Optimal. Leaf size=38 \[ \frac {4 d^{3/2} \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}-3 d x-\frac {e x^3}{3} \]
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Rubi [A] time = 0.03, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1150, 390, 208} \begin {gather*} \frac {4 d^{3/2} \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}-3 d x-\frac {e x^3}{3} \end {gather*}
Antiderivative was successfully verified.
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Rule 208
Rule 390
Rule 1150
Rubi steps
\begin {align*} \int \frac {\left (d+e x^2\right )^3}{d^2-e^2 x^4} \, dx &=\int \frac {\left (d+e x^2\right )^2}{d-e x^2} \, dx\\ &=\int \left (-3 d-e x^2+\frac {4 d^2}{d-e x^2}\right ) \, dx\\ &=-3 d x-\frac {e x^3}{3}+\left (4 d^2\right ) \int \frac {1}{d-e x^2} \, dx\\ &=-3 d x-\frac {e x^3}{3}+\frac {4 d^{3/2} \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 38, normalized size = 1.00 \begin {gather*} \frac {4 d^{3/2} \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}-3 d x-\frac {e x^3}{3} \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 {\left (d+e x^2\right )^3}{d^2-e^2 x^4} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.87, size = 90, normalized size = 2.37 \begin {gather*} \left [-\frac {1}{3} \, e x^{3} + 2 \, d \sqrt {\frac {d}{e}} \log \left (\frac {e x^{2} + 2 \, e x \sqrt {\frac {d}{e}} + d}{e x^{2} - d}\right ) - 3 \, d x, -\frac {1}{3} \, e x^{3} - 4 \, d \sqrt {-\frac {d}{e}} \arctan \left (\frac {e x \sqrt {-\frac {d}{e}}}{d}\right ) - 3 \, d x\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.23, size = 123, normalized size = 3.24 \begin {gather*} 2 \, {\left ({\left (d^{2}\right )}^{\frac {1}{4}} d e^{\frac {11}{2}} - {\left (d^{2}\right )}^{\frac {1}{4}} {\left | d \right |} e^{\frac {11}{2}}\right )} \arctan \left (\frac {x e^{\frac {1}{2}}}{{\left (d^{2}\right )}^{\frac {1}{4}}}\right ) e^{\left (-6\right )} + {\left ({\left (d^{2}\right )}^{\frac {1}{4}} d e^{\frac {15}{2}} + {\left (d^{2}\right )}^{\frac {3}{4}} e^{\frac {15}{2}}\right )} e^{\left (-8\right )} \log \left ({\left | {\left (d^{2}\right )}^{\frac {1}{4}} e^{\left (-\frac {1}{2}\right )} + x \right |}\right ) - {\left ({\left (d^{2}\right )}^{\frac {1}{4}} d e^{\frac {11}{2}} + {\left (d^{2}\right )}^{\frac {1}{4}} {\left | d \right |} e^{\frac {11}{2}}\right )} e^{\left (-6\right )} \log \left ({\left | -{\left (d^{2}\right )}^{\frac {1}{4}} e^{\left (-\frac {1}{2}\right )} + x \right |}\right ) - \frac {1}{3} \, {\left (x^{3} e^{7} + 9 \, d x e^{6}\right )} e^{\left (-6\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 31, normalized size = 0.82 \begin {gather*} -\frac {e \,x^{3}}{3}+\frac {4 d^{2} \arctanh \left (\frac {e x}{\sqrt {d e}}\right )}{\sqrt {d e}}-3 d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.45, size = 45, normalized size = 1.18 \begin {gather*} -\frac {1}{3} \, e x^{3} - \frac {2 \, d^{2} \log \left (\frac {e x - \sqrt {d e}}{e x + \sqrt {d e}}\right )}{\sqrt {d e}} - 3 \, d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 28, normalized size = 0.74 \begin {gather*} \frac {4\,d^{3/2}\,\mathrm {atanh}\left (\frac {\sqrt {e}\,x}{\sqrt {d}}\right )}{\sqrt {e}}-\frac {e\,x^3}{3}-3\,d\,x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.20, size = 58, normalized size = 1.53 \begin {gather*} - 3 d x - \frac {e x^{3}}{3} - 2 \sqrt {\frac {d^{3}}{e}} \log {\left (x - \frac {\sqrt {\frac {d^{3}}{e}}}{d} \right )} + 2 \sqrt {\frac {d^{3}}{e}} \log {\left (x + \frac {\sqrt {\frac {d^{3}}{e}}}{d} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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