3.7.31 \(\int \log (c (d+e (f+g x))^q) \, dx\) [631]

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

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 35, normalized size of antiderivative = 1.00, 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 = {2494, 2436, 2332} \begin {gather*} \frac {(d+e f+e g x) \log \left (c (d+e (f+g x))^q\right )}{e g}-q x \end {gather*}

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 2332

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

Rule 2436

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 2494

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]])

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 {\text {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*}

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Mathematica [A]
time = 0.02, size = 44, normalized size = 1.26 \begin {gather*} \frac {(d+e f) q \log (d+e f+e g x)}{e g}+x \left (-q+\log \left (c (d+e f+e g x)^q\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.22, size = 57, normalized size = 1.63

method	result	size
norman	\(x \ln \left (c \,{\mathrm e}^{q \ln \left (d +\left (g x +f \right ) e \right )}\right )+\frac {q \left (e f +d \right ) \ln \left (d +\left (g x +f \right ) e \right )}{e g}-q x\)	\(47\)
default	\(\ln \left (c \left (e g x +e f +d \right )^{q}\right ) x -e g q \left (\frac {x}{e g}+\frac {\left (-e f -d \right ) \ln \left (e g x +e f +d \right )}{e^{2} g^{2}}\right )\)	\(57\)
risch	\(x \ln \left (\left (e g x +e f +d \right )^{q}\right )-\frac {i \pi x \,\mathrm {csgn}\left (i \left (e g x +e f +d \right )^{q}\right ) \mathrm {csgn}\left (i c \left (e g x +e f +d \right )^{q}\right ) \mathrm {csgn}\left (i c \right )}{2}+\frac {i \pi x \mathrm {csgn}\left (i c \left (e g x +e f +d \right )^{q}\right )^{2} \mathrm {csgn}\left (i c \right )}{2}+\frac {i \pi x \,\mathrm {csgn}\left (i \left (e g x +e f +d \right )^{q}\right ) \mathrm {csgn}\left (i c \left (e g x +e f +d \right )^{q}\right )^{2}}{2}-\frac {i \pi x \mathrm {csgn}\left (i c \left (e g x +e f +d \right )^{q}\right )^{3}}{2}+x \ln \left (c \right )+\frac {\ln \left (e g x +e f +d \right ) f q}{g}-q x +\frac {\ln \left (e g x +e f +d \right ) d q}{e g}\)	\(189\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.30, size = 57, normalized size = 1.63 \begin {gather*} -g q {\left (\frac {x e^{\left (-1\right )}}{g} - \frac {{\left (f e + d\right )} e^{\left (-2\right )} \log \left (g x e + f e + d\right )}{g^{2}}\right )} e + x \log \left ({\left ({\left (g x + f\right )} e + d\right )}^{q} c\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-g*q*(x*e^(-1)/g - (f*e + d)*e^(-2)*log(g*x*e + f*e + d)/g^2)*e + x*log(((g*x + f)*e + d)^q*c)

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Fricas [A]
time = 0.35, size = 50, normalized size = 1.43 \begin {gather*} -\frac {{\left (g q x e - g x e \log \left (c\right ) - {\left (d q + {\left (g q x + f q\right )} e\right )} \log \left ({\left (g x + f\right )} e + d\right )\right )} e^{\left (-1\right )}}{g} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 80 vs. \(2 (29) = 58\).
time = 0.37, size = 80, normalized size = 2.29 \begin {gather*} \begin {cases} x \log {\left (c d^{q} \right )} & \text {for}\: e = 0 \wedge \left (e = 0 \vee g = 0\right ) \\x \log {\left (c \left (d + e f\right )^{q} \right )} & \text {for}\: g = 0 \\\frac {d \log {\left (c \left (d + e f + e g x\right )^{q} \right )}}{e g} + \frac {f \log {\left (c \left (d + e f + e g x\right )^{q} \right )}}{g} - q x + x \log {\left (c \left (d + e f + e g x\right )^{q} \right )} & \text {otherwise} \end {cases} \end {gather*}

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

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Giac [A]
time = 5.05, size = 69, normalized size = 1.97 \begin {gather*} \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 {gather*}

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

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Mupad [B]
time = 0.31, size = 46, normalized size = 1.31 \begin {gather*} x\,\ln \left (c\,{\left (d+e\,\left (f+g\,x\right )\right )}^q\right )-q\,x+\frac {\ln \left (d+e\,f+e\,g\,x\right )\,\left (d\,q+e\,f\,q\right )}{e\,g} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(c*(d + e*(f + g*x))^q),x)

[Out]

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

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