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

Optimal. Leaf size=350 \[ \frac{4 a^{9/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{5 b^{3/4} \sqrt{a x+b \sqrt [3]{x}}}-\frac{8 a^{9/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{5 b^{3/4} \sqrt{a x+b \sqrt [3]{x}}}+\frac{8 a^{5/2} \sqrt [3]{x} \left (a x^{2/3}+b\right )}{5 b \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{a x+b \sqrt [3]{x}}}-\frac{8 a^2 \sqrt{a x+b \sqrt [3]{x}}}{5 b \sqrt [3]{x}}-\frac{2 \left (a x+b \sqrt [3]{x}\right )^{3/2}}{3 x^2}-\frac{4 a \sqrt{a x+b \sqrt [3]{x}}}{5 x} \]

[Out]

(8*a^(5/2)*(b + a*x^(2/3))*x^(1/3))/(5*b*(Sqrt[b] + Sqrt[a]*x^(1/3))*Sqrt[b*x^(1
/3) + a*x]) - (4*a*Sqrt[b*x^(1/3) + a*x])/(5*x) - (8*a^2*Sqrt[b*x^(1/3) + a*x])/
(5*b*x^(1/3)) - (2*(b*x^(1/3) + a*x)^(3/2))/(3*x^2) - (8*a^(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)*Ellipti
cE[2*ArcTan[(a^(1/4)*x^(1/6))/b^(1/4)], 1/2])/(5*b^(3/4)*Sqrt[b*x^(1/3) + a*x])
+ (4*a^(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])/(5*b^(
3/4)*Sqrt[b*x^(1/3) + a*x])

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Rubi [A]  time = 0.796356, antiderivative size = 350, 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 \[ \frac{4 a^{9/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{5 b^{3/4} \sqrt{a x+b \sqrt [3]{x}}}-\frac{8 a^{9/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{5 b^{3/4} \sqrt{a x+b \sqrt [3]{x}}}+\frac{8 a^{5/2} \sqrt [3]{x} \left (a x^{2/3}+b\right )}{5 b \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{a x+b \sqrt [3]{x}}}-\frac{8 a^2 \sqrt{a x+b \sqrt [3]{x}}}{5 b \sqrt [3]{x}}-\frac{2 \left (a x+b \sqrt [3]{x}\right )^{3/2}}{3 x^2}-\frac{4 a \sqrt{a x+b \sqrt [3]{x}}}{5 x} \]

Antiderivative was successfully verified.

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

[Out]

(8*a^(5/2)*(b + a*x^(2/3))*x^(1/3))/(5*b*(Sqrt[b] + Sqrt[a]*x^(1/3))*Sqrt[b*x^(1
/3) + a*x]) - (4*a*Sqrt[b*x^(1/3) + a*x])/(5*x) - (8*a^2*Sqrt[b*x^(1/3) + a*x])/
(5*b*x^(1/3)) - (2*(b*x^(1/3) + a*x)^(3/2))/(3*x^2) - (8*a^(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)*Ellipti
cE[2*ArcTan[(a^(1/4)*x^(1/6))/b^(1/4)], 1/2])/(5*b^(3/4)*Sqrt[b*x^(1/3) + a*x])
+ (4*a^(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])/(5*b^(
3/4)*Sqrt[b*x^(1/3) + a*x])

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Rubi in Sympy [A]  time = 72.9508, size = 320, normalized size = 0.91 \[ - \frac{8 a^{\frac{9}{4}} \sqrt{\frac{a x^{\frac{2}{3}} + b}{\left (\sqrt{a} \sqrt [3]{x} + \sqrt{b}\right )^{2}}} \left (\sqrt{a} \sqrt [3]{x} + \sqrt{b}\right ) \sqrt{a x + b \sqrt [3]{x}} E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}} \right )}\middle | \frac{1}{2}\right )}{5 b^{\frac{3}{4}} \sqrt [6]{x} \left (a x^{\frac{2}{3}} + b\right )} + \frac{4 a^{\frac{9}{4}} \sqrt{\frac{a x^{\frac{2}{3}} + b}{\left (\sqrt{a} \sqrt [3]{x} + \sqrt{b}\right )^{2}}} \left (\sqrt{a} \sqrt [3]{x} + \sqrt{b}\right ) \sqrt{a x + b \sqrt [3]{x}} F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}} \right )}\middle | \frac{1}{2}\right )}{5 b^{\frac{3}{4}} \sqrt [6]{x} \left (a x^{\frac{2}{3}} + b\right )} + \frac{8 a^{\frac{5}{2}} \sqrt{a x + b \sqrt [3]{x}}}{5 b \left (\sqrt{a} \sqrt [3]{x} + \sqrt{b}\right )} - \frac{8 a^{2} \sqrt{a x + b \sqrt [3]{x}}}{5 b \sqrt [3]{x}} - \frac{4 a \sqrt{a x + b \sqrt [3]{x}}}{5 x} - \frac{2 \left (a x + b \sqrt [3]{x}\right )^{\frac{3}{2}}}{3 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-8*a**(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_e(2*atan(a**(1/4)*x**(1/6)/b**(1
/4)), 1/2)/(5*b**(3/4)*x**(1/6)*(a*x**(2/3) + b)) + 4*a**(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)/(5*b**(3/4)*x**(1/
6)*(a*x**(2/3) + b)) + 8*a**(5/2)*sqrt(a*x + b*x**(1/3))/(5*b*(sqrt(a)*x**(1/3)
+ sqrt(b))) - 8*a**2*sqrt(a*x + b*x**(1/3))/(5*b*x**(1/3)) - 4*a*sqrt(a*x + b*x*
*(1/3))/(5*x) - 2*(a*x + b*x**(1/3))**(3/2)/(3*x**2)

