Optimal. Leaf size=296 \[ \frac {4 x^{3/2} \sqrt {d+e x^2}}{25 \sqrt {-e}}+\frac {12 d \sqrt {-e} \sqrt {x} \sqrt {d+e x^2}}{25 e^{3/2} \left (\sqrt {d}+\sqrt {e} x\right )}+\frac {2}{5} x^{5/2} \text {ArcTan}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )-\frac {12 d^{5/4} \sqrt {-e} \left (\sqrt {d}+\sqrt {e} x\right ) \sqrt {\frac {d+e x^2}{\left (\sqrt {d}+\sqrt {e} x\right )^2}} E\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right )|\frac {1}{2}\right )}{25 e^{7/4} \sqrt {d+e x^2}}+\frac {6 d^{5/4} \sqrt {-e} \left (\sqrt {d}+\sqrt {e} x\right ) \sqrt {\frac {d+e x^2}{\left (\sqrt {d}+\sqrt {e} x\right )^2}} F\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right )|\frac {1}{2}\right )}{25 e^{7/4} \sqrt {d+e x^2}} \]
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Rubi [A]
time = 0.13, antiderivative size = 296, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {5259, 327, 335,
311, 226, 1210} \begin {gather*} \frac {6 d^{5/4} \sqrt {-e} \left (\sqrt {d}+\sqrt {e} x\right ) \sqrt {\frac {d+e x^2}{\left (\sqrt {d}+\sqrt {e} x\right )^2}} F\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right )|\frac {1}{2}\right )}{25 e^{7/4} \sqrt {d+e x^2}}-\frac {12 d^{5/4} \sqrt {-e} \left (\sqrt {d}+\sqrt {e} x\right ) \sqrt {\frac {d+e x^2}{\left (\sqrt {d}+\sqrt {e} x\right )^2}} E\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right )|\frac {1}{2}\right )}{25 e^{7/4} \sqrt {d+e x^2}}+\frac {2}{5} x^{5/2} \text {ArcTan}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )+\frac {12 d \sqrt {-e} \sqrt {x} \sqrt {d+e x^2}}{25 e^{3/2} \left (\sqrt {d}+\sqrt {e} x\right )}+\frac {4 x^{3/2} \sqrt {d+e x^2}}{25 \sqrt {-e}} \end {gather*}
Antiderivative was successfully verified.
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Rule 226
Rule 311
Rule 327
Rule 335
Rule 1210
Rule 5259
Rubi steps
\begin {align*} \int x^{3/2} \tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right ) \, dx &=\frac {2}{5} x^{5/2} \tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )-\frac {1}{5} \left (2 \sqrt {-e}\right ) \int \frac {x^{5/2}}{\sqrt {d+e x^2}} \, dx\\ &=\frac {4 x^{3/2} \sqrt {d+e x^2}}{25 \sqrt {-e}}+\frac {2}{5} x^{5/2} \tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )-\frac {(6 d) \int \frac {\sqrt {x}}{\sqrt {d+e x^2}} \, dx}{25 \sqrt {-e}}\\ &=\frac {4 x^{3/2} \sqrt {d+e x^2}}{25 \sqrt {-e}}+\frac {2}{5} x^{5/2} \tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )-\frac {(12 d) \text {Subst}\left (\int \frac {x^2}{\sqrt {d+e x^4}} \, dx,x,\sqrt {x}\right )}{25 \sqrt {-e}}\\ &=\frac {4 x^{3/2} \sqrt {d+e x^2}}{25 \sqrt {-e}}+\frac {2}{5} x^{5/2} \tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )-\frac {\left (12 d^{3/2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {d+e x^4}} \, dx,x,\sqrt {x}\right )}{25 \sqrt {-e^2}}+\frac {\left (12 d^{3/2}\right ) \text {Subst}\left (\int \frac {1-\frac {\sqrt {e} x^2}{\sqrt {d}}}{\sqrt {d+e x^4}} \, dx,x,\sqrt {x}\right )}{25 \sqrt {-e^2}}\\ &=\frac {4 x^{3/2} \sqrt {d+e x^2}}{25 \sqrt {-e}}-\frac {12 d \sqrt {x} \sqrt {d+e x^2}}{25 \sqrt {-e^2} \left (\sqrt {d}+\sqrt {e} x\right )}+\frac {2}{5} x^{5/2} \tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )+\frac {12 d^{5/4} \left (\sqrt {d}+\sqrt {e} x\right ) \sqrt {\frac {d+e x^2}{\left (\sqrt {d}+\sqrt {e} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right )|\frac {1}{2}\right )}{25 \sqrt [4]{e} \sqrt {-e^2} \sqrt {d+e x^2}}-\frac {6 d^{5/4} \left (\sqrt {d}+\sqrt {e} x\right ) \sqrt {\frac {d+e x^2}{\left (\sqrt {d}+\sqrt {e} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right )|\frac {1}{2}\right )}{25 \sqrt [4]{e} \sqrt {-e^2} \sqrt {d+e x^2}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 0.08, size = 119, normalized size = 0.40 \begin {gather*} \frac {2 x^{3/2} \left (-2 \sqrt {-e} \left (d+e x^2\right )+5 e x \sqrt {d+e x^2} \text {ArcTan}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )+2 d \sqrt {-e} \sqrt {1+\frac {e x^2}{d}} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};-\frac {e x^2}{d}\right )\right )}{25 e \sqrt {d+e x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int x^{\frac {3}{2}} \arctan \left (\frac {x \sqrt {-e}}{\sqrt {e \,x^{2}+d}}\right )\, dx\]
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 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 1.42, size = 76, normalized size = 0.26 \begin {gather*} \frac {1}{25} \, {\left (5 i \, x^{\frac {5}{2}} e \log \left (\frac {2 \, x^{2} e + 2 \, \sqrt {x^{2} e + d} x e^{\frac {1}{2}} + d}{d}\right ) - 4 i \, \sqrt {x^{2} e + d} x^{\frac {3}{2}} e^{\frac {1}{2}} - 12 i \, d {\rm weierstrassZeta}\left (-4 \, d e^{\left (-1\right )}, 0, {\rm weierstrassPInverse}\left (-4 \, d e^{\left (-1\right )}, 0, x\right )\right )\right )} e^{\left (-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 6.08, size = 75, normalized size = 0.25 \begin {gather*} \frac {2 x^{\frac {5}{2}} \operatorname {atan}{\left (\frac {x \sqrt {- e}}{\sqrt {d + e x^{2}}} \right )}}{5} - \frac {x^{\frac {7}{2}} \sqrt {- e} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {e x^{2} e^{i \pi }}{d}} \right )}}{5 \sqrt {d} \Gamma \left (\frac {11}{4}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {-e}\,x}{\sqrt {e\,x^2+d}}\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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