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

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

[Out]

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

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Rubi [A]  time = 0.03, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {1593, 446, 72} \[ \frac {\log \left (b+c x^n\right )}{n}+\log (x) \]

Antiderivative was successfully verified.

[In]

Int[(b + 2*c*x^n)/(b*x + c*x^(1 + 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]]

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rubi steps

\begin {align*} \int \frac {b+2 c x^n}{b x+c x^{1+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*}

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Mathematica [A]  time = 0.01, size = 15, normalized size = 1.00 \[ \frac {\log \left (b+c x^n\right )}{n}+\log (x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

Log[x] + Log[b + c*x^n]/n

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fricas [A]  time = 0.80, size = 23, normalized size = 1.53 \[ \frac {{\left (n - 1\right )} \log \relax (x) + \log \left (b x + c x^{n + 1}\right )}{n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {2 \, c x^{n} + b}{b x + c x^{n + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.02, size = 18, normalized size = 1.20 \[ \ln \relax (x )+\frac {\ln \left (c \,{\mathrm e}^{n \ln \relax (x )}+b \right )}{n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.58, size = 47, normalized size = 3.13 \[ b {\left (\frac {\log \relax (x)}{b} - \frac {\log \left (\frac {c x^{n} + b}{c}\right )}{b n}\right )} + \frac {2 \, \log \left (\frac {c x^{n} + b}{c}\right )}{n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b+2*c*x^n)/(b*x+c*x^(1+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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mupad [F]  time = 0.00, size = -1, normalized size = -0.07 \[ \int \frac {b+2\,c\,x^n}{b\,x+c\,x^{n+1}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 1.48, size = 29, normalized size = 1.93 \[ \begin {cases} \log {\relax (x )} & \text {for}\: c = 0 \wedge n = 0 \\\frac {\left (b + 2 c\right ) \log {\relax (x )}}{b + c} & \text {for}\: n = 0 \\\log {\relax (x )} & \text {for}\: c = 0 \\\log {\relax (x )} + \frac {\log {\left (\frac {b}{c} + x^{n} \right )}}{n} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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