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Mathematica [C]  time = 0.0869826, size = 108, normalized size = 0.31 \[ -\frac{2 \left (-12 a^3 x^2 \sqrt{\frac{b}{a x^{2/3}}+1} \, _2F_1\left (-\frac{1}{4},\frac{1}{2};\frac{3}{4};-\frac{b}{a x^{2/3}}\right )+12 a^3 x^2+23 a^2 b x^{4/3}+16 a b^2 x^{2/3}+5 b^3\right )}{15 b x^{4/3} \sqrt{a x+b \sqrt [3]{x}}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(5*b^3 + 16*a*b^2*x^(2/3) + 23*a^2*b*x^(4/3) + 12*a^3*x^2 - 12*a^3*Sqrt[1 +
b/(a*x^(2/3))]*x^2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(b/(a*x^(2/3)))]))/(15*b*x
^(4/3)*Sqrt[b*x^(1/3) + a*x])

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Maple [A]  time = 0.038, size = 339, normalized size = 1. \[{\frac{2}{15\,b{x}^{3}} \left ( 12\,{a}^{2}b\sqrt{{\frac{a\sqrt [3]{x}+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-2\,{\frac{a\sqrt [3]{x}-\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{a\sqrt [3]{x}}{\sqrt{-ab}}}}{x}^{8/3}\sqrt{\sqrt [3]{x} \left ( b+a{x}^{2/3} \right ) }{\it EllipticE} \left ( \sqrt{{\frac{a\sqrt [3]{x}+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) -6\,{a}^{2}b\sqrt{{\frac{a\sqrt [3]{x}+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-2\,{\frac{a\sqrt [3]{x}-\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{a\sqrt [3]{x}}{\sqrt{-ab}}}}{x}^{8/3}\sqrt{\sqrt [3]{x} \left ( b+a{x}^{2/3} \right ) }{\it EllipticF} \left ( \sqrt{{\frac{a\sqrt [3]{x}+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) -12\,\sqrt{b\sqrt [3]{x}+ax}{x}^{10/3}{a}^{3}-12\,\sqrt{b\sqrt [3]{x}+ax}{x}^{8/3}{a}^{2}b-16\,\sqrt{\sqrt [3]{x} \left ( b+a{x}^{2/3} \right ) }{x}^{2}a{b}^{2}-11\,\sqrt{\sqrt [3]{x} \left ( b+a{x}^{2/3} \right ) }{x}^{8/3}{a}^{2}b-5\,\sqrt{\sqrt [3]{x} \left ( b+a{x}^{2/3} \right ) }{x}^{4/3}{b}^{3} \right ) \left ( b+a{x}^{{\frac{2}{3}}} \right ) ^{-1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15*(12*a^2*b*((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^(8/3)*(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))-6*a^2*b*((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^(8/3)*(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))-12*(b*x^(1/3)+a*x)^(1/2)*x^(10/3)*a^3-12*(b*x^(1/3)+a*x)^(1/2)*x^(8/3
)*a^2*b-16*(x^(1/3)*(b+a*x^(2/3)))^(1/2)*x^2*a*b^2-11*(x^(1/3)*(b+a*x^(2/3)))^(1
/2)*x^(8/3)*a^2*b-5*(x^(1/3)*(b+a*x^(2/3)))^(1/2)*x^(4/3)*b^3)/b/x^3/(b+a*x^(2/3
))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: RuntimeError