3.2553 \(\int x^{-1-5 n} \left (a+b x^n\right )^5 \, dx\)

Optimal. Leaf size=86 \[ -\frac{a^5 x^{-5 n}}{5 n}-\frac{5 a^4 b x^{-4 n}}{4 n}-\frac{10 a^3 b^2 x^{-3 n}}{3 n}-\frac{5 a^2 b^3 x^{-2 n}}{n}-\frac{5 a b^4 x^{-n}}{n}+b^5 \log (x) \]

[Out]

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

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Rubi [A]  time = 0.0956464, antiderivative size = 86, 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 \[ -\frac{a^5 x^{-5 n}}{5 n}-\frac{5 a^4 b x^{-4 n}}{4 n}-\frac{10 a^3 b^2 x^{-3 n}}{3 n}-\frac{5 a^2 b^3 x^{-2 n}}{n}-\frac{5 a b^4 x^{-n}}{n}+b^5 \log (x) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 16.4995, size = 80, normalized size = 0.93 \[ - \frac{a^{5} x^{- 5 n}}{5 n} - \frac{5 a^{4} b x^{- 4 n}}{4 n} - \frac{10 a^{3} b^{2} x^{- 3 n}}{3 n} - \frac{5 a^{2} b^{3} x^{- 2 n}}{n} - \frac{5 a b^{4} x^{- n}}{n} + \frac{b^{5} \log{\left (x^{n} \right )}}{n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0659197, size = 69, normalized size = 0.8 \[ b^5 \log (x)-\frac{a x^{-5 n} \left (12 a^4+75 a^3 b x^n+200 a^2 b^2 x^{2 n}+300 a b^3 x^{3 n}+300 b^4 x^{4 n}\right )}{60 n} \]

Antiderivative was successfully verified.

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

[Out]

-(a*(12*a^4 + 75*a^3*b*x^n + 200*a^2*b^2*x^(2*n) + 300*a*b^3*x^(3*n) + 300*b^4*x
^(4*n)))/(60*n*x^(5*n)) + b^5*Log[x]

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Maple [A]  time = 0.029, size = 97, normalized size = 1.1 \[{\frac{1}{ \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{5}} \left ({b}^{5}\ln \left ( x \right ) \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{5}-{\frac{{a}^{5}}{5\,n}}-5\,{\frac{a{b}^{4} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{4}}{n}}-5\,{\frac{{a}^{2}{b}^{3} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}}{n}}-{\frac{10\,{a}^{3}{b}^{2} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{3\,n}}-{\frac{5\,{a}^{4}b{{\rm e}^{n\ln \left ( x \right ) }}}{4\,n}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1-5*n)*(a+b*x^n)^5,x)

[Out]

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

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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^(-5*n - 1),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.226253, size = 104, normalized size = 1.21 \[ \frac{60 \, b^{5} n x^{5 \, n} \log \left (x\right ) - 300 \, a b^{4} x^{4 \, n} - 300 \, a^{2} b^{3} x^{3 \, n} - 200 \, a^{3} b^{2} x^{2 \, n} - 75 \, a^{4} b x^{n} - 12 \, a^{5}}{60 \, n x^{5 \, n}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/60*(60*b^5*n*x^(5*n)*log(x) - 300*a*b^4*x^(4*n) - 300*a^2*b^3*x^(3*n) - 200*a^
3*b^2*x^(2*n) - 75*a^4*b*x^n - 12*a^5)/(n*x^(5*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**(-1-5*n)*(a+b*x**n)**5,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.226778, size = 111, normalized size = 1.29 \[ \frac{{\left (60 \, b^{5} n e^{\left (5 \, n{\rm ln}\left (x\right )\right )}{\rm ln}\left (x\right ) - 300 \, a b^{4} e^{\left (4 \, n{\rm ln}\left (x\right )\right )} - 300 \, a^{2} b^{3} e^{\left (3 \, n{\rm ln}\left (x\right )\right )} - 200 \, a^{3} b^{2} e^{\left (2 \, n{\rm ln}\left (x\right )\right )} - 75 \, a^{4} b e^{\left (n{\rm ln}\left (x\right )\right )} - 12 \, a^{5}\right )} e^{\left (-5 \, n{\rm ln}\left (x\right )\right )}}{60 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/60*(60*b^5*n*e^(5*n*ln(x))*ln(x) - 300*a*b^4*e^(4*n*ln(x)) - 300*a^2*b^3*e^(3*
n*ln(x)) - 200*a^3*b^2*e^(2*n*ln(x)) - 75*a^4*b*e^(n*ln(x)) - 12*a^5)*e^(-5*n*ln
(x))/n