Optimal. Leaf size=142 \[ \frac {2 d^{3/4} \left (\sqrt {d}+\sqrt {e} x\right ) \sqrt {\frac {d+e x^2}{\left (\sqrt {d}+\sqrt {e} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right )|\frac {1}{2}\right )}{9 e^{3/4} \sqrt {d+e x^2}}-\frac {4 \sqrt {x} \sqrt {d+e x^2}}{9 \sqrt {e}}+\frac {2}{3} x^{3/2} \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) \]
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Rubi [A] time = 0.07, antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {6221, 321, 329, 220} \[ \frac {2 d^{3/4} \left (\sqrt {d}+\sqrt {e} x\right ) \sqrt {\frac {d+e x^2}{\left (\sqrt {d}+\sqrt {e} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right )|\frac {1}{2}\right )}{9 e^{3/4} \sqrt {d+e x^2}}-\frac {4 \sqrt {x} \sqrt {d+e x^2}}{9 \sqrt {e}}+\frac {2}{3} x^{3/2} \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) \]
Antiderivative was successfully verified.
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Rule 220
Rule 321
Rule 329
Rule 6221
Rubi steps
\begin {align*} \int \sqrt {x} \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) \, dx &=\frac {2}{3} x^{3/2} \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )-\frac {1}{3} \left (2 \sqrt {e}\right ) \int \frac {x^{3/2}}{\sqrt {d+e x^2}} \, dx\\ &=-\frac {4 \sqrt {x} \sqrt {d+e x^2}}{9 \sqrt {e}}+\frac {2}{3} x^{3/2} \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )+\frac {(2 d) \int \frac {1}{\sqrt {x} \sqrt {d+e x^2}} \, dx}{9 \sqrt {e}}\\ &=-\frac {4 \sqrt {x} \sqrt {d+e x^2}}{9 \sqrt {e}}+\frac {2}{3} x^{3/2} \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )+\frac {(4 d) \operatorname {Subst}\left (\int \frac {1}{\sqrt {d+e x^4}} \, dx,x,\sqrt {x}\right )}{9 \sqrt {e}}\\ &=-\frac {4 \sqrt {x} \sqrt {d+e x^2}}{9 \sqrt {e}}+\frac {2}{3} x^{3/2} \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )+\frac {2 d^{3/4} \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 )}{9 e^{3/4} \sqrt {d+e x^2}}\\ \end {align*}
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Mathematica [C] time = 0.31, size = 135, normalized size = 0.95 \[ \frac {2}{9} \sqrt {x} \left (3 x \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )-\frac {2 \sqrt {d+e x^2}}{\sqrt {e}}\right )+\frac {4 \sqrt {d} x \sqrt {\frac {i \sqrt {d}}{\sqrt {e}}} \sqrt {\frac {d}{e x^2}+1} F\left (\left .i \sinh ^{-1}\left (\frac {\sqrt {\frac {i \sqrt {d}}{\sqrt {e}}}}{\sqrt {x}}\right )\right |-1\right )}{9 \sqrt {d+e x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.81, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {x} \operatorname {artanh}\left (\frac {\sqrt {e} x}{\sqrt {e x^{2} + d}}\right ), x\right ) \]
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 \[ \int \sqrt {x}\, \arctanh \left (\frac {x \sqrt {e}}{\sqrt {e \,x^{2}+d}}\right )\, dx \]
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 \[ -2 \, d \sqrt {e} \int -\frac {x e^{\left (\frac {1}{2} \, \log \left (e x^{2} + d\right ) + \frac {1}{2} \, \log \relax (x)\right )}}{3 \, {\left (e^{2} x^{4} + d e x^{2} - {\left (e x^{2} + d\right )}^{2}\right )}}\,{d x} + \frac {1}{3} \, x^{\frac {3}{2}} \log \left (\sqrt {e} x + \sqrt {e x^{2} + d}\right ) - \frac {1}{3} \, x^{\frac {3}{2}} \log \left (-\sqrt {e} x + \sqrt {e x^{2} + d}\right ) \]
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 \[ \int \sqrt {x}\,\mathrm {atanh}\left (\frac {\sqrt {e}\,x}{\sqrt {e\,x^2+d}}\right ) \,d x \]
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 \[ \int \sqrt {x} \operatorname {atanh}{\left (\frac {\sqrt {e} x}{\sqrt {d + e x^{2}}} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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