3.2034 \(\int \frac{1}{\sqrt{a+\frac{b}{x^3}} x^{12}} \, dx\)

Optimal. Leaf size=568 \[ \frac{1280 \sqrt{2} a^{10/3} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right ) \sqrt{\frac{a^{2/3}-\frac{\sqrt [3]{a} \sqrt [3]{b}}{x}+\frac{b^{2/3}}{x^2}}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}}\right )|-7-4 \sqrt{3}\right )}{1729 \sqrt [4]{3} b^{11/3} \sqrt{a+\frac{b}{x^3}} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )^2}}}-\frac{640 \sqrt [4]{3} \sqrt{2-\sqrt{3}} a^{10/3} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right ) \sqrt{\frac{a^{2/3}-\frac{\sqrt [3]{a} \sqrt [3]{b}}{x}+\frac{b^{2/3}}{x^2}}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}}\right )|-7-4 \sqrt{3}\right )}{1729 b^{11/3} \sqrt{a+\frac{b}{x^3}} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )^2}}}+\frac{1280 a^3 \sqrt{a+\frac{b}{x^3}}}{1729 b^{11/3} \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )}-\frac{320 a^2 \sqrt{a+\frac{b}{x^3}}}{1729 b^3 x^2}+\frac{32 a \sqrt{a+\frac{b}{x^3}}}{247 b^2 x^5}-\frac{2 \sqrt{a+\frac{b}{x^3}}}{19 b x^8} \]

[Out]

(1280*a^3*Sqrt[a + b/x^3])/(1729*b^(11/3)*((1 + Sqrt[3])*a^(1/3) + b^(1/3)/x)) -
(2*Sqrt[a + b/x^3])/(19*b*x^8) + (32*a*Sqrt[a + b/x^3])/(247*b^2*x^5) - (320*a^
2*Sqrt[a + b/x^3])/(1729*b^3*x^2) - (640*3^(1/4)*Sqrt[2 - Sqrt[3]]*a^(10/3)*(a^(
1/3) + b^(1/3)/x)*Sqrt[(a^(2/3) + b^(2/3)/x^2 - (a^(1/3)*b^(1/3))/x)/((1 + Sqrt[
3])*a^(1/3) + b^(1/3)/x)^2]*EllipticE[ArcSin[((1 - Sqrt[3])*a^(1/3) + b^(1/3)/x)
/((1 + Sqrt[3])*a^(1/3) + b^(1/3)/x)], -7 - 4*Sqrt[3]])/(1729*b^(11/3)*Sqrt[a +
b/x^3]*Sqrt[(a^(1/3)*(a^(1/3) + b^(1/3)/x))/((1 + Sqrt[3])*a^(1/3) + b^(1/3)/x)^
2]) + (1280*Sqrt[2]*a^(10/3)*(a^(1/3) + b^(1/3)/x)*Sqrt[(a^(2/3) + b^(2/3)/x^2 -
(a^(1/3)*b^(1/3))/x)/((1 + Sqrt[3])*a^(1/3) + b^(1/3)/x)^2]*EllipticF[ArcSin[((
1 - Sqrt[3])*a^(1/3) + b^(1/3)/x)/((1 + Sqrt[3])*a^(1/3) + b^(1/3)/x)], -7 - 4*S
qrt[3]])/(1729*3^(1/4)*b^(11/3)*Sqrt[a + b/x^3]*Sqrt[(a^(1/3)*(a^(1/3) + b^(1/3)
/x))/((1 + Sqrt[3])*a^(1/3) + b^(1/3)/x)^2])

