Optimal. Leaf size=153 \[ \frac{2 d^{3/4} \sqrt{-e} \left (\sqrt{d}+\sqrt{e} x\right ) \sqrt{\frac{d+e x^2}{\left (\sqrt{d}+\sqrt{e} x\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{e} \sqrt{x}}{\sqrt [4]{d}}\right ),\frac{1}{2}\right )}{9 e^{5/4} \sqrt{d+e x^2}}+\frac{4 \sqrt{x} \sqrt{d+e x^2}}{9 \sqrt{-e}}+\frac{2}{3} x^{3/2} \tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right ) \]
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Rubi [A] time = 0.0743981, antiderivative size = 153, 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, 321, 329, 220} \[ \frac{2 d^{3/4} \sqrt{-e} \left (\sqrt{d}+\sqrt{e} x\right ) \sqrt{\frac{d+e x^2}{\left (\sqrt{d}+\sqrt{e} x\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{e} \sqrt{x}}{\sqrt [4]{d}}\right ),\frac{1}{2}\right )}{9 e^{5/4} \sqrt{d+e x^2}}+\frac{4 \sqrt{x} \sqrt{d+e x^2}}{9 \sqrt{-e}}+\frac{2}{3} x^{3/2} \tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right ) \]
Antiderivative was successfully verified.
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Rule 5151
Rule 321
Rule 329
Rule 220
Rubi steps
\begin{align*} \int \sqrt{x} \tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right ) \, dx &=\frac{2}{3} x^{3/2} \tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right )-\frac{1}{3} \left (2 \sqrt{-e}\right ) \int \frac{x^{3/2}}{\sqrt{d+e x^2}} \, dx\\ &=\frac{4 \sqrt{x} \sqrt{d+e x^2}}{9 \sqrt{-e}}+\frac{2}{3} x^{3/2} \tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right )-\frac{(2 d) \int \frac{1}{\sqrt{x} \sqrt{d+e x^2}} \, dx}{9 \sqrt{-e}}\\ &=\frac{4 \sqrt{x} \sqrt{d+e x^2}}{9 \sqrt{-e}}+\frac{2}{3} x^{3/2} \tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right )-\frac{(4 d) \operatorname{Subst}\left (\int \frac{1}{\sqrt{d+e x^4}} \, dx,x,\sqrt{x}\right )}{9 \sqrt{-e}}\\ &=\frac{4 \sqrt{x} \sqrt{d+e x^2}}{9 \sqrt{-e}}+\frac{2}{3} x^{3/2} \tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right )-\frac{2 d^{3/4} \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 )}{9 \sqrt{-e} \sqrt [4]{e} \sqrt{d+e x^2}}\\ \end{align*}
Mathematica [C] time = 0.284602, size = 147, normalized size = 0.96 \[ -\frac{4 i d x \sqrt{\frac{d}{e x^2}+1} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{d}}{\sqrt{e}}}}{\sqrt{x}}\right ),-1\right )}{9 \sqrt{-e} \sqrt{\frac{i \sqrt{d}}{\sqrt{e}}} \sqrt{d+e x^2}}+\frac{4 \sqrt{x} \sqrt{d+e x^2}}{9 \sqrt{-e}}+\frac{2}{3} x^{3/2} \tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.289, size = 0, normalized size = 0. \begin{align*} \int \sqrt{x}\arctan \left ({x\sqrt{-e}{\frac{1}{\sqrt{e{x}^{2}+d}}}} \right ) \, 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}{3} \, x^{\frac{3}{2}} \arctan \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{1}{2} \, \log \left (x\right )\right )}}{3 \,{\left (e^{2} x^{4} + d e x^{2} -{\left (e x^{2} + d\right )}^{2}\right )}}\,{d x} \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 (\sqrt{x} \arctan \left (\frac{\sqrt{-e} x}{\sqrt{e x^{2} + d}}\right ), x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 3.61452, size = 75, normalized size = 0.49 \begin{align*} \frac{2 x^{\frac{3}{2}} \operatorname{atan}{\left (\frac{x \sqrt{- e}}{\sqrt{d + e x^{2}}} \right )}}{3} - \frac{x^{\frac{5}{2}} \sqrt{- e} \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{5}{4} \\ \frac{9}{4} \end{matrix}\middle |{\frac{e x^{2} e^{i \pi }}{d}} \right )}}{3 \sqrt{d} \Gamma \left (\frac{9}{4}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.26849, size = 1, normalized size = 0.01 \begin{align*} +\infty \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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