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

Optimal. Leaf size=45 \[ \frac{(f+g x) \log \left (c \left (d+\frac{e}{f+g x}\right )^q\right )}{g}+\frac{e q \log (d (f+g x)+e)}{d g} \]

[Out]

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

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Rubi [A]  time = 0.0267869, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2483, 2448, 263, 31} \[ \frac{(f+g x) \log \left (c \left (d+\frac{e}{f+g x}\right )^q\right )}{g}+\frac{e q \log (d (f+g x)+e)}{d g} \]

Antiderivative was successfully verified.

[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 2483

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])

Rule 2448

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 263

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 31

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

Rubi steps

\begin{align*} \int \log \left (c \left (d+\frac{e}{f+g x}\right )^q\right ) \, dx &=\frac{\operatorname{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) \operatorname{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) \operatorname{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]  time = 0.0468237, size = 56, normalized size = 1.24 \[ \frac{d g x \log \left (c \left (d+\frac{e}{f+g x}\right )^q\right )+q (d f+e) \log (d f+d g x+e)-d f q \log (f+g x)}{d g} \]

Antiderivative was successfully verified.

[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)

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Maple [A]  time = 0.131, size = 74, normalized size = 1.6 \begin{align*} \ln \left ( c \left ({\frac{dgx+df+e}{gx+f}} \right ) ^{q} \right ) x-{\frac{qf\ln \left ( gx+f \right ) }{g}}+{\frac{q\ln \left ( dgx+df+e \right ) f}{g}}+{\frac{eq\ln \left ( dgx+df+e \right ) }{dg}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln(c*((d*g*x+d*f+e)/(g*x+f))^q)*x-1/g*q*f*ln(g*x+f)+1/g*q*ln(d*g*x+d*f+e)*f+1/g*e*q/d*ln(d*g*x+d*f+e)

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Maxima [A]  time = 1.02127, size = 88, normalized size = 1.96 \begin{align*} -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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [A]  time = 1.602, size = 163, normalized size = 3.62 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [A]  time = 2.37966, size = 109, normalized size = 2.42 \begin{align*} \begin{cases} x \log{\left (c \left (\frac{e}{f}\right )^{q} \right )} & \text{for}\: d = 0 \wedge g = 0 \\x \log{\left (c \left (d + \frac{e}{f}\right )^{q} \right )} & \text{for}\: g = 0 \\- \frac{f q \log{\left (f + g x \right )}}{g} + q x \log{\left (e \right )} - q x \log{\left (f + g x \right )} + q x + x \log{\left (c \right )} & \text{for}\: d = 0 \\\frac{f q \log{\left (d + \frac{e}{f + g x} \right )}}{g} + q x \log{\left (d + \frac{e}{f + g x} \right )} + x \log{\left (c \right )} + \frac{e q \log{\left (d f + d g x + e \right )}}{d g} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

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

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Giac [A]  time = 1.30012, size = 116, normalized size = 2.58 \begin{align*} \frac{d g q x \log \left (d g x + d f + e\right ) - d g q x \log \left (g x + f\right ) + d f q \log \left (d g x + d f + e\right ) - d f q \log \left (-g x - f\right ) + d g x \log \left (c\right ) + q e \log \left (d g x + d f + e\right )}{d g} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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