Optimal. Leaf size=63 \[ -\frac {a (a e+c d) \log (a-c x)}{2 c^4}+\frac {a (c d-a e) \log (a+c x)}{2 c^4}-\frac {d x}{c^2}-\frac {e x^2}{2 c^2} \]
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Rubi [A] time = 0.06, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {801, 633, 31} \begin {gather*} -\frac {a (a e+c d) \log (a-c x)}{2 c^4}+\frac {a (c d-a e) \log (a+c x)}{2 c^4}-\frac {d x}{c^2}-\frac {e x^2}{2 c^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 633
Rule 801
Rubi steps
\begin {align*} \int \frac {x^2 (d+e x)}{a^2-c^2 x^2} \, dx &=\int \left (-\frac {d}{c^2}-\frac {e x}{c^2}+\frac {a^2 d+a^2 e x}{c^2 \left (a^2-c^2 x^2\right )}\right ) \, dx\\ &=-\frac {d x}{c^2}-\frac {e x^2}{2 c^2}+\frac {\int \frac {a^2 d+a^2 e x}{a^2-c^2 x^2} \, dx}{c^2}\\ &=-\frac {d x}{c^2}-\frac {e x^2}{2 c^2}-\frac {(a (c d-a e)) \int \frac {1}{-a c-c^2 x} \, dx}{2 c^2}+\frac {(a (c d+a e)) \int \frac {1}{a c-c^2 x} \, dx}{2 c^2}\\ &=-\frac {d x}{c^2}-\frac {e x^2}{2 c^2}-\frac {a (c d+a e) \log (a-c x)}{2 c^4}+\frac {a (c d-a e) \log (a+c x)}{2 c^4}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 56, normalized size = 0.89 \begin {gather*} -\frac {a^2 e \log \left (a^2-c^2 x^2\right )}{2 c^4}+\frac {a d \tanh ^{-1}\left (\frac {c x}{a}\right )}{c^3}-\frac {d x}{c^2}-\frac {e x^2}{2 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^2 (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 = 59, normalized size = 0.94 \begin {gather*} -\frac {c^{2} e x^{2} + 2 \, c^{2} d x - {\left (a c d - a^{2} e\right )} \log \left (c x + a\right ) + {\left (a c d + a^{2} e\right )} \log \left (c x - a\right )}{2 \, c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 72, normalized size = 1.14 \begin {gather*} \frac {{\left (a c d - a^{2} e\right )} \log \left ({\left | c x + a \right |}\right )}{2 \, c^{4}} - \frac {{\left (a c d + a^{2} e\right )} \log \left ({\left | c x - a \right |}\right )}{2 \, c^{4}} - \frac {c^{2} x^{2} e + 2 \, c^{2} d x}{2 \, c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 78, normalized size = 1.24 \begin {gather*} -\frac {e \,x^{2}}{2 c^{2}}-\frac {a^{2} e \ln \left (c x -a \right )}{2 c^{4}}-\frac {a^{2} e \ln \left (c x +a \right )}{2 c^{4}}-\frac {a d \ln \left (c x -a \right )}{2 c^{3}}+\frac {a d \ln \left (c x +a \right )}{2 c^{3}}-\frac {d x}{c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.53, size = 61, normalized size = 0.97 \begin {gather*} -\frac {e x^{2} + 2 \, d x}{2 \, c^{2}} + \frac {{\left (a c d - a^{2} e\right )} \log \left (c x + a\right )}{2 \, c^{4}} - \frac {{\left (a c d + a^{2} e\right )} \log \left (c x - a\right )}{2 \, c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 61, normalized size = 0.97 \begin {gather*} -\frac {e\,x^2}{2\,c^2}-\frac {\ln \left (a+c\,x\right )\,\left (a^2\,e-a\,c\,d\right )}{2\,c^4}-\frac {\ln \left (a-c\,x\right )\,\left (e\,a^2+c\,d\,a\right )}{2\,c^4}-\frac {d\,x}{c^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.47, size = 88, normalized size = 1.40 \begin {gather*} - \frac {a \left (a e - c d\right ) \log {\left (x + \frac {a^{2} e - a \left (a e - c d\right )}{c^{2} d} \right )}}{2 c^{4}} - \frac {a \left (a e + c d\right ) \log {\left (x + \frac {a^{2} e - a \left (a e + c d\right )}{c^{2} d} \right )}}{2 c^{4}} - \frac {d x}{c^{2}} - \frac {e x^{2}}{2 c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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