3.2554 \(\int x^{-1-6 n} \left (a+b x^n\right )^5 \, dx\)

Optimal. Leaf size=24 \[ -\frac{x^{-6 n} \left (a+b x^n\right )^6}{6 a n} \]

[Out]

-(a + b*x^n)^6/(6*a*n*x^(6*n))

_______________________________________________________________________________________

Rubi [A]  time = 0.0212702, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059 \[ -\frac{x^{-6 n} \left (a+b x^n\right )^6}{6 a n} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 - 6*n)*(a + b*x^n)^5,x]

[Out]

-(a + b*x^n)^6/(6*a*n*x^(6*n))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 3.09178, size = 19, normalized size = 0.79 \[ - \frac{x^{- 6 n} \left (a + b x^{n}\right )^{6}}{6 a n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(-1-6*n)*(a+b*x**n)**5,x)

[Out]

-x**(-6*n)*(a + b*x**n)**6/(6*a*n)

_______________________________________________________________________________________

Mathematica [B]  time = 0.0377001, size = 72, normalized size = 3. \[ -\frac{x^{-6 n} \left (a^5+6 a^4 b x^n+15 a^3 b^2 x^{2 n}+20 a^2 b^3 x^{3 n}+15 a b^4 x^{4 n}+6 b^5 x^{5 n}\right )}{6 n} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(-1 - 6*n)*(a + b*x^n)^5,x]

[Out]

-(a^5 + 6*a^4*b*x^n + 15*a^3*b^2*x^(2*n) + 20*a^2*b^3*x^(3*n) + 15*a*b^4*x^(4*n)
+ 6*b^5*x^(5*n))/(6*n*x^(6*n))

_______________________________________________________________________________________

Maple [B]  time = 0.036, size = 88, normalized size = 3.7 \[ -{\frac{{b}^{5}}{n{x}^{n}}}-{\frac{5\,a{b}^{4}}{2\,n \left ({x}^{n} \right ) ^{2}}}-{\frac{10\,{a}^{2}{b}^{3}}{3\,n \left ({x}^{n} \right ) ^{3}}}-{\frac{5\,{a}^{3}{b}^{2}}{2\,n \left ({x}^{n} \right ) ^{4}}}-{\frac{{a}^{4}b}{n \left ({x}^{n} \right ) ^{5}}}-{\frac{{a}^{5}}{6\,n \left ({x}^{n} \right ) ^{6}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1-6*n)*(a+b*x^n)^5,x)

[Out]

-b^5/n/(x^n)-5/2*a*b^4/n/(x^n)^2-10/3*a^2*b^3/n/(x^n)^3-5/2*a^3*b^2/n/(x^n)^4-a^
4*b/n/(x^n)^5-1/6*a^5/n/(x^n)^6

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^5*x^(-6*n - 1),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.226407, size = 97, normalized size = 4.04 \[ -\frac{6 \, b^{5} x^{5 \, n} + 15 \, a b^{4} x^{4 \, n} + 20 \, a^{2} b^{3} x^{3 \, n} + 15 \, a^{3} b^{2} x^{2 \, n} + 6 \, a^{4} b x^{n} + a^{5}}{6 \, n x^{6 \, n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^5*x^(-6*n - 1),x, algorithm="fricas")

[Out]

-1/6*(6*b^5*x^(5*n) + 15*a*b^4*x^(4*n) + 20*a^2*b^3*x^(3*n) + 15*a^3*b^2*x^(2*n)
+ 6*a^4*b*x^n + a^5)/(n*x^(6*n))

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(-1-6*n)*(a+b*x**n)**5,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.228178, size = 104, normalized size = 4.33 \[ -\frac{{\left (6 \, b^{5} e^{\left (5 \, n{\rm ln}\left (x\right )\right )} + 15 \, a b^{4} e^{\left (4 \, n{\rm ln}\left (x\right )\right )} + 20 \, a^{2} b^{3} e^{\left (3 \, n{\rm ln}\left (x\right )\right )} + 15 \, a^{3} b^{2} e^{\left (2 \, n{\rm ln}\left (x\right )\right )} + 6 \, a^{4} b e^{\left (n{\rm ln}\left (x\right )\right )} + a^{5}\right )} e^{\left (-6 \, n{\rm ln}\left (x\right )\right )}}{6 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^5*x^(-6*n - 1),x, algorithm="giac")

[Out]

-1/6*(6*b^5*e^(5*n*ln(x)) + 15*a*b^4*e^(4*n*ln(x)) + 20*a^2*b^3*e^(3*n*ln(x)) +
15*a^3*b^2*e^(2*n*ln(x)) + 6*a^4*b*e^(n*ln(x)) + a^5)*e^(-6*n*ln(x))/n