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

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

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {446, 72} \begin {gather*} \frac {\log \left (b+c x^n\right )}{n}+\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(b + 2*c*x^n)/(x*(b + c*x^n)),x]

[Out]

Log[x] + Log[b + c*x^n]/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]

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]]

Rubi steps

\begin {align*} \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 = 0.02, size = 15, normalized size = 1.00 \begin {gather*} \frac {\log \left (b+c x^n\right )}{n}+\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Log[x] + Log[b + c*x^n]/n

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.03, size = 24, normalized size = 1.60 \begin {gather*} \frac {\log \left (b n+c n x^n\right )}{n}+\frac {\log \left (x^n\right )}{n} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(b + 2*c*x^n)/(x*(b + c*x^n)),x]

[Out]

Log[x^n]/n + Log[b*n + c*n*x^n]/n

________________________________________________________________________________________

fricas [A]  time = 0.44, size = 17, normalized size = 1.13 \begin {gather*} \frac {n \log \relax (x) + \log \left (c x^{n} + b\right )}{n} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(n*log(x) + log(c*x^n + b))/n

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 \, c x^{n} + b}{{\left (c x^{n} + b\right )} x}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.05, size = 17, normalized size = 1.13 \begin {gather*} \frac {\ln \left (\left (c \,x^{n}+b \right ) x^{n}\right )}{n} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [B]  time = 0.49, size = 47, normalized size = 3.13 \begin {gather*} b {\left (\frac {\log \relax (x)}{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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b+2*c*x^n)/x/(b+c*x^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

________________________________________________________________________________________

mupad [B]  time = 4.72, size = 15, normalized size = 1.00 \begin {gather*} \ln \relax (x)+\frac {\ln \left (b+c\,x^n\right )}{n} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b + 2*c*x^n)/(x*(b + c*x^n)),x)

[Out]

log(x) + log(b + c*x^n)/n

________________________________________________________________________________________

sympy [A]  time = 0.62, size = 29, normalized size = 1.93 \begin {gather*} \begin {cases} \log {\relax (x )} & \text {for}\: c = 0 \wedge n = 0 \\\frac {\left (b + 2 c\right ) \log {\relax (x )}}{b + c} & \text {for}\: n = 0 \\\log {\relax (x )} & \text {for}\: c = 0 \\\log {\relax (x )} + \frac {\log {\left (\frac {b}{c} + x^{n} \right )}}{n} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________