Optimal. Leaf size=75 \[ \frac {3 \sqrt {a x^{23}} \sinh ^{-1}\left (x^{5/2}\right )}{20 x^{23/2}}-\frac {3 \sqrt {x^5+1} \sqrt {a x^{23}}}{20 x^9}+\frac {\sqrt {x^5+1} \sqrt {a x^{23}}}{10 x^4} \]
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Rubi [A] time = 0.02, 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, 321, 329, 275, 215} \begin {gather*} \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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 215
Rule 275
Rule 321
Rule 329
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*}
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Mathematica [A] time = 0.02, size = 49, normalized size = 0.65 \begin {gather*} \frac {\sqrt {a x^{23}} \left (3 \sinh ^{-1}\left (x^{5/2}\right )+\sqrt {x^5+1} \left (2 x^5-3\right ) x^{5/2}\right )}{20 x^{23/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.43, size = 84, normalized size = 1.12 \begin {gather*} \frac {\sqrt {a} x^{23/2} \left (\frac {1}{20} \sqrt {x^5+1} \left (2 \sqrt {a} x^{15/2}-3 \sqrt {a} x^{5/2}\right )+\frac {3}{20} \sqrt {a} \log \left (x^{5/2}+\sqrt {x^5+1}\right )\right )}{\sqrt {a x^{23}}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.80, 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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giac [F] time = 0.00, size = 0, normalized size = 0.00 \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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maple [A] time = 0.07, size = 64, normalized size = 0.85 \begin {gather*} \frac {\left (2 x^{5}-3\right ) \sqrt {x^{5}+1}\, \sqrt {a \,x^{23}}}{20 x^{9}}+\frac {3 \sqrt {a \,x^{23}}\, \sqrt {\left (x^{5}+1\right ) a x}\, \arcsinh \left (x^{\frac {5}{2}}\right )}{20 \sqrt {x^{5}+1}\, \sqrt {a}\, x^{12}} \end {gather*}
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*} \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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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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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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