3.2054 \(\int \frac{1}{\left (a+\frac{b}{x^3}\right )^{3/2} x^{12}} \, dx\)

Optimal. Leaf size=565 \[ -\frac{1280 \sqrt{2} a^{7/3} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right ) \sqrt{\frac{a^{2/3}-\frac{\sqrt [3]{a} \sqrt [3]{b}}{x}+\frac{b^{2/3}}{x^2}}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}}\right )|-7-4 \sqrt{3}\right )}{273 \sqrt [4]{3} b^{11/3} \sqrt{a+\frac{b}{x^3}} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )^2}}}+\frac{640 \sqrt{2-\sqrt{3}} a^{7/3} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right ) \sqrt{\frac{a^{2/3}-\frac{\sqrt [3]{a} \sqrt [3]{b}}{x}+\frac{b^{2/3}}{x^2}}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}}\right )|-7-4 \sqrt{3}\right )}{91\ 3^{3/4} b^{11/3} \sqrt{a+\frac{b}{x^3}} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )^2}}}-\frac{1280 a^2 \sqrt{a+\frac{b}{x^3}}}{273 b^{11/3} \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )}+\frac{320 a \sqrt{a+\frac{b}{x^3}}}{273 b^3 x^2}-\frac{32 \sqrt{a+\frac{b}{x^3}}}{39 b^2 x^5}+\frac{2}{3 b x^8 \sqrt{a+\frac{b}{x^3}}} \]

[Out]

(-1280*a^2*Sqrt[a + b/x^3])/(273*b^(11/3)*((1 + Sqrt[3])*a^(1/3) + b^(1/3)/x)) +
2/(3*b*Sqrt[a + b/x^3]*x^8) - (32*Sqrt[a + b/x^3])/(39*b^2*x^5) + (320*a*Sqrt[a
+ b/x^3])/(273*b^3*x^2) + (640*Sqrt[2 - Sqrt[3]]*a^(7/3)*(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]])/(91*3^(3/4)*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]) - (1280*
Sqrt[2]*a^(7/3)*(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*Sqrt[3]])/(273
*3^(1/4)*b^(11/3)*Sqrt[a + b/x^3]*Sqrt[(a^(1/3)*(a^(1/3) + b^(1/3)/x))/((1 + Sqr
t[3])*a^(1/3) + b^(1/3)/x)^2])

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Rubi [A]  time = 0.950395, antiderivative size = 565, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ -\frac{1280 \sqrt{2} a^{7/3} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right ) \sqrt{\frac{a^{2/3}-\frac{\sqrt [3]{a} \sqrt [3]{b}}{x}+\frac{b^{2/3}}{x^2}}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}}\right )|-7-4 \sqrt{3}\right )}{273 \sqrt [4]{3} b^{11/3} \sqrt{a+\frac{b}{x^3}} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )^2}}}+\frac{640 \sqrt{2-\sqrt{3}} a^{7/3} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right ) \sqrt{\frac{a^{2/3}-\frac{\sqrt [3]{a} \sqrt [3]{b}}{x}+\frac{b^{2/3}}{x^2}}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}}\right )|-7-4 \sqrt{3}\right )}{91\ 3^{3/4} b^{11/3} \sqrt{a+\frac{b}{x^3}} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )^2}}}-\frac{1280 a^2 \sqrt{a+\frac{b}{x^3}}}{273 b^{11/3} \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )}+\frac{320 a \sqrt{a+\frac{b}{x^3}}}{273 b^3 x^2}-\frac{32 \sqrt{a+\frac{b}{x^3}}}{39 b^2 x^5}+\frac{2}{3 b x^8 \sqrt{a+\frac{b}{x^3}}} \]

Antiderivative was successfully verified.

