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

Optimal. Leaf size=326 \[ \frac{7 b^{9/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{5 a^{11/4} \sqrt{a x+b \sqrt [3]{x}}}-\frac{14 b^{9/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{5 a^{11/4} \sqrt{a x+b \sqrt [3]{x}}}+\frac{14 b^2 \sqrt [3]{x} \left (a x^{2/3}+b\right )}{5 a^{5/2} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{a x+b \sqrt [3]{x}}}-\frac{14 b \sqrt [3]{x} \sqrt{a x+b \sqrt [3]{x}}}{15 a^2}+\frac{2 x \sqrt{a x+b \sqrt [3]{x}}}{3 a} \]

[Out]

(14*b^2*(b + a*x^(2/3))*x^(1/3))/(5*a^(5/2)*(Sqrt[b] + Sqrt[a]*x^(1/3))*Sqrt[b*x
^(1/3) + a*x]) - (14*b*x^(1/3)*Sqrt[b*x^(1/3) + a*x])/(15*a^2) + (2*x*Sqrt[b*x^(
1/3) + a*x])/(3*a) - (14*b^(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)*EllipticE[2*ArcTan[(a^(1/4)*x^(1/6))/b^
(1/4)], 1/2])/(5*a^(11/4)*Sqrt[b*x^(1/3) + a*x]) + (7*b^(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*a^(11/4)*Sqrt[b*x^(1/3) + a*x])

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Rubi [A]  time = 0.684039, antiderivative size = 326, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.412 \[ \frac{7 b^{9/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{5 a^{11/4} \sqrt{a x+b \sqrt [3]{x}}}-\frac{14 b^{9/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{5 a^{11/4} \sqrt{a x+b \sqrt [3]{x}}}+\frac{14 b^2 \sqrt [3]{x} \left (a x^{2/3}+b\right )}{5 a^{5/2} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{a x+b \sqrt [3]{x}}}-\frac{14 b \sqrt [3]{x} \sqrt{a x+b \sqrt [3]{x}}}{15 a^2}+\frac{2 x \sqrt{a x+b \sqrt [3]{x}}}{3 a} \]

Antiderivative was successfully verified.

[In]  Int[x/Sqrt[b*x^(1/3) + a*x],x]

[Out]

(14*b^2*(b + a*x^(2/3))*x^(1/3))/(5*a^(5/2)*(Sqrt[b] + Sqrt[a]*x^(1/3))*Sqrt[b*x
^(1/3) + a*x]) - (14*b*x^(1/3)*Sqrt[b*x^(1/3) + a*x])/(15*a^2) + (2*x*Sqrt[b*x^(
1/3) + a*x])/(3*a) - (14*b^(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)*EllipticE[2*ArcTan[(a^(1/4)*x^(1/6))/b^
(1/4)], 1/2])/(5*a^(11/4)*Sqrt[b*x^(1/3) + a*x]) + (7*b^(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*a^(11/4)*Sqrt[b*x^(1/3) + a*x])

