Optimal. Leaf size=93 \[ -\frac {c^2 (a e+c d) \log (a-c x)}{2 a^5}+\frac {c^2 (c d-a e) \log (a+c x)}{2 a^5}-\frac {c^2 d}{a^4 x}+\frac {c^2 e \log (x)}{a^4}-\frac {d}{3 a^2 x^3}-\frac {e}{2 a^2 x^2} \]
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Rubi [A] time = 0.07, antiderivative size = 93, 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} \begin {gather*} -\frac {c^2 (a e+c d) \log (a-c x)}{2 a^5}+\frac {c^2 (c d-a e) \log (a+c x)}{2 a^5}-\frac {c^2 d}{a^4 x}+\frac {c^2 e \log (x)}{a^4}-\frac {d}{3 a^2 x^3}-\frac {e}{2 a^2 x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 801
Rubi steps
\begin {align*} \int \frac {d+e x}{x^4 \left (a^2-c^2 x^2\right )} \, dx &=\int \left (\frac {d}{a^2 x^4}+\frac {e}{a^2 x^3}+\frac {c^2 d}{a^4 x^2}+\frac {c^2 e}{a^4 x}+\frac {c^3 (c d+a e)}{2 a^5 (a-c x)}-\frac {c^3 (-c d+a e)}{2 a^5 (a+c x)}\right ) \, dx\\ &=-\frac {d}{3 a^2 x^3}-\frac {e}{2 a^2 x^2}-\frac {c^2 d}{a^4 x}+\frac {c^2 e \log (x)}{a^4}-\frac {c^2 (c d+a e) \log (a-c x)}{2 a^5}+\frac {c^2 (c d-a e) \log (a+c x)}{2 a^5}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 84, normalized size = 0.90 \begin {gather*} \frac {c^3 d \tanh ^{-1}\left (\frac {c x}{a}\right )}{a^5}-\frac {c^2 d}{a^4 x}+\frac {c^2 e \log (x)}{a^4}-\frac {d}{3 a^2 x^3}-\frac {e}{2 a^2 x^2}-\frac {c^2 e \log \left (a^2-c^2 x^2\right )}{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^4 \left (a^2-c^2 x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.41, size = 93, normalized size = 1.00 \begin {gather*} \frac {6 \, a c^{2} e x^{3} \log \relax (x) - 6 \, a c^{2} d x^{2} - 3 \, a^{3} e x + 3 \, {\left (c^{3} d - a c^{2} e\right )} x^{3} \log \left (c x + a\right ) - 3 \, {\left (c^{3} d + a c^{2} e\right )} x^{3} \log \left (c x - a\right ) - 2 \, a^{3} d}{6 \, a^{5} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 104, normalized size = 1.12 \begin {gather*} \frac {c^{2} e \log \left ({\left | x \right |}\right )}{a^{4}} + \frac {{\left (c^{4} d - a c^{3} e\right )} \log \left ({\left | c x + a \right |}\right )}{2 \, a^{5} c} - \frac {{\left (c^{4} d + a c^{3} e\right )} \log \left ({\left | c x - a \right |}\right )}{2 \, a^{5} c} - \frac {6 \, c^{2} d x^{2} + 3 \, a^{2} x e + 2 \, a^{2} d}{6 \, a^{4} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 106, normalized size = 1.14 \begin {gather*} \frac {c^{2} e \ln \relax (x )}{a^{4}}-\frac {c^{2} e \ln \left (c x -a \right )}{2 a^{4}}-\frac {c^{2} e \ln \left (c x +a \right )}{2 a^{4}}-\frac {c^{3} d \ln \left (c x -a \right )}{2 a^{5}}+\frac {c^{3} d \ln \left (c x +a \right )}{2 a^{5}}-\frac {c^{2} d}{a^{4} x}-\frac {e}{2 a^{2} x^{2}}-\frac {d}{3 a^{2} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.50, size = 91, normalized size = 0.98 \begin {gather*} \frac {c^{2} e \log \relax (x)}{a^{4}} + \frac {{\left (c^{3} d - a c^{2} e\right )} \log \left (c x + a\right )}{2 \, a^{5}} - \frac {{\left (c^{3} d + a c^{2} e\right )} \log \left (c x - a\right )}{2 \, a^{5}} - \frac {6 \, c^{2} d x^{2} + 3 \, a^{2} e x + 2 \, a^{2} d}{6 \, a^{4} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.11, size = 89, normalized size = 0.96 \begin {gather*} \frac {\ln \left (a+c\,x\right )\,\left (c^3\,d-a\,c^2\,e\right )}{2\,a^5}-\frac {\frac {d}{3\,a^2}+\frac {e\,x}{2\,a^2}+\frac {c^2\,d\,x^2}{a^4}}{x^3}-\frac {\ln \left (a-c\,x\right )\,\left (d\,c^3+a\,e\,c^2\right )}{2\,a^5}+\frac {c^2\,e\,\ln \relax (x)}{a^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 2.00, size = 279, normalized size = 3.00 \begin {gather*} \frac {c^{2} e \log {\relax (x )}}{a^{4}} - \frac {2 a^{2} d + 3 a^{2} e x + 6 c^{2} d x^{2}}{6 a^{4} x^{3}} - \frac {c^{2} \left (a e - c d\right ) \log {\left (x + \frac {6 a^{4} c^{4} e^{3} - 3 a^{3} c^{4} e^{2} \left (a e - c d\right ) + 2 a^{2} c^{6} d^{2} e - 3 a^{2} c^{4} e \left (a e - c d\right )^{2} + a c^{6} d^{2} \left (a e - c d\right )}{9 a^{2} c^{6} d e^{2} - c^{8} d^{3}} \right )}}{2 a^{5}} - \frac {c^{2} \left (a e + c d\right ) \log {\left (x + \frac {6 a^{4} c^{4} e^{3} - 3 a^{3} c^{4} e^{2} \left (a e + c d\right ) + 2 a^{2} c^{6} d^{2} e - 3 a^{2} c^{4} e \left (a e + c d\right )^{2} + a c^{6} d^{2} \left (a e + c d\right )}{9 a^{2} c^{6} d e^{2} - c^{8} d^{3}} \right )}}{2 a^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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