Optimal. Leaf size=84 \[ -\frac {(a e+2 c d) \log (a-c x)}{4 a^4 c}-\frac {(2 c d-a e) \log (a+c x)}{4 a^4 c}+\frac {d \log (x)}{a^4}+\frac {d+e x}{2 a^2 \left (a^2-c^2 x^2\right )} \]
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Rubi [A] time = 0.07, antiderivative size = 84, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {d+e x}{2 a^2 \left (a^2-c^2 x^2\right )}-\frac {(a e+2 c d) \log (a-c x)}{4 a^4 c}-\frac {(2 c d-a e) \log (a+c x)}{4 a^4 c}+\frac {d \log (x)}{a^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 801
Rule 823
Rubi steps
\begin {align*} \int \frac {d+e x}{x \left (a^2-c^2 x^2\right )^2} \, dx &=\frac {d+e x}{2 a^2 \left (a^2-c^2 x^2\right )}+\frac {\int \frac {2 a^2 c^2 d+a^2 c^2 e x}{x \left (a^2-c^2 x^2\right )} \, dx}{2 a^4 c^2}\\ &=\frac {d+e x}{2 a^2 \left (a^2-c^2 x^2\right )}+\frac {\int \left (\frac {2 c^2 d}{x}+\frac {c^2 (2 c d+a e)}{2 (a-c x)}-\frac {c^2 (2 c d-a e)}{2 (a+c x)}\right ) \, dx}{2 a^4 c^2}\\ &=\frac {d+e x}{2 a^2 \left (a^2-c^2 x^2\right )}+\frac {d \log (x)}{a^4}-\frac {(2 c d+a e) \log (a-c x)}{4 a^4 c}-\frac {(2 c d-a e) \log (a+c x)}{4 a^4 c}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 65, normalized size = 0.77 \begin {gather*} \frac {\frac {a^2 (d+e x)}{a^2-c^2 x^2}-d \log \left (a^2-c^2 x^2\right )+\frac {a e \tanh ^{-1}\left (\frac {c x}{a}\right )}{c}+2 d \log (x)}{2 a^4} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {d+e x}{x \left (a^2-c^2 x^2\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.43, size = 139, normalized size = 1.65 \begin {gather*} -\frac {2 \, a^{2} c e x + 2 \, a^{2} c d - {\left (2 \, a^{2} c d - a^{3} e - {\left (2 \, c^{3} d - a c^{2} e\right )} x^{2}\right )} \log \left (c x + a\right ) - {\left (2 \, a^{2} c d + a^{3} e - {\left (2 \, c^{3} d + a c^{2} e\right )} x^{2}\right )} \log \left (c x - a\right ) - 4 \, {\left (c^{3} d x^{2} - a^{2} c d\right )} \log \relax (x)}{4 \, {\left (a^{4} c^{3} x^{2} - a^{6} c\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 94, normalized size = 1.12 \begin {gather*} \frac {d \log \left ({\left | x \right |}\right )}{a^{4}} - \frac {{\left (2 \, c d - a e\right )} \log \left ({\left | c x + a \right |}\right )}{4 \, a^{4} c} - \frac {{\left (2 \, c d + a e\right )} \log \left ({\left | c x - a \right |}\right )}{4 \, a^{4} c} - \frac {a^{2} x e + a^{2} d}{2 \, {\left (c x + a\right )} {\left (c x - a\right )} a^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 129, normalized size = 1.54 \begin {gather*} -\frac {e}{4 \left (c x +a \right ) a^{2} c}-\frac {e}{4 \left (c x -a \right ) a^{2} c}-\frac {e \ln \left (c x -a \right )}{4 a^{3} c}+\frac {e \ln \left (c x +a \right )}{4 a^{3} c}+\frac {d}{4 \left (c x +a \right ) a^{3}}-\frac {d}{4 \left (c x -a \right ) a^{3}}+\frac {d \ln \relax (x )}{a^{4}}-\frac {d \ln \left (c x -a \right )}{2 a^{4}}-\frac {d \ln \left (c x +a \right )}{2 a^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.69, size = 80, normalized size = 0.95 \begin {gather*} -\frac {e x + d}{2 \, {\left (a^{2} c^{2} x^{2} - a^{4}\right )}} + \frac {d \log \relax (x)}{a^{4}} - \frac {{\left (2 \, c d - a e\right )} \log \left (c x + a\right )}{4 \, a^{4} c} - \frac {{\left (2 \, c d + a e\right )} \log \left (c x - a\right )}{4 \, a^{4} c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.11, size = 82, normalized size = 0.98 \begin {gather*} \frac {\frac {d}{2\,a^2}+\frac {e\,x}{2\,a^2}}{a^2-c^2\,x^2}+\frac {d\,\ln \relax (x)}{a^4}+\frac {\ln \left (a+c\,x\right )\,\left (a\,e-2\,c\,d\right )}{4\,a^4\,c}-\frac {\ln \left (a-c\,x\right )\,\left (a\,e+2\,c\,d\right )}{4\,a^4\,c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.83, size = 231, normalized size = 2.75 \begin {gather*} \frac {- d - e x}{- 2 a^{4} + 2 a^{2} c^{2} x^{2}} + \frac {d \log {\relax (x )}}{a^{4}} + \frac {\left (a e - 2 c d\right ) \log {\left (x + \frac {- 4 a^{2} d e^{2} + \frac {a^{2} e^{2} \left (a e - 2 c d\right )}{c} - 48 c^{2} d^{3} - 12 c d^{2} \left (a e - 2 c d\right ) + 6 d \left (a e - 2 c d\right )^{2}}{a^{2} e^{3} - 36 c^{2} d^{2} e} \right )}}{4 a^{4} c} - \frac {\left (a e + 2 c d\right ) \log {\left (x + \frac {- 4 a^{2} d e^{2} - \frac {a^{2} e^{2} \left (a e + 2 c d\right )}{c} - 48 c^{2} d^{3} + 12 c d^{2} \left (a e + 2 c d\right ) + 6 d \left (a e + 2 c d\right )^{2}}{a^{2} e^{3} - 36 c^{2} d^{2} e} \right )}}{4 a^{4} c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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