_______________________________________________________________________________________

Rubi [A]  time = 0.945351, antiderivative size = 568, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{1280 \sqrt{2} a^{10/3} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right ) \sqrt{\frac{a^{2/3}-\frac{\sqrt [3]{a} \sqrt [3]{b}}{x}+\frac{b^{2/3}}{x^2}}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}}\right )|-7-4 \sqrt{3}\right )}{1729 \sqrt [4]{3} b^{11/3} \sqrt{a+\frac{b}{x^3}} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )^2}}}-\frac{640 \sqrt [4]{3} \sqrt{2-\sqrt{3}} a^{10/3} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right ) \sqrt{\frac{a^{2/3}-\frac{\sqrt [3]{a} \sqrt [3]{b}}{x}+\frac{b^{2/3}}{x^2}}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}}\right )|-7-4 \sqrt{3}\right )}{1729 b^{11/3} \sqrt{a+\frac{b}{x^3}} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )^2}}}+\frac{1280 a^3 \sqrt{a+\frac{b}{x^3}}}{1729 b^{11/3} \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )}-\frac{320 a^2 \sqrt{a+\frac{b}{x^3}}}{1729 b^3 x^2}+\frac{32 a \sqrt{a+\frac{b}{x^3}}}{247 b^2 x^5}-\frac{2 \sqrt{a+\frac{b}{x^3}}}{19 b x^8} \]

Antiderivative was successfully verified.

[In]  Int[1/(Sqrt[a + b/x^3]*x^12),x]

[Out]

(1280*a^3*Sqrt[a + b/x^3])/(1729*b^(11/3)*((1 + Sqrt[3])*a^(1/3) + b^(1/3)/x)) -
(2*Sqrt[a + b/x^3])/(19*b*x^8) + (32*a*Sqrt[a + b/x^3])/(247*b^2*x^5) - (320*a^
2*Sqrt[a + b/x^3])/(1729*b^3*x^2) - (640*3^(1/4)*Sqrt[2 - Sqrt[3]]*a^(10/3)*(a^(
1/3) + b^(1/3)/x)*Sqrt[(a^(2/3) + b^(2/3)/x^2 - (a^(1/3)*b^(1/3))/x)/((1 + Sqrt[
3])*a^(1/3) + b^(1/3)/x)^2]*EllipticE[ArcSin[((1 - Sqrt[3])*a^(1/3) + b^(1/3)/x)
/((1 + Sqrt[3])*a^(1/3) + b^(1/3)/x)], -7 - 4*Sqrt[3]])/(1729*b^(11/3)*Sqrt[a +
b/x^3]*Sqrt[(a^(1/3)*(a^(1/3) + b^(1/3)/x))/((1 + Sqrt[3])*a^(1/3) + b^(1/3)/x)^
2]) + (1280*Sqrt[2]*a^(10/3)*(a^(1/3) + b^(1/3)/x)*Sqrt[(a^(2/3) + b^(2/3)/x^2 -
(a^(1/3)*b^(1/3))/x)/((1 + Sqrt[3])*a^(1/3) + b^(1/3)/x)^2]*EllipticF[ArcSin[((
1 - Sqrt[3])*a^(1/3) + b^(1/3)/x)/((1 + Sqrt[3])*a^(1/3) + b^(1/3)/x)], -7 - 4*S
qrt[3]])/(1729*3^(1/4)*b^(11/3)*Sqrt[a + b/x^3]*Sqrt[(a^(1/3)*(a^(1/3) + b^(1/3)
/x))/((1 + Sqrt[3])*a^(1/3) + b^(1/3)/x)^2])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 57.74, size = 479, normalized size = 0.84 \[ - \frac{640 \sqrt [4]{3} a^{\frac{10}{3}} \sqrt{\frac{a^{\frac{2}{3}} - \frac{\sqrt [3]{a} \sqrt [3]{b}}{x} + \frac{b^{\frac{2}{3}}}{x^{2}}}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \frac{\sqrt [3]{b}}{x}\right )^{2}}} \sqrt{- \sqrt{3} + 2} \left (\sqrt [3]{a} + \frac{\sqrt [3]{b}}{x}\right ) E\left (\operatorname{asin}{\left (\frac{- \sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \frac{\sqrt [3]{b}}{x}}{\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \frac{\sqrt [3]{b}}{x}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{1729 b^{\frac{11}{3}} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a} + \frac{\sqrt [3]{b}}{x}\right )}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \frac{\sqrt [3]{b}}{x}\right )^{2}}} \sqrt{a + \frac{b}{x^{3}}}} + \frac{1280 \sqrt{2} \cdot 3^{\frac{3}{4}} a^{\frac{10}{3}} \sqrt{\frac{a^{\frac{2}{3}} - \frac{\sqrt [3]{a} \sqrt [3]{b}}{x} + \frac{b^{\frac{2}{3}}}{x^{2}}}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \frac{\sqrt [3]{b}}{x}\right )^{2}}} \left (\sqrt [3]{a} + \frac{\sqrt [3]{b}}{x}\right ) F\left (\operatorname{asin}{\left (\frac{- \sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \frac{\sqrt [3]{b}}{x}}{\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \frac{\sqrt [3]{b}}{x}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{5187 b^{\frac{11}{3}} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a} + \frac{\sqrt [3]{b}}{x}\right )}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \frac{\sqrt [3]{b}}{x}\right )^{2}}} \sqrt{a + \frac{b}{x^{3}}}} + \frac{1280 a^{3} \sqrt{a + \frac{b}{x^{3}}}}{1729 b^{\frac{11}{3}} \left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \frac{\sqrt [3]{b}}{x}\right )} - \frac{320 a^{2} \sqrt{a + \frac{b}{x^{3}}}}{1729 b^{3} x^{2}} + \frac{32 a \sqrt{a + \frac{b}{x^{3}}}}{247 b^{2} x^{5}} - \frac{2 \sqrt{a + \frac{b}{x^{3}}}}{19 b x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**12/(a+b/x**3)**(1/2),x)

