Optimal. Leaf size=298 \[ \frac{2 \sqrt{-e} \sqrt [4]{e} \left (\sqrt{d}+\sqrt{e} x\right ) \sqrt{\frac{d+e x^2}{\left (\sqrt{d}+\sqrt{e} x\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{e} \sqrt{x}}{\sqrt [4]{d}}\right ),\frac{1}{2}\right )}{3 d^{3/4} \sqrt{d+e x^2}}-\frac{4 \sqrt{-e} \sqrt [4]{e} \left (\sqrt{d}+\sqrt{e} x\right ) \sqrt{\frac{d+e x^2}{\left (\sqrt{d}+\sqrt{e} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{e} \sqrt{x}}{\sqrt [4]{d}}\right )|\frac{1}{2}\right )}{3 d^{3/4} \sqrt{d+e x^2}}+\frac{4 \sqrt{-e^2} \sqrt{x} \sqrt{d+e x^2}}{3 d \left (\sqrt{d}+\sqrt{e} x\right )}-\frac{4 \sqrt{-e} \sqrt{d+e x^2}}{3 d \sqrt{x}}-\frac{2 \tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right )}{3 x^{3/2}} \]
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Rubi [A] time = 0.166054, antiderivative size = 298, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {5151, 325, 329, 305, 220, 1196} \[ \frac{2 \sqrt{-e} \sqrt [4]{e} \left (\sqrt{d}+\sqrt{e} x\right ) \sqrt{\frac{d+e x^2}{\left (\sqrt{d}+\sqrt{e} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{e} \sqrt{x}}{\sqrt [4]{d}}\right )|\frac{1}{2}\right )}{3 d^{3/4} \sqrt{d+e x^2}}-\frac{4 \sqrt{-e} \sqrt [4]{e} \left (\sqrt{d}+\sqrt{e} x\right ) \sqrt{\frac{d+e x^2}{\left (\sqrt{d}+\sqrt{e} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{e} \sqrt{x}}{\sqrt [4]{d}}\right )|\frac{1}{2}\right )}{3 d^{3/4} \sqrt{d+e x^2}}+\frac{4 \sqrt{-e^2} \sqrt{x} \sqrt{d+e x^2}}{3 d \left (\sqrt{d}+\sqrt{e} x\right )}-\frac{4 \sqrt{-e} \sqrt{d+e x^2}}{3 d \sqrt{x}}-\frac{2 \tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right )}{3 x^{3/2}} \]
Antiderivative was successfully verified.
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Rule 5151
Rule 325
Rule 329
Rule 305
Rule 220
Rule 1196
Rubi steps
\begin{align*} \int \frac{\tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right )}{x^{5/2}} \, dx &=-\frac{2 \tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right )}{3 x^{3/2}}+\frac{1}{3} \left (2 \sqrt{-e}\right ) \int \frac{1}{x^{3/2} \sqrt{d+e x^2}} \, dx\\ &=-\frac{4 \sqrt{-e} \sqrt{d+e x^2}}{3 d \sqrt{x}}-\frac{2 \tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right )}{3 x^{3/2}}-\frac{\left (2 (-e)^{3/2}\right ) \int \frac{\sqrt{x}}{\sqrt{d+e x^2}} \, dx}{3 d}\\ &=-\frac{4 \sqrt{-e} \sqrt{d+e x^2}}{3 d \sqrt{x}}-\frac{2 \tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right )}{3 x^{3/2}}-\frac{\left (4 (-e)^{3/2}\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{d+e x^4}} \, dx,x,\sqrt{x}\right )}{3 d}\\ &=-\frac{4 \sqrt{-e} \sqrt{d+e x^2}}{3 d \sqrt{x}}-\frac{2 \tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right )}{3 x^{3/2}}-\frac{\left (4 (-e)^{3/2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{d+e x^4}} \, dx,x,\sqrt{x}\right )}{3 \sqrt{d} \sqrt{e}}+\frac{\left (4 (-e)^{3/2}\right ) \operatorname{Subst}\left (\int \frac{1-\frac{\sqrt{e} x^2}{\sqrt{d}}}{\sqrt{d+e x^4}} \, dx,x,\sqrt{x}\right )}{3 \sqrt{d} \sqrt{e}}\\ &=-\frac{4 \sqrt{-e} \sqrt{d+e x^2}}{3 d \sqrt{x}}-\frac{4 (-e)^{3/2} \sqrt{x} \sqrt{d+e x^2}}{3 d \sqrt{e} \left (\sqrt{d}+\sqrt{e} x\right )}-\frac{2 \tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right )}{3 x^{3/2}}+\frac{4 (-e)^{3/2} \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 )}{3 d^{3/4} e^{3/4} \sqrt{d+e x^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 )}{3 d^{3/4} e^{3/4} \sqrt{d+e x^2}}\\ \end{align*}
Mathematica [C] time = 0.132899, size = 121, normalized size = 0.41 \[ -\frac{2 \left (2 (-e)^{3/2} x^3 \sqrt{\frac{e x^2}{d}+1} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{7}{4};-\frac{e x^2}{d}\right )+6 \sqrt{-e} x \left (d+e x^2\right )+3 d \sqrt{d+e x^2} \tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right )\right )}{9 d x^{3/2} \sqrt{d+e x^2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.296, size = 0, normalized size = 0. \begin{align*} \int{\arctan \left ({x\sqrt{-e}{\frac{1}{\sqrt{e{x}^{2}+d}}}} \right ){x}^{-{\frac{5}{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{3}{2}} \int -\frac{\sqrt{e x^{2} + d} x}{{\left (e x^{2} + d\right )}^{2} x^{\frac{5}{2}} -{\left (e^{2} x^{4} + d e x^{2}\right )} x^{\frac{5}{2}}}\,{d x} - \arctan \left (\sqrt{-e} x, \sqrt{e x^{2} + d}\right )\right )}}{3 \, x^{\frac{3}{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{5}{2}}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 37.5289, size = 78, normalized size = 0.26 \begin{align*} - \frac{2 \operatorname{atan}{\left (\frac{x \sqrt{- e}}{\sqrt{d + e x^{2}}} \right )}}{3 x^{\frac{3}{2}}} + \frac{\sqrt{- e} \Gamma \left (- \frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{4}, \frac{1}{2} \\ \frac{3}{4} \end{matrix}\middle |{\frac{e x^{2} e^{i \pi }}{d}} \right )}}{3 \sqrt{d} \sqrt{x} \Gamma \left (\frac{3}{4}\right )} \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{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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