Optimal. Leaf size=94 \[ \frac {x^3 (d+e x)}{2 c^2 \left (a^2-c^2 x^2\right )}+\frac {a (4 a e+3 c d) \log (a-c x)}{4 c^6}-\frac {a (3 c d-4 a e) \log (a+c x)}{4 c^6}+\frac {3 d x}{2 c^4}+\frac {e x^2}{c^4} \]
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Rubi [A] time = 0.10, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {819, 801, 633, 31} \begin {gather*} \frac {x^3 (d+e x)}{2 c^2 \left (a^2-c^2 x^2\right )}+\frac {a (4 a e+3 c d) \log (a-c x)}{4 c^6}-\frac {a (3 c d-4 a e) \log (a+c x)}{4 c^6}+\frac {3 d x}{2 c^4}+\frac {e x^2}{c^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 633
Rule 801
Rule 819
Rubi steps
\begin {align*} \int \frac {x^4 (d+e x)}{\left (a^2-c^2 x^2\right )^2} \, dx &=\frac {x^3 (d+e x)}{2 c^2 \left (a^2-c^2 x^2\right )}-\frac {\int \frac {x^2 \left (3 a^2 d+4 a^2 e x\right )}{a^2-c^2 x^2} \, dx}{2 a^2 c^2}\\ &=\frac {x^3 (d+e x)}{2 c^2 \left (a^2-c^2 x^2\right )}-\frac {\int \left (-\frac {3 a^2 d}{c^2}-\frac {4 a^2 e x}{c^2}+\frac {3 a^4 d+4 a^4 e x}{c^2 \left (a^2-c^2 x^2\right )}\right ) \, dx}{2 a^2 c^2}\\ &=\frac {3 d x}{2 c^4}+\frac {e x^2}{c^4}+\frac {x^3 (d+e x)}{2 c^2 \left (a^2-c^2 x^2\right )}-\frac {\int \frac {3 a^4 d+4 a^4 e x}{a^2-c^2 x^2} \, dx}{2 a^2 c^4}\\ &=\frac {3 d x}{2 c^4}+\frac {e x^2}{c^4}+\frac {x^3 (d+e x)}{2 c^2 \left (a^2-c^2 x^2\right )}+\frac {(a (3 c d-4 a e)) \int \frac {1}{-a c-c^2 x} \, dx}{4 c^4}-\frac {(a (3 c d+4 a e)) \int \frac {1}{a c-c^2 x} \, dx}{4 c^4}\\ &=\frac {3 d x}{2 c^4}+\frac {e x^2}{c^4}+\frac {x^3 (d+e x)}{2 c^2 \left (a^2-c^2 x^2\right )}+\frac {a (3 c d+4 a e) \log (a-c x)}{4 c^6}-\frac {a (3 c d-4 a e) \log (a+c x)}{4 c^6}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 84, normalized size = 0.89 \begin {gather*} \frac {2 a^2 e \log \left (a^2-c^2 x^2\right )+\frac {a^4 e+a^2 c^2 d x}{a^2-c^2 x^2}-3 a c d \tanh ^{-1}\left (\frac {c x}{a}\right )+2 c^2 d x+c^2 e x^2}{2 c^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 {x^4 (d+e x)}{\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 = 0.40, size = 156, normalized size = 1.66 \begin {gather*} \frac {2 \, c^{4} e x^{4} + 4 \, c^{4} d x^{3} - 2 \, a^{2} c^{2} e x^{2} - 6 \, a^{2} c^{2} d x - 2 \, a^{4} e + {\left (3 \, a^{3} c d - 4 \, a^{4} e - {\left (3 \, a c^{3} d - 4 \, a^{2} c^{2} e\right )} x^{2}\right )} \log \left (c x + a\right ) - {\left (3 \, a^{3} c d + 4 \, a^{4} e - {\left (3 \, a c^{3} d + 4 \, a^{2} c^{2} e\right )} x^{2}\right )} \log \left (c x - a\right )}{4 \, {\left (c^{8} x^{2} - a^{2} c^{6}\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 = 1.19 \begin {gather*} -\frac {{\left (3 \, a c d - 4 \, a^{2} e\right )} \log \left ({\left | c x + a \right |}\right )}{4 \, c^{6}} + \frac {{\left (3 \, a c d + 4 \, a^{2} e\right )} \log \left ({\left | c x - a \right |}\right )}{4 \, c^{6}} + \frac {c^{4} x^{2} e + 2 \, c^{4} d x}{2 \, c^{8}} - \frac {a^{2} c^{2} d x + a^{4} e}{2 \, {\left (c x + a\right )} {\left (c x - a\right )} c^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 143, normalized size = 1.52 \begin {gather*} \frac {e \,x^{2}}{2 c^{4}}+\frac {a^{3} e}{4 \left (c x +a \right ) c^{6}}-\frac {a^{3} e}{4 \left (c x -a \right ) c^{6}}-\frac {a^{2} d}{4 \left (c x +a \right ) c^{5}}-\frac {a^{2} d}{4 \left (c x -a \right ) c^{5}}+\frac {a^{2} e \ln \left (c x -a \right )}{c^{6}}+\frac {a^{2} e \ln \left (c x +a \right )}{c^{6}}+\frac {3 a d \ln \left (c x -a \right )}{4 c^{5}}-\frac {3 a d \ln \left (c x +a \right )}{4 c^{5}}+\frac {d x}{c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.54, size = 99, normalized size = 1.05 \begin {gather*} -\frac {a^{2} c^{2} d x + a^{4} e}{2 \, {\left (c^{8} x^{2} - a^{2} c^{6}\right )}} + \frac {e x^{2} + 2 \, d x}{2 \, c^{4}} - \frac {{\left (3 \, a c d - 4 \, a^{2} e\right )} \log \left (c x + a\right )}{4 \, c^{6}} + \frac {{\left (3 \, a c d + 4 \, a^{2} e\right )} \log \left (c x - a\right )}{4 \, c^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.09, size = 99, normalized size = 1.05 \begin {gather*} \frac {\frac {a^4\,e}{2\,c^2}+\frac {a^2\,d\,x}{2}}{a^2\,c^4-c^6\,x^2}+\frac {e\,x^2}{2\,c^4}+\frac {\ln \left (a+c\,x\right )\,\left (4\,a^2\,e-3\,a\,c\,d\right )}{4\,c^6}+\frac {\ln \left (a-c\,x\right )\,\left (4\,e\,a^2+3\,c\,d\,a\right )}{4\,c^6}+\frac {d\,x}{c^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.80, size = 141, normalized size = 1.50 \begin {gather*} \frac {a \left (4 a e - 3 c d\right ) \log {\left (x + \frac {4 a^{2} e - a \left (4 a e - 3 c d\right )}{3 c^{2} d} \right )}}{4 c^{6}} + \frac {a \left (4 a e + 3 c d\right ) \log {\left (x + \frac {4 a^{2} e - a \left (4 a e + 3 c d\right )}{3 c^{2} d} \right )}}{4 c^{6}} + \frac {- a^{4} e - a^{2} c^{2} d x}{- 2 a^{2} c^{6} + 2 c^{8} x^{2}} + \frac {d x}{c^{4}} + \frac {e x^{2}}{2 c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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