Optimal. Leaf size=99 \[ \frac{d^2 \sqrt{d+e x^2}}{5 (-e)^{5/2}}+\frac{\left (d+e x^2\right )^{5/2}}{25 (-e)^{5/2}}-\frac{2 d \left (d+e x^2\right )^{3/2}}{15 (-e)^{5/2}}+\frac{1}{5} x^5 \tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right ) \]
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Rubi [A] time = 0.0505896, antiderivative size = 99, 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} \[ \frac{d^2 \sqrt{d+e x^2}}{5 (-e)^{5/2}}+\frac{\left (d+e x^2\right )^{5/2}}{25 (-e)^{5/2}}-\frac{2 d \left (d+e x^2\right )^{3/2}}{15 (-e)^{5/2}}+\frac{1}{5} x^5 \tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right ) \]
Antiderivative was successfully verified.
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Rule 5151
Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^4 \tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right ) \, dx &=\frac{1}{5} x^5 \tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right )-\frac{1}{5} \sqrt{-e} \int \frac{x^5}{\sqrt{d+e x^2}} \, dx\\ &=\frac{1}{5} x^5 \tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right )-\frac{1}{10} \sqrt{-e} \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{d+e x}} \, dx,x,x^2\right )\\ &=\frac{1}{5} x^5 \tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right )-\frac{1}{10} \sqrt{-e} \operatorname{Subst}\left (\int \left (\frac{d^2}{e^2 \sqrt{d+e x}}-\frac{2 d \sqrt{d+e x}}{e^2}+\frac{(d+e x)^{3/2}}{e^2}\right ) \, dx,x,x^2\right )\\ &=\frac{d^2 \sqrt{d+e x^2}}{5 (-e)^{5/2}}-\frac{2 d \left (d+e x^2\right )^{3/2}}{15 (-e)^{5/2}}+\frac{\left (d+e x^2\right )^{5/2}}{25 (-e)^{5/2}}+\frac{1}{5} x^5 \tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right )\\ \end{align*}
Mathematica [A] time = 0.0887031, size = 72, normalized size = 0.73 \[ \frac{\sqrt{d+e x^2} \left (8 d^2-4 d e x^2+3 e^2 x^4\right )}{75 (-e)^{5/2}}+\frac{1}{5} x^5 \tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.043, size = 180, normalized size = 1.8 \begin{align*}{\frac{{x}^{5}}{5}\arctan \left ({x\sqrt{-e}{\frac{1}{\sqrt{e{x}^{2}+d}}}} \right ) }+{\frac{{x}^{6}}{35\,d}\sqrt{-e}\sqrt{e{x}^{2}+d}}-{\frac{6\,{x}^{4}}{175\,e}\sqrt{-e}\sqrt{e{x}^{2}+d}}+{\frac{8\,d{x}^{2}}{175\,{e}^{2}}\sqrt{-e}\sqrt{e{x}^{2}+d}}-{\frac{16\,{d}^{2}}{175\,{e}^{3}}\sqrt{-e}\sqrt{e{x}^{2}+d}}-{\frac{{x}^{4}}{35\,de}\sqrt{-e} \left ( e{x}^{2}+d \right ) ^{{\frac{3}{2}}}}+{\frac{4\,{x}^{2}}{175\,{e}^{2}}\sqrt{-e} \left ( e{x}^{2}+d \right ) ^{{\frac{3}{2}}}}-{\frac{8\,d}{525\,{e}^{3}}\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 = 0.999257, size = 188, normalized size = 1.9 \begin{align*} \frac{1}{5} \, x^{5} \arctan \left (\frac{\sqrt{-e} x}{\sqrt{e x^{2} + d}}\right ) - \frac{{\left (15 \,{\left (e x^{2} + d\right )}^{\frac{7}{2}} - 42 \,{\left (e x^{2} + d\right )}^{\frac{5}{2}} d + 35 \,{\left (e x^{2} + d\right )}^{\frac{3}{2}} d^{2}\right )} \sqrt{-e}}{525 \, d e^{3}} + \frac{{\left (5 \,{\left (e x^{2} + d\right )}^{\frac{7}{2}} - 21 \,{\left (e x^{2} + d\right )}^{\frac{5}{2}} d + 35 \,{\left (e x^{2} + d\right )}^{\frac{3}{2}} d^{2} - 35 \, \sqrt{e x^{2} + d} d^{3}\right )} \sqrt{-e}}{175 \, d e^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.21825, size = 158, normalized size = 1.6 \begin{align*} \frac{15 \, e^{3} x^{5} \arctan \left (\frac{\sqrt{-e} x}{\sqrt{e x^{2} + d}}\right ) -{\left (3 \, e^{2} x^{4} - 4 \, d e x^{2} + 8 \, d^{2}\right )} \sqrt{e x^{2} + d} \sqrt{-e}}{75 \, e^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.50107, size = 97, normalized size = 0.98 \begin{align*} \begin{cases} - \frac{8 i d^{2} \sqrt{d + e x^{2}}}{75 e^{\frac{5}{2}}} + \frac{4 i d x^{2} \sqrt{d + e x^{2}}}{75 e^{\frac{3}{2}}} + \frac{i x^{5} \operatorname{atanh}{\left (\frac{\sqrt{e} x}{\sqrt{d + e x^{2}}} \right )}}{5} - \frac{i x^{4} \sqrt{d + e x^{2}}}{25 \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 = 1.20283, size = 138, normalized size = 1.39 \begin{align*} \frac{1}{5} \, x^{5} \arctan \left (\frac{x \sqrt{-e}}{\sqrt{x^{2} e + d}}\right ) - \frac{1}{75} \,{\left (15 \, \sqrt{-x^{2} e^{2} - d e} d^{2} e^{2} + 10 \,{\left (-x^{2} e^{2} - d e\right )}^{\frac{3}{2}} d e + 3 \,{\left (x^{2} e^{2} + d e\right )}^{2} \sqrt{-x^{2} e^{2} - d e}\right )} e^{\left (-5\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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