Optimal. Leaf size=26 \[ \text {Int}\left (\frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{x},x\right ) \]
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Rubi [A]
time = 0.08, 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 = {}
\begin {gather*} \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{x} \, dx \end {gather*}
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 \csc ^{-1}(c x)\right )}{x} \, dx &=\int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{x} \, dx\\ \end {align*}
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Mathematica [A]
time = 3.75, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{x} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [A]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\left (e \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \,\mathrm {arccsc}\left (c x \right )\right )}{x}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\left (a + b \operatorname {acsc}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{\frac {3}{2}}}{x}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {{\left (e\,x^2+d\right )}^{3/2}\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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