[Out]

-640*3**(1/4)*a**(10/3)*sqrt((a**(2/3) - a**(1/3)*b**(1/3)/x + b**(2/3)/x**2)/(a
**(1/3)*(1 + sqrt(3)) + b**(1/3)/x)**2)*sqrt(-sqrt(3) + 2)*(a**(1/3) + b**(1/3)/
x)*elliptic_e(asin((-a**(1/3)*(-1 + sqrt(3)) + b**(1/3)/x)/(a**(1/3)*(1 + sqrt(3
)) + b**(1/3)/x)), -7 - 4*sqrt(3))/(1729*b**(11/3)*sqrt(a**(1/3)*(a**(1/3) + b**
(1/3)/x)/(a**(1/3)*(1 + sqrt(3)) + b**(1/3)/x)**2)*sqrt(a + b/x**3)) + 1280*sqrt
(2)*3**(3/4)*a**(10/3)*sqrt((a**(2/3) - a**(1/3)*b**(1/3)/x + b**(2/3)/x**2)/(a*
*(1/3)*(1 + sqrt(3)) + b**(1/3)/x)**2)*(a**(1/3) + b**(1/3)/x)*elliptic_f(asin((
-a**(1/3)*(-1 + sqrt(3)) + b**(1/3)/x)/(a**(1/3)*(1 + sqrt(3)) + b**(1/3)/x)), -
7 - 4*sqrt(3))/(5187*b**(11/3)*sqrt(a**(1/3)*(a**(1/3) + b**(1/3)/x)/(a**(1/3)*(
1 + sqrt(3)) + b**(1/3)/x)**2)*sqrt(a + b/x**3)) + 1280*a**3*sqrt(a + b/x**3)/(1
729*b**(11/3)*(a**(1/3)*(1 + sqrt(3)) + b**(1/3)/x)) - 320*a**2*sqrt(a + b/x**3)
/(1729*b**3*x**2) + 32*a*sqrt(a + b/x**3)/(247*b**2*x**5) - 2*sqrt(a + b/x**3)/(
19*b*x**8)

_______________________________________________________________________________________

Mathematica [C]  time = 1.8892, size = 387, normalized size = 0.68 \[ \frac{2 \left (-640 a^{10/3} x \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )-\frac{320 (-1)^{2/3} a^3 \sqrt [3]{b} \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )^2 \sqrt{\frac{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a} x \left (\sqrt [3]{b}-\sqrt [3]{-1} \sqrt [3]{a} x\right )}{\left (\sqrt [3]{a} x+\sqrt [3]{b}\right )^2}} \sqrt{\frac{(-1)^{2/3} \sqrt [3]{a} x+\sqrt [3]{b}}{\sqrt [3]{a} x+\sqrt [3]{b}}} \left (\left (1+i \sqrt{3}\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{\left (3+i \sqrt{3}\right ) \sqrt [3]{a} x}{\sqrt [3]{a} x+\sqrt [3]{b}}}}{\sqrt{2}}\right )|\frac{-i+\sqrt{3}}{i+\sqrt{3}}\right )+\left (-3-i \sqrt{3}\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{\left (3+i \sqrt{3}\right ) \sqrt [3]{a} x}{\sqrt [3]{a} x+\sqrt [3]{b}}}}{\sqrt{2}}\right )|\frac{-i+\sqrt{3}}{i+\sqrt{3}}\right )\right )}{(-1)^{2/3}-1}+\frac{\left (a x^3+b\right ) \left (640 a^3 x^9-160 a^2 b x^6+112 a b^2 x^3-91 b^3\right )}{x^9}\right )}{1729 b^4 x^2 \sqrt{a+\frac{b}{x^3}}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[1/(Sqrt[a + b/x^3]*x^12),x]

