Optimal. Leaf size=108 \[ \frac {2 c^2 d \log (x)}{a^6}-\frac {c (3 a e+4 c d) \log (a-c x)}{4 a^6}-\frac {c (4 c d-3 a e) \log (a+c x)}{4 a^6}-\frac {d}{a^4 x^2}-\frac {3 e}{2 a^4 x}+\frac {d+e x}{2 a^2 x^2 \left (a^2-c^2 x^2\right )} \]
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Rubi [A] time = 0.11, antiderivative size = 108, 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 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} \end {gather*}
Antiderivative was successfully verified.
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Rule 801
Rule 823
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*}
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Mathematica [A] time = 0.07, size = 91, normalized size = 0.84 \begin {gather*} \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} \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^3 \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 = 184, normalized size = 1.70 \begin {gather*} -\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 \relax (x)}{4 \, {\left (a^{6} c^{2} x^{4} - a^{8} x^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 139, normalized size = 1.29 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 155, normalized size = 1.44 \begin {gather*} -\frac {c e}{4 \left (c x +a \right ) a^{4}}-\frac {c e}{4 \left (c x -a \right ) a^{4}}+\frac {c^{2} d}{4 \left (c x +a \right ) a^{5}}-\frac {c^{2} d}{4 \left (c x -a \right ) a^{5}}-\frac {3 c e \ln \left (c x -a \right )}{4 a^{5}}+\frac {3 c e \ln \left (c x +a \right )}{4 a^{5}}+\frac {2 c^{2} d \ln \relax (x )}{a^{6}}-\frac {c^{2} d \ln \left (c x -a \right )}{a^{6}}-\frac {c^{2} d \ln \left (c x +a \right )}{a^{6}}-\frac {e}{a^{4} x}-\frac {d}{2 a^{4} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.53, size = 115, normalized size = 1.06 \begin {gather*} -\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 \relax (x)}{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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.11, size = 116, normalized size = 1.07 \begin {gather*} \frac {2\,c^2\,d\,\ln \relax (x)}{a^6}-\frac {\ln \left (a+c\,x\right )\,\left (4\,c^2\,d-3\,a\,c\,e\right )}{4\,a^6}-\frac {\ln \left (a-c\,x\right )\,\left (4\,d\,c^2+3\,a\,e\,c\right )}{4\,a^6}-\frac {\frac {d}{2\,a^2}+\frac {e\,x}{a^2}-\frac {c^2\,d\,x^2}{a^4}-\frac {3\,c^2\,e\,x^3}{2\,a^4}}{a^2\,x^2-c^2\,x^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 2.30, size = 311, normalized size = 2.88 \begin {gather*} \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 {\relax (x )}}{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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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