3.138 \(\int \frac{\sqrt{b \sqrt [3]{x}+a x}}{x^4} \, dx\)

Optimal. Leaf size=413 \[ -\frac{154 a^{17/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{1105 b^{15/4} \sqrt{a x+b \sqrt [3]{x}}}+\frac{308 a^{17/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{1105 b^{15/4} \sqrt{a x+b \sqrt [3]{x}}}-\frac{308 a^{9/2} \sqrt [3]{x} \left (a x^{2/3}+b\right )}{1105 b^4 \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{a x+b \sqrt [3]{x}}}+\frac{308 a^4 \sqrt{a x+b \sqrt [3]{x}}}{1105 b^4 \sqrt [3]{x}}-\frac{308 a^3 \sqrt{a x+b \sqrt [3]{x}}}{3315 b^3 x}+\frac{44 a^2 \sqrt{a x+b \sqrt [3]{x}}}{663 b^2 x^{5/3}}-\frac{12 a \sqrt{a x+b \sqrt [3]{x}}}{221 b x^{7/3}}-\frac{6 \sqrt{a x+b \sqrt [3]{x}}}{17 x^3} \]

[Out]

(-308*a^(9/2)*(b + a*x^(2/3))*x^(1/3))/(1105*b^4*(Sqrt[b] + Sqrt[a]*x^(1/3))*Sqr
t[b*x^(1/3) + a*x]) - (6*Sqrt[b*x^(1/3) + a*x])/(17*x^3) - (12*a*Sqrt[b*x^(1/3)
+ a*x])/(221*b*x^(7/3)) + (44*a^2*Sqrt[b*x^(1/3) + a*x])/(663*b^2*x^(5/3)) - (30
8*a^3*Sqrt[b*x^(1/3) + a*x])/(3315*b^3*x) + (308*a^4*Sqrt[b*x^(1/3) + a*x])/(110
5*b^4*x^(1/3)) + (308*a^(17/4)*(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])/(1105*b^(15/4)*Sqrt[b*x^(1/3) + a*x]) - (154*a^(17/4)*(Sqrt[b] + Sqr
t[a]*x^(1/3))*Sqrt[(b + a*x^(2/3))/(Sqrt[b] + Sqrt[a]*x^(1/3))^2]*x^(1/6)*Ellipt
icF[2*ArcTan[(a^(1/4)*x^(1/6))/b^(1/4)], 1/2])/(1105*b^(15/4)*Sqrt[b*x^(1/3) + a
*x])

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Rubi [A]  time = 1.03751, antiderivative size = 413, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.421 \[ -\frac{154 a^{17/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{1105 b^{15/4} \sqrt{a x+b \sqrt [3]{x}}}+\frac{308 a^{17/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{1105 b^{15/4} \sqrt{a x+b \sqrt [3]{x}}}-\frac{308 a^{9/2} \sqrt [3]{x} \left (a x^{2/3}+b\right )}{1105 b^4 \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{a x+b \sqrt [3]{x}}}+\frac{308 a^4 \sqrt{a x+b \sqrt [3]{x}}}{1105 b^4 \sqrt [3]{x}}-\frac{308 a^3 \sqrt{a x+b \sqrt [3]{x}}}{3315 b^3 x}+\frac{44 a^2 \sqrt{a x+b \sqrt [3]{x}}}{663 b^2 x^{5/3}}-\frac{12 a \sqrt{a x+b \sqrt [3]{x}}}{221 b x^{7/3}}-\frac{6 \sqrt{a x+b \sqrt [3]{x}}}{17 x^3} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[b*x^(1/3) + a*x]/x^4,x]

[Out]

(-308*a^(9/2)*(b + a*x^(2/3))*x^(1/3))/(1105*b^4*(Sqrt[b] + Sqrt[a]*x^(1/3))*Sqr
t[b*x^(1/3) + a*x]) - (6*Sqrt[b*x^(1/3) + a*x])/(17*x^3) - (12*a*Sqrt[b*x^(1/3)
+ a*x])/(221*b*x^(7/3)) + (44*a^2*Sqrt[b*x^(1/3) + a*x])/(663*b^2*x^(5/3)) - (30
8*a^3*Sqrt[b*x^(1/3) + a*x])/(3315*b^3*x) + (308*a^4*Sqrt[b*x^(1/3) + a*x])/(110
5*b^4*x^(1/3)) + (308*a^(17/4)*(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])/(1105*b^(15/4)*Sqrt[b*x^(1/3) + a*x]) - (154*a^(17/4)*(Sqrt[b] + Sqr
t[a]*x^(1/3))*Sqrt[(b + a*x^(2/3))/(Sqrt[b] + Sqrt[a]*x^(1/3))^2]*x^(1/6)*Ellipt
icF[2*ArcTan[(a^(1/4)*x^(1/6))/b^(1/4)], 1/2])/(1105*b^(15/4)*Sqrt[b*x^(1/3) + a
*x])

