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

Optimal. Leaf size=188 \[ \frac{10 a^{11/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{77 b^{9/4} \sqrt{a x+b \sqrt [3]{x}}}+\frac{20 a^2 \sqrt{a x+b \sqrt [3]{x}}}{77 b^2 x^{2/3}}-\frac{12 a \sqrt{a x+b \sqrt [3]{x}}}{77 b x^{4/3}}-\frac{6 \sqrt{a x+b \sqrt [3]{x}}}{11 x^2} \]

[Out]

(-6*Sqrt[b*x^(1/3) + a*x])/(11*x^2) - (12*a*Sqrt[b*x^(1/3) + a*x])/(77*b*x^(4/3)
) + (20*a^2*Sqrt[b*x^(1/3) + a*x])/(77*b^2*x^(2/3)) + (10*a^(11/4)*(Sqrt[b] + Sq
rt[a]*x^(1/3))*Sqrt[(b + a*x^(2/3))/(Sqrt[b] + Sqrt[a]*x^(1/3))^2]*x^(1/6)*Ellip
ticF[2*ArcTan[(a^(1/4)*x^(1/6))/b^(1/4)], 1/2])/(77*b^(9/4)*Sqrt[b*x^(1/3) + a*x
])

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Rubi [A]  time = 0.470098, antiderivative size = 188, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316 \[ \frac{10 a^{11/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{77 b^{9/4} \sqrt{a x+b \sqrt [3]{x}}}+\frac{20 a^2 \sqrt{a x+b \sqrt [3]{x}}}{77 b^2 x^{2/3}}-\frac{12 a \sqrt{a x+b \sqrt [3]{x}}}{77 b x^{4/3}}-\frac{6 \sqrt{a x+b \sqrt [3]{x}}}{11 x^2} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[b*x^(1/3) + a*x]/x^3,x]

[Out]

(-6*Sqrt[b*x^(1/3) + a*x])/(11*x^2) - (12*a*Sqrt[b*x^(1/3) + a*x])/(77*b*x^(4/3)
) + (20*a^2*Sqrt[b*x^(1/3) + a*x])/(77*b^2*x^(2/3)) + (10*a^(11/4)*(Sqrt[b] + Sq
rt[a]*x^(1/3))*Sqrt[(b + a*x^(2/3))/(Sqrt[b] + Sqrt[a]*x^(1/3))^2]*x^(1/6)*Ellip
ticF[2*ArcTan[(a^(1/4)*x^(1/6))/b^(1/4)], 1/2])/(77*b^(9/4)*Sqrt[b*x^(1/3) + a*x
])

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Rubi in Sympy [A]  time = 40.0056, size = 180, normalized size = 0.96 \[ \frac{10 a^{\frac{11}{4}} \sqrt{\frac{a x^{\frac{2}{3}} + b}{\left (\sqrt{a} \sqrt [3]{x} + \sqrt{b}\right )^{2}}} \left (\sqrt{a} \sqrt [3]{x} + \sqrt{b}\right ) \sqrt{a x + b \sqrt [3]{x}} F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}} \right )}\middle | \frac{1}{2}\right )}{77 b^{\frac{9}{4}} \sqrt [6]{x} \left (a x^{\frac{2}{3}} + b\right )} + \frac{20 a^{2} \sqrt{a x + b \sqrt [3]{x}}}{77 b^{2} x^{\frac{2}{3}}} - \frac{12 a \sqrt{a x + b \sqrt [3]{x}}}{77 b x^{\frac{4}{3}}} - \frac{6 \sqrt{a x + b \sqrt [3]{x}}}{11 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**(1/3)+a*x)**(1/2)/x**3,x)

[Out]

10*a**(11/4)*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)/(77*b**(9/4)*x**(1/6)*(a*x**(2/3) + b)) + 20*a**2*sqrt(a*x + b*x**(1
/3))/(77*b**2*x**(2/3)) - 12*a*sqrt(a*x + b*x**(1/3))/(77*b*x**(4/3)) - 6*sqrt(a
*x + b*x**(1/3))/(11*x**2)

