3.2551 \(\int x^{-1-3 n} \left (a+b x^n\right )^5 \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0962733, 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 \[ -\frac{a^5 x^{-3 n}}{3 n}-\frac{5 a^4 b x^{-2 n}}{2 n}-\frac{10 a^3 b^2 x^{-n}}{n}+10 a^2 b^3 \log (x)+\frac{5 a b^4 x^n}{n}+\frac{b^5 x^{2 n}}{2 n} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 - 3*n)*(a + b*x^n)^5,x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(-1-3*n)*(a+b*x**n)**5,x)

[Out]

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

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Mathematica [A]  time = 0.0825211, size = 72, normalized size = 0.85 \[ -\frac{2 a^5 x^{-3 n}+15 a^4 b x^{-2 n}+60 a^3 b^2 x^{-n}-60 a^2 b^3 n \log (x)-30 a b^4 x^n-3 b^5 x^{2 n}}{6 n} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(-1 - 3*n)*(a + b*x^n)^5,x]

[Out]

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

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Maple [A]  time = 0.028, size = 98, normalized size = 1.2 \[{\frac{1}{ \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}} \left ( 10\,{a}^{2}{b}^{3}\ln \left ( x \right ) \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}-{\frac{{a}^{5}}{3\,n}}+{\frac{{b}^{5} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{5}}{2\,n}}+5\,{\frac{a{b}^{4} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{4}}{n}}-10\,{\frac{{a}^{3}{b}^{2} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{n}}-{\frac{5\,{a}^{4}b{{\rm e}^{n\ln \left ( x \right ) }}}{2\,n}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1-3*n)*(a+b*x^n)^5,x)

[Out]

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

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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^(-3*n - 1),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.227236, size = 104, normalized size = 1.22 \[ \frac{60 \, a^{2} b^{3} n x^{3 \, n} \log \left (x\right ) + 3 \, b^{5} x^{5 \, n} + 30 \, a b^{4} x^{4 \, n} - 60 \, a^{3} b^{2} x^{2 \, n} - 15 \, a^{4} b x^{n} - 2 \, a^{5}}{6 \, n x^{3 \, n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^5*x^(-3*n - 1),x, algorithm="fricas")

[Out]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.226243, size = 111, normalized size = 1.31 \[ \frac{{\left (60 \, a^{2} b^{3} n e^{\left (3 \, n{\rm ln}\left (x\right )\right )}{\rm ln}\left (x\right ) + 3 \, b^{5} e^{\left (5 \, n{\rm ln}\left (x\right )\right )} + 30 \, a b^{4} e^{\left (4 \, n{\rm ln}\left (x\right )\right )} - 60 \, a^{3} b^{2} e^{\left (2 \, n{\rm ln}\left (x\right )\right )} - 15 \, a^{4} b e^{\left (n{\rm ln}\left (x\right )\right )} - 2 \, a^{5}\right )} e^{\left (-3 \, n{\rm ln}\left (x\right )\right )}}{6 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^5*x^(-3*n - 1),x, algorithm="giac")

[Out]

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