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

Optimal. Leaf size=239 \[ \frac{663 b^{15/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{154 a^{21/4} \sqrt{a x+b \sqrt [3]{x}}}-\frac{663 b^3 \sqrt{a x+b \sqrt [3]{x}}}{77 a^5}+\frac{1989 b^2 x^{2/3} \sqrt{a x+b \sqrt [3]{x}}}{385 a^4}-\frac{221 b x^{4/3} \sqrt{a x+b \sqrt [3]{x}}}{55 a^3}+\frac{17 x^2 \sqrt{a x+b \sqrt [3]{x}}}{5 a^2}-\frac{3 x^3}{a \sqrt{a x+b \sqrt [3]{x}}} \]

[Out]

(-3*x^3)/(a*Sqrt[b*x^(1/3) + a*x]) - (663*b^3*Sqrt[b*x^(1/3) + a*x])/(77*a^5) +
(1989*b^2*x^(2/3)*Sqrt[b*x^(1/3) + a*x])/(385*a^4) - (221*b*x^(4/3)*Sqrt[b*x^(1/
3) + a*x])/(55*a^3) + (17*x^2*Sqrt[b*x^(1/3) + a*x])/(5*a^2) + (663*b^(15/4)*(Sq
rt[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])/(154*a^(21/4)*Sqrt[b*x
^(1/3) + a*x])

_______________________________________________________________________________________

Rubi [A]  time = 0.684, antiderivative size = 239, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316 \[ \frac{663 b^{15/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{154 a^{21/4} \sqrt{a x+b \sqrt [3]{x}}}-\frac{663 b^3 \sqrt{a x+b \sqrt [3]{x}}}{77 a^5}+\frac{1989 b^2 x^{2/3} \sqrt{a x+b \sqrt [3]{x}}}{385 a^4}-\frac{221 b x^{4/3} \sqrt{a x+b \sqrt [3]{x}}}{55 a^3}+\frac{17 x^2 \sqrt{a x+b \sqrt [3]{x}}}{5 a^2}-\frac{3 x^3}{a \sqrt{a x+b \sqrt [3]{x}}} \]

Antiderivative was successfully verified.

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

[Out]

(-3*x^3)/(a*Sqrt[b*x^(1/3) + a*x]) - (663*b^3*Sqrt[b*x^(1/3) + a*x])/(77*a^5) +
(1989*b^2*x^(2/3)*Sqrt[b*x^(1/3) + a*x])/(385*a^4) - (221*b*x^(4/3)*Sqrt[b*x^(1/
3) + a*x])/(55*a^3) + (17*x^2*Sqrt[b*x^(1/3) + a*x])/(5*a^2) + (663*b^(15/4)*(Sq
rt[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])/(154*a^(21/4)*Sqrt[b*x
^(1/3) + a*x])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 59.2346, size = 230, normalized size = 0.96 \[ - \frac{3 x^{3}}{a \sqrt{a x + b \sqrt [3]{x}}} + \frac{17 x^{2} \sqrt{a x + b \sqrt [3]{x}}}{5 a^{2}} - \frac{221 b x^{\frac{4}{3}} \sqrt{a x + b \sqrt [3]{x}}}{55 a^{3}} + \frac{1989 b^{2} x^{\frac{2}{3}} \sqrt{a x + b \sqrt [3]{x}}}{385 a^{4}} - \frac{663 b^{3} \sqrt{a x + b \sqrt [3]{x}}}{77 a^{5}} + \frac{663 b^{\frac{15}{4}} \sqrt{\frac{a x^{\frac{2}{3}} + b}{\left (\sqrt{a} \sqrt [3]{x} + \sqrt{b}\right )^{2}}} \left (\sqrt{a} \sqrt [3]{x} + \sqrt{b}\right ) \sqrt{a x + b \sqrt [3]{x}} F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}} \right )}\middle | \frac{1}{2}\right )}{154 a^{\frac{21}{4}} \sqrt [6]{x} \left (a x^{\frac{2}{3}} + b\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*x**3/(a*sqrt(a*x + b*x**(1/3))) + 17*x**2*sqrt(a*x + b*x**(1/3))/(5*a**2) - 2
21*b*x**(4/3)*sqrt(a*x + b*x**(1/3))/(55*a**3) + 1989*b**2*x**(2/3)*sqrt(a*x + b
*x**(1/3))/(385*a**4) - 663*b**3*sqrt(a*x + b*x**(1/3))/(77*a**5) + 663*b**(15/4
)*sqrt((a*x**(2/3) + b)/(sqrt(a)*x**(1/3) + sqrt(b))**2)*(sqrt(a)*x**(1/3) + sqr
t(b))*sqrt(a*x + b*x**(1/3))*elliptic_f(2*atan(a**(1/4)*x**(1/6)/b**(1/4)), 1/2)
/(154*a**(21/4)*x**(1/6)*(a*x**(2/3) + b))

