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

Optimal. Leaf size=301 \[ \frac{442 b^{27/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{14421 a^{25/4} \sqrt{a x+b \sqrt [3]{x}}}-\frac{884 b^6 \sqrt{a x+b \sqrt [3]{x}}}{14421 a^6}+\frac{884 b^5 x^{2/3} \sqrt{a x+b \sqrt [3]{x}}}{24035 a^5}-\frac{6188 b^4 x^{4/3} \sqrt{a x+b \sqrt [3]{x}}}{216315 a^4}+\frac{476 b^3 x^2 \sqrt{a x+b \sqrt [3]{x}}}{19665 a^3}-\frac{28 b^2 x^{8/3} \sqrt{a x+b \sqrt [3]{x}}}{1311 a^2}+\frac{4 b x^{10/3} \sqrt{a x+b \sqrt [3]{x}}}{207 a}+\frac{2}{9} x^4 \sqrt{a x+b \sqrt [3]{x}} \]

[Out]

(-884*b^6*Sqrt[b*x^(1/3) + a*x])/(14421*a^6) + (884*b^5*x^(2/3)*Sqrt[b*x^(1/3) +
a*x])/(24035*a^5) - (6188*b^4*x^(4/3)*Sqrt[b*x^(1/3) + a*x])/(216315*a^4) + (47
6*b^3*x^2*Sqrt[b*x^(1/3) + a*x])/(19665*a^3) - (28*b^2*x^(8/3)*Sqrt[b*x^(1/3) +
a*x])/(1311*a^2) + (4*b*x^(10/3)*Sqrt[b*x^(1/3) + a*x])/(207*a) + (2*x^4*Sqrt[b*
x^(1/3) + a*x])/9 + (442*b^(27/4)*(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])/(14421*a^(25/4)*Sqrt[b*x^(1/3) + a*x])

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Rubi [A]  time = 0.895757, antiderivative size = 301, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316 \[ \frac{442 b^{27/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{14421 a^{25/4} \sqrt{a x+b \sqrt [3]{x}}}-\frac{884 b^6 \sqrt{a x+b \sqrt [3]{x}}}{14421 a^6}+\frac{884 b^5 x^{2/3} \sqrt{a x+b \sqrt [3]{x}}}{24035 a^5}-\frac{6188 b^4 x^{4/3} \sqrt{a x+b \sqrt [3]{x}}}{216315 a^4}+\frac{476 b^3 x^2 \sqrt{a x+b \sqrt [3]{x}}}{19665 a^3}-\frac{28 b^2 x^{8/3} \sqrt{a x+b \sqrt [3]{x}}}{1311 a^2}+\frac{4 b x^{10/3} \sqrt{a x+b \sqrt [3]{x}}}{207 a}+\frac{2}{9} x^4 \sqrt{a x+b \sqrt [3]{x}} \]

Antiderivative was successfully verified.

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

[Out]

(-884*b^6*Sqrt[b*x^(1/3) + a*x])/(14421*a^6) + (884*b^5*x^(2/3)*Sqrt[b*x^(1/3) +
a*x])/(24035*a^5) - (6188*b^4*x^(4/3)*Sqrt[b*x^(1/3) + a*x])/(216315*a^4) + (47
6*b^3*x^2*Sqrt[b*x^(1/3) + a*x])/(19665*a^3) - (28*b^2*x^(8/3)*Sqrt[b*x^(1/3) +
a*x])/(1311*a^2) + (4*b*x^(10/3)*Sqrt[b*x^(1/3) + a*x])/(207*a) + (2*x^4*Sqrt[b*
x^(1/3) + a*x])/9 + (442*b^(27/4)*(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])/(14421*a^(25/4)*Sqrt[b*x^(1/3) + a*x])

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Rubi in Sympy [A]  time = 80.9983, size = 289, normalized size = 0.96 \[ \frac{2 x^{4} \sqrt{a x + b \sqrt [3]{x}}}{9} + \frac{4 b x^{\frac{10}{3}} \sqrt{a x + b \sqrt [3]{x}}}{207 a} - \frac{28 b^{2} x^{\frac{8}{3}} \sqrt{a x + b \sqrt [3]{x}}}{1311 a^{2}} + \frac{476 b^{3} x^{2} \sqrt{a x + b \sqrt [3]{x}}}{19665 a^{3}} - \frac{6188 b^{4} x^{\frac{4}{3}} \sqrt{a x + b \sqrt [3]{x}}}{216315 a^{4}} + \frac{884 b^{5} x^{\frac{2}{3}} \sqrt{a x + b \sqrt [3]{x}}}{24035 a^{5}} - \frac{884 b^{6} \sqrt{a x + b \sqrt [3]{x}}}{14421 a^{6}} + \frac{442 b^{\frac{27}{4}} \sqrt{\frac{a x^{\frac{2}{3}} + b}{\left (\sqrt{a} \sqrt [3]{x} + \sqrt{b}\right )^{2}}} \left (\sqrt{a} \sqrt [3]{x} + \sqrt{b}\right ) \sqrt{a x + b \sqrt [3]{x}} F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}} \right )}\middle | \frac{1}{2}\right )}{14421 a^{\frac{25}{4}} \sqrt [6]{x} \left (a x^{\frac{2}{3}} + b\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*x**4*sqrt(a*x + b*x**(1/3))/9 + 4*b*x**(10/3)*sqrt(a*x + b*x**(1/3))/(207*a) -
28*b**2*x**(8/3)*sqrt(a*x + b*x**(1/3))/(1311*a**2) + 476*b**3*x**2*sqrt(a*x +
b*x**(1/3))/(19665*a**3) - 6188*b**4*x**(4/3)*sqrt(a*x + b*x**(1/3))/(216315*a**
4) + 884*b**5*x**(2/3)*sqrt(a*x + b*x**(1/3))/(24035*a**5) - 884*b**6*sqrt(a*x +
b*x**(1/3))/(14421*a**6) + 442*b**(27/4)*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)/(14421*a**(25/4)*x**(1/6)*(a*x**(2/3) +
b))

