3.2550 \(\int x^{-1-2 n} \left (a+b x^n\right )^5 \, dx\)

Optimal. Leaf size=85 \[ -\frac{a^5 x^{-2 n}}{2 n}-\frac{5 a^4 b x^{-n}}{n}+10 a^3 b^2 \log (x)+\frac{10 a^2 b^3 x^n}{n}+\frac{5 a b^4 x^{2 n}}{2 n}+\frac{b^5 x^{3 n}}{3 n} \]

[Out]

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

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Rubi [A]  time = 0.0961271, antiderivative size = 85, 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 \[ -\frac{a^5 x^{-2 n}}{2 n}-\frac{5 a^4 b x^{-n}}{n}+10 a^3 b^2 \log (x)+\frac{10 a^2 b^3 x^n}{n}+\frac{5 a b^4 x^{2 n}}{2 n}+\frac{b^5 x^{3 n}}{3 n} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 - 2*n)*(a + b*x^n)^5,x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(-1-2*n)*(a+b*x**n)**5,x)

[Out]

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

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Mathematica [A]  time = 0.073103, size = 72, normalized size = 0.85 \[ -\frac{3 a^5 x^{-2 n}+30 a^4 b x^{-n}-60 a^3 b^2 n \log (x)-60 a^2 b^3 x^n-15 a b^4 x^{2 n}-2 b^5 x^{3 n}}{6 n} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(-1 - 2*n)*(a + b*x^n)^5,x]

[Out]

-((3*a^5)/x^(2*n) + (30*a^4*b)/x^n - 60*a^2*b^3*x^n - 15*a*b^4*x^(2*n) - 2*b^5*x
^(3*n) - 60*a^3*b^2*n*Log[x])/(6*n)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1-2*n)*(a+b*x^n)^5,x)

[Out]

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

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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^(-2*n - 1),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.228342, size = 104, normalized size = 1.22 \[ \frac{60 \, a^{3} b^{2} n x^{2 \, n} \log \left (x\right ) + 2 \, b^{5} x^{5 \, n} + 15 \, a b^{4} x^{4 \, n} + 60 \, a^{2} b^{3} x^{3 \, n} - 30 \, a^{4} b x^{n} - 3 \, a^{5}}{6 \, n x^{2 \, n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^5*x^(-2*n - 1),x, algorithm="fricas")

[Out]

1/6*(60*a^3*b^2*n*x^(2*n)*log(x) + 2*b^5*x^(5*n) + 15*a*b^4*x^(4*n) + 60*a^2*b^3
*x^(3*n) - 30*a^4*b*x^n - 3*a^5)/(n*x^(2*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**(-1-2*n)*(a+b*x**n)**5,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.226677, size = 111, normalized size = 1.31 \[ \frac{{\left (60 \, a^{3} b^{2} n e^{\left (2 \, n{\rm ln}\left (x\right )\right )}{\rm ln}\left (x\right ) + 2 \, b^{5} e^{\left (5 \, n{\rm ln}\left (x\right )\right )} + 15 \, a b^{4} e^{\left (4 \, n{\rm ln}\left (x\right )\right )} + 60 \, a^{2} b^{3} e^{\left (3 \, n{\rm ln}\left (x\right )\right )} - 30 \, a^{4} b e^{\left (n{\rm ln}\left (x\right )\right )} - 3 \, a^{5}\right )} e^{\left (-2 \, n{\rm ln}\left (x\right )\right )}}{6 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^5*x^(-2*n - 1),x, algorithm="giac")

[Out]

1/6*(60*a^3*b^2*n*e^(2*n*ln(x))*ln(x) + 2*b^5*e^(5*n*ln(x)) + 15*a*b^4*e^(4*n*ln
(x)) + 60*a^2*b^3*e^(3*n*ln(x)) - 30*a^4*b*e^(n*ln(x)) - 3*a^5)*e^(-2*n*ln(x))/n