Optimal. Leaf size=23 \[ \frac {2 x^{5/2} \left (a+\frac {b}{x}\right )^{5/2}}{5 a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {264} \[ \frac {2 x^{5/2} \left (a+\frac {b}{x}\right )^{5/2}}{5 a} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 264
Rubi steps
\begin {align*} \int \left (a+\frac {b}{x}\right )^{3/2} x^{3/2} \, dx &=\frac {2 \left (a+\frac {b}{x}\right )^{5/2} x^{5/2}}{5 a}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 23, normalized size = 1.00 \[ \frac {2 x^{5/2} \left (a+\frac {b}{x}\right )^{5/2}}{5 a} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.79, size = 35, normalized size = 1.52 \[ \frac {2 \, {\left (a^{2} x^{2} + 2 \, a b x + b^{2}\right )} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{5 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.17, size = 64, normalized size = 2.78 \[ \frac {2}{3} \, b {\left (\frac {{\left (a x + b\right )}^{\frac {3}{2}}}{a} - \frac {b^{\frac {3}{2}}}{a}\right )} \mathrm {sgn}\relax (x) + \frac {2}{15} \, a {\left (\frac {2 \, b^{\frac {5}{2}}}{a^{2}} + \frac {3 \, {\left (a x + b\right )}^{\frac {5}{2}} - 5 \, {\left (a x + b\right )}^{\frac {3}{2}} b}{a^{2}}\right )} \mathrm {sgn}\relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.00, size = 25, normalized size = 1.09 \[ \frac {2 \left (a x +b \right ) \left (\frac {a x +b}{x}\right )^{\frac {3}{2}} x^{\frac {3}{2}}}{5 a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.08, size = 17, normalized size = 0.74 \[ \frac {2 \, {\left (a + \frac {b}{x}\right )}^{\frac {5}{2}} x^{\frac {5}{2}}}{5 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.38, size = 34, normalized size = 1.48 \[ \sqrt {a+\frac {b}{x}}\,\left (\frac {2\,a\,x^{5/2}}{5}+\frac {4\,b\,x^{3/2}}{5}+\frac {2\,b^2\,\sqrt {x}}{5\,a}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 9.46, size = 63, normalized size = 2.74 \[ \frac {2 a \sqrt {b} x^{2} \sqrt {\frac {a x}{b} + 1}}{5} + \frac {4 b^{\frac {3}{2}} x \sqrt {\frac {a x}{b} + 1}}{5} + \frac {2 b^{\frac {5}{2}} \sqrt {\frac {a x}{b} + 1}}{5 a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________