3.160 \(\int \frac{x^2}{\left (b \sqrt [3]{x}+a x\right )^{3/2}} \, dx\)

Optimal. Leaf size=349 \[ \frac{77 b^{9/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{10 a^{15/4} \sqrt{a x+b \sqrt [3]{x}}}-\frac{77 b^{9/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{5 a^{15/4} \sqrt{a x+b \sqrt [3]{x}}}+\frac{77 b^2 \sqrt [3]{x} \left (a x^{2/3}+b\right )}{5 a^{7/2} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{a x+b \sqrt [3]{x}}}-\frac{77 b \sqrt [3]{x} \sqrt{a x+b \sqrt [3]{x}}}{15 a^3}+\frac{11 x \sqrt{a x+b \sqrt [3]{x}}}{3 a^2}-\frac{3 x^2}{a \sqrt{a x+b \sqrt [3]{x}}} \]

[Out]

(77*b^2*(b + a*x^(2/3))*x^(1/3))/(5*a^(7/2)*(Sqrt[b] + Sqrt[a]*x^(1/3))*Sqrt[b*x
^(1/3) + a*x]) - (3*x^2)/(a*Sqrt[b*x^(1/3) + a*x]) - (77*b*x^(1/3)*Sqrt[b*x^(1/3
) + a*x])/(15*a^3) + (11*x*Sqrt[b*x^(1/3) + a*x])/(3*a^2) - (77*b^(9/4)*(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])/(5*a^(15/4)*Sqrt[b*x^(1/3)
+ a*x]) + (77*b^(9/4)*(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
])/(10*a^(15/4)*Sqrt[b*x^(1/3) + a*x])

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Rubi [A]  time = 0.842271, antiderivative size = 349, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.421 \[ \frac{77 b^{9/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{10 a^{15/4} \sqrt{a x+b \sqrt [3]{x}}}-\frac{77 b^{9/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{5 a^{15/4} \sqrt{a x+b \sqrt [3]{x}}}+\frac{77 b^2 \sqrt [3]{x} \left (a x^{2/3}+b\right )}{5 a^{7/2} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{a x+b \sqrt [3]{x}}}-\frac{77 b \sqrt [3]{x} \sqrt{a x+b \sqrt [3]{x}}}{15 a^3}+\frac{11 x \sqrt{a x+b \sqrt [3]{x}}}{3 a^2}-\frac{3 x^2}{a \sqrt{a x+b \sqrt [3]{x}}} \]

Antiderivative was successfully verified.

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

[Out]

(77*b^2*(b + a*x^(2/3))*x^(1/3))/(5*a^(7/2)*(Sqrt[b] + Sqrt[a]*x^(1/3))*Sqrt[b*x
^(1/3) + a*x]) - (3*x^2)/(a*Sqrt[b*x^(1/3) + a*x]) - (77*b*x^(1/3)*Sqrt[b*x^(1/3
) + a*x])/(15*a^3) + (11*x*Sqrt[b*x^(1/3) + a*x])/(3*a^2) - (77*b^(9/4)*(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])/(5*a^(15/4)*Sqrt[b*x^(1/3)
+ a*x]) + (77*b^(9/4)*(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
])/(10*a^(15/4)*Sqrt[b*x^(1/3) + a*x])

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Rubi in Sympy [A]  time = 72.2189, size = 323, normalized size = 0.93 \[ - \frac{3 x^{2}}{a \sqrt{a x + b \sqrt [3]{x}}} + \frac{11 x \sqrt{a x + b \sqrt [3]{x}}}{3 a^{2}} - \frac{77 b \sqrt [3]{x} \sqrt{a x + b \sqrt [3]{x}}}{15 a^{3}} + \frac{77 b^{2} \sqrt{a x + b \sqrt [3]{x}}}{5 a^{\frac{7}{2}} \left (\sqrt{a} \sqrt [3]{x} + \sqrt{b}\right )} - \frac{77 b^{\frac{9}{4}} \sqrt{\frac{a x^{\frac{2}{3}} + b}{\left (\sqrt{a} \sqrt [3]{x} + \sqrt{b}\right )^{2}}} \left (\sqrt{a} \sqrt [3]{x} + \sqrt{b}\right ) \sqrt{a x + b \sqrt [3]{x}} E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}} \right )}\middle | \frac{1}{2}\right )}{5 a^{\frac{15}{4}} \sqrt [6]{x} \left (a x^{\frac{2}{3}} + b\right )} + \frac{77 b^{\frac{9}{4}} \sqrt{\frac{a x^{\frac{2}{3}} + b}{\left (\sqrt{a} \sqrt [3]{x} + \sqrt{b}\right )^{2}}} \left (\sqrt{a} \sqrt [3]{x} + \sqrt{b}\right ) \sqrt{a x + b \sqrt [3]{x}} F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}} \right )}\middle | \frac{1}{2}\right )}{10 a^{\frac{15}{4}} \sqrt [6]{x} \left (a x^{\frac{2}{3}} + b\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*x**2/(a*sqrt(a*x + b*x**(1/3))) + 11*x*sqrt(a*x + b*x**(1/3))/(3*a**2) - 77*b
*x**(1/3)*sqrt(a*x + b*x**(1/3))/(15*a**3) + 77*b**2*sqrt(a*x + b*x**(1/3))/(5*a
**(7/2)*(sqrt(a)*x**(1/3) + sqrt(b))) - 77*b**(9/4)*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))*e
lliptic_e(2*atan(a**(1/4)*x**(1/6)/b**(1/4)), 1/2)/(5*a**(15/4)*x**(1/6)*(a*x**(
2/3) + b)) + 77*b**(9/4)*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)/(10*a**(15/4)*x**(1/6)*(a*x**(2/3) + b))

