Optimal. Leaf size=101 \[ -\frac{3 d^2 \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{32 e^2}+\frac{3 d x \sqrt{d+e x^2}}{32 e^{3/2}}-\frac{x^3 \sqrt{d+e x^2}}{16 \sqrt{e}}+\frac{1}{4} x^4 \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right ) \]
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Rubi [A] time = 0.0388202, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {6221, 321, 217, 206} \[ -\frac{3 d^2 \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{32 e^2}+\frac{3 d x \sqrt{d+e x^2}}{32 e^{3/2}}-\frac{x^3 \sqrt{d+e x^2}}{16 \sqrt{e}}+\frac{1}{4} x^4 \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right ) \]
Antiderivative was successfully verified.
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Rule 6221
Rule 321
Rule 217
Rule 206
Rubi steps
\begin{align*} \int x^3 \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right ) \, dx &=\frac{1}{4} x^4 \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )-\frac{1}{4} \sqrt{e} \int \frac{x^4}{\sqrt{d+e x^2}} \, dx\\ &=-\frac{x^3 \sqrt{d+e x^2}}{16 \sqrt{e}}+\frac{1}{4} x^4 \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )+\frac{(3 d) \int \frac{x^2}{\sqrt{d+e x^2}} \, dx}{16 \sqrt{e}}\\ &=\frac{3 d x \sqrt{d+e x^2}}{32 e^{3/2}}-\frac{x^3 \sqrt{d+e x^2}}{16 \sqrt{e}}+\frac{1}{4} x^4 \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )-\frac{\left (3 d^2\right ) \int \frac{1}{\sqrt{d+e x^2}} \, dx}{32 e^{3/2}}\\ &=\frac{3 d x \sqrt{d+e x^2}}{32 e^{3/2}}-\frac{x^3 \sqrt{d+e x^2}}{16 \sqrt{e}}+\frac{1}{4} x^4 \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )-\frac{\left (3 d^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-e x^2} \, dx,x,\frac{x}{\sqrt{d+e x^2}}\right )}{32 e^{3/2}}\\ &=\frac{3 d x \sqrt{d+e x^2}}{32 e^{3/2}}-\frac{x^3 \sqrt{d+e x^2}}{16 \sqrt{e}}-\frac{3 d^2 \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{32 e^2}+\frac{1}{4} x^4 \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )\\ \end{align*}
Mathematica [A] time = 0.0481845, size = 88, normalized size = 0.87 \[ \frac{-3 d^2 \log \left (\sqrt{d+e x^2}+\sqrt{e} x\right )+8 e^2 x^4 \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )+\sqrt{e} x \left (3 d-2 e x^2\right ) \sqrt{d+e x^2}}{32 e^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.031, size = 134, normalized size = 1.3 \begin{align*}{\frac{{x}^{4}}{4}{\it Artanh} \left ({x\sqrt{e}{\frac{1}{\sqrt{e{x}^{2}+d}}}} \right ) }+{\frac{{x}^{5}}{24\,d}\sqrt{e}\sqrt{e{x}^{2}+d}}-{\frac{5\,{x}^{3}}{96}\sqrt{e{x}^{2}+d}{\frac{1}{\sqrt{e}}}}+{\frac{dx}{16}\sqrt{e{x}^{2}+d}{e}^{-{\frac{3}{2}}}}-{\frac{3\,{d}^{2}}{32\,{e}^{2}}\ln \left ( x\sqrt{e}+\sqrt{e{x}^{2}+d} \right ) }-{\frac{{x}^{3}}{24\,d} \left ( e{x}^{2}+d \right ) ^{{\frac{3}{2}}}{\frac{1}{\sqrt{e}}}}+{\frac{x}{32} \left ( e{x}^{2}+d \right ) ^{{\frac{3}{2}}}{e}^{-{\frac{3}{2}}}} \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}{8} \, x^{4} \log \left (\sqrt{e} x + \sqrt{e x^{2} + d}\right ) - \frac{1}{8} \, x^{4} \log \left (-\sqrt{e} x + \sqrt{e x^{2} + d}\right ) - \frac{1}{2} \, d \sqrt{e} \int -\frac{\sqrt{e x^{2} + d} x^{4}}{2 \,{\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 [A] time = 2.10243, size = 176, normalized size = 1.74 \begin{align*} -\frac{2 \,{\left (2 \, e x^{3} - 3 \, d x\right )} \sqrt{e x^{2} + d} \sqrt{e} -{\left (8 \, e^{2} x^{4} - 3 \, d^{2}\right )} \log \left (\frac{2 \, e x^{2} + 2 \, \sqrt{e x^{2} + d} \sqrt{e} x + d}{d}\right )}{64 \, e^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.84909, size = 95, normalized size = 0.94 \begin{align*} \begin{cases} - \frac{3 d^{2} \operatorname{atanh}{\left (\frac{\sqrt{e} x}{\sqrt{d + e x^{2}}} \right )}}{32 e^{2}} + \frac{3 d x \sqrt{d + e x^{2}}}{32 e^{\frac{3}{2}}} + \frac{x^{4} \operatorname{atanh}{\left (\frac{\sqrt{e} x}{\sqrt{d + e x^{2}}} \right )}}{4} - \frac{x^{3} \sqrt{d + e x^{2}}}{16 \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \left [\mathit{undef}, \mathit{undef}, \mathit{undef}, -\frac{1}{2} \, d e^{\frac{1}{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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