Optimal. Leaf size=302 \[ -\frac {6 e^{7/4} \left (\sqrt {d}+\sqrt {e} x\right ) \sqrt {\frac {d+e x^2}{\left (\sqrt {d}+\sqrt {e} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right )|\frac {1}{2}\right )}{35 d^{7/4} \sqrt {d+e x^2}}+\frac {12 e^{7/4} \left (\sqrt {d}+\sqrt {e} x\right ) \sqrt {\frac {d+e x^2}{\left (\sqrt {d}+\sqrt {e} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right )|\frac {1}{2}\right )}{35 d^{7/4} \sqrt {d+e x^2}}+\frac {12 e^{3/2} \sqrt {d+e x^2}}{35 d^2 \sqrt {x}}-\frac {12 e^2 \sqrt {x} \sqrt {d+e x^2}}{35 d^2 \left (\sqrt {d}+\sqrt {e} x\right )}-\frac {4 \sqrt {e} \sqrt {d+e x^2}}{35 d x^{5/2}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{7 x^{7/2}} \]
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Rubi [A] time = 0.18, antiderivative size = 302, 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, 325, 329, 305, 220, 1196} \[ -\frac {12 e^2 \sqrt {x} \sqrt {d+e x^2}}{35 d^2 \left (\sqrt {d}+\sqrt {e} x\right )}+\frac {12 e^{3/2} \sqrt {d+e x^2}}{35 d^2 \sqrt {x}}-\frac {6 e^{7/4} \left (\sqrt {d}+\sqrt {e} x\right ) \sqrt {\frac {d+e x^2}{\left (\sqrt {d}+\sqrt {e} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right )|\frac {1}{2}\right )}{35 d^{7/4} \sqrt {d+e x^2}}+\frac {12 e^{7/4} \left (\sqrt {d}+\sqrt {e} x\right ) \sqrt {\frac {d+e x^2}{\left (\sqrt {d}+\sqrt {e} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right )|\frac {1}{2}\right )}{35 d^{7/4} \sqrt {d+e x^2}}-\frac {4 \sqrt {e} \sqrt {d+e x^2}}{35 d x^{5/2}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{7 x^{7/2}} \]
Antiderivative was successfully verified.
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Rule 220
Rule 305
Rule 325
Rule 329
Rule 1196
Rule 6221
Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{x^{9/2}} \, dx &=-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{7 x^{7/2}}+\frac {1}{7} \left (2 \sqrt {e}\right ) \int \frac {1}{x^{7/2} \sqrt {d+e x^2}} \, dx\\ &=-\frac {4 \sqrt {e} \sqrt {d+e x^2}}{35 d x^{5/2}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{7 x^{7/2}}-\frac {\left (6 e^{3/2}\right ) \int \frac {1}{x^{3/2} \sqrt {d+e x^2}} \, dx}{35 d}\\ &=-\frac {4 \sqrt {e} \sqrt {d+e x^2}}{35 d x^{5/2}}+\frac {12 e^{3/2} \sqrt {d+e x^2}}{35 d^2 \sqrt {x}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{7 x^{7/2}}-\frac {\left (6 e^{5/2}\right ) \int \frac {\sqrt {x}}{\sqrt {d+e x^2}} \, dx}{35 d^2}\\ &=-\frac {4 \sqrt {e} \sqrt {d+e x^2}}{35 d x^{5/2}}+\frac {12 e^{3/2} \sqrt {d+e x^2}}{35 d^2 \sqrt {x}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{7 x^{7/2}}-\frac {\left (12 e^{5/2}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {d+e x^4}} \, dx,x,\sqrt {x}\right )}{35 d^2}\\ &=-\frac {4 \sqrt {e} \sqrt {d+e x^2}}{35 d x^{5/2}}+\frac {12 e^{3/2} \sqrt {d+e x^2}}{35 d^2 \sqrt {x}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{7 x^{7/2}}-\frac {\left (12 e^2\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {d+e x^4}} \, dx,x,\sqrt {x}\right )}{35 d^{3/2}}+\frac {\left (12 e^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 )}{35 d^{3/2}}\\ &=-\frac {4 \sqrt {e} \sqrt {d+e x^2}}{35 d x^{5/2}}+\frac {12 e^{3/2} \sqrt {d+e x^2}}{35 d^2 \sqrt {x}}-\frac {12 e^2 \sqrt {x} \sqrt {d+e x^2}}{35 d^2 \left (\sqrt {d}+\sqrt {e} x\right )}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{7 x^{7/2}}+\frac {12 e^{7/4} \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 )}{35 d^{7/4} \sqrt {d+e x^2}}-\frac {6 e^{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 )}{35 d^{7/4} \sqrt {d+e x^2}}\\ \end {align*}
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Mathematica [C] time = 0.11, size = 131, normalized size = 0.43 \[ \frac {4 \sqrt {e} x \left (-d^2+2 d e x^2+3 e^2 x^4\right )-10 d^2 \sqrt {d+e x^2} \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )-4 e^{5/2} x^5 \sqrt {\frac {e x^2}{d}+1} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};-\frac {e x^2}{d}\right )}{35 d^2 x^{7/2} \sqrt {d+e x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.58, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {artanh}\left (\frac {\sqrt {e} x}{\sqrt {e x^{2} + d}}\right )}{x^{\frac {9}{2}}}, x\right ) \]
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 \[ \int \frac {\operatorname {artanh}\left (\frac {\sqrt {e} x}{\sqrt {e x^{2} + d}}\right )}{x^{\frac {9}{2}}}\,{d x} \]
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 \[ \int \frac {\arctanh \left (\frac {x \sqrt {e}}{\sqrt {e \,x^{2}+d}}\right )}{x^{\frac {9}{2}}}\, dx \]
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 \[ 2 \, d \sqrt {e} \int -\frac {\sqrt {e x^{2} + d} x}{7 \, {\left ({\left (e^{2} x^{4} + d e x^{2}\right )} x^{\frac {9}{2}} - {\left (e x^{2} + d\right )} e^{\left (\log \left (e x^{2} + d\right ) + \frac {9}{2} \, \log \relax (x)\right )}\right )}}\,{d x} - \frac {\log \left (\sqrt {e} x + \sqrt {e x^{2} + d}\right )}{7 \, x^{\frac {7}{2}}} + \frac {\log \left (-\sqrt {e} x + \sqrt {e x^{2} + d}\right )}{7 \, x^{\frac {7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\mathrm {atanh}\left (\frac {\sqrt {e}\,x}{\sqrt {e\,x^2+d}}\right )}{x^{9/2}} \,d x \]
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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