Optimal. Leaf size=297 \[ -\frac {14 d^{9/4} \left (\sqrt {d}+\sqrt {e} x\right ) \sqrt {\frac {d+e x^2}{\left (\sqrt {d}+\sqrt {e} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right )|\frac {1}{2}\right )}{135 e^{9/4} \sqrt {d+e x^2}}+\frac {28 d^{9/4} \left (\sqrt {d}+\sqrt {e} x\right ) \sqrt {\frac {d+e x^2}{\left (\sqrt {d}+\sqrt {e} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right )|\frac {1}{2}\right )}{135 e^{9/4} \sqrt {d+e x^2}}-\frac {28 d^2 \sqrt {x} \sqrt {d+e x^2}}{135 e^2 \left (\sqrt {d}+\sqrt {e} x\right )}+\frac {28 d x^{3/2} \sqrt {d+e x^2}}{405 e^{3/2}}-\frac {4 x^{7/2} \sqrt {d+e x^2}}{81 \sqrt {e}}+\frac {2}{9} x^{9/2} \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) \]
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Rubi [A] time = 0.19, 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 = {6221, 321, 329, 305, 220, 1196} \[ -\frac {28 d^2 \sqrt {x} \sqrt {d+e x^2}}{135 e^2 \left (\sqrt {d}+\sqrt {e} x\right )}-\frac {14 d^{9/4} \left (\sqrt {d}+\sqrt {e} x\right ) \sqrt {\frac {d+e x^2}{\left (\sqrt {d}+\sqrt {e} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right )|\frac {1}{2}\right )}{135 e^{9/4} \sqrt {d+e x^2}}+\frac {28 d^{9/4} \left (\sqrt {d}+\sqrt {e} x\right ) \sqrt {\frac {d+e x^2}{\left (\sqrt {d}+\sqrt {e} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right )|\frac {1}{2}\right )}{135 e^{9/4} \sqrt {d+e x^2}}+\frac {28 d x^{3/2} \sqrt {d+e x^2}}{405 e^{3/2}}-\frac {4 x^{7/2} \sqrt {d+e x^2}}{81 \sqrt {e}}+\frac {2}{9} x^{9/2} \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) \]
Antiderivative was successfully verified.
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Rule 220
Rule 305
Rule 321
Rule 329
Rule 1196
Rule 6221
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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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] time = 0.15, size = 124, normalized size = 0.42 \[ \frac {2 x^{3/2} \left (-14 d^2 \sqrt {\frac {e x^2}{d}+1} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};-\frac {e x^2}{d}\right )+14 d^2+45 e^{3/2} x^3 \sqrt {d+e x^2} \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )+4 d e x^2-10 e^2 x^4\right )}{405 e^{3/2} \sqrt {d+e x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.58, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{\frac {7}{2}} \operatorname {artanh}\left (\frac {\sqrt {e} x}{\sqrt {e x^{2} + d}}\right ), x\right ) \]
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 \[ \int x^{\frac {7}{2}} \arctanh \left (\frac {x \sqrt {e}}{\sqrt {e \,x^{2}+d}}\right )\, dx \]
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 \[ \frac {1}{9} \, x^{\frac {9}{2}} \log \left (\sqrt {e} x + \sqrt {e x^{2} + d}\right ) - \frac {1}{9} \, x^{\frac {9}{2}} \log \left (-\sqrt {e} x + \sqrt {e x^{2} + d}\right ) - 2 \, d \sqrt {e} \int -\frac {x e^{\left (\frac {1}{2} \, \log \left (e x^{2} + d\right ) + \frac {7}{2} \, \log \relax (x)\right )}}{9 \, {\left (e^{2} x^{4} + d e x^{2} - {\left (e x^{2} + d\right )}^{2}\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int x^{7/2}\,\mathrm {atanh}\left (\frac {\sqrt {e}\,x}{\sqrt {e\,x^2+d}}\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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