3.154 \(\int \frac{1}{x \sqrt{b \sqrt [3]{x}+a x}} \, dx\)

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]

(6*Sqrt[a]*(b + a*x^(2/3))*x^(1/3))/(b*(Sqrt[b] + Sqrt[a]*x^(1/3))*Sqrt[b*x^(1/3
) + a*x]) - (6*Sqrt[b*x^(1/3) + a*x])/(b*x^(1/3)) - (6*a^(1/4)*(Sqrt[b] + Sqrt[a
]*x^(1/3))*Sqrt[(b + a*x^(2/3))/(Sqrt[b] + Sqrt[a]*x^(1/3))^2]*x^(1/6)*EllipticE
[2*ArcTan[(a^(1/4)*x^(1/6))/b^(1/4)], 1/2])/(b^(3/4)*Sqrt[b*x^(1/3) + a*x]) + (3
*a^(1/4)*(Sqrt[b] + Sqrt[a]*x^(1/3))*Sqrt[(b + a*x^(2/3))/(Sqrt[b] + Sqrt[a]*x^(
1/3))^2]*x^(1/6)*EllipticF[2*ArcTan[(a^(1/4)*x^(1/6))/b^(1/4)], 1/2])/(b^(3/4)*S
qrt[b*x^(1/3) + a*x])

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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]

(6*Sqrt[a]*(b + a*x^(2/3))*x^(1/3))/(b*(Sqrt[b] + Sqrt[a]*x^(1/3))*Sqrt[b*x^(1/3
) + a*x]) - (6*Sqrt[b*x^(1/3) + a*x])/(b*x^(1/3)) - (6*a^(1/4)*(Sqrt[b] + Sqrt[a
]*x^(1/3))*Sqrt[(b + a*x^(2/3))/(Sqrt[b] + Sqrt[a]*x^(1/3))^2]*x^(1/6)*EllipticE
[2*ArcTan[(a^(1/4)*x^(1/6))/b^(1/4)], 1/2])/(b^(3/4)*Sqrt[b*x^(1/3) + a*x]) + (3
*a^(1/4)*(Sqrt[b] + Sqrt[a]*x^(1/3))*Sqrt[(b + a*x^(2/3))/(Sqrt[b] + Sqrt[a]*x^(
1/3))^2]*x^(1/6)*EllipticF[2*ArcTan[(a^(1/4)*x^(1/6))/b^(1/4)], 1/2])/(b^(3/4)*S
qrt[b*x^(1/3) + a*x])

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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]

-6*a**(1/4)*sqrt((a*x**(2/3) + b)/(sqrt(a)*x**(1/3) + sqrt(b))**2)*(sqrt(a)*x**(
1/3) + sqrt(b))*sqrt(a*x + b*x**(1/3))*elliptic_e(2*atan(a**(1/4)*x**(1/6)/b**(1
/4)), 1/2)/(b**(3/4)*x**(1/6)*(a*x**(2/3) + b)) + 3*a**(1/4)*sqrt((a*x**(2/3) +
b)/(sqrt(a)*x**(1/3) + sqrt(b))**2)*(sqrt(a)*x**(1/3) + sqrt(b))*sqrt(a*x + b*x*
*(1/3))*elliptic_f(2*atan(a**(1/4)*x**(1/6)/b**(1/4)), 1/2)/(b**(3/4)*x**(1/6)*(
a*x**(2/3) + b)) + 6*sqrt(a)*sqrt(a*x + b*x**(1/3))/(b*(sqrt(a)*x**(1/3) + sqrt(
b))) - 6*sqrt(a*x + b*x**(1/3))/(b*x**(1/3))

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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]

(-6*(b + a*x^(2/3) - a*Sqrt[1 + b/(a*x^(2/3))]*x^(2/3)*Hypergeometric2F1[-1/4, 1
/2, 3/4, -(b/(a*x^(2/3)))]))/(b*Sqrt[b*x^(1/3) + a*x])

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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]

3*(2*(x^(1/3)*(b+a*x^(2/3)))^(1/2)*((a*x^(1/3)+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2)
*(-2*(a*x^(1/3)-(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2)*(-x^(1/3)/(-a*b)^(1/2)*a)^(1/2
)*EllipticE(((a*x^(1/3)+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2),1/2*2^(1/2))*b-(x^(1/3
)*(b+a*x^(2/3)))^(1/2)*((a*x^(1/3)+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2)*(-2*(a*x^(1
/3)-(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2)*(-x^(1/3)/(-a*b)^(1/2)*a)^(1/2)*EllipticF(
((a*x^(1/3)+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2),1/2*2^(1/2))*b-2*x^(2/3)*(b*x^(1/3
)+a*x)^(1/2)*a-2*(b*x^(1/3)+a*x)^(1/2)*b)/x^(1/3)/(b+a*x^(2/3))/b

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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]

integrate(1/(sqrt(a*x + b*x^(1/3))*x), x)

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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]

integral(1/(sqrt(a*x + b*x^(1/3))*x), x)

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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]

Integral(1/(x*sqrt(a*x + b*x**(1/3))), x)

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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]

integrate(1/(sqrt(a*x + b*x^(1/3))*x), x)