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Rubi in Sympy [A]  time = 103.393, size = 382, normalized size = 0.92 \[ \frac{308 a^{\frac{17}{4}} \sqrt{\frac{a x^{\frac{2}{3}} + b}{\left (\sqrt{a} \sqrt [3]{x} + \sqrt{b}\right )^{2}}} \left (\sqrt{a} \sqrt [3]{x} + \sqrt{b}\right ) \sqrt{a x + b \sqrt [3]{x}} E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}} \right )}\middle | \frac{1}{2}\right )}{1105 b^{\frac{15}{4}} \sqrt [6]{x} \left (a x^{\frac{2}{3}} + b\right )} - \frac{154 a^{\frac{17}{4}} \sqrt{\frac{a x^{\frac{2}{3}} + b}{\left (\sqrt{a} \sqrt [3]{x} + \sqrt{b}\right )^{2}}} \left (\sqrt{a} \sqrt [3]{x} + \sqrt{b}\right ) \sqrt{a x + b \sqrt [3]{x}} F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}} \right )}\middle | \frac{1}{2}\right )}{1105 b^{\frac{15}{4}} \sqrt [6]{x} \left (a x^{\frac{2}{3}} + b\right )} - \frac{308 a^{\frac{9}{2}} \sqrt{a x + b \sqrt [3]{x}}}{1105 b^{4} \left (\sqrt{a} \sqrt [3]{x} + \sqrt{b}\right )} + \frac{308 a^{4} \sqrt{a x + b \sqrt [3]{x}}}{1105 b^{4} \sqrt [3]{x}} - \frac{308 a^{3} \sqrt{a x + b \sqrt [3]{x}}}{3315 b^{3} x} + \frac{44 a^{2} \sqrt{a x + b \sqrt [3]{x}}}{663 b^{2} x^{\frac{5}{3}}} - \frac{12 a \sqrt{a x + b \sqrt [3]{x}}}{221 b x^{\frac{7}{3}}} - \frac{6 \sqrt{a x + b \sqrt [3]{x}}}{17 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**(1/3)+a*x)**(1/2)/x**4,x)

[Out]

308*a**(17/4)*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)/(1105*b**(15/4)*x**(1/6)*(a*x**(2/3) + b)) - 154*a**(17/4)*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)/(1105*b**
(15/4)*x**(1/6)*(a*x**(2/3) + b)) - 308*a**(9/2)*sqrt(a*x + b*x**(1/3))/(1105*b*
*4*(sqrt(a)*x**(1/3) + sqrt(b))) + 308*a**4*sqrt(a*x + b*x**(1/3))/(1105*b**4*x*
*(1/3)) - 308*a**3*sqrt(a*x + b*x**(1/3))/(3315*b**3*x) + 44*a**2*sqrt(a*x + b*x
**(1/3))/(663*b**2*x**(5/3)) - 12*a*sqrt(a*x + b*x**(1/3))/(221*b*x**(7/3)) - 6*
sqrt(a*x + b*x**(1/3))/(17*x**3)

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Mathematica [C]  time = 0.0867621, size = 136, normalized size = 0.33 \[ -\frac{2 \left (462 a^5 x^{10/3} \sqrt{\frac{b}{a x^{2/3}}+1} \, _2F_1\left (-\frac{1}{4},\frac{1}{2};\frac{3}{4};-\frac{b}{a x^{2/3}}\right )-462 a^5 x^{10/3}-308 a^4 b x^{8/3}+44 a^3 b^2 x^2-20 a^2 b^3 x^{4/3}+675 a b^4 x^{2/3}+585 b^5\right )}{3315 b^4 x^{8/3} \sqrt{a x+b \sqrt [3]{x}}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[b*x^(1/3) + a*x]/x^4,x]

