Optimal. Leaf size=27 \[ \frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x^5}}}{\sqrt {a}}\right )}{5 \sqrt {a}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {266, 63, 208} \[ \frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x^5}}}{\sqrt {a}}\right )}{5 \sqrt {a}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 63
Rule 208
Rule 266
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {a+\frac {b}{x^5}} x} \, dx &=-\left (\frac {1}{5} \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\frac {1}{x^5}\right )\right )\\ &=-\frac {2 \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+\frac {b}{x^5}}\right )}{5 b}\\ &=\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x^5}}}{\sqrt {a}}\right )}{5 \sqrt {a}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] time = 0.02, size = 59, normalized size = 2.19 \[ \frac {2 \sqrt {a x^5+b} \tanh ^{-1}\left (\frac {\sqrt {a} x^{5/2}}{\sqrt {a x^5+b}}\right )}{5 \sqrt {a} x^{5/2} \sqrt {a+\frac {b}{x^5}}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 2.14, size = 102, normalized size = 3.78 \[ \left [\frac {\log \left (-8 \, a^{2} x^{10} - 8 \, a b x^{5} - b^{2} - 4 \, {\left (2 \, a x^{10} + b x^{5}\right )} \sqrt {a} \sqrt {\frac {a x^{5} + b}{x^{5}}}\right )}{10 \, \sqrt {a}}, -\frac {\sqrt {-a} \arctan \left (\frac {2 \, \sqrt {-a} x^{5} \sqrt {\frac {a x^{5} + b}{x^{5}}}}{2 \, a x^{5} + b}\right )}{5 \, a}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a + \frac {b}{x^{5}}} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.07, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a +\frac {b}{x^{5}}}\, x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.93, size = 37, normalized size = 1.37 \[ -\frac {\log \left (\frac {\sqrt {a + \frac {b}{x^{5}}} - \sqrt {a}}{\sqrt {a + \frac {b}{x^{5}}} + \sqrt {a}}\right )}{5 \, \sqrt {a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.35, size = 19, normalized size = 0.70 \[ \frac {2\,\mathrm {atanh}\left (\frac {\sqrt {a+\frac {b}{x^5}}}{\sqrt {a}}\right )}{5\,\sqrt {a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 1.44, size = 24, normalized size = 0.89 \[ \frac {2 \operatorname {asinh}{\left (\frac {\sqrt {a} x^{\frac {5}{2}}}{\sqrt {b}} \right )}}{5 \sqrt {a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________