\(\int \log (c (d+\frac {e}{(f+g x)^3})^q) \, dx\) [721]

Optimal result

Integrand size = 16, antiderivative size = 165 \[ \int \log \left (c \left (d+\frac {e}{(f+g x)^3}\right )^q\right ) \, dx=-\frac {\sqrt {3} \sqrt [3]{e} q \arctan \left (\frac {\sqrt [3]{e}-2 \sqrt [3]{d} (f+g x)}{\sqrt {3} \sqrt [3]{e}}\right )}{\sqrt [3]{d} g}+\frac {(f+g x) \log \left (c \left (d+\frac {e}{(f+g x)^3}\right )^q\right )}{g}+\frac {\sqrt [3]{e} q \log \left (\sqrt [3]{e}+\sqrt [3]{d} (f+g x)\right )}{\sqrt [3]{d} g}-\frac {\sqrt [3]{e} q \log \left (e^{2/3}-\sqrt [3]{d} \sqrt [3]{e} (f+g x)+d^{2/3} (f+g x)^2\right )}{2 \sqrt [3]{d} g} \] Output:

-3^(1/2)*e^(1/3)*q*arctan(1/3*(e^(1/3)-2*d^(1/3)*(g*x+f))*3^(1/2)/e^(1/3)) 
/d^(1/3)/g+(g*x+f)*ln(c*(d+e/(g*x+f)^3)^q)/g+e^(1/3)*q*ln(e^(1/3)+d^(1/3)* 
(g*x+f))/d^(1/3)/g-1/2*e^(1/3)*q*ln(e^(2/3)-d^(1/3)*e^(1/3)*(g*x+f)+d^(2/3 
)*(g*x+f)^2)/d^(1/3)/g
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 0.60 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.40 \[ \int \log \left (c \left (d+\frac {e}{(f+g x)^3}\right )^q\right ) \, dx=-\frac {3 e q \operatorname {Hypergeometric2F1}\left (\frac {2}{3},1,\frac {5}{3},-\frac {e}{d (f+g x)^3}\right )}{2 d g (f+g x)^2}+\frac {(f+g x) \log \left (c \left (d+\frac {e}{(f+g x)^3}\right )^q\right )}{g} \] Input:

Integrate[Log[c*(d + e/(f + g*x)^3)^q],x]
 

Output:

(-3*e*q*Hypergeometric2F1[2/3, 1, 5/3, -(e/(d*(f + g*x)^3))])/(2*d*g*(f + 
g*x)^2) + ((f + g*x)*Log[c*(d + e/(f + g*x)^3)^q])/g
 

Rubi [A] (verified)

Time = 0.60 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.96, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.688, Rules used = {2933, 2898, 795, 750, 16, 1142, 25, 27, 1082, 217, 1103}

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.

\(\displaystyle \int \log \left (c \left (d+\frac {e}{(f+g x)^3}\right )^q\right ) \, dx\)

\(\Big \downarrow \) 2933

\(\displaystyle \frac {\int \log \left (c \left (d+\frac {e}{(f+g x)^3}\right )^q\right )d(f+g x)}{g}\)

\(\Big \downarrow \) 2898

\(\displaystyle \frac {3 e q \int \frac {1}{(f+g x)^3 \left (d+\frac {e}{(f+g x)^3}\right )}d(f+g x)+(f+g x) \log \left (c \left (d+\frac {e}{(f+g x)^3}\right )^q\right )}{g}\)

\(\Big \downarrow \) 795

\(\displaystyle \frac {3 e q \int \frac {1}{d (f+g x)^3+e}d(f+g x)+(f+g x) \log \left (c \left (d+\frac {e}{(f+g x)^3}\right )^q\right )}{g}\)

\(\Big \downarrow \) 750

\(\displaystyle \frac {3 e q \left (\frac {\int \frac {2 \sqrt [3]{e}-\sqrt [3]{d} (f+g x)}{d^{2/3} (f+g x)^2-\sqrt [3]{d} \sqrt [3]{e} (f+g x)+e^{2/3}}d(f+g x)}{3 e^{2/3}}+\frac {\int \frac {1}{\sqrt [3]{d} (f+g x)+\sqrt [3]{e}}d(f+g x)}{3 e^{2/3}}\right )+(f+g x) \log \left (c \left (d+\frac {e}{(f+g x)^3}\right )^q\right )}{g}\)

