Optimal. Leaf size=50 \[ -\frac {(a e+c d) \log (a-c x)}{2 c^3}-\frac {(c d-a e) \log (a+c x)}{2 c^3}-\frac {e x}{c^2} \]
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Rubi [A] time = 0.04, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {774, 633, 31} \begin {gather*} -\frac {(a e+c d) \log (a-c x)}{2 c^3}-\frac {(c d-a e) \log (a+c x)}{2 c^3}-\frac {e x}{c^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 633
Rule 774
Rubi steps
\begin {align*} \int \frac {x (d+e x)}{a^2-c^2 x^2} \, dx &=-\frac {e x}{c^2}-\frac {\int \frac {-a^2 e-c^2 d x}{a^2-c^2 x^2} \, dx}{c^2}\\ &=-\frac {e x}{c^2}+\frac {(c d-a e) \int \frac {1}{-a c-c^2 x} \, dx}{2 c}+\frac {(c d+a e) \int \frac {1}{a c-c^2 x} \, dx}{2 c}\\ &=-\frac {e x}{c^2}-\frac {(c d+a e) \log (a-c x)}{2 c^3}-\frac {(c d-a e) \log (a+c x)}{2 c^3}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 42, normalized size = 0.84 \begin {gather*} -\frac {d \log \left (a^2-c^2 x^2\right )}{2 c^2}+\frac {a e \tanh ^{-1}\left (\frac {c x}{a}\right )}{c^3}-\frac {e x}{c^2} \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 {x (d+e x)}{a^2-c^2 x^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.41, size = 42, normalized size = 0.84 \begin {gather*} -\frac {2 \, c e x + {\left (c d - a e\right )} \log \left (c x + a\right ) + {\left (c d + a e\right )} \log \left (c x - a\right )}{2 \, c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 52, normalized size = 1.04 \begin {gather*} -\frac {x e}{c^{2}} - \frac {{\left (c d - a e\right )} \log \left ({\left | c x + a \right |}\right )}{2 \, c^{3}} - \frac {{\left (c d + a e\right )} \log \left ({\left | c x - a \right |}\right )}{2 \, c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 63, normalized size = 1.26 \begin {gather*} -\frac {a e \ln \left (c x -a \right )}{2 c^{3}}+\frac {a e \ln \left (c x +a \right )}{2 c^{3}}-\frac {d \ln \left (c x -a \right )}{2 c^{2}}-\frac {d \ln \left (c x +a \right )}{2 c^{2}}-\frac {e x}{c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.50, size = 47, normalized size = 0.94 \begin {gather*} -\frac {e x}{c^{2}} - \frac {{\left (c d - a e\right )} \log \left (c x + a\right )}{2 \, c^{3}} - \frac {{\left (c d + a e\right )} \log \left (c x - a\right )}{2 \, c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.10, size = 60, normalized size = 1.20 \begin {gather*} \frac {a\,e\,\ln \left (a+c\,x\right )}{2\,c^3}-\frac {d\,\ln \left (a-c\,x\right )}{2\,c^2}-\frac {e\,x}{c^2}-\frac {d\,\ln \left (a+c\,x\right )}{2\,c^2}-\frac {a\,e\,\ln \left (a-c\,x\right )}{2\,c^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.35, size = 60, normalized size = 1.20 \begin {gather*} - \frac {e x}{c^{2}} + \frac {\left (a e - c d\right ) \log {\left (x + \frac {d + \frac {a e - c d}{c}}{e} \right )}}{2 c^{3}} - \frac {\left (a e + c d\right ) \log {\left (x + \frac {d - \frac {a e + c d}{c}}{e} \right )}}{2 c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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