Optimal. Leaf size=36 \[ -\frac {\left (a+b x^3\right )^{5/3} \, _2F_1\left (1,\frac {4}{3};\frac {2}{3};-\frac {b x^3}{a}\right )}{a x} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 49, normalized size of antiderivative = 1.36, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {365, 364} \[ -\frac {\left (a+b x^3\right )^{2/3} \, _2F_1\left (-\frac {2}{3},-\frac {1}{3};\frac {2}{3};-\frac {b x^3}{a}\right )}{x \left (\frac {b x^3}{a}+1\right )^{2/3}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 364
Rule 365
Rubi steps
\begin {align*} \int \frac {\left (a+b x^3\right )^{2/3}}{x^2} \, dx &=\frac {\left (a+b x^3\right )^{2/3} \int \frac {\left (1+\frac {b x^3}{a}\right )^{2/3}}{x^2} \, dx}{\left (1+\frac {b x^3}{a}\right )^{2/3}}\\ &=-\frac {\left (a+b x^3\right )^{2/3} \, _2F_1\left (-\frac {2}{3},-\frac {1}{3};\frac {2}{3};-\frac {b x^3}{a}\right )}{x \left (1+\frac {b x^3}{a}\right )^{2/3}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 49, normalized size = 1.36 \[ -\frac {\left (a+b x^3\right )^{2/3} \, _2F_1\left (-\frac {2}{3},-\frac {1}{3};\frac {2}{3};-\frac {b x^3}{a}\right )}{x \left (\frac {b x^3}{a}+1\right )^{2/3}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.71, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.11, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.26, size = 39, normalized size = 1.08 \[ \frac {{\left (b\,x^3+a\right )}^{2/3}\,{{}}_2{\mathrm {F}}_1\left (-\frac {2}{3},-\frac {1}{3};\ \frac {2}{3};\ -\frac {a}{b\,x^3}\right )}{x\,{\left (\frac {a}{b\,x^3}+1\right )}^{2/3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [C] time = 1.56, size = 41, normalized size = 1.14 \[ \frac {a^{\frac {2}{3}} \Gamma \left (- \frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, - \frac {1}{3} \\ \frac {2}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 x \Gamma \left (\frac {2}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________