Optimal. Leaf size=50 \[ \frac {\sqrt {x^5+1} \sqrt {a x^{13}}}{5 x^4}-\frac {\sqrt {a x^{13}} \sinh ^{-1}\left (x^{5/2}\right )}{5 x^{13/2}} \]
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Rubi [A] time = 0.01, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 5, 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} \[ \frac {\sqrt {x^5+1} \sqrt {a x^{13}}}{5 x^4}-\frac {\sqrt {a x^{13}} \sinh ^{-1}\left (x^{5/2}\right )}{5 x^{13/2}} \]
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^{13}}}{\sqrt {1+x^5}} \, dx &=\frac {\sqrt {a x^{13}} \int \frac {x^{13/2}}{\sqrt {1+x^5}} \, dx}{x^{13/2}}\\ &=\frac {\sqrt {a x^{13}} \sqrt {1+x^5}}{5 x^4}-\frac {\sqrt {a x^{13}} \int \frac {x^{3/2}}{\sqrt {1+x^5}} \, dx}{2 x^{13/2}}\\ &=\frac {\sqrt {a x^{13}} \sqrt {1+x^5}}{5 x^4}-\frac {\sqrt {a x^{13}} \operatorname {Subst}\left (\int \frac {x^4}{\sqrt {1+x^{10}}} \, dx,x,\sqrt {x}\right )}{x^{13/2}}\\ &=\frac {\sqrt {a x^{13}} \sqrt {1+x^5}}{5 x^4}-\frac {\sqrt {a x^{13}} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,x^{5/2}\right )}{5 x^{13/2}}\\ &=\frac {\sqrt {a x^{13}} \sqrt {1+x^5}}{5 x^4}-\frac {\sqrt {a x^{13}} \sinh ^{-1}\left (x^{5/2}\right )}{5 x^{13/2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 42, normalized size = 0.84 \[ \frac {\sqrt {a x^{13}} \left (x^{5/2} \sqrt {x^5+1}-\sinh ^{-1}\left (x^{5/2}\right )\right )}{5 x^{13/2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.58, size = 153, normalized size = 3.06 \[ \left [\frac {\sqrt {a} x^{4} \log \left (-\frac {8 \, a x^{14} + 8 \, a x^{9} + a x^{4} - 4 \, \sqrt {a x^{13}} {\left (2 \, x^{5} + 1\right )} \sqrt {x^{5} + 1} \sqrt {a}}{x^{4}}\right ) + 4 \, \sqrt {a x^{13}} \sqrt {x^{5} + 1}}{20 \, x^{4}}, \frac {\sqrt {-a} x^{4} \arctan \left (\frac {\sqrt {a x^{13}} {\left (2 \, x^{5} + 1\right )} \sqrt {x^{5} + 1} \sqrt {-a}}{2 \, {\left (a x^{14} + a x^{9}\right )}}\right ) + 2 \, \sqrt {a x^{13}} \sqrt {x^{5} + 1}}{10 \, x^{4}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.32, size = 68, normalized size = 1.36 \[ \frac {a^{\frac {11}{2}} \log \left (-\sqrt {a x} a^{\frac {5}{2}} x^{2} + \sqrt {a^{6} x^{5} + a^{6}}\right )}{5 \, {\left | a \right |}^{5}} + \frac {\sqrt {a^{6} x^{5} + a^{6}} \sqrt {a x} x^{2}}{5 \, a^{2} {\left | a \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 57, normalized size = 1.14 \[ \frac {\sqrt {a \,x^{13}}\, \sqrt {x^{5}+1}}{5 x^{4}}-\frac {\sqrt {a \,x^{13}}\, \sqrt {\left (x^{5}+1\right ) a x}\, \arcsinh \left (x^{\frac {5}{2}}\right )}{5 \sqrt {x^{5}+1}\, \sqrt {a}\, x^{7}} \]
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 \[ \int \frac {\sqrt {a x^{13}}}{\sqrt {x^{5} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\sqrt {a\,x^{13}}}{\sqrt {x^5+1}} \,d x \]
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 \[ \int \frac {\sqrt {a x^{13}}}{\sqrt {\left (x + 1\right ) \left (x^{4} - x^{3} + x^{2} - x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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