3.2529 \(\int x^{-1-4 n} \left (a+b x^n\right )^2 \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0527223, 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 \[ -\frac{a^2 x^{-4 n}}{4 n}-\frac{2 a b x^{-3 n}}{3 n}-\frac{b^2 x^{-2 n}}{2 n} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 - 4*n)*(a + b*x^n)^2,x]

[Out]

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

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Rubi in Sympy [A]  time = 8.10305, size = 37, normalized size = 0.82 \[ - \frac{a^{2} x^{- 4 n}}{4 n} - \frac{2 a b x^{- 3 n}}{3 n} - \frac{b^{2} x^{- 2 n}}{2 n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(-1-4*n)*(a+b*x**n)**2,x)

[Out]

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

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Mathematica [A]  time = 0.0234455, size = 35, normalized size = 0.78 \[ -\frac{x^{-4 n} \left (3 a^2+8 a b x^n+6 b^2 x^{2 n}\right )}{12 n} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(-1 - 4*n)*(a + b*x^n)^2,x]

[Out]

-(3*a^2 + 8*a*b*x^n + 6*b^2*x^(2*n))/(12*n*x^(4*n))

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Maple [A]  time = 0.025, size = 45, normalized size = 1. \[{\frac{1}{ \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{4}} \left ( -{\frac{{a}^{2}}{4\,n}}-{\frac{{b}^{2} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{2\,n}}-{\frac{2\,a{{\rm e}^{n\ln \left ( x \right ) }}b}{3\,n}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1-4*n)*(a+b*x^n)^2,x)

[Out]

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

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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^(-4*n - 1),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.224183, size = 47, normalized size = 1.04 \[ -\frac{6 \, b^{2} x^{2 \, n} + 8 \, a b x^{n} + 3 \, a^{2}}{12 \, n x^{4 \, n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^2*x^(-4*n - 1),x, algorithm="fricas")

[Out]

-1/12*(6*b^2*x^(2*n) + 8*a*b*x^n + 3*a^2)/(n*x^(4*n))

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Sympy [A]  time = 38.6359, size = 46, normalized size = 1.02 \[ \begin{cases} - \frac{a^{2} x^{- 4 n}}{4 n} - \frac{2 a b x^{- 3 n}}{3 n} - \frac{b^{2} x^{- 2 n}}{2 n} & \text{for}\: n \neq 0 \\\left (a + b\right )^{2} \log{\left (x \right )} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(-1-4*n)*(a+b*x**n)**2,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.218653, size = 50, normalized size = 1.11 \[ -\frac{{\left (6 \, b^{2} e^{\left (2 \, n{\rm ln}\left (x\right )\right )} + 8 \, a b e^{\left (n{\rm ln}\left (x\right )\right )} + 3 \, a^{2}\right )} e^{\left (-4 \, n{\rm ln}\left (x\right )\right )}}{12 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^2*x^(-4*n - 1),x, algorithm="giac")

[Out]

-1/12*(6*b^2*e^(2*n*ln(x)) + 8*a*b*e^(n*ln(x)) + 3*a^2)*e^(-4*n*ln(x))/n