Optimal. Leaf size=168 \[ \frac {20 d \sqrt {x} \sqrt {d+e x^2}}{147 e^{3/2}}-\frac {4 x^{5/2} \sqrt {d+e x^2}}{49 \sqrt {e}}+\frac {2}{7} x^{7/2} \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )-\frac {10 d^{7/4} \left (\sqrt {d}+\sqrt {e} x\right ) \sqrt {\frac {d+e x^2}{\left (\sqrt {d}+\sqrt {e} x\right )^2}} F\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right )|\frac {1}{2}\right )}{147 e^{7/4} \sqrt {d+e x^2}} \]
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Rubi [A]
time = 0.06, antiderivative size = 168, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {6356, 327, 335,
226} \begin {gather*} -\frac {10 d^{7/4} \left (\sqrt {d}+\sqrt {e} x\right ) \sqrt {\frac {d+e x^2}{\left (\sqrt {d}+\sqrt {e} x\right )^2}} F\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right )|\frac {1}{2}\right )}{147 e^{7/4} \sqrt {d+e x^2}}+\frac {20 d \sqrt {x} \sqrt {d+e x^2}}{147 e^{3/2}}-\frac {4 x^{5/2} \sqrt {d+e x^2}}{49 \sqrt {e}}+\frac {2}{7} x^{7/2} \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 226
Rule 327
Rule 335
Rule 6356
Rubi steps
\begin {align*} \int x^{5/2} \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) \, dx &=\frac {2}{7} x^{7/2} \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )-\frac {1}{7} \left (2 \sqrt {e}\right ) \int \frac {x^{7/2}}{\sqrt {d+e x^2}} \, dx\\ &=-\frac {4 x^{5/2} \sqrt {d+e x^2}}{49 \sqrt {e}}+\frac {2}{7} x^{7/2} \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )+\frac {(10 d) \int \frac {x^{3/2}}{\sqrt {d+e x^2}} \, dx}{49 \sqrt {e}}\\ &=\frac {20 d \sqrt {x} \sqrt {d+e x^2}}{147 e^{3/2}}-\frac {4 x^{5/2} \sqrt {d+e x^2}}{49 \sqrt {e}}+\frac {2}{7} x^{7/2} \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )-\frac {\left (10 d^2\right ) \int \frac {1}{\sqrt {x} \sqrt {d+e x^2}} \, dx}{147 e^{3/2}}\\ &=\frac {20 d \sqrt {x} \sqrt {d+e x^2}}{147 e^{3/2}}-\frac {4 x^{5/2} \sqrt {d+e x^2}}{49 \sqrt {e}}+\frac {2}{7} x^{7/2} \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )-\frac {\left (20 d^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {d+e x^4}} \, dx,x,\sqrt {x}\right )}{147 e^{3/2}}\\ &=\frac {20 d \sqrt {x} \sqrt {d+e x^2}}{147 e^{3/2}}-\frac {4 x^{5/2} \sqrt {d+e x^2}}{49 \sqrt {e}}+\frac {2}{7} x^{7/2} \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )-\frac {10 d^{7/4} \left (\sqrt {d}+\sqrt {e} x\right ) \sqrt {\frac {d+e x^2}{\left (\sqrt {d}+\sqrt {e} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right )|\frac {1}{2}\right )}{147 e^{7/4} \sqrt {d+e x^2}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.28, size = 147, normalized size = 0.88 \begin {gather*} \frac {2}{147} \sqrt {x} \left (\frac {2 \left (5 d-3 e x^2\right ) \sqrt {d+e x^2}}{e^{3/2}}+21 x^3 \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )\right )+\frac {20 \sqrt {d} \left (\frac {i \sqrt {d}}{\sqrt {e}}\right )^{5/2} \sqrt {1+\frac {d}{e x^2}} x F\left (\left .i \sinh ^{-1}\left (\frac {\sqrt {\frac {i \sqrt {d}}{\sqrt {e}}}}{\sqrt {x}}\right )\right |-1\right )}{147 \sqrt {d+e x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.22, size = 0, normalized size = 0.00 \[\int x^{\frac {5}{2}} \arctanh \left (\frac {x \sqrt {e}}{\sqrt {e \,x^{2}+d}}\right )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.11, size = 288, normalized size = 1.71 \begin {gather*} -\frac {20 \, d^{2} {\rm weierstrassPInverse}\left (-\frac {4 \, d}{\cosh \left (\frac {1}{2}\right )^{2} + 2 \, \cosh \left (\frac {1}{2}\right ) \sinh \left (\frac {1}{2}\right ) + \sinh \left (\frac {1}{2}\right )^{2}}, 0, x\right ) - 21 \, {\left (x^{3} \cosh \left (\frac {1}{2}\right )^{4} + 4 \, x^{3} \cosh \left (\frac {1}{2}\right )^{3} \sinh \left (\frac {1}{2}\right ) + 6 \, x^{3} \cosh \left (\frac {1}{2}\right )^{2} \sinh \left (\frac {1}{2}\right )^{2} + 4 \, x^{3} \cosh \left (\frac {1}{2}\right ) \sinh \left (\frac {1}{2}\right )^{3} + x^{3} \sinh \left (\frac {1}{2}\right )^{4}\right )} \sqrt {x} \log \left (\frac {2 \, x^{2} \cosh \left (\frac {1}{2}\right )^{2} + 4 \, x^{2} \cosh \left (\frac {1}{2}\right ) \sinh \left (\frac {1}{2}\right ) + 2 \, x^{2} \sinh \left (\frac {1}{2}\right )^{2} + 2 \, {\left (x \cosh \left (\frac {1}{2}\right ) + x \sinh \left (\frac {1}{2}\right )\right )} \sqrt {\frac {{\left (x^{2} + d\right )} \cosh \left (\frac {1}{2}\right ) + {\left (x^{2} - d\right )} \sinh \left (\frac {1}{2}\right )}{\cosh \left (\frac {1}{2}\right ) - \sinh \left (\frac {1}{2}\right )}} + d}{d}\right ) + 4 \, {\left (3 \, x^{2} \cosh \left (\frac {1}{2}\right )^{3} + 9 \, x^{2} \cosh \left (\frac {1}{2}\right ) \sinh \left (\frac {1}{2}\right )^{2} + 3 \, x^{2} \sinh \left (\frac {1}{2}\right )^{3} - 5 \, d \cosh \left (\frac {1}{2}\right ) + {\left (9 \, x^{2} \cosh \left (\frac {1}{2}\right )^{2} - 5 \, d\right )} \sinh \left (\frac {1}{2}\right )\right )} \sqrt {x} \sqrt {\frac {{\left (x^{2} + d\right )} \cosh \left (\frac {1}{2}\right ) + {\left (x^{2} - d\right )} \sinh \left (\frac {1}{2}\right )}{\cosh \left (\frac {1}{2}\right ) - \sinh \left (\frac {1}{2}\right )}}}{147 \, {\left (\cosh \left (\frac {1}{2}\right )^{4} + 4 \, \cosh \left (\frac {1}{2}\right )^{3} \sinh \left (\frac {1}{2}\right ) + 6 \, \cosh \left (\frac {1}{2}\right )^{2} \sinh \left (\frac {1}{2}\right )^{2} + 4 \, \cosh \left (\frac {1}{2}\right ) \sinh \left (\frac {1}{2}\right )^{3} + \sinh \left (\frac {1}{2}\right )^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{\frac {5}{2}} \operatorname {atanh}{\left (\frac {\sqrt {e} x}{\sqrt {d + e x^{2}}} \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^{5/2}\,\mathrm {atanh}\left (\frac {\sqrt {e}\,x}{\sqrt {e\,x^2+d}}\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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