Optimal. Leaf size=156 \[ -\frac{\sqrt [3]{a+b x^3}}{a d x}+\frac{\sqrt [3]{b} \log \left (a d-b d x^3\right )}{3\ 2^{2/3} a d}-\frac{\sqrt [3]{b} \log \left (\sqrt [3]{2} \sqrt [3]{b} x-\sqrt [3]{a+b x^3}\right )}{2^{2/3} a d}-\frac{\sqrt [3]{2} \sqrt [3]{b} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{2} \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{\sqrt{3} a d} \]
[Out]
________________________________________________________________________________________
Rubi [C] time = 0.0700063, antiderivative size = 77, normalized size of antiderivative = 0.49, number of steps used = 2, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {511, 510} \[ -\frac{\sqrt [3]{a+b x^3} \sqrt [3]{1-\frac{b x^3}{a}} \, _2F_1\left (-\frac{1}{3},-\frac{1}{3};\frac{2}{3};-\frac{2 b x^3}{a-b x^3}\right )}{a d x \sqrt [3]{\frac{b x^3}{a}+1}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
Rule 511
Rule 510
Rubi steps
\begin{align*} \int \frac{\sqrt [3]{a+b x^3}}{x^2 \left (a d-b d x^3\right )} \, dx &=\frac{\sqrt [3]{a+b x^3} \int \frac{\sqrt [3]{1+\frac{b x^3}{a}}}{x^2 \left (a d-b d x^3\right )} \, dx}{\sqrt [3]{1+\frac{b x^3}{a}}}\\ &=-\frac{\sqrt [3]{a+b x^3} \sqrt [3]{1-\frac{b x^3}{a}} \, _2F_1\left (-\frac{1}{3},-\frac{1}{3};\frac{2}{3};-\frac{2 b x^3}{a-b x^3}\right )}{a d x \sqrt [3]{1+\frac{b x^3}{a}}}\\ \end{align*}
Mathematica [C] time = 0.0166263, size = 45, normalized size = 0.29 \[ -\frac{\sqrt [3]{a+b x^3} \, _2F_1\left (-\frac{1}{3},1;\frac{2}{3};\frac{2 b x^3}{b x^3+a}\right )}{a d x} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.046, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2} \left ( -bd{x}^{3}+ad \right ) }\sqrt [3]{b{x}^{3}+a}}\, 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^{3} + a\right )}^{\frac{1}{3}}}{{\left (b d x^{3} - a d\right )} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} - \frac{\int \frac{\sqrt [3]{a + b x^{3}}}{- a x^{2} + b x^{5}}\, dx}{d} \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^{3} + a\right )}^{\frac{1}{3}}}{{\left (b d x^{3} - a d\right )} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]