Optimal. Leaf size=168 \[ -\frac{10 d^{7/4} \left (\sqrt{d}+\sqrt{e} x\right ) \sqrt{\frac{d+e x^2}{\left (\sqrt{d}+\sqrt{e} x\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{e} \sqrt{x}}{\sqrt [4]{d}}\right ),\frac{1}{2}\right )}{147 e^{7/4} \sqrt{d+e x^2}}+\frac{20 d \sqrt{x} \sqrt{d+e x^2}}{147 e^{3/2}}-\frac{4 x^{5/2} \sqrt{d+e x^2}}{49 \sqrt{e}}+\frac{2}{7} x^{7/2} \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right ) \]
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Rubi [A] time = 0.0863851, antiderivative size = 168, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {6221, 321, 329, 220} \[ -\frac{10 d^{7/4} \left (\sqrt{d}+\sqrt{e} x\right ) \sqrt{\frac{d+e x^2}{\left (\sqrt{d}+\sqrt{e} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{e} \sqrt{x}}{\sqrt [4]{d}}\right )|\frac{1}{2}\right )}{147 e^{7/4} \sqrt{d+e x^2}}+\frac{20 d \sqrt{x} \sqrt{d+e x^2}}{147 e^{3/2}}-\frac{4 x^{5/2} \sqrt{d+e x^2}}{49 \sqrt{e}}+\frac{2}{7} x^{7/2} \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right ) \]
Antiderivative was successfully verified.
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Rule 6221
Rule 321
Rule 329
Rule 220
Rubi steps
\begin{align*} \int x^{5/2} \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right ) \, dx &=\frac{2}{7} x^{7/2} \tanh ^{-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} \tanh ^{-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} \tanh ^{-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} \tanh ^{-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} \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )-\frac{10 d^{7/4} \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 )}{147 e^{7/4} \sqrt{d+e x^2}}\\ \end{align*}
Mathematica [C] time = 0.372306, size = 147, normalized size = 0.88 \[ \frac{2}{147} \sqrt{x} \left (\frac{2 \left (5 d-3 e x^2\right ) \sqrt{d+e x^2}}{e^{3/2}}+21 x^3 \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )\right )+\frac{20 \sqrt{d} x \left (\frac{i \sqrt{d}}{\sqrt{e}}\right )^{5/2} \sqrt{\frac{d}{e x^2}+1} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{d}}{\sqrt{e}}}}{\sqrt{x}}\right ),-1\right )}{147 \sqrt{d+e x^2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.865, size = 0, normalized size = 0. \begin{align*} \int{x}^{{\frac{5}{2}}}{\it Artanh} \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{1}{7} \, x^{\frac{7}{2}} \log \left (\sqrt{e} x + \sqrt{e x^{2} + d}\right ) - \frac{1}{7} \, x^{\frac{7}{2}} \log \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.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (x^{\frac{5}{2}} \operatorname{artanh}\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 [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*} \left [\mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, 4 \, d e^{\frac{1}{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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