\(\int \log (c (d+\frac {e}{f+g x})^q) \, dx\) [632]

Optimal result

Integrand size = 16, antiderivative size = 45 \[ \int \log \left (c \left (d+\frac {e}{f+g x}\right )^q\right ) \, dx=\frac {(f+g x) \log \left (c \left (d+\frac {e}{f+g x}\right )^q\right )}{g}+\frac {e q \log (e+d (f+g x))}{d g} \]

[Out]

(g*x+f)*ln(c*(d+e/(g*x+f))^q)/g+e*q*ln(e+d*(g*x+f))/d/g

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2533, 2498, 269, 31} \[ \int \log \left (c \left (d+\frac {e}{f+g x}\right )^q\right ) \, dx=\frac {(f+g x) \log \left (c \left (d+\frac {e}{f+g x}\right )^q\right )}{g}+\frac {e q \log (d (f+g x)+e)}{d g} \]

[In]

Int[Log[c*(d + e/(f + g*x))^q],x]

[Out]

((f + g*x)*Log[c*(d + e/(f + g*x))^q])/g + (e*q*Log[e + d*(f + g*x)])/(d*g)

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 269

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(m + n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b, m
, n}, x] && IntegerQ[p] && NegQ[n]

Rule 2498

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)], x_Symbol] :> Simp[x*Log[c*(d + e*x^n)^p], x] - Dist[e*n*p, Int[
x^n/(d + e*x^n), x], x] /; FreeQ[{c, d, e, n, p}, x]

Rule 2533

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*((f_.) + (g_.)*(x_))^(n_))^(p_.)]*(b_.))^(q_.), x_Symbol] :> Dist[1/g, Su
bst[Int[(a + b*Log[c*(d + e*x^n)^p])^q, x], x, f + g*x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && IGtQ[q
, 0] && (EqQ[q, 1] || IntegerQ[n])

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \log \left (c \left (d+\frac {e}{x}\right )^q\right ) \, dx,x,f+g x\right )}{g} \\ & = \frac {(f+g x) \log \left (c \left (d+\frac {e}{f+g x}\right )^q\right )}{g}+\frac {(e q) \text {Subst}\left (\int \frac {1}{\left (d+\frac {e}{x}\right ) x} \, dx,x,f+g x\right )}{g} \\ & = \frac {(f+g x) \log \left (c \left (d+\frac {e}{f+g x}\right )^q\right )}{g}+\frac {(e q) \text {Subst}\left (\int \frac {1}{e+d x} \, dx,x,f+g x\right )}{g} \\ & = \frac {(f+g x) \log \left (c \left (d+\frac {e}{f+g x}\right )^q\right )}{g}+\frac {e q \log (e+d (f+g x))}{d g} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.24 \[ \int \log \left (c \left (d+\frac {e}{f+g x}\right )^q\right ) \, dx=\frac {-d f q \log (f+g x)+(e+d f) q \log (e+d f+d g x)+d g x \log \left (c \left (d+\frac {e}{f+g x}\right )^q\right )}{d g} \]

[In]

Integrate[Log[c*(d + e/(f + g*x))^q],x]

[Out]

(-(d*f*q*Log[f + g*x]) + (e + d*f)*q*Log[e + d*f + d*g*x] + d*g*x*Log[c*(d + e/(f + g*x))^q])/(d*g)

Maple [A] (verified)

Time = 0.31 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.44

method	result	size
parts	\(\ln \left (c \left (d +\frac {e}{g x +f}\right )^{q}\right ) x +q e g \left (-\frac {f \ln \left (g x +f \right )}{g^{2} e}+\frac {\left (d f +e \right ) \ln \left (d g x +d f +e \right )}{e \,g^{2} d}\right )\)	\(65\)
default	\(\ln \left (c \left (\frac {d g x +d f +e}{g x +f}\right )^{q}\right ) x +q e g \left (-\frac {f \ln \left (g x +f \right )}{g^{2} e}+\frac {\left (d f +e \right ) \ln \left (d g x +d f +e \right )}{e \,g^{2} d}\right )\)	\(71\)
parallelrisch	\(-\frac {-x \ln \left (c \left (\frac {d g x +d f +e}{g x +f}\right )^{q}\right ) d \,g^{2} q -\ln \left (g x +f \right ) e g \,q^{2}-\ln \left (c \left (\frac {d g x +d f +e}{g x +f}\right )^{q}\right ) d f g q -\ln \left (c \left (\frac {d g x +d f +e}{g x +f}\right )^{q}\right ) e g q}{d \,g^{2} q}\)	\(111\)

[In]

int(ln(c*(d+e/(g*x+f))^q),x,method=_RETURNVERBOSE)

[Out]

ln(c*(d+e/(g*x+f))^q)*x+q*e*g*(-f/g^2/e*ln(g*x+f)+(d*f+e)/e/g^2/d*ln(d*g*x+d*f+e))

Fricas [A] (verification not implemented)

none

Time = 0.35 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.44 \[ \int \log \left (c \left (d+\frac {e}{f+g x}\right )^q\right ) \, dx=\frac {d g q x \log \left (\frac {d g x + d f + e}{g x + f}\right ) - d f q \log \left (g x + f\right ) + d g x \log \left (c\right ) + {\left (d f + e\right )} q \log \left (d g x + d f + e\right )}{d g} \]

[In]

integrate(log(c*(d+e/(g*x+f))^q),x, algorithm="fricas")

[Out]

(d*g*q*x*log((d*g*x + d*f + e)/(g*x + f)) - d*f*q*log(g*x + f) + d*g*x*log(c) + (d*f + e)*q*log(d*g*x + d*f +
e))/(d*g)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 102 vs. \(2 (36) = 72\).

