Optimal. Leaf size=27 \[ -\frac {2 \sqrt {a x^2+b x^5}}{3 a x^{5/2}} \]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {2014} \begin {gather*} -\frac {2 \sqrt {a x^2+b x^5}}{3 a x^{5/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2014
Rubi steps
\begin {align*} \int \frac {1}{x^{3/2} \sqrt {a x^2+b x^5}} \, dx &=-\frac {2 \sqrt {a x^2+b x^5}}{3 a x^{5/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 27, normalized size = 1.00 \begin {gather*} -\frac {2 \sqrt {x^2 \left (a+b x^3\right )}}{3 a x^{5/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.24, size = 27, normalized size = 1.00 \begin {gather*} -\frac {2 \sqrt {a x^2+b x^5}}{3 a x^{5/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.41, size = 21, normalized size = 0.78 \begin {gather*} -\frac {2 \, \sqrt {b x^{5} + a x^{2}}}{3 \, a x^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.20, size = 23, normalized size = 0.85 \begin {gather*} -\frac {2 \, \sqrt {b + \frac {a}{x^{3}}}}{3 \, a} + \frac {2 \, \sqrt {b}}{3 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.04, size = 29, normalized size = 1.07 \begin {gather*} -\frac {2 \left (b \,x^{3}+a \right )}{3 \sqrt {b \,x^{5}+a \,x^{2}}\, a \sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.42, size = 26, normalized size = 0.96 \begin {gather*} -\frac {2 \, {\left (b x^{4} + a x\right )}}{3 \, \sqrt {b x^{3} + a} a x^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {1}{x^{3/2}\,\sqrt {b\,x^5+a\,x^2}} \,d x \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 {1}{x^{\frac {3}{2}} \sqrt {x^{2} \left (a + b x^{3}\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________