Optimal. Leaf size=116 \[ -\frac {3 d^2 \sqrt {-e} \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{32 e^{5/2}}+\frac {3 d x \sqrt {d+e x^2}}{32 (-e)^{3/2}}+\frac {1}{4} x^4 \tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )+\frac {x^3 \sqrt {d+e x^2}}{16 \sqrt {-e}} \]
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Rubi [A] time = 0.04, antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {5151, 321, 217, 206} \[ -\frac {3 d^2 \sqrt {-e} \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{32 e^{5/2}}+\frac {x^3 \sqrt {d+e x^2}}{16 \sqrt {-e}}+\frac {3 d x \sqrt {d+e x^2}}{32 (-e)^{3/2}}+\frac {1}{4} x^4 \tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right ) \]
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rule 321
Rule 5151
Rubi steps
\begin {align*} \int x^3 \tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right ) \, dx &=\frac {1}{4} x^4 \tan ^{-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 \tan ^{-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 \tan ^{-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 \tan ^{-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 {1}{4} x^4 \tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )-\frac {3 d^2 \sqrt {-e} \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{32 e^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 74, normalized size = 0.64 \[ \frac {\left (8 e^2 x^4-3 d^2\right ) \tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )+\sqrt {-e} x \sqrt {d+e x^2} \left (3 d-2 e x^2\right )}{32 e^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 65, normalized size = 0.56 \[ -\frac {{\left (2 \, e x^{3} - 3 \, d x\right )} \sqrt {e x^{2} + d} \sqrt {-e} - {\left (8 \, e^{2} x^{4} - 3 \, d^{2}\right )} \arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right )}{32 \, e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 75, normalized size = 0.65 \[ \frac {1}{4} \, x^{4} \arctan \left (\frac {x \sqrt {-e}}{\sqrt {x^{2} e + d}}\right ) + \frac {3}{32} \, d^{2} \arcsin \left (\frac {x e}{\sqrt {-d e}}\right ) e^{\left (-2\right )} - \frac {1}{32} \, \sqrt {-x^{2} e^{2} - d e} {\left (2 \, x^{2} e^{\left (-1\right )} - 3 \, d e^{\left (-2\right )}\right )} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 163, normalized size = 1.41 \[ \frac {x^{4} \arctan \left (\frac {x \sqrt {-e}}{\sqrt {e \,x^{2}+d}}\right )}{4}+\frac {\sqrt {-e}\, x^{5} \sqrt {e \,x^{2}+d}}{24 d}-\frac {5 \sqrt {-e}\, x^{3} \sqrt {e \,x^{2}+d}}{96 e}+\frac {\sqrt {-e}\, d x \sqrt {e \,x^{2}+d}}{16 e^{2}}-\frac {3 \sqrt {-e}\, d^{2} \ln \left (x \sqrt {e}+\sqrt {e \,x^{2}+d}\right )}{32 e^{\frac {5}{2}}}-\frac {\sqrt {-e}\, x^{3} \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{24 d e}+\frac {\sqrt {-e}\, x \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{32 e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
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 \[ \int x^3\,\mathrm {atan}\left (\frac {\sqrt {-e}\,x}{\sqrt {e\,x^2+d}}\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.46, size = 102, normalized size = 0.88 \[ \begin {cases} - \frac {3 i d^{2} \operatorname {atanh}{\left (\frac {\sqrt {e} x}{\sqrt {d + e x^{2}}} \right )}}{32 e^{2}} + \frac {3 i d x \sqrt {d + e x^{2}}}{32 e^{\frac {3}{2}}} + \frac {i x^{4} \operatorname {atanh}{\left (\frac {\sqrt {e} x}{\sqrt {d + e x^{2}}} \right )}}{4} - \frac {i x^{3} \sqrt {d + e x^{2}}}{16 \sqrt {e}} & \text {for}\: e \neq 0 \\0 & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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