[Out]

(2*(-640*a^(10/3)*x*(b^(2/3) - a^(1/3)*b^(1/3)*x + a^(2/3)*x^2) + ((b + a*x^3)*(
-91*b^3 + 112*a*b^2*x^3 - 160*a^2*b*x^6 + 640*a^3*x^9))/x^9 - (320*(-1)^(2/3)*a^
3*b^(1/3)*(b^(1/3) + a^(1/3)*x)^2*Sqrt[((1 + (-1)^(1/3))*a^(1/3)*x*(b^(1/3) - (-
1)^(1/3)*a^(1/3)*x))/(b^(1/3) + a^(1/3)*x)^2]*Sqrt[(b^(1/3) + (-1)^(2/3)*a^(1/3)
*x)/(b^(1/3) + a^(1/3)*x)]*((-3 - I*Sqrt[3])*EllipticE[ArcSin[Sqrt[((3 + I*Sqrt[
3])*a^(1/3)*x)/(b^(1/3) + a^(1/3)*x)]/Sqrt[2]], (-I + Sqrt[3])/(I + Sqrt[3])] +
(1 + I*Sqrt[3])*EllipticF[ArcSin[Sqrt[((3 + I*Sqrt[3])*a^(1/3)*x)/(b^(1/3) + a^(
1/3)*x)]/Sqrt[2]], (-I + Sqrt[3])/(I + Sqrt[3])]))/(-1 + (-1)^(2/3))))/(1729*b^4
*Sqrt[a + b/x^3]*x^2)

_______________________________________________________________________________________

Maple [B]  time = 0.029, size = 3779, normalized size = 6.7 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^12/(a+b/x^3)^(1/2),x)

[Out]

