3.108 \(\int \frac{x^5}{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx\)

Optimal. Leaf size=78 \[ \frac{a}{12 b^2 \left (a+b x^3\right )^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{1}{9 b^2 \left (a+b x^3\right )^2 \sqrt{a^2+2 a b x^3+b^2 x^6}} \]

[Out]

a/(12*b^2*(a + b*x^3)^3*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) - 1/(9*b^2*(a + b*x^3)^
2*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])

_______________________________________________________________________________________

Rubi [A]  time = 0.124285, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ \frac{a}{12 b^2 \left (a+b x^3\right )^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{1}{9 b^2 \left (a+b x^3\right )^2 \sqrt{a^2+2 a b x^3+b^2 x^6}} \]

Antiderivative was successfully verified.

[In]  Int[x^5/(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2),x]

[Out]

a/(12*b^2*(a + b*x^3)^3*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) - 1/(9*b^2*(a + b*x^3)^
2*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 14.879, size = 66, normalized size = 0.85 \[ \frac{a \left (2 a + 2 b x^{3}\right )}{24 b^{2} \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{5}{2}}} - \frac{1}{9 b^{2} \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5/(b**2*x**6+2*a*b*x**3+a**2)**(5/2),x)

[Out]

a*(2*a + 2*b*x**3)/(24*b**2*(a**2 + 2*a*b*x**3 + b**2*x**6)**(5/2)) - 1/(9*b**2*
(a**2 + 2*a*b*x**3 + b**2*x**6)**(3/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0335301, size = 39, normalized size = 0.5 \[ \frac{-a-4 b x^3}{36 b^2 \left (a+b x^3\right )^3 \sqrt{\left (a+b x^3\right )^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^5/(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2),x]

[Out]

(-a - 4*b*x^3)/(36*b^2*(a + b*x^3)^3*Sqrt[(a + b*x^3)^2])

_______________________________________________________________________________________

Maple [A]  time = 0.012, size = 32, normalized size = 0.4 \[ -{\frac{ \left ( b{x}^{3}+a \right ) \left ( 4\,b{x}^{3}+a \right ) }{36\,{b}^{2}} \left ( \left ( b{x}^{3}+a \right ) ^{2} \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5/(b^2*x^6+2*a*b*x^3+a^2)^(5/2),x)

[Out]

-1/36*(b*x^3+a)*(4*b*x^3+a)/b^2/((b*x^3+a)^2)^(5/2)

_______________________________________________________________________________________

Maxima [A]  time = 0.78543, size = 65, normalized size = 0.83 \[ -\frac{1}{9 \,{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac{3}{2}} b^{2}} + \frac{a}{12 \,{\left (x^{3} + \frac{a}{b}\right )}^{4}{\left (b^{2}\right )}^{\frac{5}{2}} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(b^2*x^6 + 2*a*b*x^3 + a^2)^(5/2),x, algorithm="maxima")

[Out]

-1/9/((b^2*x^6 + 2*a*b*x^3 + a^2)^(3/2)*b^2) + 1/12*a/((x^3 + a/b)^4*(b^2)^(5/2)
*b)

_______________________________________________________________________________________

Fricas [A]  time = 0.262302, size = 78, normalized size = 1. \[ -\frac{4 \, b x^{3} + a}{36 \,{\left (b^{6} x^{12} + 4 \, a b^{5} x^{9} + 6 \, a^{2} b^{4} x^{6} + 4 \, a^{3} b^{3} x^{3} + a^{4} b^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(b^2*x^6 + 2*a*b*x^3 + a^2)^(5/2),x, algorithm="fricas")

[Out]

-1/36*(4*b*x^3 + a)/(b^6*x^12 + 4*a*b^5*x^9 + 6*a^2*b^4*x^6 + 4*a^3*b^3*x^3 + a^
4*b^2)

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{5}}{\left (\left (a + b x^{3}\right )^{2}\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5/(b**2*x**6+2*a*b*x**3+a**2)**(5/2),x)

[Out]

Integral(x**5/((a + b*x**3)**2)**(5/2), x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.734401, size = 4, normalized size = 0.05 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(b^2*x^6 + 2*a*b*x^3 + a^2)^(5/2),x, algorithm="giac")

[Out]

sage0*x