Optimal. Leaf size=168 \[ -\frac {10 d^{7/4} \left (\sqrt {d}+\sqrt {e} x\right ) \sqrt {\frac {d+e x^2}{\left (\sqrt {d}+\sqrt {e} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right )|\frac {1}{2}\right )}{147 e^{7/4} \sqrt {d+e x^2}}+\frac {20 d \sqrt {x} \sqrt {d+e x^2}}{147 e^{3/2}}-\frac {4 x^{5/2} \sqrt {d+e x^2}}{49 \sqrt {e}}+\frac {2}{7} x^{7/2} \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) \]
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Rubi [A] time = 0.09, 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 = {6221, 321, 329, 220} \[ -\frac {10 d^{7/4} \left (\sqrt {d}+\sqrt {e} x\right ) \sqrt {\frac {d+e x^2}{\left (\sqrt {d}+\sqrt {e} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right )|\frac {1}{2}\right )}{147 e^{7/4} \sqrt {d+e x^2}}+\frac {20 d \sqrt {x} \sqrt {d+e x^2}}{147 e^{3/2}}-\frac {4 x^{5/2} \sqrt {d+e x^2}}{49 \sqrt {e}}+\frac {2}{7} x^{7/2} \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) \]
Antiderivative was successfully verified.
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Rule 220
Rule 321
Rule 329
Rule 6221
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 ) \operatorname {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] time = 0.50, size = 147, normalized size = 0.88 \[ \frac {2}{147} \sqrt {x} \left (\frac {2 \left (5 d-3 e x^2\right ) \sqrt {d+e x^2}}{e^{3/2}}+21 x^3 \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )\right )+\frac {20 \sqrt {d} x \left (\frac {i \sqrt {d}}{\sqrt {e}}\right )^{5/2} \sqrt {\frac {d}{e x^2}+1} F\left (\left .i \sinh ^{-1}\left (\frac {\sqrt {\frac {i \sqrt {d}}{\sqrt {e}}}}{\sqrt {x}}\right )\right |-1\right )}{147 \sqrt {d+e x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{\frac {5}{2}} \operatorname {artanh}\left (\frac {\sqrt {e} x}{\sqrt {e x^{2} + d}}\right ), x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F(-2)] time = 180.00, 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 \[ \frac {1}{7} \, x^{\frac {7}{2}} \log \left (\sqrt {e} x + \sqrt {e x^{2} + d}\right ) - \frac {1}{7} \, x^{\frac {7}{2}} \log \left (-\sqrt {e} x + \sqrt {e x^{2} + d}\right ) - 2 \, d \sqrt {e} \int -\frac {x e^{\left (\frac {1}{2} \, \log \left (e x^{2} + d\right ) + \frac {5}{2} \, \log \relax (x)\right )}}{7 \, {\left (e^{2} x^{4} + d e x^{2} - {\left (e x^{2} + d\right )}^{2}\right )}}\,{d x} \]
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 \[ \int x^{5/2}\,\mathrm {atanh}\left (\frac {\sqrt {e}\,x}{\sqrt {e\,x^2+d}}\right ) \,d x \]
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 \[ \int x^{\frac {5}{2}} \operatorname {atanh}{\left (\frac {\sqrt {e} x}{\sqrt {d + e x^{2}}} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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