - \frac {2 \, a^{2} x e + a^{2} d}{2 \, a^{4} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 90, normalized size = 1.20 \[ -\frac {c e \ln \left (c x -a \right )}{2 a^{3}}+\frac {c e \ln \left (c x +a \right )}{2 a^{3}}+\frac {c^{2} d \ln \relax (x )}{a^{4}}-\frac {c^{2} d \ln \left (c x -a \right )}{2 a^{4}}-\frac {c^{2} d \ln \left (c x +a \right )}{2 a^{4}}-\frac {e}{a^{2} x}-\frac {d}{2 a^{2} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.56, size = 70, normalized size = 0.93 \[ \frac {c^{2} d \log \relax (x)}{a^{4}} - \frac {{\left (c^{2} d - a c e\right )} \log \left (c x + a\right )}{2 \, a^{4}} - \frac {{\left (c^{2} d + a c e\right )} \log \left (c x - a\right )}{2 \, a^{4}} - \frac {2 \, e x + d}{2 \, a^{2} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.11, size = 73, normalized size = 0.97 \[ \frac {c^2\,d\,\ln \relax (x)}{a^4}-\frac {\ln \left (a+c\,x\right )\,\left (c^2\,d-a\,c\,e\right )}{2\,a^4}-\frac {\ln \left (a-c\,x\right )\,\left (d\,c^2+a\,e\,c\right )}{2\,a^4}-\frac {\frac {d}{2\,a^2}+\frac {e\,x}{a^2}}{x^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.86, size = 236, normalized size = 3.15 \[ - \frac {d + 2 e x}{2 a^{2} x^{2}} + \frac {c^{2} d \log {\relax (x )}}{a^{4}} + \frac {c \left (a e - c d\right ) \log {\left (x + \frac {- 2 a^{2} c^{2} d e^{2} + a^{2} c e^{2} \left (a e - c d\right ) - 6 c^{4} d^{3} - 3 c^{3} d^{2} \left (a e - c d\right ) + 3 c^{2} d \left (a e - c d\right )^{2}}{a^{2} c^{2} e^{3} - 9 c^{4} d^{2} e} \right )}}{2 a^{4}} - \frac {c \left (a e + c d\right ) \log {\left (x + \frac {- 2 a^{2} c^{2} d e^{2} - a^{2} c e^{2} \left (a e + c d\right ) - 6 c^{4} d^{3} + 3 c^{3} d^{2} \left (a e + c d\right ) + 3 c^{2} d \left (a e + c d\right )^{2}}{a^{2} c^{2} e^{3} - 9 c^{4} d^{2} e} \right )}}{2 a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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