\(\int \log (c (d+e (f+g x)^p)^q) \, dx\) [715]

Optimal result

Integrand size = 16, antiderivative size = 76 \[ \int \log \left (c \left (d+e (f+g x)^p\right )^q\right ) \, dx=-\frac {e p q (f+g x)^{1+p} \operatorname {Hypergeometric2F1}\left (1,1+\frac {1}{p},2+\frac {1}{p},-\frac {e (f+g x)^p}{d}\right )}{d g (1+p)}+\frac {(f+g x) \log \left (c \left (d+e (f+g x)^p\right )^q\right )}{g} \] Output:

-e*p*q*(g*x+f)^(p+1)*hypergeom([1, 1+1/p],[2+1/p],-e*(g*x+f)^p/d)/d/g/(p+1 
)+(g*x+f)*ln(c*(d+e*(g*x+f)^p)^q)/g
 

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.86 \[ \int \log \left (c \left (d+e (f+g x)^p\right )^q\right ) \, dx=-p q x+\frac {p q (f+g x) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{p},1+\frac {1}{p},-\frac {e (f+g x)^p}{d}\right )}{g}+\frac {(f+g x) \log \left (c \left (d+e (f+g x)^p\right )^q\right )}{g} \] Input:

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

Output:

-(p*q*x) + (p*q*(f + g*x)*Hypergeometric2F1[1, p^(-1), 1 + p^(-1), -((e*(f 
 + g*x)^p)/d)])/g + ((f + g*x)*Log[c*(d + e*(f + g*x)^p)^q])/g
 

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.97, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {2933, 2898, 888}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \log \left (c \left (d+e (f+g x)^p\right )^q\right ) \, dx\)

\(\Big \downarrow \) 2933

\(\displaystyle \frac {\int \log \left (c \left (e (f+g x)^p+d\right )^q\right )d(f+g x)}{g}\)

\(\Big \downarrow \) 2898

\(\displaystyle \frac {(f+g x) \log \left (c \left (d+e (f+g x)^p\right )^q\right )-e p q \int \frac {(f+g x)^p}{e (f+g x)^p+d}d(f+g x)}{g}\)

\(\Big \downarrow \) 888

\(\displaystyle \frac {(f+g x) \log \left (c \left (d+e (f+g x)^p\right )^q\right )-\frac {e p q (f+g x)^{p+1} \operatorname {Hypergeometric2F1}\left (1,1+\frac {1}{p},2+\frac {1}{p},-\frac {e (f+g x)^p}{d}\right )}{d (p+1)}}{g}\)

Input:

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

Output:

(-((e*p*q*(f + g*x)^(1 + p)*Hypergeometric2F1[1, 1 + p^(-1), 2 + p^(-1), - 
((e*(f + g*x)^p)/d)])/(d*(1 + p))) + (f + g*x)*Log[c*(d + e*(f + g*x)^p)^q 
])/g
 

Defintions of rubi rules used

rule 888

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p 
*((c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/n, (m + 1)/n + 1 
, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] && (ILt 
Q[p, 0] || GtQ[a, 0])
 

rule 2898

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

rule 2933

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*((f_.) + (g_.)*(x_))^(n_))^(p_.)]*(b_. 
))^(q_.), x_Symbol] :> Simp[1/g   Subst[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])
 

Maple [F]

\[\int \ln \left (c \left (d +e \left (g x +f \right )^{p}\right )^{q}\right )d x\]

Input:

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

Output:

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

Fricas [F]

\[ \int \log \left (c \left (d+e (f+g x)^p\right )^q\right ) \, dx=\int { \log \left ({\left ({\left (g x + f\right )}^{p} e + d\right )}^{q} c\right ) \,d x } \] Input:

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

Output:

integral(log(((g*x + f)^p*e + d)^q*c), x)
 

Sympy [F]

\[ \int \log \left (c \left (d+e (f+g x)^p\right )^q\right ) \, dx=\int \log {\left (c \left (d + e \left (f + g x\right )^{p}\right )^{q} \right )}\, dx \] Input:

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

Output:

Integral(log(c*(d + e*(f + g*x)**p)**q), x)
 

Maxima [F]

\[ \int \log \left (c \left (d+e (f+g x)^p\right )^q\right ) \, dx=\int { \log \left ({\left ({\left (g x + f\right )}^{p} e + d\right )}^{q} c\right ) \,d x } \] Input:

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

Output:

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

Giac [F]

\[ \int \log \left (c \left (d+e (f+g x)^p\right )^q\right ) \, dx=\int { \log \left ({\left ({\left (g x + f\right )}^{p} e + d\right )}^{q} c\right ) \,d x } \] Input:

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

Output:

integrate(log(((g*x + f)^p*e + d)^q*c), x)
 

Mupad [F(-1)]

Timed out. \[ \int \log \left (c \left (d+e (f+g x)^p\right )^q\right ) \, dx=\int \ln \left (c\,{\left (d+e\,{\left (f+g\,x\right )}^p\right )}^q\right ) \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \log \left (c \left (d+e (f+g x)^p\right )^q\right ) \, dx=\frac {\left (\int \frac {x}{\left (g x +f \right )^{p} e f +\left (g x +f \right )^{p} e g x +d f +d g x}d x \right ) d \,g^{2} p q +\mathrm {log}\left (g x +f \right ) f p q +\mathrm {log}\left (\left (\left (g x +f \right )^{p} e +d \right )^{q} c \right ) g x -g p q x}{g} \] Input:

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

Output:

(int(x/((f + g*x)**p*e*f + (f + g*x)**p*e*g*x + d*f + d*g*x),x)*d*g**2*p*q 
 + log(f + g*x)*f*p*q + log(((f + g*x)**p*e + d)**q*c)*g*x - g*p*q*x)/g