Optimal. Leaf size=74 \[ -\frac {\left (d+e x^2\right )^{3/2}}{9 (-e)^{3/2}}+\frac {d \sqrt {d+e x^2}}{3 (-e)^{3/2}}+\frac {1}{3} x^3 \tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right ) \]
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Rubi [A] time = 0.04, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {5151, 266, 43} \[ -\frac {\left (d+e x^2\right )^{3/2}}{9 (-e)^{3/2}}+\frac {d \sqrt {d+e x^2}}{3 (-e)^{3/2}}+\frac {1}{3} x^3 \tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right ) \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rule 5151
Rubi steps
\begin {align*} \int x^2 \tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right ) \, dx &=\frac {1}{3} x^3 \tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )-\frac {1}{3} \sqrt {-e} \int \frac {x^3}{\sqrt {d+e x^2}} \, dx\\ &=\frac {1}{3} x^3 \tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )-\frac {1}{6} \sqrt {-e} \operatorname {Subst}\left (\int \frac {x}{\sqrt {d+e x}} \, dx,x,x^2\right )\\ &=\frac {1}{3} x^3 \tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )-\frac {1}{6} \sqrt {-e} \operatorname {Subst}\left (\int \left (-\frac {d}{e \sqrt {d+e x}}+\frac {\sqrt {d+e x}}{e}\right ) \, dx,x,x^2\right )\\ &=\frac {d \sqrt {d+e x^2}}{3 (-e)^{3/2}}-\frac {\left (d+e x^2\right )^{3/2}}{9 (-e)^{3/2}}+\frac {1}{3} x^3 \tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )\\ \end {align*}
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Mathematica [A] time = 0.09, size = 60, normalized size = 0.81 \[ \frac {1}{9} \left (\frac {\left (2 d-e x^2\right ) \sqrt {d+e x^2}}{(-e)^{3/2}}+3 x^3 \tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 56, normalized size = 0.76 \[ \frac {3 \, e^{2} x^{3} \arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right ) - \sqrt {e x^{2} + d} {\left (e x^{2} - 2 \, d\right )} \sqrt {-e}}{9 \, e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 64, normalized size = 0.86 \[ \frac {1}{3} \, x^{3} \arctan \left (\frac {x \sqrt {-e}}{\sqrt {x^{2} e + d}}\right ) + \frac {1}{3} \, \sqrt {-x^{2} e^{2} - d e} d e^{\left (-2\right )} + \frac {1}{9} \, {\left (-x^{2} e^{2} - d e\right )}^{\frac {3}{2}} e^{\left (-3\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 132, normalized size = 1.78 \[ \frac {x^{3} \arctan \left (\frac {x \sqrt {-e}}{\sqrt {e \,x^{2}+d}}\right )}{3}+\frac {\sqrt {-e}\, x^{4} \sqrt {e \,x^{2}+d}}{15 d}-\frac {4 \sqrt {-e}\, x^{2} \sqrt {e \,x^{2}+d}}{45 e}+\frac {8 \sqrt {-e}\, d \sqrt {e \,x^{2}+d}}{45 e^{2}}-\frac {\sqrt {-e}\, x^{2} \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{15 d e}+\frac {2 \sqrt {-e}\, \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{45 e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 111, normalized size = 1.50 \[ \frac {1}{3} \, x^{3} \arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right ) - \frac {{\left (3 \, {\left (e x^{2} + d\right )}^{\frac {5}{2}} - 5 \, {\left (e x^{2} + d\right )}^{\frac {3}{2}} d\right )} \sqrt {-e}}{45 \, d e^{2}} + \frac {{\left (3 \, {\left (e x^{2} + d\right )}^{\frac {5}{2}} - 10 \, {\left (e x^{2} + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {e x^{2} + d} d^{2}\right )} \sqrt {-e}}{45 \, d e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^2\,\mathrm {atan}\left (\frac {\sqrt {-e}\,x}{\sqrt {e\,x^2+d}}\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.24, size = 70, normalized size = 0.95 \[ \begin {cases} \frac {2 i d \sqrt {d + e x^{2}}}{9 e^{\frac {3}{2}}} + \frac {i x^{3} \operatorname {atanh}{\left (\frac {\sqrt {e} x}{\sqrt {d + e x^{2}}} \right )}}{3} - \frac {i x^{2} \sqrt {d + e x^{2}}}{9 \sqrt {e}} & \text {for}\: e \neq 0 \\0 & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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