Optimal. Leaf size=82 \[ \frac{x \left (\frac{b x^3}{a}+1\right )^{2/3} (2 b c-a d) \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{4}{3};-\frac{b x^3}{a}\right )}{2 b \left (a+b x^3\right )^{2/3}}+\frac{d x \sqrt [3]{a+b x^3}}{2 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0219141, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {388, 246, 245} \[ \frac{x \left (\frac{b x^3}{a}+1\right )^{2/3} (2 b c-a d) \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{4}{3};-\frac{b x^3}{a}\right )}{2 b \left (a+b x^3\right )^{2/3}}+\frac{d x \sqrt [3]{a+b x^3}}{2 b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 388
Rule 246
Rule 245
Rubi steps
\begin{align*} \int \frac{c+d x^3}{\left (a+b x^3\right )^{2/3}} \, dx &=\frac{d x \sqrt [3]{a+b x^3}}{2 b}-\frac{(-2 b c+a d) \int \frac{1}{\left (a+b x^3\right )^{2/3}} \, dx}{2 b}\\ &=\frac{d x \sqrt [3]{a+b x^3}}{2 b}-\frac{\left ((-2 b c+a d) \left (1+\frac{b x^3}{a}\right )^{2/3}\right ) \int \frac{1}{\left (1+\frac{b x^3}{a}\right )^{2/3}} \, dx}{2 b \left (a+b x^3\right )^{2/3}}\\ &=\frac{d x \sqrt [3]{a+b x^3}}{2 b}+\frac{(2 b c-a d) x \left (1+\frac{b x^3}{a}\right )^{2/3} \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{4}{3};-\frac{b x^3}{a}\right )}{2 b \left (a+b x^3\right )^{2/3}}\\ \end{align*}
Mathematica [A] time = 0.0375045, size = 73, normalized size = 0.89 \[ \frac{x \left (\frac{b x^3}{a}+1\right )^{2/3} (2 b c-a d) \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{4}{3};-\frac{b x^3}{a}\right )+d x \left (a+b x^3\right )}{2 b \left (a+b x^3\right )^{2/3}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.212, size = 0, normalized size = 0. \begin{align*} \int{(d{x}^{3}+c) \left ( b{x}^{3}+a \right ) ^{-{\frac{2}{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{d x^{3} + c}{{\left (b x^{3} + a\right )}^{\frac{2}{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{d x^{3} + c}{{\left (b x^{3} + a\right )}^{\frac{2}{3}}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] time = 1.73721, size = 78, normalized size = 0.95 \begin{align*} \frac{c x \Gamma \left (\frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{3}, \frac{2}{3} \\ \frac{4}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac{2}{3}} \Gamma \left (\frac{4}{3}\right )} + \frac{d x^{4} \Gamma \left (\frac{4}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{2}{3}, \frac{4}{3} \\ \frac{7}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac{2}{3}} \Gamma \left (\frac{7}{3}\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{d x^{3} + c}{{\left (b x^{3} + a\right )}^{\frac{2}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]