\(\Big \downarrow \) 16

\(\displaystyle \frac {3 e q \left (\frac {\int \frac {2 \sqrt [3]{e}-\sqrt [3]{d} (f+g x)}{d^{2/3} (f+g x)^2-\sqrt [3]{d} \sqrt [3]{e} (f+g x)+e^{2/3}}d(f+g x)}{3 e^{2/3}}+\frac {\log \left (\sqrt [3]{d} (f+g x)+\sqrt [3]{e}\right )}{3 \sqrt [3]{d} e^{2/3}}\right )+(f+g x) \log \left (c \left (d+\frac {e}{(f+g x)^3}\right )^q\right )}{g}\)

\(\Big \downarrow \) 1142

\(\displaystyle \frac {3 e q \left (\frac {\frac {3}{2} \sqrt [3]{e} \int \frac {1}{d^{2/3} (f+g x)^2-\sqrt [3]{d} \sqrt [3]{e} (f+g x)+e^{2/3}}d(f+g x)-\frac {\int -\frac {\sqrt [3]{d} \left (\sqrt [3]{e}-2 \sqrt [3]{d} (f+g x)\right )}{d^{2/3} (f+g x)^2-\sqrt [3]{d} \sqrt [3]{e} (f+g x)+e^{2/3}}d(f+g x)}{2 \sqrt [3]{d}}}{3 e^{2/3}}+\frac {\log \left (\sqrt [3]{d} (f+g x)+\sqrt [3]{e}\right )}{3 \sqrt [3]{d} e^{2/3}}\right )+(f+g x) \log \left (c \left (d+\frac {e}{(f+g x)^3}\right )^q\right )}{g}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {3 e q \left (\frac {\frac {3}{2} \sqrt [3]{e} \int \frac {1}{d^{2/3} (f+g x)^2-\sqrt [3]{d} \sqrt [3]{e} (f+g x)+e^{2/3}}d(f+g x)+\frac {\int \frac {\sqrt [3]{d} \left (\sqrt [3]{e}-2 \sqrt [3]{d} (f+g x)\right )}{d^{2/3} (f+g x)^2-\sqrt [3]{d} \sqrt [3]{e} (f+g x)+e^{2/3}}d(f+g x)}{2 \sqrt [3]{d}}}{3 e^{2/3}}+\frac {\log \left (\sqrt [3]{d} (f+g x)+\sqrt [3]{e}\right )}{3 \sqrt [3]{d} e^{2/3}}\right )+(f+g x) \log \left (c \left (d+\frac {e}{(f+g x)^3}\right )^q\right )}{g}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {3 e q \left (\frac {\frac {3}{2} \sqrt [3]{e} \int \frac {1}{d^{2/3} (f+g x)^2-\sqrt [3]{d} \sqrt [3]{e} (f+g x)+e^{2/3}}d(f+g x)+\frac {1}{2} \int \frac {\sqrt [3]{e}-2 \sqrt [3]{d} (f+g x)}{d^{2/3} (f+g x)^2-\sqrt [3]{d} \sqrt [3]{e} (f+g x)+e^{2/3}}d(f+g x)}{3 e^{2/3}}+\frac {\log \left (\sqrt [3]{d} (f+g x)+\sqrt [3]{e}\right )}{3 \sqrt [3]{d} e^{2/3}}\right )+(f+g x) \log \left (c \left (d+\frac {e}{(f+g x)^3}\right )^q\right )}{g}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {3 e q \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{e}-2 \sqrt [3]{d} (f+g x)}{d^{2/3} (f+g x)^2-\sqrt [3]{d} \sqrt [3]{e} (f+g x)+e^{2/3}}d(f+g x)+\frac {3 \int \frac {1}{-\left (1-\frac {2 \sqrt [3]{d} (f+g x)}{\sqrt [3]{e}}\right )^2-3}d\left (1-\frac {2 \sqrt [3]{d} (f+g x)}{\sqrt [3]{e}}\right )}{\sqrt [3]{d}}}{3 e^{2/3}}+\frac {\log \left (\sqrt [3]{d} (f+g x)+\sqrt [3]{e}\right )}{3 \sqrt [3]{d} e^{2/3}}\right )+(f+g x) \log \left (c \left (d+\frac {e}{(f+g x)^3}\right )^q\right )}{g}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {3 e q \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{e}-2 \sqrt [3]{d} (f+g x)}{d^{2/3} (f+g x)^2-\sqrt [3]{d} \sqrt [3]{e} (f+g x)+e^{2/3}}d(f+g x)-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{d} (f+g x)}{\sqrt [3]{e}}}{\sqrt {3}}\right )}{\sqrt [3]{d}}}{3 e^{2/3}}+\frac {\log \left (\sqrt [3]{d} (f+g x)+\sqrt [3]{e}\right )}{3 \sqrt [3]{d} e^{2/3}}\right )+(f+g x) \log \left (c \left (d+\frac {e}{(f+g x)^3}\right )^q\right )}{g}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {3 e q \left (\frac {-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{d} (f+g x)}{\sqrt [3]{e}}}{\sqrt {3}}\right )}{\sqrt [3]{d}}-\frac {\log \left (d^{2/3} (f+g x)^2-\sqrt [3]{d} \sqrt [3]{e} (f+g x)+e^{2/3}\right )}{2 \sqrt [3]{d}}}{3 e^{2/3}}+\frac {\log \left (\sqrt [3]{d} (f+g x)+\sqrt [3]{e}\right )}{3 \sqrt [3]{d} e^{2/3}}\right )+(f+g x) \log \left (c \left (d+\frac {e}{(f+g x)^3}\right )^q\right )}{g}\)