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Rubi in Sympy [A]  time = 59.4459, size = 301, normalized size = 0.92 \[ \frac{2 x \sqrt{a x + b \sqrt [3]{x}}}{3 a} - \frac{14 b \sqrt [3]{x} \sqrt{a x + b \sqrt [3]{x}}}{15 a^{2}} + \frac{14 b^{2} \sqrt{a x + b \sqrt [3]{x}}}{5 a^{\frac{5}{2}} \left (\sqrt{a} \sqrt [3]{x} + \sqrt{b}\right )} - \frac{14 b^{\frac{9}{4}} \sqrt{\frac{a x^{\frac{2}{3}} + b}{\left (\sqrt{a} \sqrt [3]{x} + \sqrt{b}\right )^{2}}} \left (\sqrt{a} \sqrt [3]{x} + \sqrt{b}\right ) \sqrt{a x + b \sqrt [3]{x}} E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}} \right )}\middle | \frac{1}{2}\right )}{5 a^{\frac{11}{4}} \sqrt [6]{x} \left (a x^{\frac{2}{3}} + b\right )} + \frac{7 b^{\frac{9}{4}} \sqrt{\frac{a x^{\frac{2}{3}} + b}{\left (\sqrt{a} \sqrt [3]{x} + \sqrt{b}\right )^{2}}} \left (\sqrt{a} \sqrt [3]{x} + \sqrt{b}\right ) \sqrt{a x + b \sqrt [3]{x}} F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}} \right )}\middle | \frac{1}{2}\right )}{5 a^{\frac{11}{4}} \sqrt [6]{x} \left (a x^{\frac{2}{3}} + b\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*x*sqrt(a*x + b*x**(1/3))/(3*a) - 14*b*x**(1/3)*sqrt(a*x + b*x**(1/3))/(15*a**2
) + 14*b**2*sqrt(a*x + b*x**(1/3))/(5*a**(5/2)*(sqrt(a)*x**(1/3) + sqrt(b))) - 1
4*b**(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*a**(11/4)*x**(1/6)*(a*x**(2/3) + b)) + 7*b**(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*a**(11/4)*x**(1
/6)*(a*x**(2/3) + b))

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Mathematica [C]  time = 0.0744242, size = 94, normalized size = 0.29 \[ \frac{2 x^{2/3} \left (5 a^2 x^{4/3}+21 b^2 \sqrt{\frac{b}{a x^{2/3}}+1} \, _2F_1\left (-\frac{1}{4},\frac{1}{2};\frac{3}{4};-\frac{b}{a x^{2/3}}\right )-2 a b x^{2/3}-7 b^2\right )}{15 a^2 \sqrt{a x+b \sqrt [3]{x}}} \]

Antiderivative was successfully verified.

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

[Out]

(2*x^(2/3)*(-7*b^2 - 2*a*b*x^(2/3) + 5*a^2*x^(4/3) + 21*b^2*Sqrt[1 + b/(a*x^(2/3
))]*Hypergeometric2F1[-1/4, 1/2, 3/4, -(b/(a*x^(2/3)))]))/(15*a^2*Sqrt[b*x^(1/3)
+ a*x])

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Maple [A]  time = 0.013, size = 228, normalized size = 0.7 \[ -{\frac{1}{15\,{a}^{3}} \left ( -42\,{b}^{3}\sqrt{{\frac{a\sqrt [3]{x}+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-2\,{\frac{a\sqrt [3]{x}-\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{a\sqrt [3]{x}}{\sqrt{-ab}}}}{\it EllipticE} \left ( \sqrt{{\frac{a\sqrt [3]{x}+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) +21\,{b}^{3}\sqrt{{\frac{a\sqrt [3]{x}+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-2\,{\frac{a\sqrt [3]{x}-\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{a\sqrt [3]{x}}{\sqrt{-ab}}}}{\it EllipticF} \left ( \sqrt{{\frac{a\sqrt [3]{x}+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) +14\,a{b}^{2}{x}^{2/3}+4\,{a}^{2}b{x}^{4/3}-10\,{x}^{2}{a}^{3} \right ){\frac{1}{\sqrt{\sqrt [3]{x} \left ( b+a{x}^{{\frac{2}{3}}} \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(b*x^(1/3)+a*x)^(1/2),x)

[Out]

-1/15/a^3*(-42*b^3*((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)*EllipticE(((a*
x^(1/3)+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2),1/2*2^(1/2))+21*b^3*((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))+14*a*b^2*x^(2/3)+4*a^2*b*x^(4/3)-10*x^2*a^3)/(x^(1/3)*(b+a*x^(
2/3)))^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x/sqrt(a*x + b*x^(1/3)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(x/sqrt(a*x + b*x^(1/3)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x/sqrt(a*x + b*x**(1/3)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x/sqrt(a*x + b*x^(1/3)), x)