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Mathematica [C]  time = 0.0782301, size = 94, normalized size = 0.27 \[ \frac{x^{2/3} \left (10 a^2 x^{4/3}+231 b^2 \sqrt{\frac{b}{a x^{2/3}}+1} \, _2F_1\left (-\frac{1}{4},\frac{1}{2};\frac{3}{4};-\frac{b}{a x^{2/3}}\right )-22 a b x^{2/3}-77 b^2\right )}{15 a^3 \sqrt{a x+b \sqrt [3]{x}}} \]

Antiderivative was successfully verified.

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

[Out]

(x^(2/3)*(-77*b^2 - 22*a*b*x^(2/3) + 10*a^2*x^(4/3) + 231*b^2*Sqrt[1 + b/(a*x^(2
/3))]*Hypergeometric2F1[-1/4, 1/2, 3/4, -(b/(a*x^(2/3)))]))/(15*a^3*Sqrt[b*x^(1/
3) + a*x])

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Maple [A]  time = 0.017, size = 312, normalized size = 0.9 \[ -{\frac{1}{30\,{a}^{4}} \left ( -462\,{b}^{3}\sqrt{{\frac{a\sqrt [3]{x}+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-2\,{\frac{a\sqrt [3]{x}-\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{a\sqrt [3]{x}}{\sqrt{-ab}}}}\sqrt{\sqrt [3]{x} \left ( b+a{x}^{2/3} \right ) }{\it EllipticE} \left ( \sqrt{{\frac{a\sqrt [3]{x}+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) +231\,{b}^{3}\sqrt{{\frac{a\sqrt [3]{x}+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-2\,{\frac{a\sqrt [3]{x}-\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{a\sqrt [3]{x}}{\sqrt{-ab}}}}\sqrt{\sqrt [3]{x} \left ( b+a{x}^{2/3} \right ) }{\it EllipticF} \left ( \sqrt{{\frac{a\sqrt [3]{x}+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) +64\,\sqrt{\sqrt [3]{x} \left ( b+a{x}^{2/3} \right ) }{x}^{2/3}a{b}^{2}+44\,\sqrt{\sqrt [3]{x} \left ( b+a{x}^{2/3} \right ) }{x}^{4/3}{a}^{2}b-20\,\sqrt{\sqrt [3]{x} \left ( b+a{x}^{2/3} \right ) }{x}^{2}{a}^{3}+90\,\sqrt{b\sqrt [3]{x}+ax}{x}^{2/3}a{b}^{2} \right ){\frac{1}{\sqrt [3]{x}}} \left ( b+a{x}^{{\frac{2}{3}}} \right ) ^{-1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/30/a^4*(-462*b^3*((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)*(x^(1/3)*(b+a
*x^(2/3)))^(1/2)*EllipticE(((a*x^(1/3)+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2),1/2*2^(
1/2))+231*b^3*((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)*(x^(1/3)*(b+a*x^(2/
3)))^(1/2)*EllipticF(((a*x^(1/3)+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2),1/2*2^(1/2))+
64*(x^(1/3)*(b+a*x^(2/3)))^(1/2)*x^(2/3)*a*b^2+44*(x^(1/3)*(b+a*x^(2/3)))^(1/2)*
x^(4/3)*a^2*b-20*(x^(1/3)*(b+a*x^(2/3)))^(1/2)*x^2*a^3+90*(b*x^(1/3)+a*x)^(1/2)*
x^(2/3)*a*b^2)/x^(1/3)/(b+a*x^(2/3))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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