Optimal. Leaf size=26 \[ \text {Int}\left (\frac {\left (d+e x^2\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right )}{x},x\right ) \]
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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 = {} \[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right )}{x} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right )}{x} \, dx &=\int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right )}{x} \, dx\\ \end {align*}
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Mathematica [A] time = 4.97, size = 0, normalized size = 0.00 \[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right )}{x} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a e x^{2} + a d + {\left (b e x^{2} + b d\right )} \operatorname {arcsec}\left (c x\right )\right )} \sqrt {e x^{2} + d}}{x}, x\right ) \]
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 \[ \int \frac {{\left (e x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arcsec}\left (c x\right ) + a\right )}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 2.20, size = 0, normalized size = 0.00 \[ \int \frac {\left (e \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \,\mathrm {arcsec}\left (c x \right )\right )}{x}\, dx \]
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 \[ -\frac {1}{3} \, {\left (3 \, d^{\frac {3}{2}} \operatorname {arsinh}\left (\frac {d}{\sqrt {d e} {\left | x \right |}}\right ) - {\left (e x^{2} + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {e x^{2} + d} d\right )} a + {\left (e \int \sqrt {e x^{2} + d} x \arctan \left (\sqrt {c x + 1} \sqrt {c x - 1}\right )\,{d x} + d \int \frac {\sqrt {e x^{2} + d} \arctan \left (\sqrt {c x + 1} \sqrt {c x - 1}\right )}{x}\,{d x}\right )} b \]
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 \[ \int \frac {{\left (e\,x^2+d\right )}^{3/2}\,\left (a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )\right )}{x} \,d x \]
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 \[ \int \frac {\left (a + b \operatorname {asec}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{\frac {3}{2}}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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