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Mathematica [C]  time = 0.0794998, size = 108, normalized size = 0.57 \[ \frac{-20 a^3 x^2 \sqrt{\frac{b}{a x^{2/3}}+1} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{b}{a x^{2/3}}\right )+20 a^3 x^2+8 a^2 b x^{4/3}-54 a b^2 x^{2/3}-42 b^3}{77 b^2 x^{5/3} \sqrt{a x+b \sqrt [3]{x}}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[b*x^(1/3) + a*x]/x^3,x]

[Out]

(-42*b^3 - 54*a*b^2*x^(2/3) + 8*a^2*b*x^(4/3) + 20*a^3*x^2 - 20*a^3*Sqrt[1 + b/(
a*x^(2/3))]*x^2*Hypergeometric2F1[1/4, 1/2, 5/4, -(b/(a*x^(2/3)))])/(77*b^2*x^(5
/3)*Sqrt[b*x^(1/3) + a*x])

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Maple [A]  time = 0.034, size = 179, normalized size = 1. \[ -{\frac{6}{11\,{x}^{2}}\sqrt{b\sqrt [3]{x}+ax}}-{\frac{12\,a}{77\,b}\sqrt{b\sqrt [3]{x}+ax}{x}^{-{\frac{4}{3}}}}+{\frac{20\,{a}^{2}}{77\,{b}^{2}}\sqrt{b\sqrt [3]{x}+ax}{x}^{-{\frac{2}{3}}}}+{\frac{10\,{a}^{2}}{77\,{b}^{2}}\sqrt{-ab}\sqrt{{a \left ( \sqrt [3]{x}+{\frac{1}{a}\sqrt{-ab}} \right ){\frac{1}{\sqrt{-ab}}}}}\sqrt{-2\,{\frac{a}{\sqrt{-ab}} \left ( \sqrt [3]{x}-{\frac{\sqrt{-ab}}{a}} \right ) }}\sqrt{-{a\sqrt [3]{x}{\frac{1}{\sqrt{-ab}}}}}{\it EllipticF} \left ( \sqrt{{a \left ( \sqrt [3]{x}+{\frac{1}{a}\sqrt{-ab}} \right ){\frac{1}{\sqrt{-ab}}}}},{\frac{\sqrt{2}}{2}} \right ){\frac{1}{\sqrt{b\sqrt [3]{x}+ax}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^(1/3)+a*x)^(1/2)/x^3,x)

[Out]

-6/11*(b*x^(1/3)+a*x)^(1/2)/x^2-12/77*a*(b*x^(1/3)+a*x)^(1/2)/b/x^(4/3)+20/77*a^
2*(b*x^(1/3)+a*x)^(1/2)/b^2/x^(2/3)+10/77*a^2/b^2*(-a*b)^(1/2)*((x^(1/3)+(-a*b)^
(1/2)/a)/(-a*b)^(1/2)*a)^(1/2)*(-2*(x^(1/3)-(-a*b)^(1/2)/a)/(-a*b)^(1/2)*a)^(1/2
)*(-x^(1/3)/(-a*b)^(1/2)*a)^(1/2)/(b*x^(1/3)+a*x)^(1/2)*EllipticF(((x^(1/3)+(-a*
b)^(1/2)/a)/(-a*b)^(1/2)*a)^(1/2),1/2*2^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{a x + b x^{\frac{1}{3}}}}{x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(a*x + b*x^(1/3))/x^3,x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{a x + b x^{\frac{1}{3}}}}{x^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(a*x + b*x^(1/3))/x^3,x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{a x + b \sqrt [3]{x}}}{x^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**(1/3)+a*x)**(1/2)/x**3,x)

[Out]

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

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(a*x + b*x^(1/3))/x^3,x, algorithm="giac")

[Out]

Exception raised: RuntimeError