_______________________________________________________________________________________

Mathematica [C]  time = 0.0902771, size = 118, normalized size = 0.49 \[ \frac{\sqrt [3]{x} \left (154 a^4 x^{8/3}-238 a^3 b x^2+442 a^2 b^2 x^{4/3}-3315 b^4 \sqrt{\frac{b}{a x^{2/3}}+1} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{b}{a x^{2/3}}\right )-1326 a b^3 x^{2/3}-3315 b^4\right )}{385 a^5 \sqrt{a x+b \sqrt [3]{x}}} \]

Antiderivative was successfully verified.

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

[Out]

(x^(1/3)*(-3315*b^4 - 1326*a*b^3*x^(2/3) + 442*a^2*b^2*x^(4/3) - 238*a^3*b*x^2 +
154*a^4*x^(8/3) - 3315*b^4*Sqrt[1 + b/(a*x^(2/3))]*Hypergeometric2F1[1/4, 1/2,
5/4, -(b/(a*x^(2/3)))]))/(385*a^5*Sqrt[b*x^(1/3) + a*x])

_______________________________________________________________________________________

Maple [A]  time = 0.039, size = 260, normalized size = 1.1 \[ -{\frac{1}{770\,{a}^{6}} \left ( -884\,\sqrt{\sqrt [3]{x} \left ( b+a{x}^{2/3} \right ) }{x}^{5/3}{a}^{3}{b}^{2}+476\,\sqrt{\sqrt [3]{x} \left ( b+a{x}^{2/3} \right ) }{x}^{7/3}{a}^{4}b-3315\,\sqrt{\sqrt [3]{x} \left ( b+a{x}^{2/3} \right ) }\sqrt{{\frac{a\sqrt [3]{x}+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-2\,{\frac{a\sqrt [3]{x}-\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{a\sqrt [3]{x}}{\sqrt{-ab}}}}{\it EllipticF} \left ( \sqrt{{\frac{a\sqrt [3]{x}+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) \sqrt{-ab}{b}^{4}+2652\,\sqrt{\sqrt [3]{x} \left ( b+a{x}^{2/3} \right ) }x{a}^{2}{b}^{3}-308\,\sqrt{\sqrt [3]{x} \left ( b+a{x}^{2/3} \right ) }{x}^{3}{a}^{5}+4320\,\sqrt{\sqrt [3]{x} \left ( b+a{x}^{2/3} \right ) }\sqrt [3]{x}a{b}^{4}+2310\,\sqrt [3]{x}\sqrt{b\sqrt [3]{x}+ax}a{b}^{4} \right ){\frac{1}{\sqrt [3]{x}}} \left ( b+a{x}^{{\frac{2}{3}}} \right ) ^{-1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/770*(-884*(x^(1/3)*(b+a*x^(2/3)))^(1/2)*x^(5/3)*a^3*b^2+476*(x^(1/3)*(b+a*x^(
2/3)))^(1/2)*x^(7/3)*a^4*b-3315*(x^(1/3)*(b+a*x^(2/3)))^(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))*(-a*b)^(1/2)*b^4+2652*(x^(1/3)*(b+a*x^(2/3)))^(1/2)*x*a^2*b^3-
308*(x^(1/3)*(b+a*x^(2/3)))^(1/2)*x^3*a^5+4320*(x^(1/3)*(b+a*x^(2/3)))^(1/2)*x^(
1/3)*a*b^4+2310*x^(1/3)*(b*x^(1/3)+a*x)^(1/2)*a*b^4)/x^(1/3)/(b+a*x^(2/3))/a^6

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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