3.2535 \(\int x^{-1-3 n} (a+b x^n)^2 \, dx\)

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 = 0.0048236, 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, Rules used = {264} \[ -\frac{x^{-3 n} \left (a+b x^n\right )^3}{3 a n} \]

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))

Rule 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rubi steps

\begin{align*} \int x^{-1-3 n} \left (a+b x^n\right )^2 \, dx &=-\frac{x^{-3 n} \left (a+b x^n\right )^3}{3 a n}\\ \end{align*}

Mathematica [A]  time = 0.0057434, size = 24, normalized size = 1. \[ -\frac{x^{-3 n} \left (a+b x^n\right )^3}{3 a n} \]

Antiderivative was successfully verified.

[In]

Integrate[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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Maple [A]  time = 0.013, size = 45, normalized size = 1.9 \begin{align*}{\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 ) } \end{align*}

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(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1-3*n)*(a+b*x^n)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.24799, size = 70, normalized size = 2.92 \begin{align*} -\frac{3 \, b^{2} x^{2 \, n} + 3 \, a b x^{n} + a^{2}}{3 \, n x^{3 \, n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1-3*n)*(a+b*x^n)^2,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 = 14.2528, size = 39, normalized size = 1.62 \begin{align*} \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} \end{align*}

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 [A]  time = 1.22713, size = 45, normalized size = 1.88 \begin{align*} -\frac{3 \, b^{2} x^{2 \, n} + 3 \, a b x^{n} + a^{2}}{3 \, n x^{3 \, n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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