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

Optimal. Leaf size=213 \[ -\frac{6 b^{15/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{77 a^{13/4} \sqrt{a x+b \sqrt [3]{x}}}+\frac{12 b^3 \sqrt{a x+b \sqrt [3]{x}}}{77 a^3}-\frac{36 b^2 x^{2/3} \sqrt{a x+b \sqrt [3]{x}}}{385 a^2}+\frac{4 b x^{4/3} \sqrt{a x+b \sqrt [3]{x}}}{55 a}+\frac{2}{5} x^2 \sqrt{a x+b \sqrt [3]{x}} \]

[Out]

(12*b^3*Sqrt[b*x^(1/3) + a*x])/(77*a^3) - (36*b^2*x^(2/3)*Sqrt[b*x^(1/3) + a*x])
/(385*a^2) + (4*b*x^(4/3)*Sqrt[b*x^(1/3) + a*x])/(55*a) + (2*x^2*Sqrt[b*x^(1/3)
+ a*x])/5 - (6*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^(13/4)*Sqrt[b*x^(1/3) + a*x])

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Rubi [A]  time = 0.552869, antiderivative size = 213, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.353 \[ -\frac{6 b^{15/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{77 a^{13/4} \sqrt{a x+b \sqrt [3]{x}}}+\frac{12 b^3 \sqrt{a x+b \sqrt [3]{x}}}{77 a^3}-\frac{36 b^2 x^{2/3} \sqrt{a x+b \sqrt [3]{x}}}{385 a^2}+\frac{4 b x^{4/3} \sqrt{a x+b \sqrt [3]{x}}}{55 a}+\frac{2}{5} x^2 \sqrt{a x+b \sqrt [3]{x}} \]

Antiderivative was successfully verified.

[In]  Int[x*Sqrt[b*x^(1/3) + a*x],x]

[Out]

(12*b^3*Sqrt[b*x^(1/3) + a*x])/(77*a^3) - (36*b^2*x^(2/3)*Sqrt[b*x^(1/3) + a*x])
/(385*a^2) + (4*b*x^(4/3)*Sqrt[b*x^(1/3) + a*x])/(55*a) + (2*x^2*Sqrt[b*x^(1/3)
+ a*x])/5 - (6*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^(13/4)*Sqrt[b*x^(1/3) + a*x])

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Rubi in Sympy [A]  time = 47.4334, size = 204, normalized size = 0.96 \[ \frac{2 x^{2} \sqrt{a x + b \sqrt [3]{x}}}{5} + \frac{4 b x^{\frac{4}{3}} \sqrt{a x + b \sqrt [3]{x}}}{55 a} - \frac{36 b^{2} x^{\frac{2}{3}} \sqrt{a x + b \sqrt [3]{x}}}{385 a^{2}} + \frac{12 b^{3} \sqrt{a x + b \sqrt [3]{x}}}{77 a^{3}} - \frac{6 b^{\frac{15}{4}} \sqrt{\frac{a x^{\frac{2}{3}} + b}{\left (\sqrt{a} \sqrt [3]{x} + \sqrt{b}\right )^{2}}} \left (\sqrt{a} \sqrt [3]{x} + \sqrt{b}\right ) \sqrt{a x + b \sqrt [3]{x}} F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}} \right )}\middle | \frac{1}{2}\right )}{77 a^{\frac{13}{4}} \sqrt [6]{x} \left (a x^{\frac{2}{3}} + b\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*x**2*sqrt(a*x + b*x**(1/3))/5 + 4*b*x**(4/3)*sqrt(a*x + b*x**(1/3))/(55*a) - 3
6*b**2*x**(2/3)*sqrt(a*x + b*x**(1/3))/(385*a**2) + 12*b**3*sqrt(a*x + b*x**(1/3
))/(77*a**3) - 6*b**(15/4)*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*a**(13/4)*x**(1/6)*(a*x**(2/3) + b))

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Mathematica [C]  time = 0.0839891, size = 118, normalized size = 0.55 \[ \frac{2 \sqrt [3]{x} \left (77 a^4 x^{8/3}+91 a^3 b x^2-4 a^2 b^2 x^{4/3}+30 b^4 \sqrt{\frac{b}{a x^{2/3}}+1} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{b}{a x^{2/3}}\right )+12 a b^3 x^{2/3}+30 b^4\right )}{385 a^3 \sqrt{a x+b \sqrt [3]{x}}} \]

Antiderivative was successfully verified.

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

[Out]

(2*x^(1/3)*(30*b^4 + 12*a*b^3*x^(2/3) - 4*a^2*b^2*x^(4/3) + 91*a^3*b*x^2 + 77*a^
4*x^(8/3) + 30*b^4*Sqrt[1 + b/(a*x^(2/3))]*Hypergeometric2F1[1/4, 1/2, 5/4, -(b/
(a*x^(2/3)))]))/(385*a^3*Sqrt[b*x^(1/3) + a*x])

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Maple [A]  time = 0.023, size = 198, normalized size = 0.9 \[{\frac{2\,{x}^{2}}{5}\sqrt{b\sqrt [3]{x}+ax}}+{\frac{4\,b}{55\,a}{x}^{{\frac{4}{3}}}\sqrt{b\sqrt [3]{x}+ax}}-{\frac{36\,{b}^{2}}{385\,{a}^{2}}{x}^{{\frac{2}{3}}}\sqrt{b\sqrt [3]{x}+ax}}+{\frac{12\,{b}^{3}}{77\,{a}^{3}}\sqrt{b\sqrt [3]{x}+ax}}-{\frac{6\,{b}^{4}}{77\,{a}^{4}}\sqrt{-ab}\sqrt{{a \left ( \sqrt [3]{x}+{\frac{1}{a}\sqrt{-ab}} \right ){\frac{1}{\sqrt{-ab}}}}}\sqrt{-2\,{\frac{a}{\sqrt{-ab}} \left ( \sqrt [3]{x}-{\frac{\sqrt{-ab}}{a}} \right ) }}\sqrt{-{a\sqrt [3]{x}{\frac{1}{\sqrt{-ab}}}}}{\it EllipticF} \left ( \sqrt{{a \left ( \sqrt [3]{x}+{\frac{1}{a}\sqrt{-ab}} \right ){\frac{1}{\sqrt{-ab}}}}},{\frac{\sqrt{2}}{2}} \right ){\frac{1}{\sqrt{b\sqrt [3]{x}+ax}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(b*x^(1/3)+a*x)^(1/2),x)

[Out]

2/5*x^2*(b*x^(1/3)+a*x)^(1/2)+4/55*b*x^(4/3)*(b*x^(1/3)+a*x)^(1/2)/a-36/385*b^2*
x^(2/3)*(b*x^(1/3)+a*x)^(1/2)/a^2+12/77*b^3*(b*x^(1/3)+a*x)^(1/2)/a^3-6/77/a^4*b
^4*(-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. \[ \int \sqrt{a x + b x^{\frac{1}{3}}} x\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(a*x + b*x^(1/3))*x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(sqrt(a*x + b*x^(1/3))*x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x*sqrt(a*x + b*x**(1/3)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(a*x + b*x^(1/3))*x, x)