Optimal. Leaf size=51 \[ -7 d^2 x-\frac {4}{3} d e x^3-\frac {e^2 x^5}{5}+\frac {8 d^{5/2} \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}} \]
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Rubi [A]
time = 0.03, antiderivative size = 51, 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 = {1164, 398, 214}
\begin {gather*} \frac {8 d^{5/2} \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}-7 d^2 x-\frac {4}{3} d e x^3-\frac {1}{5} e^2 x^5 \end {gather*}
Antiderivative was successfully verified.
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Rule 214
Rule 398
Rule 1164
Rubi steps
\begin {align*} \int \frac {\left (d+e x^2\right )^4}{d^2-e^2 x^4} \, dx &=\int \frac {\left (d+e x^2\right )^3}{d-e x^2} \, dx\\ &=\int \left (-7 d^2-4 d e x^2-e^2 x^4+\frac {8 d^3}{d-e x^2}\right ) \, dx\\ &=-7 d^2 x-\frac {4}{3} d e x^3-\frac {e^2 x^5}{5}+\left (8 d^3\right ) \int \frac {1}{d-e x^2} \, dx\\ &=-7 d^2 x-\frac {4}{3} d e x^3-\frac {e^2 x^5}{5}+\frac {8 d^{5/2} \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 51, normalized size = 1.00 \begin {gather*} -7 d^2 x-\frac {4}{3} d e x^3-\frac {e^2 x^5}{5}+\frac {8 d^{5/2} \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 42, normalized size = 0.82
method | result | size |
default | \(-\frac {e^{2} x^{5}}{5}-\frac {4 x^{3} d e}{3}-7 d^{2} x +\frac {8 d^{3} \arctanh \left (\frac {e x}{\sqrt {d e}}\right )}{\sqrt {d e}}\) | \(42\) |
risch | \(-\frac {e^{2} x^{5}}{5}-\frac {4 x^{3} d e}{3}-7 d^{2} x -\frac {4 \sqrt {d e}\, d^{2} \ln \left (\sqrt {d e}\, x -d \right )}{e}+\frac {4 \sqrt {d e}\, d^{2} \ln \left (-\sqrt {d e}\, x -d \right )}{e}\) | \(74\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 56, normalized size = 1.10 \begin {gather*} -\frac {1}{5} \, x^{5} e^{2} - \frac {4}{3} \, d x^{3} e - 4 \, d^{\frac {5}{2}} e^{\left (-\frac {1}{2}\right )} \log \left (\frac {x e - \sqrt {d} e^{\frac {1}{2}}}{x e + \sqrt {d} e^{\frac {1}{2}}}\right ) - 7 \, d^{2} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 109, normalized size = 2.14 \begin {gather*} \left [-\frac {1}{5} \, x^{5} e^{2} - \frac {4}{3} \, d x^{3} e + 4 \, d^{\frac {5}{2}} e^{\left (-\frac {1}{2}\right )} \log \left (\frac {x^{2} e + 2 \, \sqrt {d} x e^{\frac {1}{2}} + d}{x^{2} e - d}\right ) - 7 \, d^{2} x, -\frac {1}{5} \, x^{5} e^{2} - \frac {4}{3} \, d x^{3} e - 8 \, \sqrt {-d e^{\left (-1\right )}} d^{2} \arctan \left (\frac {\sqrt {-d e^{\left (-1\right )}} x e}{d}\right ) - 7 \, d^{2} x\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.09, size = 75, normalized size = 1.47 \begin {gather*} - 7 d^{2} x - \frac {4 d e x^{3}}{3} - \frac {e^{2} x^{5}}{5} - 4 \sqrt {\frac {d^{5}}{e}} \log {\left (x - \frac {\sqrt {\frac {d^{5}}{e}}}{d^{2}} \right )} + 4 \sqrt {\frac {d^{5}}{e}} \log {\left (x + \frac {\sqrt {\frac {d^{5}}{e}}}{d^{2}} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.32, size = 53, normalized size = 1.04 \begin {gather*} -\frac {8 \, d^{3} \arctan \left (\frac {x e}{\sqrt {-d e}}\right )}{\sqrt {-d e}} - \frac {1}{15} \, {\left (3 \, x^{5} e^{7} + 20 \, d x^{3} e^{6} + 105 \, d^{2} x e^{5}\right )} e^{\left (-5\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.09, size = 42, normalized size = 0.82 \begin {gather*} -7\,d^2\,x-\frac {e^2\,x^5}{5}-\frac {4\,d\,e\,x^3}{3}-\frac {d^{5/2}\,\mathrm {atan}\left (\frac {\sqrt {e}\,x\,1{}\mathrm {i}}{\sqrt {d}}\right )\,8{}\mathrm {i}}{\sqrt {e}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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