Optimal. Leaf size=147 \[ \frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt [6]{2} \sqrt{3} \left (\sqrt [3]{b x^2-2}+\sqrt [3]{2}\right )}{\sqrt{b} x}\right )}{4\ 2^{5/6} \sqrt{3} d}-\frac{\sqrt{b} \tanh ^{-1}\left (\frac{\left (\sqrt [3]{b x^2-2}+\sqrt [3]{2}\right )^2}{3 \sqrt [6]{2} \sqrt{b} x}\right )}{12\ 2^{5/6} d}+\frac{\sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} x}{3 \sqrt{2}}\right )}{12\ 2^{5/6} d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0239209, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.038, Rules used = {395} \[ \frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt [6]{2} \sqrt{3} \left (\sqrt [3]{b x^2-2}+\sqrt [3]{2}\right )}{\sqrt{b} x}\right )}{4\ 2^{5/6} \sqrt{3} d}-\frac{\sqrt{b} \tanh ^{-1}\left (\frac{\left (\sqrt [3]{b x^2-2}+\sqrt [3]{2}\right )^2}{3 \sqrt [6]{2} \sqrt{b} x}\right )}{12\ 2^{5/6} d}+\frac{\sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} x}{3 \sqrt{2}}\right )}{12\ 2^{5/6} d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 395
Rubi steps
\begin{align*} \int \frac{1}{\sqrt [3]{-2+b x^2} \left (-\frac{18 d}{b}+d x^2\right )} \, dx &=\frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt [6]{2} \sqrt{3} \left (\sqrt [3]{2}+\sqrt [3]{-2+b x^2}\right )}{\sqrt{b} x}\right )}{4\ 2^{5/6} \sqrt{3} d}+\frac{\sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} x}{3 \sqrt{2}}\right )}{12\ 2^{5/6} d}-\frac{\sqrt{b} \tanh ^{-1}\left (\frac{\left (\sqrt [3]{2}+\sqrt [3]{-2+b x^2}\right )^2}{3 \sqrt [6]{2} \sqrt{b} x}\right )}{12\ 2^{5/6} d}\\ \end{align*}
Mathematica [C] time = 0.162218, size = 148, normalized size = 1.01 \[ \frac{27 b x F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};\frac{b x^2}{2},\frac{b x^2}{18}\right )}{d \left (b x^2-18\right ) \sqrt [3]{b x^2-2} \left (b x^2 \left (F_1\left (\frac{3}{2};\frac{1}{3},2;\frac{5}{2};\frac{b x^2}{2},\frac{b x^2}{18}\right )+3 F_1\left (\frac{3}{2};\frac{4}{3},1;\frac{5}{2};\frac{b x^2}{2},\frac{b x^2}{18}\right )\right )+27 F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};\frac{b x^2}{2},\frac{b x^2}{18}\right )\right )} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.043, size = 0, normalized size = 0. \begin{align*} \int{{\frac{1}{\sqrt [3]{b{x}^{2}-2}}} \left ( -18\,{\frac{d}{b}}+d{x}^{2} \right ) ^{-1}}\, 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{1}{{\left (b x^{2} - 2\right )}^{\frac{1}{3}}{\left (d x^{2} - \frac{18 \, d}{b}\right )}}\,{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{b \int \frac{1}{b x^{2} \sqrt [3]{b x^{2} - 2} - 18 \sqrt [3]{b x^{2} - 2}}\, 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{1}{{\left (b x^{2} - 2\right )}^{\frac{1}{3}}{\left (d x^{2} - \frac{18 \, d}{b}\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]