Optimal. Leaf size=123 \[ \frac{2 b^{3/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right ),\frac{1}{2}\right )}{\sqrt [4]{a} \sqrt{a x+b \sqrt [3]{x}}}+2 \sqrt{a x+b \sqrt [3]{x}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.146223, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {2018, 2021, 2011, 329, 220} \[ \frac{2 b^{3/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{\sqrt [4]{a} \sqrt{a x+b \sqrt [3]{x}}}+2 \sqrt{a x+b \sqrt [3]{x}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2018
Rule 2021
Rule 2011
Rule 329
Rule 220
Rubi steps
\begin{align*} \int \frac{\sqrt{b \sqrt [3]{x}+a x}}{x} \, dx &=3 \operatorname{Subst}\left (\int \frac{\sqrt{b x+a x^3}}{x} \, dx,x,\sqrt [3]{x}\right )\\ &=2 \sqrt{b \sqrt [3]{x}+a x}+(2 b) \operatorname{Subst}\left (\int \frac{1}{\sqrt{b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )\\ &=2 \sqrt{b \sqrt [3]{x}+a x}+\frac{\left (2 b \sqrt{b+a x^{2/3}} \sqrt [6]{x}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} \sqrt{b+a x^2}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt{b \sqrt [3]{x}+a x}}\\ &=2 \sqrt{b \sqrt [3]{x}+a x}+\frac{\left (4 b \sqrt{b+a x^{2/3}} \sqrt [6]{x}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{b+a x^4}} \, dx,x,\sqrt [6]{x}\right )}{\sqrt{b \sqrt [3]{x}+a x}}\\ &=2 \sqrt{b \sqrt [3]{x}+a x}+\frac{2 b^{3/4} \left (\sqrt{b}+\sqrt{a} \sqrt [3]{x}\right ) \sqrt{\frac{b+a x^{2/3}}{\left (\sqrt{b}+\sqrt{a} \sqrt [3]{x}\right )^2}} \sqrt [6]{x} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{\sqrt [4]{a} \sqrt{b \sqrt [3]{x}+a x}}\\ \end{align*}
Mathematica [C] time = 0.0439971, size = 54, normalized size = 0.44 \[ \frac{6 \sqrt{a x+b \sqrt [3]{x}} \, _2F_1\left (-\frac{1}{2},\frac{1}{4};\frac{5}{4};-\frac{a x^{2/3}}{b}\right )}{\sqrt{\frac{a x^{2/3}}{b}+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.012, size = 132, normalized size = 1.1 \begin{align*} 2\,\sqrt{b\sqrt [3]{x}+ax}+2\,{\frac{b\sqrt{-ab}}{a\sqrt{b\sqrt [3]{x}+ax}}\sqrt{{\frac{a}{\sqrt{-ab}} \left ( \sqrt [3]{x}+{\frac{\sqrt{-ab}}{a}} \right ) }}\sqrt{-2\,{\frac{a}{\sqrt{-ab}} \left ( \sqrt [3]{x}-{\frac{\sqrt{-ab}}{a}} \right ) }}\sqrt{-{\frac{a\sqrt [3]{x}}{\sqrt{-ab}}}}{\it EllipticF} \left ( \sqrt{{\frac{a}{\sqrt{-ab}} \left ( \sqrt [3]{x}+{\frac{\sqrt{-ab}}{a}} \right ) }},1/2\,\sqrt{2} \right ) } \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{\sqrt{a x + b x^{\frac{1}{3}}}}{x}\,{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{\sqrt{a x + b x^{\frac{1}{3}}}}{x}, x\right ) \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{\sqrt{a x + b \sqrt [3]{x}}}{x}\, 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{\sqrt{a x + b x^{\frac{1}{3}}}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]