Input:

Int[Log[c*(d + e/(f + g*x)^3)^q],x]
 

Output:

((f + g*x)*Log[c*(d + e/(f + g*x)^3)^q] + 3*e*q*(Log[e^(1/3) + d^(1/3)*(f 
+ g*x)]/(3*d^(1/3)*e^(2/3)) + (-((Sqrt[3]*ArcTan[(1 - (2*d^(1/3)*(f + g*x) 
)/e^(1/3))/Sqrt[3]])/d^(1/3)) - Log[e^(2/3) - d^(1/3)*e^(1/3)*(f + g*x) + 
d^(2/3)*(f + g*x)^2]/(2*d^(1/3)))/(3*e^(2/3))))/g
 

Defintions of rubi rules used

rule 16

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

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 217

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( 
-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & 
& (LtQ[a, 0] || LtQ[b, 0])
 

rule 750

Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Simp[1/(3*Rt[a, 3]^2)   Int[1/ 
(Rt[a, 3] + Rt[b, 3]*x), x], x] + Simp[1/(3*Rt[a, 3]^2)   Int[(2*Rt[a, 3] - 
 Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2*x^2), x], x] /; 
 FreeQ[{a, b}, x]
 

rule 795

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 1082

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S 
implify[a*(c/b^2)]}, Simp[-2/b   Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b 
)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /; Fre 
eQ[{a, b, c}, x]
 

rule 1103

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[2*c*d - b*e, 0]
 

rule 1142

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[(2*c*d - b*e)/(2*c)   Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) 
Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
 

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 [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 4.13 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.84

method	result	size
parts	\(\ln \left (c \left (d +\frac {e}{\left (g x +f \right )^{3}}\right )^{q}\right ) x +3 q e g \left (-\frac {f \ln \left (g x +f \right )}{g^{2} e}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (d \,g^{3} \textit {\_Z}^{3}+3 d f \,g^{2} \textit {\_Z}^{2}+3 d \,f^{2} g \textit {\_Z} +d \,f^{3}+e \right )}{\sum }\frac {\left (\textit {\_R}^{2} d f \,g^{2}+2 \textit {\_R} d \,f^{2} g +d \,f^{3}+e \right ) \ln \left (x -\textit {\_R} \right )}{g^{2} \textit {\_R}^{2}+2 g f \textit {\_R} +f^{2}}}{3 d \,g^{2} e}\right )\)	\(138\)
default	\(x \ln \left (c \left (\frac {d \,g^{3} x^{3}+3 d f \,g^{2} x^{2}+3 d \,f^{2} g x +d \,f^{3}+e}{\left (g x +f \right )^{3}}\right )^{q}\right )+3 q e g \left (-\frac {f \ln \left (g x +f \right )}{g^{2} e}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (d \,g^{3} \textit {\_Z}^{3}+3 d f \,g^{2} \textit {\_Z}^{2}+3 d \,f^{2} g \textit {\_Z} +d \,f^{3}+e \right )}{\sum }\frac {\left (\textit {\_R}^{2} d f \,g^{2}+2 \textit {\_R} d \,f^{2} g +d \,f^{3}+e \right ) \ln \left (x -\textit {\_R} \right )}{g^{2} \textit {\_R}^{2}+2 g f \textit {\_R} +f^{2}}}{3 d \,g^{2} e}\right )\)	\(168\)