2/5187/((a*x^3+b)/x^3)^(1/2)/x^12*(-3840*I*(-a^2*b)^(1/3)*(-(I*3^(1/2)-3)*x*a/(I
*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*((I*3^(1/2)*(-a^2*b)^(1/3)+2*a*x+(-a^2*
b)^(1/3))/(I*3^(1/2)+1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*((I*3^(1/2)*(-a^2*b)^(1/3)-
2*a*x-(-a^2*b)^(1/3))/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*EllipticE((-(I*
3^(1/2)-3)*x*a/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2),((I*3^(1/2)+3)*(I*3^(1
/2)-1)/(I*3^(1/2)+1)/(I*3^(1/2)-3))^(1/2))*3^(1/2)*(x*(a*x^3+b))^(1/2)*x^12*a^3+
7680*I*(-a^2*b)^(2/3)*(-(I*3^(1/2)-3)*x*a/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(
1/2)*((I*3^(1/2)*(-a^2*b)^(1/3)+2*a*x+(-a^2*b)^(1/3))/(I*3^(1/2)+1)/(-a*x+(-a^2*
b)^(1/3)))^(1/2)*((I*3^(1/2)*(-a^2*b)^(1/3)-2*a*x-(-a^2*b)^(1/3))/(I*3^(1/2)-1)/
(-a*x+(-a^2*b)^(1/3)))^(1/2)*EllipticE((-(I*3^(1/2)-3)*x*a/(I*3^(1/2)-1)/(-a*x+(
-a^2*b)^(1/3)))^(1/2),((I*3^(1/2)+3)*(I*3^(1/2)-1)/(I*3^(1/2)+1)/(I*3^(1/2)-3))^
(1/2))*3^(1/2)*(x*(a*x^3+b))^(1/2)*x^11*a^2+3840*I*a^3*b*(-(I*3^(1/2)-3)*x*a/(I*
3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*((I*3^(1/2)*(-a^2*b)^(1/3)+2*a*x+(-a^2*b
)^(1/3))/(I*3^(1/2)+1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*((I*3^(1/2)*(-a^2*b)^(1/3)-2
*a*x-(-a^2*b)^(1/3))/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*EllipticE((-(I*3
^(1/2)-3)*x*a/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2),((I*3^(1/2)+3)*(I*3^(1/
2)-1)/(I*3^(1/2)+1)/(I*3^(1/2)-3))^(1/2))*3^(1/2)*(x*(a*x^3+b))^(1/2)*x^10-3840*
I*(-a^2*b)^(1/3)*3^(1/2)*(x*(a*x^3+b))^(1/2)*x^12*a^3+1280*I*(1/a^2*x*(-a*x+(-a^
2*b)^(1/3))*(I*3^(1/2)*(-a^2*b)^(1/3)+2*a*x+(-a^2*b)^(1/3))*(I*3^(1/2)*(-a^2*b)^
(1/3)-2*a*x-(-a^2*b)^(1/3)))^(1/2)*3^(1/2)*x^10*a^3*b-273*I*(a*x^4+b*x)^(1/2)*(1
/a^2*x*(-a*x+(-a^2*b)^(1/3))*(I*3^(1/2)*(-a^2*b)^(1/3)+2*a*x+(-a^2*b)^(1/3))*(I*
3^(1/2)*(-a^2*b)^(1/3)-2*a*x-(-a^2*b)^(1/3)))^(1/2)*3^(1/2)*(x*(a*x^3+b))^(1/2)*
b^3-3840*I*(-a^2*b)^(2/3)*3^(1/2)*(x*(a*x^3+b))^(1/2)*x^11*a^2+11520*(-a^2*b)^(2
/3)*(x*(a*x^3+b))^(1/2)*x^11*a^2+11520*(-a^2*b)^(1/3)*(x*(a*x^3+b))^(1/2)*x^12*a
^3-3840*(1/a^2*x*(-a*x+(-a^2*b)^(1/3))*(I*3^(1/2)*(-a^2*b)^(1/3)+2*a*x+(-a^2*b)^
(1/3))*(I*3^(1/2)*(-a^2*b)^(1/3)-2*a*x-(-a^2*b)^(1/3)))^(1/2)*x^10*a^3*b+819*(a*
x^4+b*x)^(1/2)*b^3*(x*(a*x^3+b))^(1/2)*(1/a^2*x*(-a*x+(-a^2*b)^(1/3))*(I*3^(1/2)
*(-a^2*b)^(1/3)+2*a*x+(-a^2*b)^(1/3))*(I*3^(1/2)*(-a^2*b)^(1/3)-2*a*x-(-a^2*b)^(
1/3)))^(1/2)+1440*a^2*(a*x^4+b*x)^(1/2)*x^6*b*(x*(a*x^3+b))^(1/2)*(1/a^2*x*(-a*x
+(-a^2*b)^(1/3))*(I*3^(1/2)*(-a^2*b)^(1/3)+2*a*x+(-a^2*b)^(1/3))*(I*3^(1/2)*(-a^
2*b)^(1/3)-2*a*x-(-a^2*b)^(1/3)))^(1/2)-3840*(1/a^2*x*(-a*x+(-a^2*b)^(1/3))*(I*3
^(1/2)*(-a^2*b)^(1/3)+2*a*x+(-a^2*b)^(1/3))*(I*3^(1/2)*(-a^2*b)^(1/3)-2*a*x-(-a^
