3.156 \(\int \frac{1}{x^3 \sqrt{b \sqrt [3]{x}+a x}} \, dx\)

Optimal. Leaf size=388 \[ -\frac{77 a^{13/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{65 b^{15/4} \sqrt{a x+b \sqrt [3]{x}}}+\frac{154 a^{13/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{65 b^{15/4} \sqrt{a x+b \sqrt [3]{x}}}-\frac{154 a^{7/2} \sqrt [3]{x} \left (a x^{2/3}+b\right )}{65 b^4 \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{a x+b \sqrt [3]{x}}}+\frac{154 a^3 \sqrt{a x+b \sqrt [3]{x}}}{65 b^4 \sqrt [3]{x}}-\frac{154 a^2 \sqrt{a x+b \sqrt [3]{x}}}{195 b^3 x}+\frac{22 a \sqrt{a x+b \sqrt [3]{x}}}{39 b^2 x^{5/3}}-\frac{6 \sqrt{a x+b \sqrt [3]{x}}}{13 b x^{7/3}} \]

[Out]

(-154*a^(7/2)*(b + a*x^(2/3))*x^(1/3))/(65*b^4*(Sqrt[b] + Sqrt[a]*x^(1/3))*Sqrt[
b*x^(1/3) + a*x]) - (6*Sqrt[b*x^(1/3) + a*x])/(13*b*x^(7/3)) + (22*a*Sqrt[b*x^(1
/3) + a*x])/(39*b^2*x^(5/3)) - (154*a^2*Sqrt[b*x^(1/3) + a*x])/(195*b^3*x) + (15
4*a^3*Sqrt[b*x^(1/3) + a*x])/(65*b^4*x^(1/3)) + (154*a^(13/4)*(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])/(65*b^(15/4)*Sqrt[b*x^(1/3) + a*x]) -
(77*a^(13/4)*(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])/(65*b
^(15/4)*Sqrt[b*x^(1/3) + a*x])

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Rubi [A]  time = 0.932464, antiderivative size = 388, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.368 \[ -\frac{77 a^{13/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{65 b^{15/4} \sqrt{a x+b \sqrt [3]{x}}}+\frac{154 a^{13/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{65 b^{15/4} \sqrt{a x+b \sqrt [3]{x}}}-\frac{154 a^{7/2} \sqrt [3]{x} \left (a x^{2/3}+b\right )}{65 b^4 \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{a x+b \sqrt [3]{x}}}+\frac{154 a^3 \sqrt{a x+b \sqrt [3]{x}}}{65 b^4 \sqrt [3]{x}}-\frac{154 a^2 \sqrt{a x+b \sqrt [3]{x}}}{195 b^3 x}+\frac{22 a \sqrt{a x+b \sqrt [3]{x}}}{39 b^2 x^{5/3}}-\frac{6 \sqrt{a x+b \sqrt [3]{x}}}{13 b x^{7/3}} \]

Antiderivative was successfully verified.

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

[Out]

(-154*a^(7/2)*(b + a*x^(2/3))*x^(1/3))/(65*b^4*(Sqrt[b] + Sqrt[a]*x^(1/3))*Sqrt[
b*x^(1/3) + a*x]) - (6*Sqrt[b*x^(1/3) + a*x])/(13*b*x^(7/3)) + (22*a*Sqrt[b*x^(1
/3) + a*x])/(39*b^2*x^(5/3)) - (154*a^2*Sqrt[b*x^(1/3) + a*x])/(195*b^3*x) + (15
4*a^3*Sqrt[b*x^(1/3) + a*x])/(65*b^4*x^(1/3)) + (154*a^(13/4)*(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])/(65*b^(15/4)*Sqrt[b*x^(1/3) + a*x]) -
(77*a^(13/4)*(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])/(65*b
^(15/4)*Sqrt[b*x^(1/3) + a*x])

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Rubi in Sympy [A]  time = 82.6471, size = 359, normalized size = 0.93 \[ \frac{154 a^{\frac{13}{4}} \sqrt{\frac{a x^{\frac{2}{3}} + b}{\left (\sqrt{a} \sqrt [3]{x} + \sqrt{b}\right )^{2}}} \left (\sqrt{a} \sqrt [3]{x} + \sqrt{b}\right ) \sqrt{a x + b \sqrt [3]{x}} E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}} \right )}\middle | \frac{1}{2}\right )}{65 b^{\frac{15}{4}} \sqrt [6]{x} \left (a x^{\frac{2}{3}} + b\right )} - \frac{77 a^{\frac{13}{4}} \sqrt{\frac{a x^{\frac{2}{3}} + b}{\left (\sqrt{a} \sqrt [3]{x} + \sqrt{b}\right )^{2}}} \left (\sqrt{a} \sqrt [3]{x} + \sqrt{b}\right ) \sqrt{a x + b \sqrt [3]{x}} F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}} \right )}\middle | \frac{1}{2}\right )}{65 b^{\frac{15}{4}} \sqrt [6]{x} \left (a x^{\frac{2}{3}} + b\right )} - \frac{154 a^{\frac{7}{2}} \sqrt{a x + b \sqrt [3]{x}}}{65 b^{4} \left (\sqrt{a} \sqrt [3]{x} + \sqrt{b}\right )} + \frac{154 a^{3} \sqrt{a x + b \sqrt [3]{x}}}{65 b^{4} \sqrt [3]{x}} - \frac{154 a^{2} \sqrt{a x + b \sqrt [3]{x}}}{195 b^{3} x} + \frac{22 a \sqrt{a x + b \sqrt [3]{x}}}{39 b^{2} x^{\frac{5}{3}}} - \frac{6 \sqrt{a x + b \sqrt [3]{x}}}{13 b x^{\frac{7}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

