Optimal. Leaf size=65 \[ -\frac{2\ 2^{2/3} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{2} \sqrt [3]{b} x\right )}{\sqrt{a-b x^3}}\right )}{\sqrt{3} \sqrt [6]{a} \sqrt [3]{b}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.199191, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 55, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.036, Rules used = {2137, 203} \[ -\frac{2\ 2^{2/3} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{2} \sqrt [3]{b} x\right )}{\sqrt{a-b x^3}}\right )}{\sqrt{3} \sqrt [6]{a} \sqrt [3]{b}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2137
Rule 203
Rubi steps
\begin{align*} \int \frac{2^{2/3} \sqrt [3]{a}+2 \sqrt [3]{b} x}{\left (2^{2/3} \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt{a-b x^3}} \, dx &=-\frac{\left (2\ 2^{2/3} \sqrt [3]{a}\right ) \operatorname{Subst}\left (\int \frac{1}{1+3 a x^2} \, dx,x,\frac{1-\frac{\sqrt [3]{2} \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{a-b x^3}}\right )}{\sqrt [3]{b}}\\ &=-\frac{2\ 2^{2/3} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{2} \sqrt [3]{b} x\right )}{\sqrt{a-b x^3}}\right )}{\sqrt{3} \sqrt [6]{a} \sqrt [3]{b}}\\ \end{align*}
Mathematica [C] time = 1.11903, size = 336, normalized size = 5.17 \[ \frac{2 \sqrt{\frac{\sqrt [3]{a}-\sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \left (-\frac{2 \left (\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{\sqrt [3]{-1} \left (\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} F\left (\sin ^{-1}\left (\sqrt{\frac{\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}}\right )|\sqrt [3]{-1}\right )}{\sqrt{\frac{\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}}}+\frac{\sqrt [3]{-1} 2^{2/3} \left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a} \sqrt{\frac{3 b^{2/3} x^2}{a^{2/3}}+\frac{3 \sqrt [3]{b} x}{\sqrt [3]{a}}+3} \Pi \left (\frac{i \sqrt{3}}{\sqrt [3]{-1}+2^{2/3}};\sin ^{-1}\left (\sqrt{\frac{\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}}\right )|\sqrt [3]{-1}\right )}{\sqrt [3]{-1}+2^{2/3}}\right )}{\sqrt [3]{b} \sqrt{a-b x^3}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.217, size = 0, normalized size = 0. \begin{align*} \int{ \left ({2}^{{\frac{2}{3}}}\sqrt [3]{a}+2\,\sqrt [3]{b}x \right ) \left ({2}^{{\frac{2}{3}}}\sqrt [3]{a}-\sqrt [3]{b}x \right ) ^{-1}{\frac{1}{\sqrt{-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{2 \, b^{\frac{1}{3}} x + 2^{\frac{2}{3}} a^{\frac{1}{3}}}{\sqrt{-b x^{3} + a}{\left (b^{\frac{1}{3}} x - 2^{\frac{2}{3}} a^{\frac{1}{3}}\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*} - \int \frac{2^{\frac{2}{3}} \sqrt [3]{a}}{- 2^{\frac{2}{3}} \sqrt [3]{a} \sqrt{a - b x^{3}} + \sqrt [3]{b} x \sqrt{a - b x^{3}}}\, dx - \int \frac{2 \sqrt [3]{b} x}{- 2^{\frac{2}{3}} \sqrt [3]{a} \sqrt{a - b x^{3}} + \sqrt [3]{b} x \sqrt{a - b x^{3}}}\, dx \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{2 \, b^{\frac{1}{3}} x + 2^{\frac{2}{3}} a^{\frac{1}{3}}}{\sqrt{-b x^{3} + a}{\left (b^{\frac{1}{3}} x - 2^{\frac{2}{3}} a^{\frac{1}{3}}\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]