Optimal. Leaf size=49 \[ \frac {4}{15} x^6 \sqrt {x^5+1} \sqrt {\frac {a}{x^{17}}}-\frac {2}{15} x \sqrt {x^5+1} \sqrt {\frac {a}{x^{17}}} \]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {15, 271, 264} \begin {gather*} \frac {4}{15} x^6 \sqrt {x^5+1} \sqrt {\frac {a}{x^{17}}}-\frac {2}{15} x \sqrt {x^5+1} \sqrt {\frac {a}{x^{17}}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 15
Rule 264
Rule 271
Rubi steps
\begin {align*} \int \frac {\sqrt {\frac {a}{x^{17}}}}{\sqrt {1+x^5}} \, dx &=\left (\sqrt {\frac {a}{x^{17}}} x^{17/2}\right ) \int \frac {1}{x^{17/2} \sqrt {1+x^5}} \, dx\\ &=-\frac {2}{15} \sqrt {\frac {a}{x^{17}}} x \sqrt {1+x^5}-\frac {1}{3} \left (2 \sqrt {\frac {a}{x^{17}}} x^{17/2}\right ) \int \frac {1}{x^{7/2} \sqrt {1+x^5}} \, dx\\ &=-\frac {2}{15} \sqrt {\frac {a}{x^{17}}} x \sqrt {1+x^5}+\frac {4}{15} \sqrt {\frac {a}{x^{17}}} x^6 \sqrt {1+x^5}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 30, normalized size = 0.61 \begin {gather*} -\frac {2}{15} x \left (1-2 x^5\right ) \sqrt {x^5+1} \sqrt {\frac {a}{x^{17}}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 9.88, size = 35, normalized size = 0.71 \begin {gather*} \frac {2 x^{35} \sqrt {x^5+1} \left (2 x^5-1\right ) \left (\frac {a}{x^{17}}\right )^{5/2}}{15 a^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.52, size = 25, normalized size = 0.51 \begin {gather*} \frac {2}{15} \, {\left (2 \, x^{6} - x\right )} \sqrt {x^{5} + 1} \sqrt {\frac {a}{x^{17}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.00, size = 44, normalized size = 0.90 \begin {gather*} \frac {2 \left (x +1\right ) \left (x^{4}-x^{3}+x^{2}-x +1\right ) \left (2 x^{5}-1\right ) \sqrt {\frac {a}{x^{17}}}\, x}{15 \sqrt {x^{5}+1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 2.87, size = 50, normalized size = 1.02 \begin {gather*} \frac {2 \, {\left (2 \, \sqrt {a} x^{11} + \sqrt {a} x^{6} - \sqrt {a} x\right )}}{15 \, \sqrt {x^{4} - x^{3} + x^{2} - x + 1} \sqrt {x + 1} x^{\frac {17}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 2.67, size = 29, normalized size = 0.59 \begin {gather*} \frac {\sqrt {\frac {a}{x^{17}}}\,\left (\frac {4\,x^{11}}{15}+\frac {2\,x^6}{15}-\frac {2\,x}{15}\right )}{\sqrt {x^5+1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {\frac {a}{x^{17}}}}{\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.
[In]
[Out]
________________________________________________________________________________________