Input:

int(ln(c*(d+e/(g*x+f)^3)^q),x,method=_RETURNVERBOSE)
 

Output:

ln(c*(d+e/(g*x+f)^3)^q)*x+3*q*e*g*(-f/g^2/e*ln(g*x+f)+1/3/d/g^2*sum((_R^2* 
d*f*g^2+2*_R*d*f^2*g+d*f^3+e)/(_R^2*g^2+2*_R*f*g+f^2)*ln(x-_R),_R=RootOf(_ 
Z^3*d*g^3+3*_Z^2*d*f*g^2+3*_Z*d*f^2*g+d*f^3+e))/e)
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.00 (sec) , antiderivative size = 1169, normalized size of antiderivative = 7.08 \[ \int \log \left (c \left (d+\frac {e}{(f+g x)^3}\right )^q\right ) \, dx=\text {Too large to display} \] Input:

integrate(log(c*(d+e/(g*x+f)^3)^q),x, algorithm="fricas")
 

Output:

1/4*(4*g*q*x*log((d*g^3*x^3 + 3*d*f*g^2*x^2 + 3*d*f^2*g*x + d*f^3 + e)/(g^ 
3*x^3 + 3*f*g^2*x^2 + 3*f^2*g*x + f^3)) - 12*f*q*log(g*x + f) - 2*((-1/2*f 
^3*q^3/g^3 + 1/2*e*q^3/(d*g^3) + 1/2*(d*f^3*q^3 + e*q^3)/(d*g^3))^(1/3)*(I 
*sqrt(3) + 1) - 2*f*q/g)*g*log(q*x - 1/2*(-1/2*f^3*q^3/g^3 + 1/2*e*q^3/(d* 
g^3) + 1/2*(d*f^3*q^3 + e*q^3)/(d*g^3))^(1/3)*(I*sqrt(3) + 1) + f*q/g) + 4 
*g*x*log(c) + (((-1/2*f^3*q^3/g^3 + 1/2*e*q^3/(d*g^3) + 1/2*(d*f^3*q^3 + e 
*q^3)/(d*g^3))^(1/3)*(I*sqrt(3) + 1) - 2*f*q/g)*g + 6*f*q + sqrt(3)*g*sqrt 
(-(((-1/2*f^3*q^3/g^3 + 1/2*e*q^3/(d*g^3) + 1/2*(d*f^3*q^3 + e*q^3)/(d*g^3 
))^(1/3)*(I*sqrt(3) + 1) - 2*f*q/g)^2*g^2 + 4*((-1/2*f^3*q^3/g^3 + 1/2*e*q 
^3/(d*g^3) + 1/2*(d*f^3*q^3 + e*q^3)/(d*g^3))^(1/3)*(I*sqrt(3) + 1) - 2*f* 
q/g)*f*g*q + 4*f^2*q^2)/g^2))*log(2*g*q*x + 1/2*((-1/2*f^3*q^3/g^3 + 1/2*e 
*q^3/(d*g^3) + 1/2*(d*f^3*q^3 + e*q^3)/(d*g^3))^(1/3)*(I*sqrt(3) + 1) - 2* 
f*q/g)*g + 3*f*q + 1/2*sqrt(3)*g*sqrt(-(((-1/2*f^3*q^3/g^3 + 1/2*e*q^3/(d* 
g^3) + 1/2*(d*f^3*q^3 + e*q^3)/(d*g^3))^(1/3)*(I*sqrt(3) + 1) - 2*f*q/g)^2 
*g^2 + 4*((-1/2*f^3*q^3/g^3 + 1/2*e*q^3/(d*g^3) + 1/2*(d*f^3*q^3 + e*q^3)/ 
(d*g^3))^(1/3)*(I*sqrt(3) + 1) - 2*f*q/g)*f*g*q + 4*f^2*q^2)/g^2)) + (((-1 
/2*f^3*q^3/g^3 + 1/2*e*q^3/(d*g^3) + 1/2*(d*f^3*q^3 + e*q^3)/(d*g^3))^(1/3 
)*(I*sqrt(3) + 1) - 2*f*q/g)*g + 6*f*q - sqrt(3)*g*sqrt(-(((-1/2*f^3*q^3/g 
^3 + 1/2*e*q^3/(d*g^3) + 1/2*(d*f^3*q^3 + e*q^3)/(d*g^3))^(1/3)*(I*sqrt(3) 
 + 1) - 2*f*q/g)^2*g^2 + 4*((-1/2*f^3*q^3/g^3 + 1/2*e*q^3/(d*g^3) + 1/2...
 

