3.631 \(\int \log (c (d+e (f+g x))^q) \, dx\)

Optimal. Leaf size=35 \[ \frac{(d+e f+e g x) \log \left (c (d+e (f+g x))^q\right )}{e g}-q x \]

[Out]

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

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Rubi [A]  time = 0.0160286, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {2444, 2389, 2295} \[ \frac{(d+e f+e g x) \log \left (c (d+e (f+g x))^q\right )}{e g}-q x \]

Antiderivative was successfully verified.

[In]

Int[Log[c*(d + e*(f + g*x))^q],x]

[Out]

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

Rule 2444

Int[((a_.) + Log[(c_.)*(v_)^(n_.)]*(b_.))^(p_.)*(u_.), x_Symbol] :> Int[u*(a + b*Log[c*ExpandToSum[v, x]^n])^p
, x] /; FreeQ[{a, b, c, n, p}, x] && LinearQ[v, x] &&  !LinearMatchQ[v, x] &&  !(EqQ[n, 1] && MatchQ[c*v, (e_.
)*((f_) + (g_.)*x) /; FreeQ[{e, f, g}, x]])

Rule 2389

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.), x_Symbol] :> Dist[1/e, Subst[Int[(a + b*Log[c*
x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, n, p}, x]

Rule 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rubi steps

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

Mathematica [A]  time = 0.031426, size = 47, normalized size = 1.34 \[ \frac{(f+g x) \log \left (c (d+e (f+g x))^q\right )}{g}+\frac{d q \log (d+e f+e g x)}{e g}-q x \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.122, size = 57, normalized size = 1.6 \begin{align*} \ln \left ( c \left ( egx+fe+d \right ) ^{q} \right ) x-qx+{\frac{q\ln \left ( egx+fe+d \right ) f}{g}}+{\frac{q\ln \left ( egx+fe+d \right ) d}{eg}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.01673, size = 73, normalized size = 2.09 \begin{align*} -e g q{\left (\frac{x}{e g} - \frac{{\left (e f + d\right )} \log \left (e g x + e f + d\right )}{e^{2} g^{2}}\right )} + x \log \left ({\left ({\left (g x + f\right )} e + d\right )}^{q} c\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*(x/(e*g) - (e*f + d)*log(e*g*x + e*f + d)/(e^2*g^2)) + x*log(((g*x + f)*e + d)^q*c)

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Fricas [A]  time = 1.50005, size = 108, normalized size = 3.09 \begin{align*} -\frac{e g q x - e g x \log \left (c\right ) -{\left (e g q x +{\left (e f + d\right )} q\right )} \log \left (e g x + e f + d\right )}{e 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]

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

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Sympy [A]  time = 1.51682, size = 85, normalized size = 2.43 \begin{align*} \begin{cases} x \log{\left (c d^{q} \right )} & \text{for}\: e = 0 \wedge g = 0 \\x \log{\left (c \left (d + e f\right )^{q} \right )} & \text{for}\: g = 0 \\x \log{\left (c d^{q} \right )} & \text{for}\: e = 0 \\\frac{d q \log{\left (d + e f + e g x \right )}}{e g} + \frac{f q \log{\left (d + e f + e g x \right )}}{g} + q x \log{\left (d + e f + e g x \right )} - q x + x \log{\left (c \right )} & \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*d**q), Eq(e, 0) & Eq(g, 0)), (x*log(c*(d + e*f)**q), Eq(g, 0)), (x*log(c*d**q), Eq(e, 0)),
(d*q*log(d + e*f + e*g*x)/(e*g) + f*q*log(d + e*f + e*g*x)/g + q*x*log(d + e*f + e*g*x) - q*x + x*log(c), True
))

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Giac [A]  time = 1.17378, size = 93, normalized size = 2.66 \begin{align*} \frac{{\left (g x e + f e + d\right )} q e^{\left (-1\right )} \log \left (g x e + f e + d\right )}{g} - \frac{{\left (g x e + f e + d\right )} q e^{\left (-1\right )}}{g} + \frac{{\left (g x e + f e + d\right )} e^{\left (-1\right )} \log \left (c\right )}{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]

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