Optimal. Leaf size=294 \[ \frac{3 \sqrt [4]{a} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{b^{3/4} \sqrt{a x+b \sqrt [3]{x}}}-\frac{6 \sqrt [4]{a} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{b^{3/4} \sqrt{a x+b \sqrt [3]{x}}}+\frac{6 \sqrt{a} \sqrt [3]{x} \left (a x^{2/3}+b\right )}{b \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{a x+b \sqrt [3]{x}}}-\frac{6 \sqrt{a x+b \sqrt [3]{x}}}{b \sqrt [3]{x}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.594521, antiderivative size = 294, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.368 \[ \frac{3 \sqrt [4]{a} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{b^{3/4} \sqrt{a x+b \sqrt [3]{x}}}-\frac{6 \sqrt [4]{a} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{b^{3/4} \sqrt{a x+b \sqrt [3]{x}}}+\frac{6 \sqrt{a} \sqrt [3]{x} \left (a x^{2/3}+b\right )}{b \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{a x+b \sqrt [3]{x}}}-\frac{6 \sqrt{a x+b \sqrt [3]{x}}}{b \sqrt [3]{x}} \]
Antiderivative was successfully verified.
[In] Int[1/(x*Sqrt[b*x^(1/3) + a*x]),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 50.3053, size = 269, normalized size = 0.91 \[ - \frac{6 \sqrt [4]{a} \sqrt{\frac{a x^{\frac{2}{3}} + b}{\left (\sqrt{a} \sqrt [3]{x} + \sqrt{b}\right )^{2}}} \left (\sqrt{a} \sqrt [3]{x} + \sqrt{b}\right ) \sqrt{a x + b \sqrt [3]{x}} E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}} \right )}\middle | \frac{1}{2}\right )}{b^{\frac{3}{4}} \sqrt [6]{x} \left (a x^{\frac{2}{3}} + b\right )} + \frac{3 \sqrt [4]{a} \sqrt{\frac{a x^{\frac{2}{3}} + b}{\left (\sqrt{a} \sqrt [3]{x} + \sqrt{b}\right )^{2}}} \left (\sqrt{a} \sqrt [3]{x} + \sqrt{b}\right ) \sqrt{a x + b \sqrt [3]{x}} F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}} \right )}\middle | \frac{1}{2}\right )}{b^{\frac{3}{4}} \sqrt [6]{x} \left (a x^{\frac{2}{3}} + b\right )} + \frac{6 \sqrt{a} \sqrt{a x + b \sqrt [3]{x}}}{b \left (\sqrt{a} \sqrt [3]{x} + \sqrt{b}\right )} - \frac{6 \sqrt{a x + b \sqrt [3]{x}}}{b \sqrt [3]{x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x/(b*x**(1/3)+a*x)**(1/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [C] time = 0.0658071, size = 74, normalized size = 0.25 \[ -\frac{6 \left (a x^{2/3} \left (-\sqrt{\frac{b}{a x^{2/3}}+1}\right ) \, _2F_1\left (-\frac{1}{4},\frac{1}{2};\frac{3}{4};-\frac{b}{a x^{2/3}}\right )+a x^{2/3}+b\right )}{b \sqrt{a x+b \sqrt [3]{x}}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x*Sqrt[b*x^(1/3) + a*x]),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.03, size = 254, normalized size = 0.9 \[ 3\,{\frac{1}{\sqrt [3]{x} \left ( b+a{x}^{2/3} \right ) b} \left ( 2\,\sqrt{\sqrt [3]{x} \left ( b+a{x}^{2/3} \right ) }\sqrt{{\frac{a\sqrt [3]{x}+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-2\,{\frac{a\sqrt [3]{x}-\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{a\sqrt [3]{x}}{\sqrt{-ab}}}}{\it EllipticE} \left ( \sqrt{{\frac{a\sqrt [3]{x}+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) b-\sqrt{\sqrt [3]{x} \left ( b+a{x}^{2/3} \right ) }\sqrt{{\frac{a\sqrt [3]{x}+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-2\,{\frac{a\sqrt [3]{x}-\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{a\sqrt [3]{x}}{\sqrt{-ab}}}}{\it EllipticF} \left ( \sqrt{{\frac{a\sqrt [3]{x}+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) b-2\,{x}^{2/3}\sqrt{b\sqrt [3]{x}+ax}a-2\,\sqrt{b\sqrt [3]{x}+ax}b \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x/(b*x^(1/3)+a*x)^(1/2),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{a x + b x^{\frac{1}{3}}} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(a*x + b*x^(1/3))*x),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{a x + b x^{\frac{1}{3}}} x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(a*x + b*x^(1/3))*x),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x \sqrt{a x + b \sqrt [3]{x}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x/(b*x**(1/3)+a*x)**(1/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{a x + b x^{\frac{1}{3}}} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(a*x + b*x^(1/3))*x),x, algorithm="giac")
[Out]