3.67 \(\int \frac{1}{\left (a^5+5 a^4 b x+10 a^3 b^2 x^2+10 a^2 b^3 x^3+5 a b^4 x^4+b^5 x^5\right )^2} \, dx\)

Optimal. Leaf size=14 \[ -\frac{1}{9 b (a+b x)^9} \]

[Out]

-1/(9*b*(a + b*x)^9)

_______________________________________________________________________________________

Rubi [A]  time = 0.0330936, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 51, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.039 \[ -\frac{1}{9 b (a+b x)^9} \]

Antiderivative was successfully verified.

[In]  Int[(a^5 + 5*a^4*b*x + 10*a^3*b^2*x^2 + 10*a^2*b^3*x^3 + 5*a*b^4*x^4 + b^5*x^5)^(-2),x]

[Out]

-1/(9*b*(a + b*x)^9)

_______________________________________________________________________________________

Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b**5*x**5+5*a*b**4*x**4+10*a**2*b**3*x**3+10*a**3*b**2*x**2+5*a**4*b*x+a**5)**2,x)

[Out]

Timed out

_______________________________________________________________________________________

Mathematica [A]  time = 0.00541347, size = 14, normalized size = 1. \[ -\frac{1}{9 b (a+b x)^9} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^5 + 5*a^4*b*x + 10*a^3*b^2*x^2 + 10*a^2*b^3*x^3 + 5*a*b^4*x^4 + b^5*x^5)^(-2),x]

[Out]

-1/(9*b*(a + b*x)^9)

_______________________________________________________________________________________

Maple [A]  time = 0.004, size = 13, normalized size = 0.9 \[ -{\frac{1}{9\,b \left ( bx+a \right ) ^{9}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b^5*x^5+5*a*b^4*x^4+10*a^2*b^3*x^3+10*a^3*b^2*x^2+5*a^4*b*x+a^5)^2,x)

[Out]

-1/9/b/(b*x+a)^9

_______________________________________________________________________________________

Maxima [A]  time = 0.823871, size = 136, normalized size = 9.71 \[ -\frac{1}{9 \,{\left (b^{10} x^{9} + 9 \, a b^{9} x^{8} + 36 \, a^{2} b^{8} x^{7} + 84 \, a^{3} b^{7} x^{6} + 126 \, a^{4} b^{6} x^{5} + 126 \, a^{5} b^{5} x^{4} + 84 \, a^{6} b^{4} x^{3} + 36 \, a^{7} b^{3} x^{2} + 9 \, a^{8} b^{2} x + a^{9} b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^5*x^5 + 5*a*b^4*x^4 + 10*a^2*b^3*x^3 + 10*a^3*b^2*x^2 + 5*a^4*b*x + a^5)^(-2),x, algorithm="maxima")

[Out]

-1/9/(b^10*x^9 + 9*a*b^9*x^8 + 36*a^2*b^8*x^7 + 84*a^3*b^7*x^6 + 126*a^4*b^6*x^5
+ 126*a^5*b^5*x^4 + 84*a^6*b^4*x^3 + 36*a^7*b^3*x^2 + 9*a^8*b^2*x + a^9*b)

_______________________________________________________________________________________

Fricas [A]  time = 0.24729, size = 136, normalized size = 9.71 \[ -\frac{1}{9 \,{\left (b^{10} x^{9} + 9 \, a b^{9} x^{8} + 36 \, a^{2} b^{8} x^{7} + 84 \, a^{3} b^{7} x^{6} + 126 \, a^{4} b^{6} x^{5} + 126 \, a^{5} b^{5} x^{4} + 84 \, a^{6} b^{4} x^{3} + 36 \, a^{7} b^{3} x^{2} + 9 \, a^{8} b^{2} x + a^{9} b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^5*x^5 + 5*a*b^4*x^4 + 10*a^2*b^3*x^3 + 10*a^3*b^2*x^2 + 5*a^4*b*x + a^5)^(-2),x, algorithm="fricas")

[Out]

-1/9/(b^10*x^9 + 9*a*b^9*x^8 + 36*a^2*b^8*x^7 + 84*a^3*b^7*x^6 + 126*a^4*b^6*x^5
+ 126*a^5*b^5*x^4 + 84*a^6*b^4*x^3 + 36*a^7*b^3*x^2 + 9*a^8*b^2*x + a^9*b)

_______________________________________________________________________________________

Sympy [A]  time = 3.07586, size = 109, normalized size = 7.79 \[ - \frac{1}{9 a^{9} b + 81 a^{8} b^{2} x + 324 a^{7} b^{3} x^{2} + 756 a^{6} b^{4} x^{3} + 1134 a^{5} b^{5} x^{4} + 1134 a^{4} b^{6} x^{5} + 756 a^{3} b^{7} x^{6} + 324 a^{2} b^{8} x^{7} + 81 a b^{9} x^{8} + 9 b^{10} x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b**5*x**5+5*a*b**4*x**4+10*a**2*b**3*x**3+10*a**3*b**2*x**2+5*a**4*b*x+a**5)**2,x)

[Out]

-1/(9*a**9*b + 81*a**8*b**2*x + 324*a**7*b**3*x**2 + 756*a**6*b**4*x**3 + 1134*a
**5*b**5*x**4 + 1134*a**4*b**6*x**5 + 756*a**3*b**7*x**6 + 324*a**2*b**8*x**7 +
81*a*b**9*x**8 + 9*b**10*x**9)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.258385, size = 16, normalized size = 1.14 \[ -\frac{1}{9 \,{\left (b x + a\right )}^{9} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^5*x^5 + 5*a*b^4*x^4 + 10*a^2*b^3*x^3 + 10*a^3*b^2*x^2 + 5*a^4*b*x + a^5)^(-2),x, algorithm="giac")

[Out]

-1/9/((b*x + a)^9*b)