Sympy [F(-1)]

Timed out. \[ \int \log \left (c \left (d+\frac {e}{(f+g x)^3}\right )^q\right ) \, dx=\text {Timed out} \] Input:

integrate(ln(c*(d+e/(g*x+f)**3)**q),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \log \left (c \left (d+\frac {e}{(f+g x)^3}\right )^q\right ) \, dx=\int { \log \left (c {\left (d + \frac {e}{{\left (g x + f\right )}^{3}}\right )}^{q}\right ) \,d x } \] Input:

integrate(log(c*(d+e/(g*x+f)^3)^q),x, algorithm="maxima")
 

Output:

3*q*integrate((d*f*g^2*x^2 + 2*d*f^2*g*x + d*f^3 + e)/(d*g^3*x^3 + 3*d*f*g 
^2*x^2 + 3*d*f^2*g*x + d*f^3 + e), x) - (3*f*q*log(g*x + f) - g*x*log((d*g 
^3*x^3 + 3*d*f*g^2*x^2 + 3*d*f^2*g*x + d*f^3 + e)^q) + 3*g*x*log((g*x + f) 
^q) - g*x*log(c))/g
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 319 vs. \(2 (132) = 264\).

Time = 0.46 (sec) , antiderivative size = 319, normalized size of antiderivative = 1.93 \[ \int \log \left (c \left (d+\frac {e}{(f+g x)^3}\right )^q\right ) \, dx=\frac {1}{2} \, d e g^{5} q {\left (\frac {2 \, f \log \left ({\left | d g^{3} x^{3} + 3 \, d f g^{2} x^{2} + 3 \, d f^{2} g x + d f^{3} + e \right |}\right )}{d e g^{6}} - \frac {6 \, f \log \left ({\left | g x + f \right |}\right )}{d e g^{6}} + \frac {2 \, \sqrt {3} \left (d^{5} e^{4} g^{21}\right )^{\frac {1}{3}} \arctan \left (-\frac {d g x + d f + \left (d^{2} e\right )^{\frac {1}{3}}}{\sqrt {3} d g x + \sqrt {3} d f - \sqrt {3} \left (d^{2} e\right )^{\frac {1}{3}}}\right ) - \left (d^{5} e^{4} g^{21}\right )^{\frac {1}{3}} \log \left (4 \, {\left (\sqrt {3} d g x + \sqrt {3} d f - \sqrt {3} \left (d^{2} e\right )^{\frac {1}{3}}\right )}^{2} + 4 \, {\left (d g x + d f + \left (d^{2} e\right )^{\frac {1}{3}}\right )}^{2}\right ) + 2 \, \left (d^{5} e^{4} g^{21}\right )^{\frac {1}{3}} \log \left ({\left | d g x + d f + \left (d^{2} e\right )^{\frac {1}{3}} \right |}\right )}{d^{3} e^{2} g^{13}}\right )} + q x \log \left (d g^{3} x^{3} + 3 \, d f g^{2} x^{2} + 3 \, d f^{2} g x + d f^{3} + e\right ) - q x \log \left (g^{3} x^{3} + 3 \, f g^{2} x^{2} + 3 \, f^{2} g x + f^{3}\right ) + x \log \left (c\right ) \] Input:

integrate(log(c*(d+e/(g*x+f)^3)^q),x, algorithm="giac")
 

Output:

1/2*d*e*g^5*q*(2*f*log(abs(d*g^3*x^3 + 3*d*f*g^2*x^2 + 3*d*f^2*g*x + d*f^3 
 + e))/(d*e*g^6) - 6*f*log(abs(g*x + f))/(d*e*g^6) + (2*sqrt(3)*(d^5*e^4*g 
^21)^(1/3)*arctan(-(d*g*x + d*f + (d^2*e)^(1/3))/(sqrt(3)*d*g*x + sqrt(3)* 
d*f - sqrt(3)*(d^2*e)^(1/3))) - (d^5*e^4*g^21)^(1/3)*log(4*(sqrt(3)*d*g*x 
+ sqrt(3)*d*f - sqrt(3)*(d^2*e)^(1/3))^2 + 4*(d*g*x + d*f + (d^2*e)^(1/3)) 
^2) + 2*(d^5*e^4*g^21)^(1/3)*log(abs(d*g*x + d*f + (d^2*e)^(1/3))))/(d^3*e 
^2*g^13)) + q*x*log(d*g^3*x^3 + 3*d*f*g^2*x^2 + 3*d*f^2*g*x + d*f^3 + e) - 
 q*x*log(g^3*x^3 + 3*f*g^2*x^2 + 3*f^2*g*x + f^3) + x*log(c)
 

