3.151 \(\int \frac{x^2}{\sqrt{b \sqrt [3]{x}+a x}} \, dx\)

Optimal. Leaf size=216 \[ \frac{39 b^{15/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{77 a^{17/4} \sqrt{a x+b \sqrt [3]{x}}}-\frac{78 b^3 \sqrt{a x+b \sqrt [3]{x}}}{77 a^4}+\frac{234 b^2 x^{2/3} \sqrt{a x+b \sqrt [3]{x}}}{385 a^3}-\frac{26 b x^{4/3} \sqrt{a x+b \sqrt [3]{x}}}{55 a^2}+\frac{2 x^2 \sqrt{a x+b \sqrt [3]{x}}}{5 a} \]

[Out]

(-78*b^3*Sqrt[b*x^(1/3) + a*x])/(77*a^4) + (234*b^2*x^(2/3)*Sqrt[b*x^(1/3) + a*x
])/(385*a^3) - (26*b*x^(4/3)*Sqrt[b*x^(1/3) + a*x])/(55*a^2) + (2*x^2*Sqrt[b*x^(
1/3) + a*x])/(5*a) + (39*b^(15/4)*(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])/(77*a^(17/4)*Sqrt[b*x^(1/3) + a*x])

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Rubi [A]  time = 0.587421, antiderivative size = 216, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263 \[ \frac{39 b^{15/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{77 a^{17/4} \sqrt{a x+b \sqrt [3]{x}}}-\frac{78 b^3 \sqrt{a x+b \sqrt [3]{x}}}{77 a^4}+\frac{234 b^2 x^{2/3} \sqrt{a x+b \sqrt [3]{x}}}{385 a^3}-\frac{26 b x^{4/3} \sqrt{a x+b \sqrt [3]{x}}}{55 a^2}+\frac{2 x^2 \sqrt{a x+b \sqrt [3]{x}}}{5 a} \]

Antiderivative was successfully verified.

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

[Out]

(-78*b^3*Sqrt[b*x^(1/3) + a*x])/(77*a^4) + (234*b^2*x^(2/3)*Sqrt[b*x^(1/3) + a*x
])/(385*a^3) - (26*b*x^(4/3)*Sqrt[b*x^(1/3) + a*x])/(55*a^2) + (2*x^2*Sqrt[b*x^(
1/3) + a*x])/(5*a) + (39*b^(15/4)*(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])/(77*a^(17/4)*Sqrt[b*x^(1/3) + a*x])

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Rubi in Sympy [A]  time = 49.2454, size = 207, normalized size = 0.96 \[ \frac{2 x^{2} \sqrt{a x + b \sqrt [3]{x}}}{5 a} - \frac{26 b x^{\frac{4}{3}} \sqrt{a x + b \sqrt [3]{x}}}{55 a^{2}} + \frac{234 b^{2} x^{\frac{2}{3}} \sqrt{a x + b \sqrt [3]{x}}}{385 a^{3}} - \frac{78 b^{3} \sqrt{a x + b \sqrt [3]{x}}}{77 a^{4}} + \frac{39 b^{\frac{15}{4}} \sqrt{\frac{a x^{\frac{2}{3}} + b}{\left (\sqrt{a} \sqrt [3]{x} + \sqrt{b}\right )^{2}}} \left (\sqrt{a} \sqrt [3]{x} + \sqrt{b}\right ) \sqrt{a x + b \sqrt [3]{x}} F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}} \right )}\middle | \frac{1}{2}\right )}{77 a^{\frac{17}{4}} \sqrt [6]{x} \left (a x^{\frac{2}{3}} + b\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*x**2*sqrt(a*x + b*x**(1/3))/(5*a) - 26*b*x**(4/3)*sqrt(a*x + b*x**(1/3))/(55*a
**2) + 234*b**2*x**(2/3)*sqrt(a*x + b*x**(1/3))/(385*a**3) - 78*b**3*sqrt(a*x +
b*x**(1/3))/(77*a**4) + 39*b**(15/4)*sqrt((a*x**(2/3) + b)/(sqrt(a)*x**(1/3) + s
qrt(b))**2)*(sqrt(a)*x**(1/3) + sqrt(b))*sqrt(a*x + b*x**(1/3))*elliptic_f(2*ata
n(a**(1/4)*x**(1/6)/b**(1/4)), 1/2)/(77*a**(17/4)*x**(1/6)*(a*x**(2/3) + b))

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Mathematica [C]  time = 0.0857378, size = 118, normalized size = 0.55 \[ \frac{2 \sqrt [3]{x} \left (77 a^4 x^{8/3}-14 a^3 b x^2+26 a^2 b^2 x^{4/3}-195 b^4 \sqrt{\frac{b}{a x^{2/3}}+1} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{b}{a x^{2/3}}\right )-78 a b^3 x^{2/3}-195 b^4\right )}{385 a^4 \sqrt{a x+b \sqrt [3]{x}}} \]

Antiderivative was successfully verified.

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

[Out]

(2*x^(1/3)*(-195*b^4 - 78*a*b^3*x^(2/3) + 26*a^2*b^2*x^(4/3) - 14*a^3*b*x^2 + 77
*a^4*x^(8/3) - 195*b^4*Sqrt[1 + b/(a*x^(2/3))]*Hypergeometric2F1[1/4, 1/2, 5/4,
-(b/(a*x^(2/3)))]))/(385*a^4*Sqrt[b*x^(1/3) + a*x])

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Maple [A]  time = 0.011, size = 163, normalized size = 0.8 \[{\frac{1}{385\,{a}^{5}} \left ( 52\,{a}^{3}{b}^{2}{x}^{5/3}-28\,{a}^{4}b{x}^{7/3}+195\,{b}^{4}\sqrt{-ab}\sqrt{{\frac{a\sqrt [3]{x}+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-2\,{\frac{a\sqrt [3]{x}-\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{a\sqrt [3]{x}}{\sqrt{-ab}}}}{\it EllipticF} \left ( \sqrt{{\frac{a\sqrt [3]{x}+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) -156\,{a}^{2}{b}^{3}x+154\,{x}^{3}{a}^{5}-390\,a{b}^{4}\sqrt [3]{x} \right ){\frac{1}{\sqrt{\sqrt [3]{x} \left ( b+a{x}^{{\frac{2}{3}}} \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/385*(52*a^3*b^2*x^(5/3)-28*a^4*b*x^(7/3)+195*b^4*(-a*b)^(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))-156*a^2*b^3*x+154*x^3*a^5-390*a*b^4*x^(1/3))/(x^(1/3)*(b+a*x
^(2/3)))^(1/2)/a^5

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{\sqrt{a x + b x^{\frac{1}{3}}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/sqrt(a*x + b*x^(1/3)),x, algorithm="maxima")

[Out]

integrate(x^2/sqrt(a*x + b*x^(1/3)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{2}}{\sqrt{a x + b x^{\frac{1}{3}}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/sqrt(a*x + b*x^(1/3)),x, algorithm="fricas")

[Out]

integral(x^2/sqrt(a*x + b*x^(1/3)), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{\sqrt{a x + b \sqrt [3]{x}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x**2/sqrt(a*x + b*x**(1/3)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{\sqrt{a x + b x^{\frac{1}{3}}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/sqrt(a*x + b*x^(1/3)),x, algorithm="giac")

[Out]

integrate(x^2/sqrt(a*x + b*x^(1/3)), x)