2*b)^(1/3)))^(1/2)*x^13*a^4+11520*(x*(a*x^3+b))^(1/2)*x^13*a^4-1920*a^3*(a*x^4+b
*x)^(1/2)*x^9*(x*(a*x^3+b))^(1/2)*(1/a^2*x*(-a*x+(-a^2*b)^(1/3))*(I*3^(1/2)*(-a^
2*b)^(1/3)+2*a*x+(-a^2*b)^(1/3))*(I*3^(1/2)*(-a^2*b)^(1/3)-2*a*x-(-a^2*b)^(1/3))
)^(1/2)-1008*a*(a*x^4+b*x)^(1/2)*x^3*b^2*(x*(a*x^3+b))^(1/2)*(1/a^2*x*(-a*x+(-a^
2*b)^(1/3))*(I*3^(1/2)*(-a^2*b)^(1/3)+2*a*x+(-a^2*b)^(1/3))*(I*3^(1/2)*(-a^2*b)^
(1/3)-2*a*x-(-a^2*b)^(1/3)))^(1/2)+640*I*(a*x^4+b*x)^(1/2)*(1/a^2*x*(-a*x+(-a^2*
b)^(1/3))*(I*3^(1/2)*(-a^2*b)^(1/3)+2*a*x+(-a^2*b)^(1/3))*(I*3^(1/2)*(-a^2*b)^(1
/3)-2*a*x-(-a^2*b)^(1/3)))^(1/2)*3^(1/2)*(x*(a*x^3+b))^(1/2)*x^9*a^3+336*I*(a*x^
4+b*x)^(1/2)*(1/a^2*x*(-a*x+(-a^2*b)^(1/3))*(I*3^(1/2)*(-a^2*b)^(1/3)+2*a*x+(-a^
2*b)^(1/3))*(I*3^(1/2)*(-a^2*b)^(1/3)-2*a*x-(-a^2*b)^(1/3)))^(1/2)*3^(1/2)*(x*(a
*x^3+b))^(1/2)*x^3*a*b^2+1280*I*(1/a^2*x*(-a*x+(-a^2*b)^(1/3))*(I*3^(1/2)*(-a^2*
b)^(1/3)+2*a*x+(-a^2*b)^(1/3))*(I*3^(1/2)*(-a^2*b)^(1/3)-2*a*x-(-a^2*b)^(1/3)))^
(1/2)*3^(1/2)*x^13*a^4-3840*I*3^(1/2)*(x*(a*x^3+b))^(1/2)*x^13*a^4-23040*(-(I*3^
(1/2)-3)*x*a/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*((I*3^(1/2)*(-a^2*b)^(1/
3)+2*a*x+(-a^2*b)^(1/3))/(I*3^(1/2)+1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*((I*3^(1/2)*
(-a^2*b)^(1/3)-2*a*x-(-a^2*b)^(1/3))/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*
EllipticE((-(I*3^(1/2)-3)*x*a/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2),((I*3^(
1/2)+3)*(I*3^(1/2)-1)/(I*3^(1/2)+1)/(I*3^(1/2)-3))^(1/2))*(-a^2*b)^(2/3)*(x*(a*x
^3+b))^(1/2)*x^11*a^2-7680*(-(I*3^(1/2)-3)*x*a/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3
)))^(1/2)*((I*3^(1/2)*(-a^2*b)^(1/3)+2*a*x+(-a^2*b)^(1/3))/(I*3^(1/2)+1)/(-a*x+(
-a^2*b)^(1/3)))^(1/2)*((I*3^(1/2)*(-a^2*b)^(1/3)-2*a*x-(-a^2*b)^(1/3))/(I*3^(1/2
)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*EllipticF((-(I*3^(1/2)-3)*x*a/(I*3^(1/2)-1)/(-
a*x+(-a^2*b)^(1/3)))^(1/2),((I*3^(1/2)+3)*(I*3^(1/2)-1)/(I*3^(1/2)+1)/(I*3^(1/2)
-3))^(1/2))*(-a^2*b)^(1/3)*(x*(a*x^3+b))^(1/2)*x^12*a^3+7680*(-(I*3^(1/2)-3)*x*a
/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*((I*3^(1/2)*(-a^2*b)^(1/3)+2*a*x+(-a
^2*b)^(1/3))/(I*3^(1/2)+1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*((I*3^(1/2)*(-a^2*b)^(1/
3)-2*a*x-(-a^2*b)^(1/3))/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*EllipticF((-
(I*3^(1/2)-3)*x*a/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2),((I*3^(1/2)+3)*(I*3
^(1/2)-1)/(I*3^(1/2)+1)/(I*3^(1/2)-3))^(1/2))*a^3*b*(x*(a*x^3+b))^(1/2)*x^10-115
20*(-(I*3^(1/2)-3)*x*a/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*((I*3^(1/2)*(-
a^2*b)^(1/3)+2*a*x+(-a^2*b)^(1/3))/(I*3^(1/2)+1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*((
I*3^(1/2)*(-a^2*b)^(1/3)-2*a*x-(-a^2*b)^(1/3))/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3