Mupad [B] (verification not implemented)

Time = 26.12 (sec) , antiderivative size = 499, normalized size of antiderivative = 3.02 \[ \int \log \left (c \left (d+\frac {e}{(f+g x)^3}\right )^q\right ) \, dx=x\,\ln \left (c\,{\left (d+\frac {e}{{\left (f+g\,x\right )}^3}\right )}^q\right )-\left (\sum _{k=1}^3\ln \left (-d^2\,e^2\,g^{11}\,\left (3\,e\,q^3\,x+\mathrm {root}\left (d\,g^3\,z^3+3\,d\,f\,g^2\,q\,z^2+3\,d\,f^2\,g\,q^2\,z+d\,f^3\,q^3+e\,q^3,z,k\right )\,e\,q^2+{\mathrm {root}\left (d\,g^3\,z^3+3\,d\,f\,g^2\,q\,z^2+3\,d\,f^2\,g\,q^2\,z+d\,f^3\,q^3+e\,q^3,z,k\right )}^3\,d\,f\,g^2\,4+\mathrm {root}\left (d\,g^3\,z^3+3\,d\,f\,g^2\,q\,z^2+3\,d\,f^2\,g\,q^2\,z+d\,f^3\,q^3+e\,q^3,z,k\right )\,d\,f^3\,q^2\,4+{\mathrm {root}\left (d\,g^3\,z^3+3\,d\,f\,g^2\,q\,z^2+3\,d\,f^2\,g\,q^2\,z+d\,f^3\,q^3+e\,q^3,z,k\right )}^3\,d\,g^3\,x\,4+{\mathrm {root}\left (d\,g^3\,z^3+3\,d\,f\,g^2\,q\,z^2+3\,d\,f^2\,g\,q^2\,z+d\,f^3\,q^3+e\,q^3,z,k\right )}^2\,d\,f^2\,g\,q\,8+\mathrm {root}\left (d\,g^3\,z^3+3\,d\,f\,g^2\,q\,z^2+3\,d\,f^2\,g\,q^2\,z+d\,f^3\,q^3+e\,q^3,z,k\right )\,d\,f^2\,g\,q^2\,x\,4+{\mathrm {root}\left (d\,g^3\,z^3+3\,d\,f\,g^2\,q\,z^2+3\,d\,f^2\,g\,q^2\,z+d\,f^3\,q^3+e\,q^3,z,k\right )}^2\,d\,f\,g^2\,q\,x\,8\right )\,9\right )\,\mathrm {root}\left (d\,g^3\,z^3+3\,d\,f\,g^2\,q\,z^2+3\,d\,f^2\,g\,q^2\,z+d\,f^3\,q^3+e\,q^3,z,k\right )\right )-\frac {3\,f\,q\,\ln \left (f+g\,x\right )}{g} \] Input:

int(log(c*(d + e/(f + g*x)^3)^q),x)
 

Output:

