Optimal. Leaf size=26 \[ \text {Int}\left (\frac {1}{\left (a^2 c x^2+c\right )^{5/2} \sqrt {\sinh ^{-1}(a x)}},x\right ) \]
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Rubi [A] time = 0.04, 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 = {} \[ \int \frac {1}{\left (c+a^2 c x^2\right )^{5/2} \sqrt {\sinh ^{-1}(a x)}} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {1}{\left (c+a^2 c x^2\right )^{5/2} \sqrt {\sinh ^{-1}(a x)}} \, dx &=\int \frac {1}{\left (c+a^2 c x^2\right )^{5/2} \sqrt {\sinh ^{-1}(a x)}} \, dx\\ \end {align*}
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Mathematica [A] time = 1.66, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (c+a^2 c x^2\right )^{5/2} \sqrt {\sinh ^{-1}(a x)}} \, dx \]
Verification is Not applicable to the result.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
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 \[ \int \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \sqrt {\operatorname {arsinh}\left (a x\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.46, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a^{2} c \,x^{2}+c \right )^{\frac {5}{2}} \sqrt {\arcsinh \left (a x \right )}}\, dx \]
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 \[ \int \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \sqrt {\operatorname {arsinh}\left (a x\right )}}\,{d x} \]
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 \[ \int \frac {1}{\sqrt {\mathrm {asinh}\left (a\,x\right )}\,{\left (c\,a^2\,x^2+c\right )}^{5/2}} \,d x \]
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 \[ \int \frac {1}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}} \sqrt {\operatorname {asinh}{\left (a x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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