Optimal. Leaf size=181 \[ -\frac{10 d^{7/4} \sqrt{-e} \left (\sqrt{d}+\sqrt{e} x\right ) \sqrt{\frac{d+e x^2}{\left (\sqrt{d}+\sqrt{e} x\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{e} \sqrt{x}}{\sqrt [4]{d}}\right ),\frac{1}{2}\right )}{147 e^{9/4} \sqrt{d+e x^2}}+\frac{4 x^{5/2} \sqrt{d+e x^2}}{49 \sqrt{-e}}+\frac{20 d \sqrt{x} \sqrt{d+e x^2}}{147 (-e)^{3/2}}+\frac{2}{7} x^{7/2} \tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0883095, antiderivative size = 181, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148, Rules used = {5151, 321, 329, 220} \[ -\frac{10 d^{7/4} \sqrt{-e} \left (\sqrt{d}+\sqrt{e} x\right ) \sqrt{\frac{d+e x^2}{\left (\sqrt{d}+\sqrt{e} x\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{e} \sqrt{x}}{\sqrt [4]{d}}\right ),\frac{1}{2}\right )}{147 e^{9/4} \sqrt{d+e x^2}}+\frac{4 x^{5/2} \sqrt{d+e x^2}}{49 \sqrt{-e}}+\frac{20 d \sqrt{x} \sqrt{d+e x^2}}{147 (-e)^{3/2}}+\frac{2}{7} x^{7/2} \tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5151
Rule 321
Rule 329
Rule 220
Rubi steps
\begin{align*} \int x^{5/2} \tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right ) \, dx &=\frac{2}{7} x^{7/2} \tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right )-\frac{1}{7} \left (2 \sqrt{-e}\right ) \int \frac{x^{7/2}}{\sqrt{d+e x^2}} \, dx\\ &=\frac{4 x^{5/2} \sqrt{d+e x^2}}{49 \sqrt{-e}}+\frac{2}{7} x^{7/2} \tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right )-\frac{(10 d) \int \frac{x^{3/2}}{\sqrt{d+e x^2}} \, dx}{49 \sqrt{-e}}\\ &=\frac{20 d \sqrt{x} \sqrt{d+e x^2}}{147 (-e)^{3/2}}+\frac{4 x^{5/2} \sqrt{d+e x^2}}{49 \sqrt{-e}}+\frac{2}{7} x^{7/2} \tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right )-\frac{\left (10 d^2\right ) \int \frac{1}{\sqrt{x} \sqrt{d+e x^2}} \, dx}{147 (-e)^{3/2}}\\ &=\frac{20 d \sqrt{x} \sqrt{d+e x^2}}{147 (-e)^{3/2}}+\frac{4 x^{5/2} \sqrt{d+e x^2}}{49 \sqrt{-e}}+\frac{2}{7} x^{7/2} \tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right )-\frac{\left (20 d^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{d+e x^4}} \, dx,x,\sqrt{x}\right )}{147 (-e)^{3/2}}\\ &=\frac{20 d \sqrt{x} \sqrt{d+e x^2}}{147 (-e)^{3/2}}+\frac{4 x^{5/2} \sqrt{d+e x^2}}{49 \sqrt{-e}}+\frac{2}{7} x^{7/2} \tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right )-\frac{10 d^{7/4} \sqrt{-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 )}{147 e^{9/4} \sqrt{d+e x^2}}\\ \end{align*}
Mathematica [C] time = 0.415558, size = 158, normalized size = 0.87 \[ \frac{2}{147} \sqrt{x} \left (\frac{2 \left (5 d-3 e x^2\right ) \sqrt{d+e x^2}}{(-e)^{3/2}}+21 x^3 \tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right )\right )-\frac{20 i d^2 x \sqrt{\frac{d}{e x^2}+1} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{d}}{\sqrt{e}}}}{\sqrt{x}}\right ),-1\right )}{147 (-e)^{3/2} \sqrt{\frac{i \sqrt{d}}{\sqrt{e}}} \sqrt{d+e x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.401, size = 0, normalized size = 0. \begin{align*} \int{x}^{{\frac{5}{2}}}\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.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{2}{7} \, x^{\frac{7}{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{5}{2} \, \log \left (x\right )\right )}}{7 \,{\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.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (x^{\frac{5}{2}} \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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \left [\mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, 2 \, e^{\frac{1}{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]