Optimal. Leaf size=75 \[ -\frac {3 \sqrt {a x^{23}} \sqrt {1+x^5}}{20 x^9}+\frac {\sqrt {a x^{23}} \sqrt {1+x^5}}{10 x^4}+\frac {3 \sqrt {a x^{23}} \sinh ^{-1}\left (x^{5/2}\right )}{20 x^{23/2}} \]
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Rubi [A]
time = 0.01, antiderivative size = 75, normalized size of antiderivative = 1.00, 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, 327, 335,
281, 221} \begin {gather*} \frac {3 \sqrt {a x^{23}} \sinh ^{-1}\left (x^{5/2}\right )}{20 x^{23/2}}-\frac {3 \sqrt {x^5+1} \sqrt {a x^{23}}}{20 x^9}+\frac {\sqrt {x^5+1} \sqrt {a x^{23}}}{10 x^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 221
Rule 281
Rule 327
Rule 335
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 ) \text {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 ) \text {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*}
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Mathematica [A]
time = 0.94, size = 59, normalized size = 0.79 \begin {gather*} \frac {\sqrt {a x^{23}} \left (x^{5/2} \sqrt {1+x^5} \left (-3+2 x^5\right )+3 \tanh ^{-1}\left (\frac {x^{5/2}}{\sqrt {1+x^5}}\right )\right )}{20 x^{23/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.23, size = 48, normalized size = 0.64
method | result | size |
meijerg | \(\frac {\sqrt {a \,x^{23}}\, \left (-\frac {\sqrt {\pi }\, x^{\frac {5}{2}} \left (-10 x^{5}+15\right ) \sqrt {x^{5}+1}}{20}+\frac {3 \sqrt {\pi }\, \arcsinh \left (x^{\frac {5}{2}}\right )}{4}\right )}{5 x^{\frac {23}{2}} \sqrt {\pi }}\) | \(48\) |
risch | \(\frac {\left (2 x^{5}-3\right ) \sqrt {x^{5}+1}\, \sqrt {a \,x^{23}}}{20 x^{9}}+\frac {3 \arcsinh \left (x^{\frac {5}{2}}\right ) \sqrt {a \,x^{23}}\, \sqrt {a x \left (x^{5}+1\right )}}{20 \sqrt {a}\, x^{12} \sqrt {x^{5}+1}}\) | \(64\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.42, size = 169, normalized size = 2.25 \begin {gather*} \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 {gather*}
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 \begin {gather*} \int \frac {\sqrt {a x^{23}}}{\sqrt {\left (x + 1\right ) \left (x^{4} - x^{3} + x^{2} - x + 1\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 {\sqrt {a\,x^{23}}}{\sqrt {x^5+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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