)))^(1/2)*EllipticE((-(I*3^(1/2)-3)*x*a/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/
2),((I*3^(1/2)+3)*(I*3^(1/2)-1)/(I*3^(1/2)+1)/(I*3^(1/2)-3))^(1/2))*a^3*b*(x*(a*
x^3+b))^(1/2)*x^10+11520*(-(I*3^(1/2)-3)*x*a/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3))
)^(1/2)*((I*3^(1/2)*(-a^2*b)^(1/3)+2*a*x+(-a^2*b)^(1/3))/(I*3^(1/2)+1)/(-a*x+(-a
^2*b)^(1/3)))^(1/2)*((I*3^(1/2)*(-a^2*b)^(1/3)-2*a*x-(-a^2*b)^(1/3))/(I*3^(1/2)-
1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*EllipticE((-(I*3^(1/2)-3)*x*a/(I*3^(1/2)-1)/(-a*
x+(-a^2*b)^(1/3)))^(1/2),((I*3^(1/2)+3)*(I*3^(1/2)-1)/(I*3^(1/2)+1)/(I*3^(1/2)-3
))^(1/2))*(-a^2*b)^(1/3)*(x*(a*x^3+b))^(1/2)*x^12*a^3+15360*(-(I*3^(1/2)-3)*x*a/
(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*((I*3^(1/2)*(-a^2*b)^(1/3)+2*a*x+(-a^
2*b)^(1/3))/(I*3^(1/2)+1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*((I*3^(1/2)*(-a^2*b)^(1/3
)-2*a*x-(-a^2*b)^(1/3))/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*EllipticF((-(
I*3^(1/2)-3)*x*a/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2),((I*3^(1/2)+3)*(I*3^
(1/2)-1)/(I*3^(1/2)+1)/(I*3^(1/2)-3))^(1/2))*(-a^2*b)^(2/3)*(x*(a*x^3+b))^(1/2)*
x^11*a^2-480*I*(a*x^4+b*x)^(1/2)*(1/a^2*x*(-a*x+(-a^2*b)^(1/3))*(I*3^(1/2)*(-a^2
*b)^(1/3)+2*a*x+(-a^2*b)^(1/3))*(I*3^(1/2)*(-a^2*b)^(1/3)-2*a*x-(-a^2*b)^(1/3)))
^(1/2)*3^(1/2)*(x*(a*x^3+b))^(1/2)*x^6*a^2*b)/b^4/(I*3^(1/2)-3)/(1/a^2*x*(-a*x+(
-a^2*b)^(1/3))*(I*3^(1/2)*(-a^2*b)^(1/3)+2*a*x+(-a^2*b)^(1/3))*(I*3^(1/2)*(-a^2*
b)^(1/3)-2*a*x-(-a^2*b)^(1/3)))^(1/2)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{a + \frac{b}{x^{3}}} x^{12}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(a + b/x^3)*x^12),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(a + b/x^3)*x^12), x)

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{x^{12} \sqrt{\frac{a x^{3} + b}{x^{3}}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(a + b/x^3)*x^12),x, algorithm="fricas")

[Out]

integral(1/(x^12*sqrt((a*x^3 + b)/x^3)), x)

_______________________________________________________________________________________

Sympy [A]  time = 18.8263, size = 39, normalized size = 0.07 \[ - \frac{\Gamma \left (\frac{11}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{11}{3} \\ \frac{14}{3} \end{matrix}\middle |{\frac{b e^{i \pi }}{a x^{3}}} \right )}}{3 \sqrt{a} x^{11} \Gamma \left (\frac{14}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**12/(a+b/x**3)**(1/2),x)

[Out]

-gamma(11/3)*hyper((1/2, 11/3), (14/3,), b*exp_polar(I*pi)/(a*x**3))/(3*sqrt(a)*
x**11*gamma(14/3))

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{a + \frac{b}{x^{3}}} x^{12}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(a + b/x^3)*x^12),x, algorithm="giac")

[Out]

integrate(1/(sqrt(a + b/x^3)*x^12), x)