Optimal. Leaf size=23 \[ \text {Int}\left (\left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right ),x\right ) \]
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Rubi [A] time = 0.04, 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 = {} \[ \int \left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right ) \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right ) \, dx &=\int \left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right ) \, dx\\ \end {align*}
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Mathematica [A] time = 4.48, size = 0, normalized size = 0.00 \[ \int \left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right ) \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a e x^{2} + a d + {\left (b e x^{2} + b d\right )} \operatorname {arsech}\left (c x\right )\right )} \sqrt {e x^{2} + d}, x\right ) \]
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 \[ \int {\left (e x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arsech}\left (c x\right ) + a\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.44, size = 0, normalized size = 0.00 \[ \int \left (e \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \,\mathrm {arcsech}\left (c x \right )\right )\, dx \]
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 \[ \frac {1}{8} \, {\left (2 \, {\left (e x^{2} + d\right )}^{\frac {3}{2}} x + 3 \, \sqrt {e x^{2} + d} d x + \frac {3 \, d^{2} \operatorname {arsinh}\left (\frac {e x}{\sqrt {d e}}\right )}{\sqrt {e}}\right )} a + b \int {\left (e x^{2} + d\right )}^{\frac {3}{2}} \log \left (\sqrt {\frac {1}{c x} + 1} \sqrt {\frac {1}{c x} - 1} + \frac {1}{c x}\right )\,{d x} \]
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 \[ \int {\left (e\,x^2+d\right )}^{3/2}\,\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \operatorname {asech}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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