3.2525 \(\int x^{-1-3 n} (a+b x^n) \, dx\)

Optimal. Leaf size=27 \[ -\frac{a x^{-3 n}}{3 n}-\frac{b x^{-2 n}}{2 n} \]

[Out]

-a/(3*n*x^(3*n)) - b/(2*n*x^(2*n))

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Rubi [A]  time = 0.0068886, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {14} \[ -\frac{a x^{-3 n}}{3 n}-\frac{b x^{-2 n}}{2 n} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 - 3*n)*(a + b*x^n),x]

[Out]

-a/(3*n*x^(3*n)) - b/(2*n*x^(2*n))

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

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

Mathematica [A]  time = 0.0095873, size = 22, normalized size = 0.81 \[ -\frac{x^{-3 n} \left (2 a+3 b x^n\right )}{6 n} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(-1 - 3*n)*(a + b*x^n),x]

[Out]

-(2*a + 3*b*x^n)/(6*n*x^(3*n))

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Maple [A]  time = 0.013, size = 27, normalized size = 1. \begin{align*}{\frac{1}{ \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}} \left ( -{\frac{a}{3\,n}}-{\frac{b{{\rm e}^{n\ln \left ( x \right ) }}}{2\,n}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(-1-3*n)*(a+b*x^n),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.06352, size = 46, normalized size = 1.7 \begin{align*} -\frac{3 \, b x^{n} + 2 \, a}{6 \, 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),x, algorithm="fricas")

[Out]

-1/6*(3*b*x^n + 2*a)/(n*x^(3*n))

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Sympy [A]  time = 4.44988, size = 27, normalized size = 1. \begin{align*} \begin{cases} - \frac{a x^{- 3 n}}{3 n} - \frac{b x^{- 2 n}}{2 n} & \text{for}\: n \neq 0 \\\left (a + b\right ) \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),x)

[Out]

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

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Giac [A]  time = 1.15993, size = 30, normalized size = 1.11 \begin{align*} -\frac{3 \, b x^{n} + 2 \, a}{6 \, 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),x, algorithm="giac")

[Out]

-1/6*(3*b*x^n + 2*a)/(n*x^(3*n))