Time = 0.78 (sec) , antiderivative size = 102, normalized size of antiderivative = 2.27 \[ \int \log \left (c \left (d+\frac {e}{f+g x}\right )^q\right ) \, dx=\begin {cases} x \log {\left (c \left (\frac {e}{f}\right )^{q} \right )} & \text {for}\: d = 0 \wedge g = 0 \\\frac {f \log {\left (c \left (\frac {e}{f + g x}\right )^{q} \right )}}{g} + q x + x \log {\left (c \left (\frac {e}{f + g x}\right )^{q} \right )} & \text {for}\: d = 0 \\x \log {\left (c \left (d + \frac {e}{f}\right )^{q} \right )} & \text {for}\: g = 0 \\\frac {f \log {\left (c \left (d + \frac {e}{f + g x}\right )^{q} \right )}}{g} + x \log {\left (c \left (d + \frac {e}{f + g x}\right )^{q} \right )} + \frac {e q \log {\left (d f + d g x + e \right )}}{d g} & \text {otherwise} \end {cases} \]

[In]

integrate(ln(c*(d+e/(g*x+f))**q),x)

[Out]

Piecewise((x*log(c*(e/f)**q), Eq(d, 0) & Eq(g, 0)), (f*log(c*(e/(f + g*x))**q)/g + q*x + x*log(c*(e/(f + g*x))
**q), Eq(d, 0)), (x*log(c*(d + e/f)**q), Eq(g, 0)), (f*log(c*(d + e/(f + g*x))**q)/g + x*log(c*(d + e/(f + g*x
))**q) + e*q*log(d*f + d*g*x + e)/(d*g), True))

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.44 \[ \int \log \left (c \left (d+\frac {e}{f+g x}\right )^q\right ) \, dx=-e g q {\left (\frac {f \log \left (g x + f\right )}{e g^{2}} - \frac {{\left (d f + e\right )} \log \left (d g x + d f + e\right )}{d e g^{2}}\right )} + x \log \left (c {\left (d + \frac {e}{g x + f}\right )}^{q}\right ) \]

[In]

integrate(log(c*(d+e/(g*x+f))^q),x, algorithm="maxima")

[Out]

-e*g*q*(f*log(g*x + f)/(e*g^2) - (d*f + e)*log(d*g*x + d*f + e)/(d*e*g^2)) + x*log(c*(d + e/(g*x + f))^q)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 171 vs. \(2 (45) = 90\).

Time = 0.35 (sec) , antiderivative size = 171, normalized size of antiderivative = 3.80 \[ \int \log \left (c \left (d+\frac {e}{f+g x}\right )^q\right ) \, dx={\left (\frac {e^{2} q \log \left (\frac {d g x + d f + e}{g x + f}\right )}{d g^{2} - \frac {{\left (d g x + d f + e\right )} g^{2}}{g x + f}} + \frac {e^{2} \log \left (c\right )}{d g^{2} - \frac {{\left (d g x + d f + e\right )} g^{2}}{g x + f}} + \frac {e^{2} q \log \left (-d + \frac {d g x + d f + e}{g x + f}\right )}{d g^{2}} - \frac {e^{2} q \log \left (\frac {d g x + d f + e}{g x + f}\right )}{d g^{2}}\right )} {\left (\frac {d f g}{e^{2}} - \frac {{\left (d f + e\right )} g}{e^{2}}\right )} \]

[In]

integrate(log(c*(d+e/(g*x+f))^q),x, algorithm="giac")

[Out]

(e^2*q*log((d*g*x + d*f + e)/(g*x + f))/(d*g^2 - (d*g*x + d*f + e)*g^2/(g*x + f)) + e^2*log(c)/(d*g^2 - (d*g*x
+ d*f + e)*g^2/(g*x + f)) + e^2*q*log(-d + (d*g*x + d*f + e)/(g*x + f))/(d*g^2) - e^2*q*log((d*g*x + d*f + e)
/(g*x + f))/(d*g^2))*(d*f*g/e^2 - (d*f + e)*g/e^2)

Mupad [B] (verification not implemented)

Time = 1.44 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.49 \[ \int \log \left (c \left (d+\frac {e}{f+g x}\right )^q\right ) \, dx=x\,\ln \left (c\,{\left (d+\frac {e}{f+g\,x}\right )}^q\right )-\frac {f\,q\,\ln \left (f+g\,x\right )}{g}+\frac {f\,q\,\ln \left (e+d\,f+d\,g\,x\right )}{g}+\frac {e\,q\,\ln \left (e+d\,f+d\,g\,x\right )}{d\,g} \]

[In]

int(log(c*(d + e/(f + g*x))^q),x)

[Out]

x*log(c*(d + e/(f + g*x))^q) - (f*q*log(f + g*x))/g + (f*q*log(e + d*f + d*g*x))/g + (e*q*log(e + d*f + d*g*x)
)/(d*g)