Optimal. Leaf size=297 \[ \frac {28 d x^{3/2} \sqrt {d+e x^2}}{405 e^{3/2}}-\frac {4 x^{7/2} \sqrt {d+e x^2}}{81 \sqrt {e}}-\frac {28 d^2 \sqrt {x} \sqrt {d+e x^2}}{135 e^2 \left (\sqrt {d}+\sqrt {e} x\right )}+\frac {2}{9} x^{9/2} \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )+\frac {28 d^{9/4} \left (\sqrt {d}+\sqrt {e} x\right ) \sqrt {\frac {d+e x^2}{\left (\sqrt {d}+\sqrt {e} x\right )^2}} E\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right )|\frac {1}{2}\right )}{135 e^{9/4} \sqrt {d+e x^2}}-\frac {14 d^{9/4} \left (\sqrt {d}+\sqrt {e} x\right ) \sqrt {\frac {d+e x^2}{\left (\sqrt {d}+\sqrt {e} x\right )^2}} F\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right )|\frac {1}{2}\right )}{135 e^{9/4} \sqrt {d+e x^2}} \]
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Rubi [A]
time = 0.14, antiderivative size = 297, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {6356, 327, 335,
311, 226, 1210} \begin {gather*} -\frac {14 d^{9/4} \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 )}{135 e^{9/4} \sqrt {d+e x^2}}+\frac {28 d^{9/4} \left (\sqrt {d}+\sqrt {e} x\right ) \sqrt {\frac {d+e x^2}{\left (\sqrt {d}+\sqrt {e} x\right )^2}} E\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right )|\frac {1}{2}\right )}{135 e^{9/4} \sqrt {d+e x^2}}-\frac {28 d^2 \sqrt {x} \sqrt {d+e x^2}}{135 e^2 \left (\sqrt {d}+\sqrt {e} x\right )}+\frac {28 d x^{3/2} \sqrt {d+e x^2}}{405 e^{3/2}}-\frac {4 x^{7/2} \sqrt {d+e x^2}}{81 \sqrt {e}}+\frac {2}{9} x^{9/2} \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 311
Rule 327
Rule 335
Rule 1210
Rule 6356
Rubi steps
\begin {align*} \int x^{7/2} \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) \, dx &=\frac {2}{9} x^{9/2} \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )-\frac {1}{9} \left (2 \sqrt {e}\right ) \int \frac {x^{9/2}}{\sqrt {d+e x^2}} \, dx\\ &=-\frac {4 x^{7/2} \sqrt {d+e x^2}}{81 \sqrt {e}}+\frac {2}{9} x^{9/2} \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )+\frac {(14 d) \int \frac {x^{5/2}}{\sqrt {d+e x^2}} \, dx}{81 \sqrt {e}}\\ &=\frac {28 d x^{3/2} \sqrt {d+e x^2}}{405 e^{3/2}}-\frac {4 x^{7/2} \sqrt {d+e x^2}}{81 \sqrt {e}}+\frac {2}{9} x^{9/2} \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )-\frac {\left (14 d^2\right ) \int \frac {\sqrt {x}}{\sqrt {d+e x^2}} \, dx}{135 e^{3/2}}\\ &=\frac {28 d x^{3/2} \sqrt {d+e x^2}}{405 e^{3/2}}-\frac {4 x^{7/2} \sqrt {d+e x^2}}{81 \sqrt {e}}+\frac {2}{9} x^{9/2} \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )-\frac {\left (28 d^2\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {d+e x^4}} \, dx,x,\sqrt {x}\right )}{135 e^{3/2}}\\ &=\frac {28 d x^{3/2} \sqrt {d+e x^2}}{405 e^{3/2}}-\frac {4 x^{7/2} \sqrt {d+e x^2}}{81 \sqrt {e}}+\frac {2}{9} x^{9/2} \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )-\frac {\left (28 d^{5/2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {d+e x^4}} \, dx,x,\sqrt {x}\right )}{135 e^2}+\frac {\left (28 d^{5/2}\right ) \text {Subst}\left (\int \frac {1-\frac {\sqrt {e} x^2}{\sqrt {d}}}{\sqrt {d+e x^4}} \, dx,x,\sqrt {x}\right )}{135 e^2}\\ &=\frac {28 d x^{3/2} \sqrt {d+e x^2}}{405 e^{3/2}}-\frac {4 x^{7/2} \sqrt {d+e x^2}}{81 \sqrt {e}}-\frac {28 d^2 \sqrt {x} \sqrt {d+e x^2}}{135 e^2 \left (\sqrt {d}+\sqrt {e} x\right )}+\frac {2}{9} x^{9/2} \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )+\frac {28 d^{9/4} \left (\sqrt {d}+\sqrt {e} x\right ) \sqrt {\frac {d+e x^2}{\left (\sqrt {d}+\sqrt {e} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right )|\frac {1}{2}\right )}{135 e^{9/4} \sqrt {d+e x^2}}-\frac {14 d^{9/4} \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 )}{135 e^{9/4} \sqrt {d+e x^2}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 0.09, size = 124, normalized size = 0.42 \begin {gather*} \frac {2 x^{3/2} \left (14 d^2+4 d e x^2-10 e^2 x^4+45 e^{3/2} x^3 \sqrt {d+e x^2} \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )-14 d^2 \sqrt {1+\frac {e x^2}{d}} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};-\frac {e x^2}{d}\right )\right )}{405 e^{3/2} \sqrt {d+e x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.31, size = 0, normalized size = 0.00 \[\int x^{\frac {7}{2}} \arctanh \left (\frac {x \sqrt {e}}{\sqrt {e \,x^{2}+d}}\right )\, dx\]
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 = 312, normalized size = 1.05 \begin {gather*} \frac {84 \, d^{2} {\rm weierstrassZeta}\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, {\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 )\right ) + 45 \, {\left (x^{4} \cosh \left (\frac {1}{2}\right )^{4} + 4 \, x^{4} \cosh \left (\frac {1}{2}\right )^{3} \sinh \left (\frac {1}{2}\right ) + 6 \, x^{4} \cosh \left (\frac {1}{2}\right )^{2} \sinh \left (\frac {1}{2}\right )^{2} + 4 \, x^{4} \cosh \left (\frac {1}{2}\right ) \sinh \left (\frac {1}{2}\right )^{3} + x^{4} \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 (5 \, x^{3} \cosh \left (\frac {1}{2}\right )^{3} + 15 \, x^{3} \cosh \left (\frac {1}{2}\right ) \sinh \left (\frac {1}{2}\right )^{2} + 5 \, x^{3} \sinh \left (\frac {1}{2}\right )^{3} - 7 \, d x \cosh \left (\frac {1}{2}\right ) + {\left (15 \, x^{3} \cosh \left (\frac {1}{2}\right )^{2} - 7 \, d x\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 )}}}{405 \, {\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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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 = -0.00 \begin {gather*} \int x^{7/2}\,\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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