Optimal. Leaf size=173 \[ \frac {10 e^{9/4} \left (\sqrt {d}+\sqrt {e} x\right ) \sqrt {\frac {d+e x^2}{\left (\sqrt {d}+\sqrt {e} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right )|\frac {1}{2}\right )}{189 d^{9/4} \sqrt {d+e x^2}}+\frac {20 e^{3/2} \sqrt {d+e x^2}}{189 d^2 x^{3/2}}-\frac {4 \sqrt {e} \sqrt {d+e x^2}}{63 d x^{7/2}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{9 x^{9/2}} \]
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Rubi [A] time = 0.09, antiderivative size = 173, 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, 325, 329, 220} \[ \frac {20 e^{3/2} \sqrt {d+e x^2}}{189 d^2 x^{3/2}}+\frac {10 e^{9/4} \left (\sqrt {d}+\sqrt {e} x\right ) \sqrt {\frac {d+e x^2}{\left (\sqrt {d}+\sqrt {e} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right )|\frac {1}{2}\right )}{189 d^{9/4} \sqrt {d+e x^2}}-\frac {4 \sqrt {e} \sqrt {d+e x^2}}{63 d x^{7/2}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{9 x^{9/2}} \]
Antiderivative was successfully verified.
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Rule 220
Rule 325
Rule 329
Rule 6221
Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{x^{11/2}} \, dx &=-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{9 x^{9/2}}+\frac {1}{9} \left (2 \sqrt {e}\right ) \int \frac {1}{x^{9/2} \sqrt {d+e x^2}} \, dx\\ &=-\frac {4 \sqrt {e} \sqrt {d+e x^2}}{63 d x^{7/2}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{9 x^{9/2}}-\frac {\left (10 e^{3/2}\right ) \int \frac {1}{x^{5/2} \sqrt {d+e x^2}} \, dx}{63 d}\\ &=-\frac {4 \sqrt {e} \sqrt {d+e x^2}}{63 d x^{7/2}}+\frac {20 e^{3/2} \sqrt {d+e x^2}}{189 d^2 x^{3/2}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{9 x^{9/2}}+\frac {\left (10 e^{5/2}\right ) \int \frac {1}{\sqrt {x} \sqrt {d+e x^2}} \, dx}{189 d^2}\\ &=-\frac {4 \sqrt {e} \sqrt {d+e x^2}}{63 d x^{7/2}}+\frac {20 e^{3/2} \sqrt {d+e x^2}}{189 d^2 x^{3/2}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{9 x^{9/2}}+\frac {\left (20 e^{5/2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {d+e x^4}} \, dx,x,\sqrt {x}\right )}{189 d^2}\\ &=-\frac {4 \sqrt {e} \sqrt {d+e x^2}}{63 d x^{7/2}}+\frac {20 e^{3/2} \sqrt {d+e x^2}}{189 d^2 x^{3/2}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{9 x^{9/2}}+\frac {10 e^{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 )}{189 d^{9/4} \sqrt {d+e x^2}}\\ \end {align*}
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Mathematica [C] time = 0.34, size = 154, normalized size = 0.89 \[ \frac {4 \sqrt {e} x \sqrt {d+e x^2} \left (5 e x^2-3 d\right )-42 d^2 \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{189 d^2 x^{9/2}}+\frac {20 e^3 x \sqrt {\frac {i \sqrt {d}}{\sqrt {e}}} \sqrt {\frac {d}{e x^2}+1} F\left (\left .i \sinh ^{-1}\left (\frac {\sqrt {\frac {i \sqrt {d}}{\sqrt {e}}}}{\sqrt {x}}\right )\right |-1\right )}{189 d^{5/2} \sqrt {d+e x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.58, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {artanh}\left (\frac {\sqrt {e} x}{\sqrt {e x^{2} + d}}\right )}{x^{\frac {11}{2}}}, x\right ) \]
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 \[ \int \frac {\operatorname {artanh}\left (\frac {\sqrt {e} x}{\sqrt {e x^{2} + d}}\right )}{x^{\frac {11}{2}}}\,{d x} \]
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 \[ \int \frac {\arctanh \left (\frac {x \sqrt {e}}{\sqrt {e \,x^{2}+d}}\right )}{x^{\frac {11}{2}}}\, dx \]
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 \[ 2 \, d \sqrt {e} \int -\frac {\sqrt {e x^{2} + d} x}{9 \, {\left ({\left (e^{2} x^{4} + d e x^{2}\right )} x^{\frac {11}{2}} - {\left (e x^{2} + d\right )} e^{\left (\log \left (e x^{2} + d\right ) + \frac {11}{2} \, \log \relax (x)\right )}\right )}}\,{d x} - \frac {\log \left (\sqrt {e} x + \sqrt {e x^{2} + d}\right )}{9 \, x^{\frac {9}{2}}} + \frac {\log \left (-\sqrt {e} x + \sqrt {e x^{2} + d}\right )}{9 \, x^{\frac {9}{2}}} \]
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 \[ \int \frac {\mathrm {atanh}\left (\frac {\sqrt {e}\,x}{\sqrt {e\,x^2+d}}\right )}{x^{11/2}} \,d x \]
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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