Optimal. Leaf size=158 \[ -\frac{5 a^{3/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right ),\frac{1}{2}\right )}{2 b^{9/4} \sqrt{a x+b \sqrt [3]{x}}}-\frac{5 \sqrt{a x+b \sqrt [3]{x}}}{b^2 x^{2/3}}+\frac{3}{b \sqrt [3]{x} \sqrt{a x+b \sqrt [3]{x}}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.205232, antiderivative size = 158, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316, Rules used = {2018, 2023, 2025, 2011, 329, 220} \[ -\frac{5 a^{3/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{2 b^{9/4} \sqrt{a x+b \sqrt [3]{x}}}-\frac{5 \sqrt{a x+b \sqrt [3]{x}}}{b^2 x^{2/3}}+\frac{3}{b \sqrt [3]{x} \sqrt{a x+b \sqrt [3]{x}}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2018
Rule 2023
Rule 2025
Rule 2011
Rule 329
Rule 220
Rubi steps
\begin{align*} \int \frac{1}{x \left (b \sqrt [3]{x}+a x\right )^{3/2}} \, dx &=3 \operatorname{Subst}\left (\int \frac{1}{x \left (b x+a x^3\right )^{3/2}} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{3}{b \sqrt [3]{x} \sqrt{b \sqrt [3]{x}+a x}}+\frac{15 \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{2 b}\\ &=\frac{3}{b \sqrt [3]{x} \sqrt{b \sqrt [3]{x}+a x}}-\frac{5 \sqrt{b \sqrt [3]{x}+a x}}{b^2 x^{2/3}}-\frac{(5 a) \operatorname{Subst}\left (\int \frac{1}{\sqrt{b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{2 b^2}\\ &=\frac{3}{b \sqrt [3]{x} \sqrt{b \sqrt [3]{x}+a x}}-\frac{5 \sqrt{b \sqrt [3]{x}+a x}}{b^2 x^{2/3}}-\frac{\left (5 a \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 )}{2 b^2 \sqrt{b \sqrt [3]{x}+a x}}\\ &=\frac{3}{b \sqrt [3]{x} \sqrt{b \sqrt [3]{x}+a x}}-\frac{5 \sqrt{b \sqrt [3]{x}+a x}}{b^2 x^{2/3}}-\frac{\left (5 a \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 )}{b^2 \sqrt{b \sqrt [3]{x}+a x}}\\ &=\frac{3}{b \sqrt [3]{x} \sqrt{b \sqrt [3]{x}+a x}}-\frac{5 \sqrt{b \sqrt [3]{x}+a x}}{b^2 x^{2/3}}-\frac{5 a^{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 )}{2 b^{9/4} \sqrt{b \sqrt [3]{x}+a x}}\\ \end{align*}
Mathematica [C] time = 0.0597793, size = 62, normalized size = 0.39 \[ -\frac{2 \sqrt{\frac{a x^{2/3}}{b}+1} \, _2F_1\left (-\frac{3}{4},\frac{3}{2};\frac{1}{4};-\frac{a x^{2/3}}{b}\right )}{b \sqrt [3]{x} \sqrt{a x+b \sqrt [3]{x}}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.012, size = 181, normalized size = 1.2 \begin{align*} -{\frac{1}{2\,{b}^{2}x} \left ( 5\,\sqrt{{\frac{a\sqrt [3]{x}+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{2}\sqrt{{\frac{-a\sqrt [3]{x}+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{a\sqrt [3]{x}}{\sqrt{-ab}}}}{\it EllipticF} \left ( \sqrt{{\frac{a\sqrt [3]{x}+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) \sqrt{-ab}{x}^{2/3}\sqrt{\sqrt [3]{x} \left ( b+a{x}^{2/3} \right ) }+6\,\sqrt{b\sqrt [3]{x}+ax}xa+4\,\sqrt [3]{x}\sqrt{\sqrt [3]{x} \left ( b+a{x}^{2/3} \right ) }b+4\,\sqrt{\sqrt [3]{x} \left ( b+a{x}^{2/3} \right ) }xa \right ) \left ( b+a{x}^{{\frac{2}{3}}} \right ) ^{-1}} \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 (a x + b x^{\frac{1}{3}}\right )}^{\frac{3}{2}} 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{{\left (a^{4} x^{3} + 3 \, a^{2} b^{2} x^{\frac{5}{3}} - 2 \, a b^{3} x -{\left (2 \, a^{3} b x^{2} - b^{4}\right )} x^{\frac{1}{3}}\right )} \sqrt{a x + b x^{\frac{1}{3}}}}{a^{6} x^{6} + 2 \, a^{3} b^{3} x^{4} + b^{6} x^{2}}, 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{1}{x \left (a x + b \sqrt [3]{x}\right )^{\frac{3}{2}}}\, 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{1}{{\left (a x + b x^{\frac{1}{3}}\right )}^{\frac{3}{2}} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]