154*a**(13/4)*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)/(65*b**(15/4)*x**(1/6)*(a*x**(2/3) + b)) - 77*a**(13/4)*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)/(65*b**(15/4
)*x**(1/6)*(a*x**(2/3) + b)) - 154*a**(7/2)*sqrt(a*x + b*x**(1/3))/(65*b**4*(sqr
t(a)*x**(1/3) + sqrt(b))) + 154*a**3*sqrt(a*x + b*x**(1/3))/(65*b**4*x**(1/3)) -
154*a**2*sqrt(a*x + b*x**(1/3))/(195*b**3*x) + 22*a*sqrt(a*x + b*x**(1/3))/(39*
b**2*x**(5/3)) - 6*sqrt(a*x + b*x**(1/3))/(13*b*x**(7/3))

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Mathematica [C]  time = 0.092773, size = 121, normalized size = 0.31 \[ \frac{-462 a^4 x^{8/3} \sqrt{\frac{b}{a x^{2/3}}+1} \, _2F_1\left (-\frac{1}{4},\frac{1}{2};\frac{3}{4};-\frac{b}{a x^{2/3}}\right )+462 a^4 x^{8/3}+308 a^3 b x^2-44 a^2 b^2 x^{4/3}+20 a b^3 x^{2/3}-90 b^4}{195 b^4 x^2 \sqrt{a x+b \sqrt [3]{x}}} \]

Antiderivative was successfully verified.

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

[Out]

(-90*b^4 + 20*a*b^3*x^(2/3) - 44*a^2*b^2*x^(4/3) + 308*a^3*b*x^2 + 462*a^4*x^(8/
3) - 462*a^4*Sqrt[1 + b/(a*x^(2/3))]*x^(8/3)*Hypergeometric2F1[-1/4, 1/2, 3/4, -
(b/(a*x^(2/3)))])/(195*b^4*x^2*Sqrt[b*x^(1/3) + a*x])

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Maple [A]  time = 0.042, size = 363, normalized size = 0.9 \[ -{\frac{1}{195\,{b}^{4}} \left ( 462\,{a}^{3}b\sqrt{{\frac{a\sqrt [3]{x}+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-2\,{\frac{a\sqrt [3]{x}-\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{a\sqrt [3]{x}}{\sqrt{-ab}}}}{x}^{10/3}\sqrt{\sqrt [3]{x} \left ( b+a{x}^{2/3} \right ) }{\it EllipticE} \left ( \sqrt{{\frac{a\sqrt [3]{x}+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) -231\,{a}^{3}b\sqrt{{\frac{a\sqrt [3]{x}+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-2\,{\frac{a\sqrt [3]{x}-\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{a\sqrt [3]{x}}{\sqrt{-ab}}}}{x}^{10/3}\sqrt{\sqrt [3]{x} \left ( b+a{x}^{2/3} \right ) }{\it EllipticF} \left ( \sqrt{{\frac{a\sqrt [3]{x}+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) -462\,\sqrt{b\sqrt [3]{x}+ax}{x}^{10/3}{a}^{3}b+44\,\sqrt{\sqrt [3]{x} \left ( b+a{x}^{2/3} \right ) }{x}^{8/3}{a}^{2}{b}^{2}+154\,\sqrt{\sqrt [3]{x} \left ( b+a{x}^{2/3} \right ) }{x}^{10/3}{a}^{3}b-20\,\sqrt{\sqrt [3]{x} \left ( b+a{x}^{2/3} \right ) }{x}^{2}a{b}^{3}-462\,\sqrt{b\sqrt [3]{x}+ax}{x}^{4}{a}^{4}+90\,\sqrt{\sqrt [3]{x} \left ( b+a{x}^{2/3} \right ) }{x}^{4/3}{b}^{4} \right ){x}^{-{\frac{11}{3}}} \left ( b+a{x}^{{\frac{2}{3}}} \right ) ^{-1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/195*(462*a^3*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^(10/3)*(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))-231*a^3*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^(10/3)*(
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))-462*(b*x^(1/3)+a*x)^(1/2)*x^(10/3)*a^3*b+44*(x^(1/3)*(b+a*x^(2
/3)))^(1/2)*x^(8/3)*a^2*b^2+154*(x^(1/3)*(b+a*x^(2/3)))^(1/2)*x^(10/3)*a^3*b-20*
(x^(1/3)*(b+a*x^(2/3)))^(1/2)*x^2*a*b^3-462*(b*x^(1/3)+a*x)^(1/2)*x^4*a^4+90*(x^
(1/3)*(b+a*x^(2/3)))^(1/2)*x^(4/3)*b^4)/x^(11/3)/(b+a*x^(2/3))/b^4

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{3} \sqrt{a x + b \sqrt [3]{x}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

[Out]

Exception raised: RuntimeError