Optimal. Leaf size=74 \[ -\frac{\left (d+e x^2\right )^{3/2}}{9 (-e)^{3/2}}+\frac{d \sqrt{d+e x^2}}{3 (-e)^{3/2}}+\frac{1}{3} x^3 \tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right ) \]
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Rubi [A] time = 0.0362828, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {5151, 266, 43} \[ -\frac{\left (d+e x^2\right )^{3/2}}{9 (-e)^{3/2}}+\frac{d \sqrt{d+e x^2}}{3 (-e)^{3/2}}+\frac{1}{3} x^3 \tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right ) \]
Antiderivative was successfully verified.
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Rule 5151
Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^2 \tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right ) \, dx &=\frac{1}{3} x^3 \tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right )-\frac{1}{3} \sqrt{-e} \int \frac{x^3}{\sqrt{d+e x^2}} \, dx\\ &=\frac{1}{3} x^3 \tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right )-\frac{1}{6} \sqrt{-e} \operatorname{Subst}\left (\int \frac{x}{\sqrt{d+e x}} \, dx,x,x^2\right )\\ &=\frac{1}{3} x^3 \tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right )-\frac{1}{6} \sqrt{-e} \operatorname{Subst}\left (\int \left (-\frac{d}{e \sqrt{d+e x}}+\frac{\sqrt{d+e x}}{e}\right ) \, dx,x,x^2\right )\\ &=\frac{d \sqrt{d+e x^2}}{3 (-e)^{3/2}}-\frac{\left (d+e x^2\right )^{3/2}}{9 (-e)^{3/2}}+\frac{1}{3} x^3 \tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right )\\ \end{align*}
Mathematica [A] time = 0.0826092, size = 60, normalized size = 0.81 \[ \frac{1}{9} \left (\frac{\left (2 d-e x^2\right ) \sqrt{d+e x^2}}{(-e)^{3/2}}+3 x^3 \tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.042, size = 132, normalized size = 1.8 \begin{align*}{\frac{{x}^{3}}{3}\arctan \left ({x\sqrt{-e}{\frac{1}{\sqrt{e{x}^{2}+d}}}} \right ) }+{\frac{{x}^{4}}{15\,d}\sqrt{-e}\sqrt{e{x}^{2}+d}}-{\frac{4\,{x}^{2}}{45\,e}\sqrt{-e}\sqrt{e{x}^{2}+d}}+{\frac{8\,d}{45\,{e}^{2}}\sqrt{-e}\sqrt{e{x}^{2}+d}}-{\frac{{x}^{2}}{15\,de}\sqrt{-e} \left ( e{x}^{2}+d \right ) ^{{\frac{3}{2}}}}+{\frac{2}{45\,{e}^{2}}\sqrt{-e} \left ( e{x}^{2}+d \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00957, size = 150, normalized size = 2.03 \begin{align*} \frac{1}{3} \, x^{3} \arctan \left (\frac{\sqrt{-e} x}{\sqrt{e x^{2} + d}}\right ) - \frac{{\left (3 \,{\left (e x^{2} + d\right )}^{\frac{5}{2}} - 5 \,{\left (e x^{2} + d\right )}^{\frac{3}{2}} d\right )} \sqrt{-e}}{45 \, d e^{2}} + \frac{{\left (3 \,{\left (e x^{2} + d\right )}^{\frac{5}{2}} - 10 \,{\left (e x^{2} + d\right )}^{\frac{3}{2}} d + 15 \, \sqrt{e x^{2} + d} d^{2}\right )} \sqrt{-e}}{45 \, d e^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.36367, size = 131, normalized size = 1.77 \begin{align*} \frac{3 \, e^{2} x^{3} \arctan \left (\frac{\sqrt{-e} x}{\sqrt{e x^{2} + d}}\right ) - \sqrt{e x^{2} + d}{\left (e x^{2} - 2 \, d\right )} \sqrt{-e}}{9 \, e^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.32682, size = 70, normalized size = 0.95 \begin{align*} \begin{cases} \frac{2 i d \sqrt{d + e x^{2}}}{9 e^{\frac{3}{2}}} + \frac{i x^{3} \operatorname{atanh}{\left (\frac{\sqrt{e} x}{\sqrt{d + e x^{2}}} \right )}}{3} - \frac{i x^{2} \sqrt{d + e x^{2}}}{9 \sqrt{e}} & \text{for}\: e \neq 0 \\0 & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16226, size = 88, normalized size = 1.19 \begin{align*} \frac{1}{3} \, x^{3} \arctan \left (\frac{x \sqrt{-e}}{\sqrt{x^{2} e + d}}\right ) + \frac{1}{9} \,{\left (3 \, \sqrt{-x^{2} e^{2} - d e} d e +{\left (-x^{2} e^{2} - d e\right )}^{\frac{3}{2}}\right )} e^{\left (-3\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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