Optimal. Leaf size=124 \[ -\frac{d^2 \left (d+e x^2\right )^{3/2}}{7 (-e)^{7/2}}+\frac{d^3 \sqrt{d+e x^2}}{7 (-e)^{7/2}}-\frac{\left (d+e x^2\right )^{7/2}}{49 (-e)^{7/2}}+\frac{3 d \left (d+e x^2\right )^{5/2}}{35 (-e)^{7/2}}+\frac{1}{7} x^7 \tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right ) \]
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Rubi [A] time = 0.0670544, antiderivative size = 124, 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} \[ -\frac{d^2 \left (d+e x^2\right )^{3/2}}{7 (-e)^{7/2}}+\frac{d^3 \sqrt{d+e x^2}}{7 (-e)^{7/2}}-\frac{\left (d+e x^2\right )^{7/2}}{49 (-e)^{7/2}}+\frac{3 d \left (d+e x^2\right )^{5/2}}{35 (-e)^{7/2}}+\frac{1}{7} x^7 \tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right ) \]
Antiderivative was successfully verified.
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Rule 5151
Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^6 \tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right ) \, dx &=\frac{1}{7} x^7 \tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right )-\frac{1}{7} \sqrt{-e} \int \frac{x^7}{\sqrt{d+e x^2}} \, dx\\ &=\frac{1}{7} x^7 \tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right )-\frac{1}{14} \sqrt{-e} \operatorname{Subst}\left (\int \frac{x^3}{\sqrt{d+e x}} \, dx,x,x^2\right )\\ &=\frac{1}{7} x^7 \tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right )-\frac{1}{14} \sqrt{-e} \operatorname{Subst}\left (\int \left (-\frac{d^3}{e^3 \sqrt{d+e x}}+\frac{3 d^2 \sqrt{d+e x}}{e^3}-\frac{3 d (d+e x)^{3/2}}{e^3}+\frac{(d+e x)^{5/2}}{e^3}\right ) \, dx,x,x^2\right )\\ &=\frac{d^3 \sqrt{d+e x^2}}{7 (-e)^{7/2}}-\frac{d^2 \left (d+e x^2\right )^{3/2}}{7 (-e)^{7/2}}+\frac{3 d \left (d+e x^2\right )^{5/2}}{35 (-e)^{7/2}}-\frac{\left (d+e x^2\right )^{7/2}}{49 (-e)^{7/2}}+\frac{1}{7} x^7 \tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right )\\ \end{align*}
Mathematica [A] time = 0.104445, size = 83, normalized size = 0.67 \[ \frac{\sqrt{d+e x^2} \left (-8 d^2 e x^2+16 d^3+6 d e^2 x^4-5 e^3 x^6\right )}{245 (-e)^{7/2}}+\frac{1}{7} x^7 \tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.046, size = 230, normalized size = 1.9 \begin{align*}{\frac{{x}^{7}}{7}\arctan \left ({x\sqrt{-e}{\frac{1}{\sqrt{e{x}^{2}+d}}}} \right ) }+{\frac{{x}^{8}}{63\,d}\sqrt{-e}\sqrt{e{x}^{2}+d}}-{\frac{8\,{x}^{6}}{441\,e}\sqrt{-e}\sqrt{e{x}^{2}+d}}+{\frac{16\,d{x}^{4}}{735\,{e}^{2}}\sqrt{-e}\sqrt{e{x}^{2}+d}}-{\frac{64\,{d}^{2}{x}^{2}}{2205\,{e}^{3}}\sqrt{-e}\sqrt{e{x}^{2}+d}}+{\frac{128\,{d}^{3}}{2205\,{e}^{4}}\sqrt{-e}\sqrt{e{x}^{2}+d}}-{\frac{{x}^{6}}{63\,de}\sqrt{-e} \left ( e{x}^{2}+d \right ) ^{{\frac{3}{2}}}}+{\frac{2\,{x}^{4}}{147\,{e}^{2}}\sqrt{-e} \left ( e{x}^{2}+d \right ) ^{{\frac{3}{2}}}}-{\frac{8\,d{x}^{2}}{735\,{e}^{3}}\sqrt{-e} \left ( e{x}^{2}+d \right ) ^{{\frac{3}{2}}}}+{\frac{16\,{d}^{2}}{2205\,{e}^{4}}\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 = 1.00712, size = 225, normalized size = 1.81 \begin{align*} \frac{1}{7} \, x^{7} \arctan \left (\frac{\sqrt{-e} x}{\sqrt{e x^{2} + d}}\right ) - \frac{{\left (35 \,{\left (e x^{2} + d\right )}^{\frac{9}{2}} - 135 \,{\left (e x^{2} + d\right )}^{\frac{7}{2}} d + 189 \,{\left (e x^{2} + d\right )}^{\frac{5}{2}} d^{2} - 105 \,{\left (e x^{2} + d\right )}^{\frac{3}{2}} d^{3}\right )} \sqrt{-e}}{2205 \, d e^{4}} + \frac{{\left (35 \,{\left (e x^{2} + d\right )}^{\frac{9}{2}} - 180 \,{\left (e x^{2} + d\right )}^{\frac{7}{2}} d + 378 \,{\left (e x^{2} + d\right )}^{\frac{5}{2}} d^{2} - 420 \,{\left (e x^{2} + d\right )}^{\frac{3}{2}} d^{3} + 315 \, \sqrt{e x^{2} + d} d^{4}\right )} \sqrt{-e}}{2205 \, d e^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.3296, size = 182, normalized size = 1.47 \begin{align*} \frac{35 \, e^{4} x^{7} \arctan \left (\frac{\sqrt{-e} x}{\sqrt{e x^{2} + d}}\right ) -{\left (5 \, e^{3} x^{6} - 6 \, d e^{2} x^{4} + 8 \, d^{2} e x^{2} - 16 \, d^{3}\right )} \sqrt{e x^{2} + d} \sqrt{-e}}{245 \, e^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 15.8235, size = 124, normalized size = 1. \begin{align*} \begin{cases} \frac{16 i d^{3} \sqrt{d + e x^{2}}}{245 e^{\frac{7}{2}}} - \frac{8 i d^{2} x^{2} \sqrt{d + e x^{2}}}{245 e^{\frac{5}{2}}} + \frac{6 i d x^{4} \sqrt{d + e x^{2}}}{245 e^{\frac{3}{2}}} + \frac{i x^{7} \operatorname{atanh}{\left (\frac{\sqrt{e} x}{\sqrt{d + e x^{2}}} \right )}}{7} - \frac{i x^{6} \sqrt{d + e x^{2}}}{49 \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 = 1.18974, size = 185, normalized size = 1.49 \begin{align*} \frac{1}{7} \, x^{7} \arctan \left (\frac{x \sqrt{-e}}{\sqrt{x^{2} e + d}}\right ) + \frac{1}{245} \,{\left (35 \, \sqrt{-x^{2} e^{2} - d e} d^{3} e^{3} + 35 \,{\left (-x^{2} e^{2} - d e\right )}^{\frac{3}{2}} d^{2} e^{2} + 21 \,{\left (x^{2} e^{2} + d e\right )}^{2} \sqrt{-x^{2} e^{2} - d e} d e - 5 \,{\left (x^{2} e^{2} + d e\right )}^{3} \sqrt{-x^{2} e^{2} - d e}\right )} e^{\left (-7\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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