3.2531 \(\int x^{-1-6 n} \left (a+b x^n\right )^2 \, dx\)

Optimal. Leaf size=45 \[ -\frac{a^2 x^{-6 n}}{6 n}-\frac{2 a b x^{-5 n}}{5 n}-\frac{b^2 x^{-4 n}}{4 n} \]

[Out]

-a^2/(6*n*x^(6*n)) - (2*a*b)/(5*n*x^(5*n)) - b^2/(4*n*x^(4*n))

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Rubi [A]  time = 0.0517531, 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 \[ -\frac{a^2 x^{-6 n}}{6 n}-\frac{2 a b x^{-5 n}}{5 n}-\frac{b^2 x^{-4 n}}{4 n} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 - 6*n)*(a + b*x^n)^2,x]

[Out]

-a^2/(6*n*x^(6*n)) - (2*a*b)/(5*n*x^(5*n)) - b^2/(4*n*x^(4*n))

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Rubi in Sympy [A]  time = 8.30522, size = 37, normalized size = 0.82 \[ - \frac{a^{2} x^{- 6 n}}{6 n} - \frac{2 a b x^{- 5 n}}{5 n} - \frac{b^{2} x^{- 4 n}}{4 n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(-1-6*n)*(a+b*x**n)**2,x)

[Out]

-a**2*x**(-6*n)/(6*n) - 2*a*b*x**(-5*n)/(5*n) - b**2*x**(-4*n)/(4*n)

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Mathematica [A]  time = 0.0242048, size = 35, normalized size = 0.78 \[ -\frac{x^{-6 n} \left (10 a^2+24 a b x^n+15 b^2 x^{2 n}\right )}{60 n} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(-1 - 6*n)*(a + b*x^n)^2,x]

[Out]

-(10*a^2 + 24*a*b*x^n + 15*b^2*x^(2*n))/(60*n*x^(6*n))

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Maple [A]  time = 0.031, size = 40, normalized size = 0.9 \[ -{\frac{{b}^{2}}{4\,n \left ({x}^{n} \right ) ^{4}}}-{\frac{2\,ab}{5\,n \left ({x}^{n} \right ) ^{5}}}-{\frac{{a}^{2}}{6\,n \left ({x}^{n} \right ) ^{6}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1-6*n)*(a+b*x^n)^2,x)

[Out]

-1/4*b^2/n/(x^n)^4-2/5*a*b/n/(x^n)^5-1/6*a^2/n/(x^n)^6

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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^(-6*n - 1),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.224991, size = 47, normalized size = 1.04 \[ -\frac{15 \, b^{2} x^{2 \, n} + 24 \, a b x^{n} + 10 \, a^{2}}{60 \, n x^{6 \, n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^2*x^(-6*n - 1),x, algorithm="fricas")

[Out]

-1/60*(15*b^2*x^(2*n) + 24*a*b*x^n + 10*a^2)/(n*x^(6*n))

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Sympy [A]  time = 38.7945, size = 46, normalized size = 1.02 \[ \begin{cases} - \frac{a^{2} x^{- 6 n}}{6 n} - \frac{2 a b x^{- 5 n}}{5 n} - \frac{b^{2} x^{- 4 n}}{4 n} & \text{for}\: n \neq 0 \\\left (a + b\right )^{2} \log{\left (x \right )} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(-1-6*n)*(a+b*x**n)**2,x)

[Out]

Piecewise((-a**2*x**(-6*n)/(6*n) - 2*a*b*x**(-5*n)/(5*n) - b**2*x**(-4*n)/(4*n),
Ne(n, 0)), ((a + b)**2*log(x), True))

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GIAC/XCAS [A]  time = 0.219009, size = 50, normalized size = 1.11 \[ -\frac{{\left (15 \, b^{2} e^{\left (2 \, n{\rm ln}\left (x\right )\right )} + 24 \, a b e^{\left (n{\rm ln}\left (x\right )\right )} + 10 \, a^{2}\right )} e^{\left (-6 \, n{\rm ln}\left (x\right )\right )}}{60 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^2*x^(-6*n - 1),x, algorithm="giac")

[Out]

-1/60*(15*b^2*e^(2*n*ln(x)) + 24*a*b*e^(n*ln(x)) + 10*a^2)*e^(-6*n*ln(x))/n