[Out]

(-2*(585*b^5 + 675*a*b^4*x^(2/3) - 20*a^2*b^3*x^(4/3) + 44*a^3*b^2*x^2 - 308*a^4
*b*x^(8/3) - 462*a^5*x^(10/3) + 462*a^5*Sqrt[1 + b/(a*x^(2/3))]*x^(10/3)*Hyperge
ometric2F1[-1/4, 1/2, 3/4, -(b/(a*x^(2/3)))]))/(3315*b^4*x^(8/3)*Sqrt[b*x^(1/3)
+ a*x])

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Maple [A]  time = 0.033, size = 281, normalized size = 0.7 \[ -{\frac{6}{17\,{x}^{3}}\sqrt{b\sqrt [3]{x}+ax}}-{\frac{12\,a}{221\,b}\sqrt{b\sqrt [3]{x}+ax}{x}^{-{\frac{7}{3}}}}+{\frac{44\,{a}^{2}}{663\,{b}^{2}}\sqrt{b\sqrt [3]{x}+ax}{x}^{-{\frac{5}{3}}}}-{\frac{308\,{a}^{3}}{3315\,{b}^{3}x}\sqrt{b\sqrt [3]{x}+ax}}+{\frac{308\,{a}^{4}}{1105\,{b}^{4}} \left ( b+a{x}^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt{\sqrt [3]{x} \left ( b+a{x}^{{\frac{2}{3}}} \right ) }}}}-{\frac{154\,{a}^{4}}{1105\,{b}^{4}}\sqrt{-ab}\sqrt{{a \left ( \sqrt [3]{x}+{\frac{1}{a}\sqrt{-ab}} \right ){\frac{1}{\sqrt{-ab}}}}}\sqrt{-2\,{\frac{a}{\sqrt{-ab}} \left ( \sqrt [3]{x}-{\frac{\sqrt{-ab}}{a}} \right ) }}\sqrt{-{a\sqrt [3]{x}{\frac{1}{\sqrt{-ab}}}}} \left ( -2\,{\frac{\sqrt{-ab}}{a}{\it EllipticE} \left ( \sqrt{{\frac{a}{\sqrt{-ab}} \left ( \sqrt [3]{x}+{\frac{\sqrt{-ab}}{a}} \right ) }},1/2\,\sqrt{2} \right ) }+{\frac{1}{a}\sqrt{-ab}{\it EllipticF} \left ( \sqrt{{a \left ( \sqrt [3]{x}+{\frac{1}{a}\sqrt{-ab}} \right ){\frac{1}{\sqrt{-ab}}}}},{\frac{\sqrt{2}}{2}} \right ) } \right ){\frac{1}{\sqrt{b\sqrt [3]{x}+ax}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^(1/3)+a*x)^(1/2)/x^4,x)

[Out]

-6/17*(b*x^(1/3)+a*x)^(1/2)/x^3-12/221*a*(b*x^(1/3)+a*x)^(1/2)/b/x^(7/3)+44/663*
a^2*(b*x^(1/3)+a*x)^(1/2)/b^2/x^(5/3)-308/3315*a^3*(b*x^(1/3)+a*x)^(1/2)/b^3/x+3
08/1105*(b+a*x^(2/3))*a^4/b^4/(x^(1/3)*(b+a*x^(2/3)))^(1/2)-154/1105*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)*(-2*(-a*b)^(1/2)/a*EllipticE(((x^(1/3)+(-a*b)^(1/2)/a)/(-a*b)^(1/2)*a)^(1/2),
1/2*2^(1/2))+(-a*b)^(1/2)/a*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. \[ \int \frac{\sqrt{a x + b x^{\frac{1}{3}}}}{x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(a*x + b*x^(1/3))/x^4,x, algorithm="maxima")

[Out]

integrate(sqrt(a*x + b*x^(1/3))/x^4, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{a x + b x^{\frac{1}{3}}}}{x^{4}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(a*x + b*x^(1/3))/x^4,x, algorithm="fricas")

[Out]

integral(sqrt(a*x + b*x^(1/3))/x^4, 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)**(1/2)/x**4,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(sqrt(a*x + b*x^(1/3))/x^4,x, algorithm="giac")

[Out]

Exception raised: RuntimeError