Optimal. Leaf size=32 \[ \frac {2 \tanh ^{-1}\left (\frac {\sqrt {b} x^{5/2}}{\sqrt {a+b x^5}}\right )}{5 \sqrt {b}} \]
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Rubi [A]
time = 0.02, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {335, 281, 223,
212} \begin {gather*} \frac {2 \tanh ^{-1}\left (\frac {\sqrt {b} x^{5/2}}{\sqrt {a+b x^5}}\right )}{5 \sqrt {b}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 223
Rule 281
Rule 335
Rubi steps
\begin {align*} \int \frac {x^{3/2}}{\sqrt {a+b x^5}} \, dx &=2 \text {Subst}\left (\int \frac {x^4}{\sqrt {a+b x^{10}}} \, dx,x,\sqrt {x}\right )\\ &=\frac {2}{5} \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,x^{5/2}\right )\\ &=\frac {2}{5} \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x^{5/2}}{\sqrt {a+b x^5}}\right )\\ &=\frac {2 \tanh ^{-1}\left (\frac {\sqrt {b} x^{5/2}}{\sqrt {a+b x^5}}\right )}{5 \sqrt {b}}\\ \end {align*}
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Mathematica [A]
time = 0.20, size = 32, normalized size = 1.00 \begin {gather*} \frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b x^5}}{\sqrt {b} x^{5/2}}\right )}{5 \sqrt {b}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {x^{\frac {3}{2}}}{\sqrt {b \,x^{5}+a}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 45 vs.
\(2 (22) = 44\).
time = 0.51, size = 45, normalized size = 1.41 \begin {gather*} -\frac {\log \left (-\frac {\sqrt {b} - \frac {\sqrt {b x^{5} + a}}{x^{\frac {5}{2}}}}{\sqrt {b} + \frac {\sqrt {b x^{5} + a}}{x^{\frac {5}{2}}}}\right )}{5 \, \sqrt {b}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.66, size = 97, normalized size = 3.03 \begin {gather*} \left [\frac {\log \left (-8 \, b^{2} x^{10} - 8 \, a b x^{5} - 4 \, {\left (2 \, b x^{7} + a x^{2}\right )} \sqrt {b x^{5} + a} \sqrt {b} \sqrt {x} - a^{2}\right )}{10 \, \sqrt {b}}, -\frac {\sqrt {-b} \arctan \left (\frac {2 \, \sqrt {b x^{5} + a} \sqrt {-b} x^{\frac {5}{2}}}{2 \, b x^{5} + a}\right )}{5 \, b}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.71, size = 24, normalized size = 0.75 \begin {gather*} \frac {2 \operatorname {asinh}{\left (\frac {\sqrt {b} x^{\frac {5}{2}}}{\sqrt {a}} \right )}}{5 \sqrt {b}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.68, size = 25, normalized size = 0.78 \begin {gather*} -\frac {2 \, \log \left ({\left | -\sqrt {b} x^{\frac {5}{2}} + \sqrt {b x^{5} + a} \right |}\right )}{5 \, \sqrt {b}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {x^{3/2}}{\sqrt {b\,x^5+a}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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