Optimal. Leaf size=57 \[ -\frac{3 a \sqrt [3]{a+b x^2} \, _2F_1\left (-\frac{4}{3},-\frac{1}{6};\frac{5}{6};-\frac{b x^2}{a}\right )}{c \sqrt [3]{c x} \sqrt [3]{\frac{b x^2}{a}+1}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0189494, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {365, 364} \[ -\frac{3 a \sqrt [3]{a+b x^2} \, _2F_1\left (-\frac{4}{3},-\frac{1}{6};\frac{5}{6};-\frac{b x^2}{a}\right )}{c \sqrt [3]{c x} \sqrt [3]{\frac{b x^2}{a}+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 365
Rule 364
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right )^{4/3}}{(c x)^{4/3}} \, dx &=\frac{\left (a \sqrt [3]{a+b x^2}\right ) \int \frac{\left (1+\frac{b x^2}{a}\right )^{4/3}}{(c x)^{4/3}} \, dx}{\sqrt [3]{1+\frac{b x^2}{a}}}\\ &=-\frac{3 a \sqrt [3]{a+b x^2} \, _2F_1\left (-\frac{4}{3},-\frac{1}{6};\frac{5}{6};-\frac{b x^2}{a}\right )}{c \sqrt [3]{c x} \sqrt [3]{1+\frac{b x^2}{a}}}\\ \end{align*}
Mathematica [A] time = 0.0126199, size = 55, normalized size = 0.96 \[ -\frac{3 a x \sqrt [3]{a+b x^2} \, _2F_1\left (-\frac{4}{3},-\frac{1}{6};\frac{5}{6};-\frac{b x^2}{a}\right )}{(c x)^{4/3} \sqrt [3]{\frac{b x^2}{a}+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.018, size = 0, normalized size = 0. \begin{align*} \int{ \left ( b{x}^{2}+a \right ) ^{{\frac{4}{3}}} \left ( cx \right ) ^{-{\frac{4}{3}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x^{2} + a\right )}^{\frac{4}{3}}}{\left (c x\right )^{\frac{4}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b x^{2} + a\right )}^{\frac{4}{3}} \left (c x\right )^{\frac{2}{3}}}{c^{2} x^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] time = 14.4061, size = 49, normalized size = 0.86 \begin{align*} \frac{a^{\frac{4}{3}} \Gamma \left (- \frac{1}{6}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{4}{3}, - \frac{1}{6} \\ \frac{5}{6} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 c^{\frac{4}{3}} \sqrt [3]{x} \Gamma \left (\frac{5}{6}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x^{2} + a\right )}^{\frac{4}{3}}}{\left (c x\right )^{\frac{4}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]