Optimal. Leaf size=83 \[ -\frac {3 a x^{5/2} \sqrt {a+b x^5}}{20 b^2}+\frac {x^{15/2} \sqrt {a+b x^5}}{10 b}+\frac {3 a^2 \tanh ^{-1}\left (\frac {\sqrt {b} x^{5/2}}{\sqrt {a+b x^5}}\right )}{20 b^{5/2}} \]
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Rubi [A]
time = 0.03, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {327, 335, 281,
223, 212} \begin {gather*} \frac {3 a^2 \tanh ^{-1}\left (\frac {\sqrt {b} x^{5/2}}{\sqrt {a+b x^5}}\right )}{20 b^{5/2}}-\frac {3 a x^{5/2} \sqrt {a+b x^5}}{20 b^2}+\frac {x^{15/2} \sqrt {a+b x^5}}{10 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 223
Rule 281
Rule 327
Rule 335
Rubi steps
\begin {align*} \int \frac {x^{23/2}}{\sqrt {a+b x^5}} \, dx &=\frac {x^{15/2} \sqrt {a+b x^5}}{10 b}-\frac {(3 a) \int \frac {x^{13/2}}{\sqrt {a+b x^5}} \, dx}{4 b}\\ &=-\frac {3 a x^{5/2} \sqrt {a+b x^5}}{20 b^2}+\frac {x^{15/2} \sqrt {a+b x^5}}{10 b}+\frac {\left (3 a^2\right ) \int \frac {x^{3/2}}{\sqrt {a+b x^5}} \, dx}{8 b^2}\\ &=-\frac {3 a x^{5/2} \sqrt {a+b x^5}}{20 b^2}+\frac {x^{15/2} \sqrt {a+b x^5}}{10 b}+\frac {\left (3 a^2\right ) \text {Subst}\left (\int \frac {x^4}{\sqrt {a+b x^{10}}} \, dx,x,\sqrt {x}\right )}{4 b^2}\\ &=-\frac {3 a x^{5/2} \sqrt {a+b x^5}}{20 b^2}+\frac {x^{15/2} \sqrt {a+b x^5}}{10 b}+\frac {\left (3 a^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,x^{5/2}\right )}{20 b^2}\\ &=-\frac {3 a x^{5/2} \sqrt {a+b x^5}}{20 b^2}+\frac {x^{15/2} \sqrt {a+b x^5}}{10 b}+\frac {\left (3 a^2\right ) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x^{5/2}}{\sqrt {a+b x^5}}\right )}{20 b^2}\\ &=-\frac {3 a x^{5/2} \sqrt {a+b x^5}}{20 b^2}+\frac {x^{15/2} \sqrt {a+b x^5}}{10 b}+\frac {3 a^2 \tanh ^{-1}\left (\frac {\sqrt {b} x^{5/2}}{\sqrt {a+b x^5}}\right )}{20 b^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 1.26, size = 71, normalized size = 0.86 \begin {gather*} \frac {\sqrt {a+b x^5} \left (-3 a x^{5/2}+2 b x^{15/2}\right )}{20 b^2}+\frac {3 a^2 \tanh ^{-1}\left (\frac {\sqrt {a+b x^5}}{\sqrt {b} x^{5/2}}\right )}{20 b^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {x^{\frac {23}{2}}}{\sqrt {b \,x^{5}+a}}\, dx\]
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. 124 vs.
\(2 (61) = 122\).
time = 0.51, size = 124, normalized size = 1.49 \begin {gather*} -\frac {3 \, a^{2} \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 )}{40 \, b^{\frac {5}{2}}} + \frac {\frac {5 \, \sqrt {b x^{5} + a} a^{2} b}{x^{\frac {5}{2}}} - \frac {3 \, {\left (b x^{5} + a\right )}^{\frac {3}{2}} a^{2}}{x^{\frac {15}{2}}}}{20 \, {\left (b^{4} - \frac {2 \, {\left (b x^{5} + a\right )} b^{3}}{x^{5}} + \frac {{\left (b x^{5} + a\right )}^{2} b^{2}}{x^{10}}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.69, size = 172, normalized size = 2.07 \begin {gather*} \left [\frac {3 \, a^{2} \sqrt {b} \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 ) + 4 \, {\left (2 \, b^{2} x^{7} - 3 \, a b x^{2}\right )} \sqrt {b x^{5} + a} \sqrt {x}}{80 \, b^{3}}, -\frac {3 \, a^{2} \sqrt {-b} \arctan \left (\frac {2 \, \sqrt {b x^{5} + a} \sqrt {-b} x^{\frac {5}{2}}}{2 \, b x^{5} + a}\right ) - 2 \, {\left (2 \, b^{2} x^{7} - 3 \, a b x^{2}\right )} \sqrt {b x^{5} + a} \sqrt {x}}{40 \, b^{3}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.16, size = 58, normalized size = 0.70 \begin {gather*} \frac {1}{20} \, \sqrt {b x^{5} + a} {\left (\frac {2 \, x^{5}}{b} - \frac {3 \, a}{b^{2}}\right )} x^{\frac {5}{2}} - \frac {3 \, a^{2} \log \left ({\left | -\sqrt {b} x^{\frac {5}{2}} + \sqrt {b x^{5} + a} \right |}\right )}{20 \, b^{\frac {5}{2}}} \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 = -0.01 \begin {gather*} \int \frac {x^{23/2}}{\sqrt {b\,x^5+a}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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