3.2521 \(\int x^{-1+n} (a+b x^n) \, dx\)

Optimal. Leaf size=22 \[ \frac{a x^n}{n}+\frac{b x^{2 n}}{2 n} \]

[Out]

(a*x^n)/n + (b*x^(2*n))/(2*n)

________________________________________________________________________________________

Rubi [A]  time = 0.0064788, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {14} \[ \frac{a x^n}{n}+\frac{b x^{2 n}}{2 n} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 + n)*(a + b*x^n),x]

[Out]

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

Mathematica [A]  time = 0.0078187, size = 19, normalized size = 0.86 \[ \frac{x^n \left (2 a+b x^n\right )}{2 n} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(-1 + n)*(a + b*x^n),x]

[Out]

(x^n*(2*a + b*x^n))/(2*n)

________________________________________________________________________________________

Maple [A]  time = 0.012, size = 25, normalized size = 1.1 \begin{align*}{\frac{a{{\rm e}^{n\ln \left ( x \right ) }}}{n}}+{\frac{b \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{2\,n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0.978572, size = 23, normalized size = 1.05 \begin{align*} \frac{{\left (b x^{n} + a\right )}^{2}}{2 \, b n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(b*x^n + a)^2/(b*n)

________________________________________________________________________________________

Fricas [A]  time = 1.05427, size = 39, normalized size = 1.77 \begin{align*} \frac{b x^{2 \, n} + 2 \, a x^{n}}{2 \, n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(b*x^(2*n) + 2*a*x^n)/n

________________________________________________________________________________________

Sympy [A]  time = 1.56788, size = 22, normalized size = 1. \begin{align*} \begin{cases} \frac{a x^{n}}{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+n)*(a+b*x**n),x)

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.16129, size = 26, normalized size = 1.18 \begin{align*} \frac{b x^{2 \, n} + 2 \, a x^{n}}{2 \, n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(b*x^(2*n) + 2*a*x^n)/n