Optimal. Leaf size=93 \[ -\frac {(2 a e+3 c d) \log (a-c x)}{4 a^5}+\frac {(3 c d-2 a e) \log (a+c x)}{4 a^5}-\frac {3 d}{2 a^4 x}+\frac {e \log (x)}{a^4}+\frac {d+e x}{2 a^2 x \left (a^2-c^2 x^2\right )} \]
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Rubi [A] time = 0.09, antiderivative size = 93, 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 \left (a^2-c^2 x^2\right )}-\frac {(2 a e+3 c d) \log (a-c x)}{4 a^5}+\frac {(3 c d-2 a e) \log (a+c x)}{4 a^5}-\frac {3 d}{2 a^4 x}+\frac {e \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^2 \left (a^2-c^2 x^2\right )^2} \, dx &=\frac {d+e x}{2 a^2 x \left (a^2-c^2 x^2\right )}+\frac {\int \frac {3 a^2 c^2 d+2 a^2 c^2 e x}{x^2 \left (a^2-c^2 x^2\right )} \, dx}{2 a^4 c^2}\\ &=\frac {d+e x}{2 a^2 x \left (a^2-c^2 x^2\right )}+\frac {\int \left (\frac {3 c^2 d}{x^2}+\frac {2 c^2 e}{x}+\frac {c^3 (3 c d+2 a e)}{2 a (a-c x)}-\frac {c^3 (-3 c d+2 a e)}{2 a (a+c x)}\right ) \, dx}{2 a^4 c^2}\\ &=-\frac {3 d}{2 a^4 x}+\frac {d+e x}{2 a^2 x \left (a^2-c^2 x^2\right )}+\frac {e \log (x)}{a^4}-\frac {(3 c d+2 a e) \log (a-c x)}{4 a^5}+\frac {(3 c d-2 a e) \log (a+c x)}{4 a^5}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 77, normalized size = 0.83 \begin {gather*} \frac {-a e \log \left (a^2-c^2 x^2\right )+\frac {a^3 e+a c^2 d x}{a^2-c^2 x^2}+3 c d \tanh ^{-1}\left (\frac {c x}{a}\right )-\frac {2 a d}{x}+2 a e \log (x)}{2 a^5} \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^2 \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 = 0.49, size = 155, normalized size = 1.67 \begin {gather*} -\frac {6 \, a c^{2} d x^{2} + 2 \, a^{3} e x - 4 \, a^{3} d - {\left ({\left (3 \, c^{3} d - 2 \, a c^{2} e\right )} x^{3} - {\left (3 \, a^{2} c d - 2 \, a^{3} e\right )} x\right )} \log \left (c x + a\right ) + {\left ({\left (3 \, c^{3} d + 2 \, a c^{2} e\right )} x^{3} - {\left (3 \, a^{2} c d + 2 \, a^{3} e\right )} x\right )} \log \left (c x - a\right ) - 4 \, {\left (a c^{2} e x^{3} - a^{3} e x\right )} \log \relax (x)}{4 \, {\left (a^{5} c^{2} x^{3} - a^{7} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 112, normalized size = 1.20 \begin {gather*} \frac {e \log \left ({\left | x \right |}\right )}{a^{4}} - \frac {3 \, c^{2} d x^{2} + a^{2} x e - 2 \, a^{2} d}{2 \, {\left (c^{2} x^{3} - a^{2} x\right )} a^{4}} + \frac {{\left (3 \, c^{2} d - 2 \, a c e\right )} \log \left ({\left | c x + a \right |}\right )}{4 \, a^{5} c} - \frac {{\left (3 \, c^{2} d + 2 \, a c e\right )} \log \left ({\left | c x - a \right |}\right )}{4 \, a^{5} c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 130, normalized size = 1.40 \begin {gather*} \frac {e}{4 \left (c x +a \right ) a^{3}}-\frac {e}{4 \left (c x -a \right ) a^{3}}-\frac {c d}{4 \left (c x +a \right ) a^{4}}-\frac {c d}{4 \left (c x -a \right ) a^{4}}+\frac {e \ln \relax (x )}{a^{4}}-\frac {e \ln \left (c x -a \right )}{2 a^{4}}-\frac {e \ln \left (c x +a \right )}{2 a^{4}}-\frac {3 c d \ln \left (c x -a \right )}{4 a^{5}}+\frac {3 c d \ln \left (c x +a \right )}{4 a^{5}}-\frac {d}{a^{4} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.47, size = 93, normalized size = 1.00 \begin {gather*} -\frac {3 \, c^{2} d x^{2} + a^{2} e x - 2 \, a^{2} d}{2 \, {\left (a^{4} c^{2} x^{3} - a^{6} x\right )}} + \frac {e \log \relax (x)}{a^{4}} + \frac {{\left (3 \, c d - 2 \, a e\right )} \log \left (c x + a\right )}{4 \, a^{5}} - \frac {{\left (3 \, c d + 2 \, a e\right )} \log \left (c x - a\right )}{4 \, a^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.11, size = 92, normalized size = 0.99 \begin {gather*} \frac {\frac {e\,x}{2\,a^2}-\frac {d}{a^2}+\frac {3\,c^2\,d\,x^2}{2\,a^4}}{a^2\,x-c^2\,x^3}-\frac {\ln \left (a+c\,x\right )\,\left (2\,a\,e-3\,c\,d\right )}{4\,a^5}-\frac {\ln \left (a-c\,x\right )\,\left (2\,a\,e+3\,c\,d\right )}{4\,a^5}+\frac {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 = 1.95, size = 291, normalized size = 3.13 \begin {gather*} \frac {2 a^{2} d - a^{2} e x - 3 c^{2} d x^{2}}{- 2 a^{6} x + 2 a^{4} c^{2} x^{3}} + \frac {e \log {\relax (x )}}{a^{4}} - \frac {\left (2 a e - 3 c d\right ) \log {\left (x + \frac {16 a^{4} e^{3} - 4 a^{3} e^{2} \left (2 a e - 3 c d\right ) + 12 a^{2} c^{2} d^{2} e - 2 a^{2} e \left (2 a e - 3 c d\right )^{2} + 3 a c^{2} d^{2} \left (2 a e - 3 c d\right )}{36 a^{2} c^{2} d e^{2} - 9 c^{4} d^{3}} \right )}}{4 a^{5}} - \frac {\left (2 a e + 3 c d\right ) \log {\left (x + \frac {16 a^{4} e^{3} - 4 a^{3} e^{2} \left (2 a e + 3 c d\right ) + 12 a^{2} c^{2} d^{2} e - 2 a^{2} e \left (2 a e + 3 c d\right )^{2} + 3 a c^{2} d^{2} \left (2 a e + 3 c d\right )}{36 a^{2} c^{2} d e^{2} - 9 c^{4} d^{3}} \right )}}{4 a^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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