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

[Out]

(-1280*a^2*Sqrt[a + b/x^3])/(273*b^(11/3)*((1 + Sqrt[3])*a^(1/3) + b^(1/3)/x)) +
2/(3*b*Sqrt[a + b/x^3]*x^8) - (32*Sqrt[a + b/x^3])/(39*b^2*x^5) + (320*a*Sqrt[a
+ b/x^3])/(273*b^3*x^2) + (640*Sqrt[2 - Sqrt[3]]*a^(7/3)*(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]])/(91*3^(3/4)*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]) - (1280*
Sqrt[2]*a^(7/3)*(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*Sqrt[3]])/(273
*3^(1/4)*b^(11/3)*Sqrt[a + b/x^3]*Sqrt[(a^(1/3)*(a^(1/3) + b^(1/3)/x))/((1 + Sqr
t[3])*a^(1/3) + b^(1/3)/x)^2])

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Rubi in Sympy [A]  time = 57.6374, size = 476, normalized size = 0.84 \[ \frac{640 \sqrt [4]{3} a^{\frac{7}{3}} \sqrt{\frac{a^{\frac{2}{3}} - \frac{\sqrt [3]{a} \sqrt [3]{b}}{x} + \frac{b^{\frac{2}{3}}}{x^{2}}}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \frac{\sqrt [3]{b}}{x}\right )^{2}}} \sqrt{- \sqrt{3} + 2} \left (\sqrt [3]{a} + \frac{\sqrt [3]{b}}{x}\right ) E\left (\operatorname{asin}{\left (\frac{- \sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \frac{\sqrt [3]{b}}{x}}{\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \frac{\sqrt [3]{b}}{x}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{273 b^{\frac{11}{3}} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a} + \frac{\sqrt [3]{b}}{x}\right )}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \frac{\sqrt [3]{b}}{x}\right )^{2}}} \sqrt{a + \frac{b}{x^{3}}}} - \frac{1280 \sqrt{2} \cdot 3^{\frac{3}{4}} a^{\frac{7}{3}} \sqrt{\frac{a^{\frac{2}{3}} - \frac{\sqrt [3]{a} \sqrt [3]{b}}{x} + \frac{b^{\frac{2}{3}}}{x^{2}}}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \frac{\sqrt [3]{b}}{x}\right )^{2}}} \left (\sqrt [3]{a} + \frac{\sqrt [3]{b}}{x}\right ) F\left (\operatorname{asin}{\left (\frac{- \sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \frac{\sqrt [3]{b}}{x}}{\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \frac{\sqrt [3]{b}}{x}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{819 b^{\frac{11}{3}} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a} + \frac{\sqrt [3]{b}}{x}\right )}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \frac{\sqrt [3]{b}}{x}\right )^{2}}} \sqrt{a + \frac{b}{x^{3}}}} - \frac{1280 a^{2} \sqrt{a + \frac{b}{x^{3}}}}{273 b^{\frac{11}{3}} \left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \frac{\sqrt [3]{b}}{x}\right )} + \frac{320 a \sqrt{a + \frac{b}{x^{3}}}}{273 b^{3} x^{2}} + \frac{2}{3 b x^{8} \sqrt{a + \frac{b}{x^{3}}}} - \frac{32 \sqrt{a + \frac{b}{x^{3}}}}{39 b^{2} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

640*3**(1/4)*a**(7/3)*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))/(273*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**(7/3)*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))/(819*b**(11/3)*sqrt(a**(1/3)*(a**(1/3) + b**(1/3)/x)/(a**(1/3)*(1 + s
qrt(3)) + b**(1/3)/x)**2)*sqrt(a + b/x**3)) - 1280*a**2*sqrt(a + b/x**3)/(273*b*
*(11/3)*(a**(1/3)*(1 + sqrt(3)) + b**(1/3)/x)) + 320*a*sqrt(a + b/x**3)/(273*b**
3*x**2) + 2/(3*b*x**8*sqrt(a + b/x**3)) - 32*sqrt(a + b/x**3)/(39*b**2*x**5)

