Optimal. Leaf size=156 \[ -\frac{2 \sqrt{-e} e^{3/4} \left (\sqrt{d}+\sqrt{e} x\right ) \sqrt{\frac{d+e x^2}{\left (\sqrt{d}+\sqrt{e} x\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{e} \sqrt{x}}{\sqrt [4]{d}}\right ),\frac{1}{2}\right )}{15 d^{5/4} \sqrt{d+e x^2}}-\frac{4 \sqrt{-e} \sqrt{d+e x^2}}{15 d x^{3/2}}-\frac{2 \tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right )}{5 x^{5/2}} \]
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Rubi [A] time = 0.0749551, antiderivative size = 156, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148, Rules used = {5151, 325, 329, 220} \[ -\frac{2 \sqrt{-e} e^{3/4} \left (\sqrt{d}+\sqrt{e} x\right ) \sqrt{\frac{d+e x^2}{\left (\sqrt{d}+\sqrt{e} x\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{e} \sqrt{x}}{\sqrt [4]{d}}\right ),\frac{1}{2}\right )}{15 d^{5/4} \sqrt{d+e x^2}}-\frac{4 \sqrt{-e} \sqrt{d+e x^2}}{15 d x^{3/2}}-\frac{2 \tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right )}{5 x^{5/2}} \]
Antiderivative was successfully verified.
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Rule 5151
Rule 325
Rule 329
Rule 220
Rubi steps
\begin{align*} \int \frac{\tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right )}{x^{7/2}} \, dx &=-\frac{2 \tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right )}{5 x^{5/2}}+\frac{1}{5} \left (2 \sqrt{-e}\right ) \int \frac{1}{x^{5/2} \sqrt{d+e x^2}} \, dx\\ &=-\frac{4 \sqrt{-e} \sqrt{d+e x^2}}{15 d x^{3/2}}-\frac{2 \tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right )}{5 x^{5/2}}+\frac{\left (2 (-e)^{3/2}\right ) \int \frac{1}{\sqrt{x} \sqrt{d+e x^2}} \, dx}{15 d}\\ &=-\frac{4 \sqrt{-e} \sqrt{d+e x^2}}{15 d x^{3/2}}-\frac{2 \tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right )}{5 x^{5/2}}+\frac{\left (4 (-e)^{3/2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{d+e x^4}} \, dx,x,\sqrt{x}\right )}{15 d}\\ &=-\frac{4 \sqrt{-e} \sqrt{d+e x^2}}{15 d x^{3/2}}-\frac{2 \tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right )}{5 x^{5/2}}+\frac{2 (-e)^{3/2} \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 )}{15 d^{5/4} \sqrt [4]{e} \sqrt{d+e x^2}}\\ \end{align*}
Mathematica [C] time = 0.266727, size = 150, normalized size = 0.96 \[ -\frac{2 \left (2 \sqrt{-e} x \sqrt{d+e x^2}+3 d \tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right )\right )}{15 d x^{5/2}}+\frac{4 i (-e)^{3/2} x \sqrt{\frac{d}{e x^2}+1} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{d}}{\sqrt{e}}}}{\sqrt{x}}\right ),-1\right )}{15 d \sqrt{\frac{i \sqrt{d}}{\sqrt{e}}} \sqrt{d+e x^2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.29, size = 0, normalized size = 0. \begin{align*} \int{\arctan \left ({x\sqrt{-e}{\frac{1}{\sqrt{e{x}^{2}+d}}}} \right ){x}^{-{\frac{7}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{2 \,{\left (-d \sqrt{-e} x^{\frac{5}{2}} \int -\frac{\sqrt{e x^{2} + d} x}{{\left (e x^{2} + d\right )}^{2} x^{\frac{7}{2}} -{\left (e^{2} x^{4} + d e x^{2}\right )} x^{\frac{7}{2}}}\,{d x} - \arctan \left (\sqrt{-e} x, \sqrt{e x^{2} + d}\right )\right )}}{5 \, x^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\arctan \left (\frac{\sqrt{-e} x}{\sqrt{e x^{2} + d}}\right )}{x^{\frac{7}{2}}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\arctan \left (\frac{\sqrt{-e} x}{\sqrt{e x^{2} + d}}\right )}{x^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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