3.2549 \(\int x^{-1-n} \left (a+b x^n\right )^5 \, dx\)

Optimal. Leaf size=83 \[ -\frac{a^5 x^{-n}}{n}+5 a^4 b \log (x)+\frac{10 a^3 b^2 x^n}{n}+\frac{5 a^2 b^3 x^{2 n}}{n}+\frac{5 a b^4 x^{3 n}}{3 n}+\frac{b^5 x^{4 n}}{4 n} \]

[Out]

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

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Rubi [A]  time = 0.0962486, antiderivative size = 83, 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 \[ -\frac{a^5 x^{-n}}{n}+5 a^4 b \log (x)+\frac{10 a^3 b^2 x^n}{n}+\frac{5 a^2 b^3 x^{2 n}}{n}+\frac{5 a b^4 x^{3 n}}{3 n}+\frac{b^5 x^{4 n}}{4 n} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 - n)*(a + b*x^n)^5,x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0424755, size = 72, normalized size = 0.87 \[ -\frac{12 a^5 x^{-n}-60 a^4 b n \log (x)-120 a^3 b^2 x^n-60 a^2 b^3 x^{2 n}-20 a b^4 x^{3 n}-3 b^5 x^{4 n}}{12 n} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.027, size = 98, normalized size = 1.2 \[{\frac{1}{{{\rm e}^{n\ln \left ( x \right ) }}} \left ( 5\,{a}^{4}b\ln \left ( x \right ){{\rm e}^{n\ln \left ( x \right ) }}-{\frac{{a}^{5}}{n}}+{\frac{{b}^{5} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{5}}{4\,n}}+{\frac{5\,a{b}^{4} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{4}}{3\,n}}+5\,{\frac{{a}^{2}{b}^{3} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}}{n}}+10\,{\frac{{a}^{3}{b}^{2} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{n}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1-n)*(a+b*x^n)^5,x)

[Out]

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

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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^(-n - 1),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.226039, size = 101, normalized size = 1.22 \[ \frac{60 \, a^{4} b n x^{n} \log \left (x\right ) + 3 \, b^{5} x^{5 \, n} + 20 \, a b^{4} x^{4 \, n} + 60 \, a^{2} b^{3} x^{3 \, n} + 120 \, a^{3} b^{2} x^{2 \, n} - 12 \, a^{5}}{12 \, n x^{n}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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-n)*(a+b*x**n)**5,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.229502, size = 111, normalized size = 1.34 \[ \frac{{\left (60 \, a^{4} b n e^{\left (n{\rm ln}\left (x\right )\right )}{\rm ln}\left (x\right ) + 3 \, b^{5} e^{\left (5 \, n{\rm ln}\left (x\right )\right )} + 20 \, a b^{4} e^{\left (4 \, n{\rm ln}\left (x\right )\right )} + 60 \, a^{2} b^{3} e^{\left (3 \, n{\rm ln}\left (x\right )\right )} + 120 \, a^{3} b^{2} e^{\left (2 \, n{\rm ln}\left (x\right )\right )} - 12 \, a^{5}\right )} e^{\left (-n{\rm ln}\left (x\right )\right )}}{12 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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