Optimal. Leaf size=75 \[ \frac{\sqrt{x^5+1} \sqrt{a x^{23}}}{10 x^4}-\frac{3 \sqrt{x^5+1} \sqrt{a x^{23}}}{20 x^9}+\frac{3 \sqrt{a x^{23}} \sinh ^{-1}\left (x^{5/2}\right )}{20 x^{23/2}} \]
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Rubi [A] time = 0.0174213, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {15, 321, 329, 275, 215} \[ \frac{\sqrt{x^5+1} \sqrt{a x^{23}}}{10 x^4}-\frac{3 \sqrt{x^5+1} \sqrt{a x^{23}}}{20 x^9}+\frac{3 \sqrt{a x^{23}} \sinh ^{-1}\left (x^{5/2}\right )}{20 x^{23/2}} \]
Antiderivative was successfully verified.
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Rule 15
Rule 321
Rule 329
Rule 275
Rule 215
Rubi steps
\begin{align*} \int \frac{\sqrt{a x^{23}}}{\sqrt{1+x^5}} \, dx &=\frac{\sqrt{a x^{23}} \int \frac{x^{23/2}}{\sqrt{1+x^5}} \, dx}{x^{23/2}}\\ &=\frac{\sqrt{a x^{23}} \sqrt{1+x^5}}{10 x^4}-\frac{\left (3 \sqrt{a x^{23}}\right ) \int \frac{x^{13/2}}{\sqrt{1+x^5}} \, dx}{4 x^{23/2}}\\ &=-\frac{3 \sqrt{a x^{23}} \sqrt{1+x^5}}{20 x^9}+\frac{\sqrt{a x^{23}} \sqrt{1+x^5}}{10 x^4}+\frac{\left (3 \sqrt{a x^{23}}\right ) \int \frac{x^{3/2}}{\sqrt{1+x^5}} \, dx}{8 x^{23/2}}\\ &=-\frac{3 \sqrt{a x^{23}} \sqrt{1+x^5}}{20 x^9}+\frac{\sqrt{a x^{23}} \sqrt{1+x^5}}{10 x^4}+\frac{\left (3 \sqrt{a x^{23}}\right ) \operatorname{Subst}\left (\int \frac{x^4}{\sqrt{1+x^{10}}} \, dx,x,\sqrt{x}\right )}{4 x^{23/2}}\\ &=-\frac{3 \sqrt{a x^{23}} \sqrt{1+x^5}}{20 x^9}+\frac{\sqrt{a x^{23}} \sqrt{1+x^5}}{10 x^4}+\frac{\left (3 \sqrt{a x^{23}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+x^2}} \, dx,x,x^{5/2}\right )}{20 x^{23/2}}\\ &=-\frac{3 \sqrt{a x^{23}} \sqrt{1+x^5}}{20 x^9}+\frac{\sqrt{a x^{23}} \sqrt{1+x^5}}{10 x^4}+\frac{3 \sqrt{a x^{23}} \sinh ^{-1}\left (x^{5/2}\right )}{20 x^{23/2}}\\ \end{align*}
Mathematica [A] time = 0.0199386, size = 49, normalized size = 0.65 \[ \frac{\sqrt{a x^{23}} \left (\sqrt{x^5+1} \left (2 x^5-3\right ) x^{5/2}+3 \sinh ^{-1}\left (x^{5/2}\right )\right )}{20 x^{23/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.055, size = 64, normalized size = 0.9 \begin{align*}{\frac{2\,{x}^{5}-3}{20\,{x}^{9}}\sqrt{{x}^{5}+1}\sqrt{a{x}^{23}}}+{\frac{3}{20\,{x}^{12}}{\it Arcsinh} \left ({x}^{{\frac{5}{2}}} \right ) \sqrt{a{x}^{23}}\sqrt{ax \left ({x}^{5}+1 \right ) }{\frac{1}{\sqrt{a}}}{\frac{1}{\sqrt{{x}^{5}+1}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{a x^{23}}}{\sqrt{x^{5} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.89538, size = 423, normalized size = 5.64 \begin{align*} \left [\frac{3 \, \sqrt{a} x^{9} \log \left (-\frac{8 \, a x^{19} + 8 \, a x^{14} + a x^{9} + 4 \, \sqrt{a x^{23}}{\left (2 \, x^{5} + 1\right )} \sqrt{x^{5} + 1} \sqrt{a}}{x^{9}}\right ) + 4 \, \sqrt{a x^{23}}{\left (2 \, x^{5} - 3\right )} \sqrt{x^{5} + 1}}{80 \, x^{9}}, -\frac{3 \, \sqrt{-a} x^{9} \arctan \left (\frac{\sqrt{a x^{23}}{\left (2 \, x^{5} + 1\right )} \sqrt{x^{5} + 1} \sqrt{-a}}{2 \,{\left (a x^{19} + a x^{14}\right )}}\right ) - 2 \, \sqrt{a x^{23}}{\left (2 \, x^{5} - 3\right )} \sqrt{x^{5} + 1}}{40 \, x^{9}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{a x^{23}}}{\sqrt{\left (x + 1\right ) \left (x^{4} - x^{3} + x^{2} - x + 1\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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