3.2530 \(\int x^{-1-5 n} \left (a+b x^n\right )^2 \, dx\)

Optimal. Leaf size=45 \[ -\frac{a^2 x^{-5 n}}{5 n}-\frac{a b x^{-4 n}}{2 n}-\frac{b^2 x^{-3 n}}{3 n} \]

[Out]

-a^2/(5*n*x^(5*n)) - (a*b)/(2*n*x^(4*n)) - b^2/(3*n*x^(3*n))

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Rubi [A]  time = 0.0525738, antiderivative size = 45, 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 \[ -\frac{a^2 x^{-5 n}}{5 n}-\frac{a b x^{-4 n}}{2 n}-\frac{b^2 x^{-3 n}}{3 n} \]

Antiderivative was successfully verified.

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

[Out]

-a^2/(5*n*x^(5*n)) - (a*b)/(2*n*x^(4*n)) - b^2/(3*n*x^(3*n))

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Rubi in Sympy [A]  time = 8.09847, size = 36, normalized size = 0.8 \[ - \frac{a^{2} x^{- 5 n}}{5 n} - \frac{a b x^{- 4 n}}{2 n} - \frac{b^{2} x^{- 3 n}}{3 n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0233207, size = 35, normalized size = 0.78 \[ -\frac{x^{-5 n} \left (6 a^2+15 a b x^n+10 b^2 x^{2 n}\right )}{30 n} \]

Antiderivative was successfully verified.

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

[Out]

-(6*a^2 + 15*a*b*x^n + 10*b^2*x^(2*n))/(30*n*x^(5*n))

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Maple [A]  time = 0.028, size = 45, normalized size = 1. \[{\frac{1}{ \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{5}} \left ( -{\frac{{a}^{2}}{5\,n}}-{\frac{{b}^{2} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{3\,n}}-{\frac{a{{\rm e}^{n\ln \left ( x \right ) }}b}{2\,n}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1-5*n)*(a+b*x^n)^2,x)

[Out]

(-1/5*a^2/n-1/3*b^2/n*exp(n*ln(x))^2-1/2*a*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)^2*x^(-5*n - 1),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.224038, size = 47, normalized size = 1.04 \[ -\frac{10 \, b^{2} x^{2 \, n} + 15 \, a b x^{n} + 6 \, a^{2}}{30 \, n x^{5 \, n}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/30*(10*b^2*x^(2*n) + 15*a*b*x^n + 6*a^2)/(n*x^(5*n))

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Sympy [A]  time = 37.4153, size = 44, normalized size = 0.98 \[ \begin{cases} - \frac{a^{2} x^{- 5 n}}{5 n} - \frac{a b x^{- 4 n}}{2 n} - \frac{b^{2} x^{- 3 n}}{3 n} & \text{for}\: n \neq 0 \\\left (a + b\right )^{2} \log{\left (x \right )} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-a**2*x**(-5*n)/(5*n) - a*b*x**(-4*n)/(2*n) - b**2*x**(-3*n)/(3*n), N
e(n, 0)), ((a + b)**2*log(x), True))

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GIAC/XCAS [A]  time = 0.221069, size = 50, normalized size = 1.11 \[ -\frac{{\left (10 \, b^{2} e^{\left (2 \, n{\rm ln}\left (x\right )\right )} + 15 \, a b e^{\left (n{\rm ln}\left (x\right )\right )} + 6 \, a^{2}\right )} e^{\left (-5 \, n{\rm ln}\left (x\right )\right )}}{30 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/30*(10*b^2*e^(2*n*ln(x)) + 15*a*b*e^(n*ln(x)) + 6*a^2)*e^(-5*n*ln(x))/n