Optimal. Leaf size=24 \[ \text {Int}\left (\frac {\sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{x},x\right ) \]
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Rubi [A] time = 0.07, 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 = {} \[ \int \frac {\sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{x} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{x} \, dx &=\int \frac {\sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{x} \, dx\\ \end {align*}
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Mathematica [A] time = 18.52, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{x} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 1.27, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {e x + d} {\left (b \operatorname {arccsc}\left (c x\right ) + a\right )}}{x}, x\right ) \]
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 \[ \int \frac {\sqrt {e x + d} {\left (b \operatorname {arccsc}\left (c x\right ) + a\right )}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 8.85, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +b \,\mathrm {arccsc}\left (c x \right )\right ) \sqrt {e x +d}}{x}\, dx \]
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 \[ a \sqrt {d} \log \left (\frac {e x}{e x + 2 \, \sqrt {e x + d} \sqrt {d} + 2 \, d}\right ) + b \int \frac {\sqrt {e x + d} \arctan \left (1, \sqrt {c x + 1} \sqrt {c x - 1}\right )}{x}\,{d x} + 2 \, \sqrt {e x + d} a \]
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 \[ \int \frac {\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )\,\sqrt {d+e\,x}}{x} \,d x \]
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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