3.676 \(\int \frac{\sqrt [3]{a+b x^2}}{x^4} \, dx\)

Optimal. Leaf size=290 \[ \frac{2 \sqrt{2-\sqrt{3}} b \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}\right )|-7+4 \sqrt{3}\right )}{9 \sqrt [4]{3} a x \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}}-\frac{2 b \sqrt [3]{a+b x^2}}{9 a x}-\frac{\sqrt [3]{a+b x^2}}{3 x^3} \]

[Out]

-(a + b*x^2)^(1/3)/(3*x^3) - (2*b*(a + b*x^2)^(1/3))/(9*a*x) + (2*Sqrt[2 - Sqrt[
3]]*b*(a^(1/3) - (a + b*x^2)^(1/3))*Sqrt[(a^(2/3) + a^(1/3)*(a + b*x^2)^(1/3) +
(a + b*x^2)^(2/3))/((1 - Sqrt[3])*a^(1/3) - (a + b*x^2)^(1/3))^2]*EllipticF[ArcS
in[((1 + Sqrt[3])*a^(1/3) - (a + b*x^2)^(1/3))/((1 - Sqrt[3])*a^(1/3) - (a + b*x
^2)^(1/3))], -7 + 4*Sqrt[3]])/(9*3^(1/4)*a*x*Sqrt[-((a^(1/3)*(a^(1/3) - (a + b*x
^2)^(1/3)))/((1 - Sqrt[3])*a^(1/3) - (a + b*x^2)^(1/3))^2)])

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Rubi [A]  time = 0.402347, antiderivative size = 290, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{2 \sqrt{2-\sqrt{3}} b \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}\right )|-7+4 \sqrt{3}\right )}{9 \sqrt [4]{3} a x \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}}-\frac{2 b \sqrt [3]{a+b x^2}}{9 a x}-\frac{\sqrt [3]{a+b x^2}}{3 x^3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)^(1/3)/x^4,x]

[Out]

-(a + b*x^2)^(1/3)/(3*x^3) - (2*b*(a + b*x^2)^(1/3))/(9*a*x) + (2*Sqrt[2 - Sqrt[
3]]*b*(a^(1/3) - (a + b*x^2)^(1/3))*Sqrt[(a^(2/3) + a^(1/3)*(a + b*x^2)^(1/3) +
(a + b*x^2)^(2/3))/((1 - Sqrt[3])*a^(1/3) - (a + b*x^2)^(1/3))^2]*EllipticF[ArcS
in[((1 + Sqrt[3])*a^(1/3) - (a + b*x^2)^(1/3))/((1 - Sqrt[3])*a^(1/3) - (a + b*x
^2)^(1/3))], -7 + 4*Sqrt[3]])/(9*3^(1/4)*a*x*Sqrt[-((a^(1/3)*(a^(1/3) - (a + b*x
^2)^(1/3)))/((1 - Sqrt[3])*a^(1/3) - (a + b*x^2)^(1/3))^2)])

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Rubi in Sympy [A]  time = 15.0519, size = 235, normalized size = 0.81 \[ - \frac{\sqrt [3]{a + b x^{2}}}{3 x^{3}} + \frac{2 \cdot 3^{\frac{3}{4}} b \sqrt{\frac{a^{\frac{2}{3}} + \sqrt [3]{a} \sqrt [3]{a + b x^{2}} + \left (a + b x^{2}\right )^{\frac{2}{3}}}{\left (\sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{a + b x^{2}}\right )^{2}}} \sqrt{- \sqrt{3} + 2} \left (\sqrt [3]{a} - \sqrt [3]{a + b x^{2}}\right ) F\left (\operatorname{asin}{\left (\frac{\sqrt [3]{a} \left (1 + \sqrt{3}\right ) - \sqrt [3]{a + b x^{2}}}{- \sqrt [3]{a} \left (-1 + \sqrt{3}\right ) - \sqrt [3]{a + b x^{2}}} \right )}\middle | -7 + 4 \sqrt{3}\right )}{27 a x \sqrt{- \frac{\sqrt [3]{a} \left (\sqrt [3]{a} - \sqrt [3]{a + b x^{2}}\right )}{\left (\sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{a + b x^{2}}\right )^{2}}}} - \frac{2 b \sqrt [3]{a + b x^{2}}}{9 a x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(a + b*x**2)**(1/3)/(3*x**3) + 2*3**(3/4)*b*sqrt((a**(2/3) + a**(1/3)*(a + b*x*
*2)**(1/3) + (a + b*x**2)**(2/3))/(a**(1/3)*(-1 + sqrt(3)) + (a + b*x**2)**(1/3)
)**2)*sqrt(-sqrt(3) + 2)*(a**(1/3) - (a + b*x**2)**(1/3))*elliptic_f(asin((a**(1
/3)*(1 + sqrt(3)) - (a + b*x**2)**(1/3))/(-a**(1/3)*(-1 + sqrt(3)) - (a + b*x**2
)**(1/3))), -7 + 4*sqrt(3))/(27*a*x*sqrt(-a**(1/3)*(a**(1/3) - (a + b*x**2)**(1/
3))/(a**(1/3)*(-1 + sqrt(3)) + (a + b*x**2)**(1/3))**2)) - 2*b*(a + b*x**2)**(1/
3)/(9*a*x)

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Mathematica [C]  time = 0.0449163, size = 88, normalized size = 0.3 \[ \left (-\frac{2 b}{9 a x}-\frac{1}{3 x^3}\right ) \sqrt [3]{a+b x^2}-\frac{2 b^2 x \left (\frac{a+b x^2}{a}\right )^{2/3} \, _2F_1\left (\frac{1}{2},\frac{2}{3};\frac{3}{2};-\frac{b x^2}{a}\right )}{27 a \left (a+b x^2\right )^{2/3}} \]

Antiderivative was successfully verified.

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

[Out]

(-1/(3*x^3) - (2*b)/(9*a*x))*(a + b*x^2)^(1/3) - (2*b^2*x*((a + b*x^2)/a)^(2/3)*
Hypergeometric2F1[1/2, 2/3, 3/2, -((b*x^2)/a)])/(27*a*(a + b*x^2)^(2/3))

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Maple [F]  time = 0.041, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{4}}\sqrt [3]{b{x}^{2}+a}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^(1/3)/x^4,x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}^{\frac{1}{3}}}{x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(1/3)/x^4,x, algorithm="maxima")

[Out]

integrate((b*x^2 + a)^(1/3)/x^4, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b x^{2} + a\right )}^{\frac{1}{3}}}{x^{4}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(1/3)/x^4,x, algorithm="fricas")

[Out]

integral((b*x^2 + a)^(1/3)/x^4, x)

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Sympy [A]  time = 2.8454, size = 34, normalized size = 0.12 \[ - \frac{\sqrt [3]{a}{{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{2}, - \frac{1}{3} \\ - \frac{1}{2} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a**(1/3)*hyper((-3/2, -1/3), (-1/2,), b*x**2*exp_polar(I*pi)/a)/(3*x**3)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}^{\frac{1}{3}}}{x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(1/3)/x^4,x, algorithm="giac")

[Out]

integrate((b*x^2 + a)^(1/3)/x^4, x)