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

Optimal. Leaf size=17 \[ \log \left (a x^{1-n}+b\right )+n \log (x) \]

[Out]

n*Log[x] + Log[b + a*x^(1 - n)]

________________________________________________________________________________________

Rubi [A]  time = 0.0377438, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103, Rules used = {514, 446, 72} \[ \log \left (a x^{1-n}+b\right )+n \log (x) \]

Antiderivative was successfully verified.

[In]

Int[(a + b*n*x^(-1 + n))/(x^n*(b + a*x^(1 - n))),x]

[Out]

n*Log[x] + Log[b + a*x^(1 - n)]

Rule 514

Int[(x_)^(m_.)*((c_) + (d_.)*(x_)^(mn_.))^(q_.)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[x^(m - n*q)*
(a + b*x^n)^p*(d + c*x^n)^q, x] /; FreeQ[{a, b, c, d, m, n, p}, x] && EqQ[mn, -n] && IntegerQ[q] && (PosQ[n] |
|  !IntegerQ[p])

Rule 446

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 72

Int[((e_.) + (f_.)*(x_))^(p_.)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Int[ExpandIntegrand[(
e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[p]

Rubi steps

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

Mathematica [A]  time = 0.0158437, size = 17, normalized size = 1. \[ \log \left (a x^{1-n}+b\right )+n \log (x) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*n*x^(-1 + n))/(x^n*(b + a*x^(1 - n))),x]

[Out]

n*Log[x] + Log[b + a*x^(1 - n)]

________________________________________________________________________________________

Maple [A]  time = 0.021, size = 13, normalized size = 0.8 \begin{align*} \ln \left ( ax+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln(a*x+b*exp(n*ln(x)))

________________________________________________________________________________________

Maxima [B]  time = 1.11918, size = 116, normalized size = 6.82 \begin{align*} b n{\left (\frac{\log \left (x\right )}{b} - \frac{n \log \left (x\right )}{b{\left (n - 1\right )}} + \frac{\log \left (\frac{a x + b x^{n}}{b}\right )}{b{\left (n - 1\right )}}\right )} + a{\left (\frac{n \log \left (x\right )}{a{\left (n - 1\right )}} - \frac{\log \left (\frac{a x + b x^{n}}{b}\right )}{a{\left (n - 1\right )}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.59771, size = 24, normalized size = 1.41 \begin{align*} \log \left (a x + b x^{n}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(a*x + b*x^n)

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*n*x**(-1+n))/(x**n)/(b+a*x**(1-n)),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b n x^{n - 1} + a}{{\left (a x^{-n + 1} + b\right )} x^{n}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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