Optimal. Leaf size=26 \[ \text {Int}\left (\frac {x^4 \left (a+b \sec ^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}},x\right ) \]
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Rubi [A] time = 0.11, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {x^4 \left (a+b \sec ^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {x^4 \left (a+b \sec ^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx &=\int \frac {x^4 \left (a+b \sec ^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx\\ \end {align*}
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Mathematica [A] time = 9.67, size = 0, normalized size = 0.00 \[ \int \frac {x^4 \left (a+b \sec ^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 1.65, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x^{4} \operatorname {arcsec}\left (c x\right ) + a x^{4}\right )} \sqrt {e x^{2} + d}}{e^{3} x^{6} + 3 \, d e^{2} x^{4} + 3 \, d^{2} e x^{2} + d^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \operatorname {arcsec}\left (c x\right ) + a\right )} x^{4}}{{\left (e x^{2} + d\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 3.36, size = 0, normalized size = 0.00 \[ \int \frac {x^{4} \left (a +b \,\mathrm {arcsec}\left (c x \right )\right )}{\left (e \,x^{2}+d \right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {1}{3} \, {\left (x {\left (\frac {3 \, x^{2}}{{\left (e x^{2} + d\right )}^{\frac {3}{2}} e} + \frac {2 \, d}{{\left (e x^{2} + d\right )}^{\frac {3}{2}} e^{2}}\right )} + \frac {x}{\sqrt {e x^{2} + d} e^{2}} - \frac {3 \, \operatorname {arsinh}\left (\frac {e x}{\sqrt {d e}}\right )}{e^{\frac {5}{2}}}\right )} a + b \int \frac {x^{4} \arctan \left (\sqrt {c x + 1} \sqrt {c x - 1}\right )}{{\left (e^{2} x^{4} + 2 \, d e x^{2} + d^{2}\right )} \sqrt {e x^{2} + d}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {x^4\,\left (a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )\right )}{{\left (e\,x^2+d\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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