Optimal. Leaf size=24 \[ \frac {2 \sqrt {a x^3} \sinh ^{-1}\left (x^{5/2}\right )}{5 x^{3/2}} \]
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Rubi [A] time = 0.01, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {15, 329, 275, 215} \[ \frac {2 \sqrt {a x^3} \sinh ^{-1}\left (x^{5/2}\right )}{5 x^{3/2}} \]
Antiderivative was successfully verified.
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Rule 15
Rule 215
Rule 275
Rule 329
Rubi steps
\begin {align*} \int \frac {\sqrt {a x^3}}{\sqrt {1+x^5}} \, dx &=\frac {\sqrt {a x^3} \int \frac {x^{3/2}}{\sqrt {1+x^5}} \, dx}{x^{3/2}}\\ &=\frac {\left (2 \sqrt {a x^3}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\sqrt {1+x^{10}}} \, dx,x,\sqrt {x}\right )}{x^{3/2}}\\ &=\frac {\left (2 \sqrt {a x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,x^{5/2}\right )}{5 x^{3/2}}\\ &=\frac {2 \sqrt {a x^3} \sinh ^{-1}\left (x^{5/2}\right )}{5 x^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 24, normalized size = 1.00 \[ \frac {2 \sqrt {a x^3} \sinh ^{-1}\left (x^{5/2}\right )}{5 x^{3/2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.55, size = 98, normalized size = 4.08 \[ \left [\frac {1}{10} \, \sqrt {a} \log \left (-8 \, a x^{10} - 8 \, a x^{5} - 4 \, {\left (2 \, x^{6} + x\right )} \sqrt {x^{5} + 1} \sqrt {a x^{3}} \sqrt {a} - a\right ), -\frac {1}{5} \, \sqrt {-a} \arctan \left (\frac {{\left (2 \, x^{5} + 1\right )} \sqrt {x^{5} + 1} \sqrt {a x^{3}} \sqrt {-a}}{2 \, {\left (a x^{9} + a x^{4}\right )}}\right )\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.19, size = 58, normalized size = 2.42 \[ -\frac {2 \, a^{\frac {3}{2}} \log \left (-\sqrt {a x} a^{\frac {5}{2}} x^{2} + \sqrt {a^{6} x^{5} + a^{6}}\right ) \mathrm {sgn}\relax (x)}{5 \, {\left | a \right |}} + \frac {2 \, a^{\frac {3}{2}} \log \left (a^{2} {\left | a \right |}\right ) \mathrm {sgn}\relax (x)}{5 \, {\left | a \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 17, normalized size = 0.71 \[ \frac {2 \sqrt {a \,x^{3}}\, \arcsinh \left (x^{\frac {5}{2}}\right )}{5 x^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {a x^{3}}}{\sqrt {x^{5} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {\sqrt {a\,x^3}}{\sqrt {x^5+1}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {a x^{3}}}{\sqrt {\left (x + 1\right ) \left (x^{4} - x^{3} + x^{2} - x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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