3.2528 \(\int x^{-1-3 n} \left (a+b x^n\right )^2 \, dx\)

Optimal. Leaf size=24 \[ -\frac{x^{-3 n} \left (a+b x^n\right )^3}{3 a n} \]

[Out]

-(a + b*x^n)^3/(3*a*n*x^(3*n))

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Rubi [A]  time = 0.0210002, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059 \[ -\frac{x^{-3 n} \left (a+b x^n\right )^3}{3 a n} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 - 3*n)*(a + b*x^n)^2,x]

[Out]

-(a + b*x^n)^3/(3*a*n*x^(3*n))

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Rubi in Sympy [A]  time = 3.11272, size = 19, normalized size = 0.79 \[ - \frac{x^{- 3 n} \left (a + b x^{n}\right )^{3}}{3 a n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(-1-3*n)*(a+b*x**n)**2,x)

[Out]

-x**(-3*n)*(a + b*x**n)**3/(3*a*n)

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Mathematica [A]  time = 0.0219348, size = 33, normalized size = 1.38 \[ -\frac{x^{-3 n} \left (a^2+3 a b x^n+3 b^2 x^{2 n}\right )}{3 n} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(-1 - 3*n)*(a + b*x^n)^2,x]

[Out]

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

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Maple [A]  time = 0.023, size = 45, normalized size = 1.9 \[{\frac{1}{ \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}} \left ( -{\frac{{a}^{2}}{3\,n}}-{\frac{{b}^{2} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{n}}-{\frac{a{{\rm e}^{n\ln \left ( x \right ) }}b}{n}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1-3*n)*(a+b*x^n)^2,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.225579, size = 45, normalized size = 1.88 \[ -\frac{3 \, b^{2} x^{2 \, n} + 3 \, a b x^{n} + a^{2}}{3 \, n x^{3 \, n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^2*x^(-3*n - 1),x, algorithm="fricas")

[Out]

-1/3*(3*b^2*x^(2*n) + 3*a*b*x^n + a^2)/(n*x^(3*n))

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Sympy [A]  time = 38.6669, size = 39, normalized size = 1.62 \[ \begin{cases} - \frac{a^{2} x^{- 3 n}}{3 n} - \frac{a b x^{- 2 n}}{n} - \frac{b^{2} x^{- n}}{n} & \text{for}\: n \neq 0 \\\left (a + b\right )^{2} \log{\left (x \right )} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(-1-3*n)*(a+b*x**n)**2,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.219769, size = 47, normalized size = 1.96 \[ -\frac{{\left (3 \, b^{2} e^{\left (2 \, n{\rm ln}\left (x\right )\right )} + 3 \, a b e^{\left (n{\rm ln}\left (x\right )\right )} + a^{2}\right )} e^{\left (-3 \, n{\rm ln}\left (x\right )\right )}}{3 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^2*x^(-3*n - 1),x, algorithm="giac")

[Out]

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