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Mathematica [C]  time = 1.92368, size = 400, normalized size = 0.71 \[ \frac{2 \left (a x^3+b\right ) \left (640 a^{7/3} x \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )-91 a^3 x^3-549 a^2 \left (a x^3+b\right )+\frac{320 (-1)^{2/3} a^2 \sqrt [3]{b} \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )^2 \sqrt{\frac{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a} x \left (\sqrt [3]{b}-\sqrt [3]{-1} \sqrt [3]{a} x\right )}{\left (\sqrt [3]{a} x+\sqrt [3]{b}\right )^2}} \sqrt{\frac{(-1)^{2/3} \sqrt [3]{a} x+\sqrt [3]{b}}{\sqrt [3]{a} x+\sqrt [3]{b}}} \left (\left (1+i \sqrt{3}\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{\left (3+i \sqrt{3}\right ) \sqrt [3]{a} x}{\sqrt [3]{a} x+\sqrt [3]{b}}}}{\sqrt{2}}\right )|\frac{-i+\sqrt{3}}{i+\sqrt{3}}\right )+\left (-3-i \sqrt{3}\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{\left (3+i \sqrt{3}\right ) \sqrt [3]{a} x}{\sqrt [3]{a} x+\sqrt [3]{b}}}}{\sqrt{2}}\right )|\frac{-i+\sqrt{3}}{i+\sqrt{3}}\right )\right )}{(-1)^{2/3}-1}-\frac{21 b^2 \left (a x^3+b\right )}{x^6}+\frac{69 a b \left (a x^3+b\right )}{x^3}\right )}{273 b^4 x^5 \left (a+\frac{b}{x^3}\right )^{3/2}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(2*(b + a*x^3)*(-91*a^3*x^3 + 640*a^(7/3)*x*(b^(2/3) - a^(1/3)*b^(1/3)*x + a^(2/
3)*x^2) - 549*a^2*(b + a*x^3) - (21*b^2*(b + a*x^3))/x^6 + (69*a*b*(b + a*x^3))/
x^3 + (320*(-1)^(2/3)*a^2*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[A
rcSin[Sqrt[((3 + I*Sqrt[3])*a^(1/3)*x)/(b^(1/3) + a^(1/3)*x)]/Sqrt[2]], (-I + Sq
rt[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))))/(273*b^4*(a + b/x^3)^(3/2)*x^5)

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Maple [B]  time = 0.032, size = 3554, normalized size = 6.3 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x^3)^(3/2)/x^12,x)

[Out]

-2/273/((a*x^3+b)/x^3)^(3/2)/x^12*(a*x^3+b)*(-1280*I*(-a^2*b)^(1/3)*3^(1/2)*(x*(
a*x^3+b))^(1/2)*x^9*a^2+457*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^7*a^2*b-1280*I*(-a^2*b)^(2/3)*3^(1/2)*(x*(a*x^3+b))^(1/2)*x^8*a+38
40*(-a^2*b)^(2/3)*(x*(a*x^3+b))^(1/2)*x^8*a+3840*(-a^2*b)^(1/3)*(x*(a*x^3+b))^(1
/2)*x^9*a^2-1371*(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^7*a^2*b
-63*(a*x^4+b*x)^(1/2)*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)+548*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^3-1280*I*3^(1/2)*(x*(a*x^3+b))^(1/2)*x^10*a^3+207*a*(a*x^4+b*x)^(
1/2)*x^3*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)+92*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^2-276*a^2*(a*x^4+b*x)^(1/2)*x^6*(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)-1644*(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+3840*(x*(a*x^3+b))^(1/2)*x^10*
a^3+21*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^2-1280*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^9*a^2+2560
*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^8*a+1280*I*a^2*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^7-2560*(-(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)*E
llipticF((-(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^9*a^2+3840*(-(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^9*a^2+5120*(-(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^
8*a-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)*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^8*a+2560*(-(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*(x*(a*x^3+b))^(1/2)*x^7-3840*(-(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*(x*(a*x^3+b))^(1/2)*x^7-69*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)/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)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (a + \frac{b}{x^{3}}\right )}^{\frac{3}{2}} x^{12}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^3)^(3/2)*x^12),x, algorithm="maxima")

[Out]

integrate(1/((a + b/x^3)^(3/2)*x^12), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (a x^{12} + b x^{9}\right )} \sqrt{\frac{a x^{3} + b}{x^{3}}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^3)^(3/2)*x^12),x, algorithm="fricas")

[Out]

integral(1/((a*x^12 + b*x^9)*sqrt((a*x^3 + b)/x^3)), x)

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Sympy [A]  time = 31.6769, size = 39, normalized size = 0.07 \[ - \frac{\Gamma \left (\frac{11}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{2}, \frac{11}{3} \\ \frac{14}{3} \end{matrix}\middle |{\frac{b e^{i \pi }}{a x^{3}}} \right )}}{3 a^{\frac{3}{2}} x^{11} \Gamma \left (\frac{14}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (a + \frac{b}{x^{3}}\right )}^{\frac{3}{2}} x^{12}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^3)^(3/2)*x^12),x, algorithm="giac")

[Out]

integrate(1/((a + b/x^3)^(3/2)*x^12), x)