Optimal. Leaf size=26 \[ \text {Int}\left (x^2 \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right ),x\right ) \]
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Rubi [A] time = 0.12, 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 = {} \[ \int x^2 \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right ) \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int x^2 \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right ) \, dx &=\int x^2 \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right ) \, dx\\ \end {align*}
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Mathematica [A] time = 9.91, size = 0, normalized size = 0.00 \[ \int x^2 \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right ) \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.86, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a e x^{4} + a d x^{2} + {\left (b e x^{4} + b d x^{2}\right )} \operatorname {arccsc}\left (c x\right )\right )} \sqrt {e x^{2} + d}, x\right ) \]
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 \[ \int {\left (e x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arccsc}\left (c x\right ) + a\right )} x^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 5.86, size = 0, normalized size = 0.00 \[ \int x^{2} \left (e \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \,\mathrm {arccsc}\left (c x \right )\right )\, dx \]
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 \[ \frac {1}{48} \, {\left (\frac {8 \, {\left (e x^{2} + d\right )}^{\frac {5}{2}} x}{e} - \frac {2 \, {\left (e x^{2} + d\right )}^{\frac {3}{2}} d x}{e} - \frac {3 \, \sqrt {e x^{2} + d} d^{2} x}{e} - \frac {3 \, d^{3} \operatorname {arsinh}\left (\frac {e x}{\sqrt {d e}}\right )}{e^{\frac {3}{2}}}\right )} a + b \int {\left (e x^{4} \arctan \left (1, \sqrt {c x + 1} \sqrt {c x - 1}\right ) + d x^{2} \arctan \left (1, \sqrt {c x + 1} \sqrt {c x - 1}\right )\right )} \sqrt {e x^{2} + d}\,{d x} \]
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 \[ \int x^2\,{\left (e\,x^2+d\right )}^{3/2}\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right ) \,d x \]
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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