3.2524 \(\int x^{-1-2 n} (a+b x^n) \, dx\)

Optimal. Leaf size=25 \[ -\frac{a x^{-2 n}}{2 n}-\frac{b x^{-n}}{n} \]

[Out]

-a/(2*n*x^(2*n)) - b/(n*x^n)

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Rubi [A]  time = 0.0072845, antiderivative size = 25, 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} \[ -\frac{a x^{-2 n}}{2 n}-\frac{b x^{-n}}{n} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 - 2*n)*(a + b*x^n),x]

[Out]

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

Mathematica [A]  time = 0.0099992, size = 20, normalized size = 0.8 \[ -\frac{x^{-2 n} \left (a+2 b x^n\right )}{2 n} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(-1 - 2*n)*(a + b*x^n),x]

[Out]

-(a + 2*b*x^n)/(2*n*x^(2*n))

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Maple [A]  time = 0.011, size = 27, normalized size = 1.1 \begin{align*}{\frac{1}{ \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}} \left ( -{\frac{a}{2\,n}}-{\frac{b{{\rm e}^{n\ln \left ( x \right ) }}}{n}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(-1-2*n)*(a+b*x^n),x)

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.01105, size = 43, normalized size = 1.72 \begin{align*} -\frac{2 \, b x^{n} + a}{2 \, n x^{2 \, n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(2*b*x^n + a)/(n*x^(2*n))

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Sympy [A]  time = 4.4273, size = 24, normalized size = 0.96 \begin{align*} \begin{cases} - \frac{a x^{- 2 n}}{2 n} - \frac{b x^{- n}}{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**(-1-2*n)*(a+b*x**n),x)

[Out]

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

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Giac [A]  time = 1.14435, size = 27, normalized size = 1.08 \begin{align*} -\frac{2 \, b x^{n} + a}{2 \, n x^{2 \, n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(2*b*x^n + a)/(n*x^(2*n))