x*log(c*(d + e/(f + g*x)^3)^q) - symsum(log(-9*d^2*e^2*g^11*(3*e*q^3*x + r 
oot(d*g^3*z^3 + 3*d*f*g^2*q*z^2 + 3*d*f^2*g*q^2*z + d*f^3*q^3 + e*q^3, z, 
k)*e*q^2 + 4*root(d*g^3*z^3 + 3*d*f*g^2*q*z^2 + 3*d*f^2*g*q^2*z + d*f^3*q^ 
3 + e*q^3, z, k)^3*d*f*g^2 + 4*root(d*g^3*z^3 + 3*d*f*g^2*q*z^2 + 3*d*f^2* 
g*q^2*z + d*f^3*q^3 + e*q^3, z, k)*d*f^3*q^2 + 4*root(d*g^3*z^3 + 3*d*f*g^ 
2*q*z^2 + 3*d*f^2*g*q^2*z + d*f^3*q^3 + e*q^3, z, k)^3*d*g^3*x + 8*root(d* 
g^3*z^3 + 3*d*f*g^2*q*z^2 + 3*d*f^2*g*q^2*z + d*f^3*q^3 + e*q^3, z, k)^2*d 
*f^2*g*q + 4*root(d*g^3*z^3 + 3*d*f*g^2*q*z^2 + 3*d*f^2*g*q^2*z + d*f^3*q^ 
3 + e*q^3, z, k)*d*f^2*g*q^2*x + 8*root(d*g^3*z^3 + 3*d*f*g^2*q*z^2 + 3*d* 
f^2*g*q^2*z + d*f^3*q^3 + e*q^3, z, k)^2*d*f*g^2*q*x))*root(d*g^3*z^3 + 3* 
d*f*g^2*q*z^2 + 3*d*f^2*g*q^2*z + d*f^3*q^3 + e*q^3, z, k), k, 1, 3) - (3* 
f*q*log(f + g*x))/g
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.84 \[ \int \log \left (c \left (d+\frac {e}{(f+g x)^3}\right )^q\right ) \, dx=\frac {2 e^{\frac {1}{3}} \sqrt {3}\, \mathit {atan} \left (\frac {2 d^{\frac {1}{3}} f +2 d^{\frac {1}{3}} g x -e^{\frac {1}{3}}}{e^{\frac {1}{3}} \sqrt {3}}\right ) q +2 d^{\frac {1}{3}} \mathrm {log}\left (\frac {\left (d \,g^{3} x^{3}+3 d f \,g^{2} x^{2}+3 d \,f^{2} g x +d \,f^{3}+e \right )^{q} c}{\left (g^{3} x^{3}+3 f \,g^{2} x^{2}+3 f^{2} g x +f^{3}\right )^{q}}\right ) f +2 d^{\frac {1}{3}} \mathrm {log}\left (\frac {\left (d \,g^{3} x^{3}+3 d f \,g^{2} x^{2}+3 d \,f^{2} g x +d \,f^{3}+e \right )^{q} c}{\left (g^{3} x^{3}+3 f \,g^{2} x^{2}+3 f^{2} g x +f^{3}\right )^{q}}\right ) g x +3 e^{\frac {1}{3}} \mathrm {log}\left (d^{\frac {1}{3}} f +d^{\frac {1}{3}} g x +e^{\frac {1}{3}}\right ) q -3 e^{\frac {1}{3}} \mathrm {log}\left (g x +f \right ) q -e^{\frac {1}{3}} \mathrm {log}\left (\frac {\left (d \,g^{3} x^{3}+3 d f \,g^{2} x^{2}+3 d \,f^{2} g x +d \,f^{3}+e \right )^{q} c}{\left (g^{3} x^{3}+3 f \,g^{2} x^{2}+3 f^{2} g x +f^{3}\right )^{q}}\right )}{2 d^{\frac {1}{3}} g} \] Input:

int(log(c*(d+e/(g*x+f)^3)^q),x)
 

Output:

(2*e**(1/3)*sqrt(3)*atan((2*d**(1/3)*f + 2*d**(1/3)*g*x - e**(1/3))/(e**(1 
/3)*sqrt(3)))*q + 2*d**(1/3)*log(((d*f**3 + 3*d*f**2*g*x + 3*d*f*g**2*x**2 
 + d*g**3*x**3 + e)**q*c)/(f**3 + 3*f**2*g*x + 3*f*g**2*x**2 + g**3*x**3)* 
*q)*f + 2*d**(1/3)*log(((d*f**3 + 3*d*f**2*g*x + 3*d*f*g**2*x**2 + d*g**3* 
x**3 + e)**q*c)/(f**3 + 3*f**2*g*x + 3*f*g**2*x**2 + g**3*x**3)**q)*g*x + 
3*e**(1/3)*log(d**(1/3)*f + d**(1/3)*g*x + e**(1/3))*q - 3*e**(1/3)*log(f 
+ g*x)*q - e**(1/3)*log(((d*f**3 + 3*d*f**2*g*x + 3*d*f*g**2*x**2 + d*g**3 
*x**3 + e)**q*c)/(f**3 + 3*f**2*g*x + 3*f*g**2*x**2 + g**3*x**3)**q))/(2*d 
**(1/3)*g)