Optimal. Leaf size=24 \[ \text {Int}\left (\frac {a+b \csc ^{-1}(c x)}{x (d+e x)^{3/2}},x\right ) \]
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Rubi [A] time = 0.09, 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 = {} \[ \int \frac {a+b \csc ^{-1}(c x)}{x (d+e x)^{3/2}} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {a+b \csc ^{-1}(c x)}{x (d+e x)^{3/2}} \, dx &=\int \frac {a+b \csc ^{-1}(c x)}{x (d+e x)^{3/2}} \, dx\\ \end {align*}
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Mathematica [A] time = 11.78, size = 0, normalized size = 0.00 \[ \int \frac {a+b \csc ^{-1}(c x)}{x (d+e x)^{3/2}} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 1.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {e x + d} {\left (b \operatorname {arccsc}\left (c x\right ) + a\right )}}{e^{2} x^{3} + 2 \, d e x^{2} + d^{2} x}, x\right ) \]
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 \[ \int \frac {b \operatorname {arccsc}\left (c x\right ) + a}{{\left (e x + d\right )}^{\frac {3}{2}} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 7.38, size = 0, normalized size = 0.00 \[ \int \frac {a +b \,\mathrm {arccsc}\left (c x \right )}{x \left (e x +d \right )^{\frac {3}{2}}}\, dx \]
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 \[ \frac {{\left (b d^{\frac {3}{2}} \int \frac {\arctan \left (1, \sqrt {c x + 1} \sqrt {c x - 1}\right )}{{\left (e x + d\right )}^{\frac {3}{2}} x}\,{d x} + a \log \left (\frac {e x}{e x + 2 \, \sqrt {e x + d} \sqrt {d} + 2 \, d}\right )\right )} \sqrt {e x + d} + 2 \, a \sqrt {d}}{\sqrt {e x + d} d^{\frac {3}{2}}} \]
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 \[ \int \frac {a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )}{x\,{\left (d+e\,x\right )}^{3/2}} \,d x \]
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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