3.2552 \(\int x^{-1-4 n} \left (a+b x^n\right )^5 \, dx\)

Optimal. Leaf size=82 \[ -\frac{a^5 x^{-4 n}}{4 n}-\frac{5 a^4 b x^{-3 n}}{3 n}-\frac{5 a^3 b^2 x^{-2 n}}{n}-\frac{10 a^2 b^3 x^{-n}}{n}+5 a b^4 \log (x)+\frac{b^5 x^n}{n} \]

[Out]

-a^5/(4*n*x^(4*n)) - (5*a^4*b)/(3*n*x^(3*n)) - (5*a^3*b^2)/(n*x^(2*n)) - (10*a^2
*b^3)/(n*x^n) + (b^5*x^n)/n + 5*a*b^4*Log[x]

_______________________________________________________________________________________

Rubi [A]  time = 0.0974764, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ -\frac{a^5 x^{-4 n}}{4 n}-\frac{5 a^4 b x^{-3 n}}{3 n}-\frac{5 a^3 b^2 x^{-2 n}}{n}-\frac{10 a^2 b^3 x^{-n}}{n}+5 a b^4 \log (x)+\frac{b^5 x^n}{n} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 - 4*n)*(a + b*x^n)^5,x]

[Out]

-a^5/(4*n*x^(4*n)) - (5*a^4*b)/(3*n*x^(3*n)) - (5*a^3*b^2)/(n*x^(2*n)) - (10*a^2
*b^3)/(n*x^n) + (b^5*x^n)/n + 5*a*b^4*Log[x]

_______________________________________________________________________________________

Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{a^{5} x^{- 4 n}}{4 n} - \frac{5 a^{4} b x^{- 3 n}}{3 n} - \frac{5 a^{3} b^{2} x^{- 2 n}}{n} - \frac{10 a^{2} b^{3} x^{- n}}{n} + \frac{5 a b^{4} \log{\left (x^{n} \right )}}{n} + \frac{\int ^{x^{n}} b^{5}\, dx}{n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(-1-4*n)*(a+b*x**n)**5,x)

[Out]

-a**5*x**(-4*n)/(4*n) - 5*a**4*b*x**(-3*n)/(3*n) - 5*a**3*b**2*x**(-2*n)/n - 10*
a**2*b**3*x**(-n)/n + 5*a*b**4*log(x**n)/n + Integral(b**5, (x, x**n))/n

_______________________________________________________________________________________

Mathematica [A]  time = 0.101281, size = 72, normalized size = 0.88 \[ 5 a b^4 \log (x)-\frac{x^{-4 n} \left (3 a^5+20 a^4 b x^n+60 a^3 b^2 x^{2 n}+120 a^2 b^3 x^{3 n}-12 b^5 x^{5 n}\right )}{12 n} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(-1 - 4*n)*(a + b*x^n)^5,x]

[Out]

-(3*a^5 + 20*a^4*b*x^n + 60*a^3*b^2*x^(2*n) + 120*a^2*b^3*x^(3*n) - 12*b^5*x^(5*
n))/(12*n*x^(4*n)) + 5*a*b^4*Log[x]

_______________________________________________________________________________________

Maple [A]  time = 0.028, size = 97, normalized size = 1.2 \[{\frac{1}{ \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{4}} \left ({\frac{{b}^{5} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{5}}{n}}+5\,a{b}^{4}\ln \left ( x \right ) \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{4}-{\frac{{a}^{5}}{4\,n}}-10\,{\frac{{a}^{2}{b}^{3} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}}{n}}-5\,{\frac{{a}^{3}{b}^{2} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{n}}-{\frac{5\,{a}^{4}b{{\rm e}^{n\ln \left ( x \right ) }}}{3\,n}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1-4*n)*(a+b*x^n)^5,x)

[Out]

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

_______________________________________________________________________________________

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^(-4*n - 1),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.226463, size = 104, normalized size = 1.27 \[ \frac{60 \, a b^{4} n x^{4 \, n} \log \left (x\right ) + 12 \, b^{5} x^{5 \, n} - 120 \, a^{2} b^{3} x^{3 \, n} - 60 \, a^{3} b^{2} x^{2 \, n} - 20 \, a^{4} b x^{n} - 3 \, a^{5}}{12 \, n x^{4 \, n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^5*x^(-4*n - 1),x, algorithm="fricas")

[Out]

1/12*(60*a*b^4*n*x^(4*n)*log(x) + 12*b^5*x^(5*n) - 120*a^2*b^3*x^(3*n) - 60*a^3*
b^2*x^(2*n) - 20*a^4*b*x^n - 3*a^5)/(n*x^(4*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**(-1-4*n)*(a+b*x**n)**5,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.228652, size = 111, normalized size = 1.35 \[ \frac{{\left (60 \, a b^{4} n e^{\left (4 \, n{\rm ln}\left (x\right )\right )}{\rm ln}\left (x\right ) + 12 \, b^{5} e^{\left (5 \, n{\rm ln}\left (x\right )\right )} - 120 \, a^{2} b^{3} e^{\left (3 \, n{\rm ln}\left (x\right )\right )} - 60 \, a^{3} b^{2} e^{\left (2 \, n{\rm ln}\left (x\right )\right )} - 20 \, a^{4} b e^{\left (n{\rm ln}\left (x\right )\right )} - 3 \, a^{5}\right )} e^{\left (-4 \, n{\rm ln}\left (x\right )\right )}}{12 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^5*x^(-4*n - 1),x, algorithm="giac")

[Out]

1/12*(60*a*b^4*n*e^(4*n*ln(x))*ln(x) + 12*b^5*e^(5*n*ln(x)) - 120*a^2*b^3*e^(3*n
*ln(x)) - 60*a^3*b^2*e^(2*n*ln(x)) - 20*a^4*b*e^(n*ln(x)) - 3*a^5)*e^(-4*n*ln(x)
)/n