Optimal. Leaf size=181 \[ -\frac {10 d^{7/4} \sqrt {-e} \left (\sqrt {d}+\sqrt {e} x\right ) \sqrt {\frac {d+e x^2}{\left (\sqrt {d}+\sqrt {e} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right )|\frac {1}{2}\right )}{147 e^{9/4} \sqrt {d+e x^2}}+\frac {20 d \sqrt {x} \sqrt {d+e x^2}}{147 (-e)^{3/2}}+\frac {4 x^{5/2} \sqrt {d+e x^2}}{49 \sqrt {-e}}+\frac {2}{7} x^{7/2} \tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right ) \]
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Rubi [A] time = 0.09, antiderivative size = 181, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {5151, 321, 329, 220} \[ -\frac {10 d^{7/4} \sqrt {-e} \left (\sqrt {d}+\sqrt {e} x\right ) \sqrt {\frac {d+e x^2}{\left (\sqrt {d}+\sqrt {e} x\right )^2}} \text {EllipticF}\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right ),\frac {1}{2}\right )}{147 e^{9/4} \sqrt {d+e x^2}}+\frac {4 x^{5/2} \sqrt {d+e x^2}}{49 \sqrt {-e}}+\frac {20 d \sqrt {x} \sqrt {d+e x^2}}{147 (-e)^{3/2}}+\frac {2}{7} x^{7/2} \tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right ) \]
Antiderivative was successfully verified.
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Rule 220
Rule 321
Rule 329
Rule 5151
Rubi steps
\begin {align*} \int x^{5/2} \tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right ) \, dx &=\frac {2}{7} x^{7/2} \tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )-\frac {1}{7} \left (2 \sqrt {-e}\right ) \int \frac {x^{7/2}}{\sqrt {d+e x^2}} \, dx\\ &=\frac {4 x^{5/2} \sqrt {d+e x^2}}{49 \sqrt {-e}}+\frac {2}{7} x^{7/2} \tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )-\frac {(10 d) \int \frac {x^{3/2}}{\sqrt {d+e x^2}} \, dx}{49 \sqrt {-e}}\\ &=\frac {20 d \sqrt {x} \sqrt {d+e x^2}}{147 (-e)^{3/2}}+\frac {4 x^{5/2} \sqrt {d+e x^2}}{49 \sqrt {-e}}+\frac {2}{7} x^{7/2} \tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )-\frac {\left (10 d^2\right ) \int \frac {1}{\sqrt {x} \sqrt {d+e x^2}} \, dx}{147 (-e)^{3/2}}\\ &=\frac {20 d \sqrt {x} \sqrt {d+e x^2}}{147 (-e)^{3/2}}+\frac {4 x^{5/2} \sqrt {d+e x^2}}{49 \sqrt {-e}}+\frac {2}{7} x^{7/2} \tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )-\frac {\left (20 d^2\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {d+e x^4}} \, dx,x,\sqrt {x}\right )}{147 (-e)^{3/2}}\\ &=\frac {20 d \sqrt {x} \sqrt {d+e x^2}}{147 (-e)^{3/2}}+\frac {4 x^{5/2} \sqrt {d+e x^2}}{49 \sqrt {-e}}+\frac {2}{7} x^{7/2} \tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )-\frac {10 d^{7/4} \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 \tan ^{-1}\left (\frac {\sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right )|\frac {1}{2}\right )}{147 e^{9/4} \sqrt {d+e x^2}}\\ \end {align*}
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Mathematica [C] time = 0.46, size = 158, normalized size = 0.87 \[ \frac {2}{147} \sqrt {x} \left (\frac {2 \left (5 d-3 e x^2\right ) \sqrt {d+e x^2}}{(-e)^{3/2}}+21 x^3 \tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )\right )-\frac {20 i d^2 x \sqrt {\frac {d}{e x^2}+1} F\left (\left .i \sinh ^{-1}\left (\frac {\sqrt {\frac {i \sqrt {d}}{\sqrt {e}}}}{\sqrt {x}}\right )\right |-1\right )}{147 (-e)^{3/2} \sqrt {\frac {i \sqrt {d}}{\sqrt {e}}} \sqrt {d+e x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.60, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{\frac {5}{2}} \arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right ), x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.24, size = 0, normalized size = 0.00 \[ \int x^{\frac {5}{2}} \arctan \left (\frac {x \sqrt {-e}}{\sqrt {e \,x^{2}+d}}\right )\, dx \]
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 \[ \text {Exception raised: RuntimeError} \]
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 \[ \int x^{5/2}\,\mathrm {atan}\left (\frac {\sqrt {-e}\,x}{\sqrt {e\,x^2+d}}\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 150.30, size = 75, normalized size = 0.41 \[ \frac {2 x^{\frac {7}{2}} \operatorname {atan}{\left (\frac {x \sqrt {- e}}{\sqrt {d + e x^{2}}} \right )}}{7} - \frac {x^{\frac {9}{2}} \sqrt {- e} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {\frac {e x^{2} e^{i \pi }}{d}} \right )}}{7 \sqrt {d} \Gamma \left (\frac {13}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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