3.124 \(\int \frac{x^{-1+n} (b+2 c x^n)}{b x^n+c x^{2 n}} \, dx\)

Optimal. Leaf size=15 \[ \frac{\log \left (b+c x^n\right )}{n}+\log (x) \]

[Out]

Log[x] + Log[b + c*x^n]/n

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Rubi [A]  time = 0.0350369, antiderivative size = 15, 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 = {1584, 446, 72} \[ \frac{\log \left (b+c x^n\right )}{n}+\log (x) \]

Antiderivative was successfully verified.

[In]

Int[(x^(-1 + n)*(b + 2*c*x^n))/(b*x^n + c*x^(2*n)),x]

[Out]

Log[x] + Log[b + c*x^n]/n

Rule 1584

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(m + n*p)*(a + b*x^(q -
p))^n, x] /; FreeQ[{a, b, m, p, q}, x] && IntegerQ[n] && PosQ[q - 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^{-1+n} \left (b+2 c x^n\right )}{b x^n+c x^{2 n}} \, dx &=\int \frac{b+2 c x^n}{x \left (b+c x^n\right )} \, dx\\ &=\frac{\operatorname{Subst}\left (\int \frac{b+2 c x}{x (b+c x)} \, dx,x,x^n\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{1}{x}+\frac{c}{b+c x}\right ) \, dx,x,x^n\right )}{n}\\ &=\log (x)+\frac{\log \left (b+c x^n\right )}{n}\\ \end{align*}

Mathematica [A]  time = 0.0124065, size = 15, normalized size = 1. \[ \frac{\log \left (b+c x^n\right )}{n}+\log (x) \]

Antiderivative was successfully verified.

[In]

Integrate[(x^(-1 + n)*(b + 2*c*x^n))/(b*x^n + c*x^(2*n)),x]

[Out]

Log[x] + Log[b + c*x^n]/n

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Maple [A]  time = 0.018, size = 18, normalized size = 1.2 \begin{align*} \ln \left ( x \right ) +{\frac{\ln \left ( c{{\rm e}^{n\ln \left ( x \right ) }}+b \right ) }{n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln(x)+1/n*ln(c*exp(n*ln(x))+b)

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Maxima [B]  time = 1.0284, size = 63, normalized size = 4.2 \begin{align*} b{\left (\frac{\log \left (x\right )}{b} - \frac{\log \left (\frac{c x^{n} + b}{c}\right )}{b n}\right )} + \frac{2 \, \log \left (\frac{c x^{n} + b}{c}\right )}{n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*(log(x)/b - log((c*x^n + b)/c)/(b*n)) + 2*log((c*x^n + b)/c)/n

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Fricas [A]  time = 1.08609, size = 42, normalized size = 2.8 \begin{align*} \frac{n \log \left (x\right ) + \log \left (c x^{n} + b\right )}{n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(n*log(x) + log(c*x^n + b))/n

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Sympy [A]  time = 134.943, size = 48, normalized size = 3.2 \begin{align*} \begin{cases} \log{\left (x \right )} & \text{for}\: c = 0 \wedge n = 0 \\\frac{\left (b + 2 c\right ) \log{\left (x \right )}}{b + c} & \text{for}\: n = 0 \\\frac{n^{2} \log{\left (x \right )}}{n^{2} - n} - \frac{n \log{\left (x \right )}}{n^{2} - n} & \text{for}\: c = 0 \\\log{\left (x \right )} + \frac{\log{\left (\frac{b}{c} + x^{n} \right )}}{n} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(-1+n)*(b+2*c*x**n)/(b*x**n+c*x**(2*n)),x)

[Out]

Piecewise((log(x), Eq(c, 0) & Eq(n, 0)), ((b + 2*c)*log(x)/(b + c), Eq(n, 0)), (n**2*log(x)/(n**2 - n) - n*log
(x)/(n**2 - n), Eq(c, 0)), (log(x) + log(b/c + x**n)/n, True))

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Giac [A]  time = 1.09055, size = 23, normalized size = 1.53 \begin{align*} \frac{\log \left ({\left | c x^{n} + b \right |}\right )}{n} + \log \left ({\left | x \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(abs(c*x^n + b))/n + log(abs(x))