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Mathematica [C]  time = 0.127625, size = 155, normalized size = 0.51 \[ \frac{2 \sqrt [3]{x} \left (24035 a^7 x^{14/3}+26125 a^6 b x^4-220 a^5 b^2 x^{10/3}+308 a^4 b^3 x^{8/3}-476 a^3 b^4 x^2+884 a^2 b^5 x^{4/3}-6630 b^7 \sqrt{\frac{b}{a x^{2/3}}+1} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{b}{a x^{2/3}}\right )-2652 a b^6 x^{2/3}-6630 b^7\right )}{216315 a^6 \sqrt{a x+b \sqrt [3]{x}}} \]

Antiderivative was successfully verified.

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

[Out]

(2*x^(1/3)*(-6630*b^7 - 2652*a*b^6*x^(2/3) + 884*a^2*b^5*x^(4/3) - 476*a^3*b^4*x
^2 + 308*a^4*b^3*x^(8/3) - 220*a^5*b^2*x^(10/3) + 26125*a^6*b*x^4 + 24035*a^7*x^
(14/3) - 6630*b^7*Sqrt[1 + b/(a*x^(2/3))]*Hypergeometric2F1[1/4, 1/2, 5/4, -(b/(
a*x^(2/3)))]))/(216315*a^6*Sqrt[b*x^(1/3) + a*x])

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Maple [A]  time = 0.05, size = 264, normalized size = 0.9 \[{\frac{2\,{x}^{4}}{9}\sqrt{b\sqrt [3]{x}+ax}}+{\frac{4\,b}{207\,a}{x}^{{\frac{10}{3}}}\sqrt{b\sqrt [3]{x}+ax}}-{\frac{28\,{b}^{2}}{1311\,{a}^{2}}{x}^{{\frac{8}{3}}}\sqrt{b\sqrt [3]{x}+ax}}+{\frac{476\,{b}^{3}{x}^{2}}{19665\,{a}^{3}}\sqrt{b\sqrt [3]{x}+ax}}-{\frac{6188\,{b}^{4}}{216315\,{a}^{4}}{x}^{{\frac{4}{3}}}\sqrt{b\sqrt [3]{x}+ax}}+{\frac{884\,{b}^{5}}{24035\,{a}^{5}}{x}^{{\frac{2}{3}}}\sqrt{b\sqrt [3]{x}+ax}}-{\frac{884\,{b}^{6}}{14421\,{a}^{6}}\sqrt{b\sqrt [3]{x}+ax}}+{\frac{442\,{b}^{7}}{14421\,{a}^{7}}\sqrt{-ab}\sqrt{{a \left ( \sqrt [3]{x}+{\frac{1}{a}\sqrt{-ab}} \right ){\frac{1}{\sqrt{-ab}}}}}\sqrt{-2\,{\frac{a}{\sqrt{-ab}} \left ( \sqrt [3]{x}-{\frac{\sqrt{-ab}}{a}} \right ) }}\sqrt{-{a\sqrt [3]{x}{\frac{1}{\sqrt{-ab}}}}}{\it EllipticF} \left ( \sqrt{{a \left ( \sqrt [3]{x}+{\frac{1}{a}\sqrt{-ab}} \right ){\frac{1}{\sqrt{-ab}}}}},{\frac{\sqrt{2}}{2}} \right ){\frac{1}{\sqrt{b\sqrt [3]{x}+ax}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/9*x^4*(b*x^(1/3)+a*x)^(1/2)+4/207*b*x^(10/3)*(b*x^(1/3)+a*x)^(1/2)/a-28/1311*b
^2*x^(8/3)*(b*x^(1/3)+a*x)^(1/2)/a^2+476/19665*b^3*x^2*(b*x^(1/3)+a*x)^(1/2)/a^3
-6188/216315*b^4*x^(4/3)*(b*x^(1/3)+a*x)^(1/2)/a^4+884/24035*b^5*x^(2/3)*(b*x^(1
/3)+a*x)^(1/2)/a^5-884/14421*b^6*(b*x^(1/3)+a*x)^(1/2)/a^6+442/14421*b^7/a^7*(-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. \[ \int \sqrt{a x + b x^{\frac{1}{3}}} x^{3}\,{d x} \]

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. \[{\rm integral}\left (\sqrt{a x + b x^{\frac{1}{3}}} x^{3}, x\right ) \]

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. \[ \int x^{3} \sqrt{a x + b \sqrt [3]{x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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