Optimal. Leaf size=26 \[ \text {Int}\left (\frac {a+b \csc ^{-1}(c x)}{x \left (d+e x^2\right )^{5/2}},x\right ) \]
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Rubi [A] time = 0.12, 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 = {} \[ \int \frac {a+b \csc ^{-1}(c x)}{x \left (d+e x^2\right )^{5/2}} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {a+b \csc ^{-1}(c x)}{x \left (d+e x^2\right )^{5/2}} \, dx &=\int \frac {a+b \csc ^{-1}(c x)}{x \left (d+e x^2\right )^{5/2}} \, dx\\ \end {align*}
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Mathematica [A] time = 37.84, size = 0, normalized size = 0.00 \[ \int \frac {a+b \csc ^{-1}(c x)}{x \left (d+e x^2\right )^{5/2}} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.82, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {e x^{2} + d} {\left (b \operatorname {arccsc}\left (c x\right ) + a\right )}}{e^{3} x^{7} + 3 \, d e^{2} x^{5} + 3 \, d^{2} e x^{3} + d^{3} x}, x\right ) \]
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 \[ \int \frac {b \operatorname {arccsc}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{\frac {5}{2}} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 3.73, size = 0, normalized size = 0.00 \[ \int \frac {a +b \,\mathrm {arccsc}\left (c x \right )}{x \left (e \,x^{2}+d \right )^{\frac {5}{2}}}\, dx \]
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 \[ -\frac {1}{3} \, a {\left (\frac {3 \, \operatorname {arsinh}\left (\frac {d}{\sqrt {d e} {\left | x \right |}}\right )}{d^{\frac {5}{2}}} - \frac {3}{\sqrt {e x^{2} + d} d^{2}} - \frac {1}{{\left (e x^{2} + d\right )}^{\frac {3}{2}} d}\right )} + b \int \frac {\arctan \left (1, \sqrt {c x + 1} \sqrt {c x - 1}\right )}{{\left (e^{2} x^{5} + 2 \, d e x^{3} + d^{2} x\right )} \sqrt {e x^{2} + d}}\,{d x} \]
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 \[ \int \frac {a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )}{x\,{\left (e\,x^2+d\right )}^{5/2}} \,d x \]
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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