Optimal. Leaf size=40 \[ \frac{\left (a+b x^{3/2}\right )^{4/3}}{2 b^2}-\frac{2 a \sqrt [3]{a+b x^{3/2}}}{b^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0205204, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {266, 43} \[ \frac{\left (a+b x^{3/2}\right )^{4/3}}{2 b^2}-\frac{2 a \sqrt [3]{a+b x^{3/2}}}{b^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^2}{\left (a+b x^{3/2}\right )^{2/3}} \, dx &=\frac{2}{3} \operatorname{Subst}\left (\int \frac{x}{(a+b x)^{2/3}} \, dx,x,x^{3/2}\right )\\ &=\frac{2}{3} \operatorname{Subst}\left (\int \left (-\frac{a}{b (a+b x)^{2/3}}+\frac{\sqrt [3]{a+b x}}{b}\right ) \, dx,x,x^{3/2}\right )\\ &=-\frac{2 a \sqrt [3]{a+b x^{3/2}}}{b^2}+\frac{\left (a+b x^{3/2}\right )^{4/3}}{2 b^2}\\ \end{align*}
Mathematica [A] time = 0.0131603, size = 31, normalized size = 0.78 \[ \frac{\left (b x^{3/2}-3 a\right ) \sqrt [3]{a+b x^{3/2}}}{2 b^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.003, size = 30, normalized size = 0.8 \begin{align*} 2\,{\frac{1/4\, \left ( a+b{x}^{3/2} \right ) ^{4/3}-a\sqrt [3]{a+b{x}^{3/2}}}{{b}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.99922, size = 41, normalized size = 1.02 \begin{align*} \frac{{\left (b x^{\frac{3}{2}} + a\right )}^{\frac{4}{3}}}{2 \, b^{2}} - \frac{2 \,{\left (b x^{\frac{3}{2}} + a\right )}^{\frac{1}{3}} a}{b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 3.52201, size = 66, normalized size = 1.65 \begin{align*} \frac{{\left (b x^{\frac{3}{2}} + a\right )}^{\frac{1}{3}}{\left (b x^{\frac{3}{2}} - 3 \, a\right )}}{2 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 2.09262, size = 49, normalized size = 1.22 \begin{align*} \begin{cases} - \frac{3 a \sqrt [3]{a + b x^{\frac{3}{2}}}}{2 b^{2}} + \frac{x^{\frac{3}{2}} \sqrt [3]{a + b x^{\frac{3}{2}}}}{2 b} & \text{for}\: b \neq 0 \\\frac{x^{3}}{3 a^{\frac{2}{3}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.1171, size = 36, normalized size = 0.9 \begin{align*} \frac{{\left (b x^{\frac{3}{2}} + a\right )}^{\frac{4}{3}} - 4 \,{\left (b x^{\frac{3}{2}} + a\right )}^{\frac{1}{3}} a}{2 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]