Integrand size = 27, antiderivative size = 149 \[ \int \left (f+\frac {g}{x}\right )^3 x^3 \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx=\frac {b (d f-e g)^3 n x}{4 e^3}-\frac {b (d f-e g)^2 n (g+f x)^2}{8 e^2 f}+\frac {b (d f-e g) n (g+f x)^3}{12 e f}-\frac {b n (g+f x)^4}{16 f}-\frac {b (d f-e g)^4 n \log (d+e x)}{4 e^4 f}+\frac {(g+f x)^4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 f} \]
1/4*b*(d*f-e*g)^3*n*x/e^3-1/8*b*(d*f-e*g)^2*n*(f*x+g)^2/e^2/f+1/12*b*(d*f- e*g)*n*(f*x+g)^3/e/f-1/16*b*n*(f*x+g)^4/f-1/4*b*(d*f-e*g)^4*n*ln(e*x+d)/e^ 4/f+1/4*(f*x+g)^4*(a+b*ln(c*(e*x+d)^n))/f
Time = 0.14 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.52 \[ \int \left (f+\frac {g}{x}\right )^3 x^3 \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx=\frac {e x \left (12 a e^3 \left (4 g^3+6 f g^2 x+4 f^2 g x^2+f^3 x^3\right )+b n \left (12 d^3 f^3-6 d^2 e f^2 (8 g+f x)+4 d e^2 f \left (18 g^2+6 f g x+f^2 x^2\right )-e^3 \left (48 g^3+36 f g^2 x+16 f^2 g x^2+3 f^3 x^3\right )\right )\right )-12 b d^2 f \left (d^2 f^2-4 d e f g+6 e^2 g^2\right ) n \log (d+e x)+12 b e^3 \left (4 d g^3+e x \left (4 g^3+6 f g^2 x+4 f^2 g x^2+f^3 x^3\right )\right ) \log \left (c (d+e x)^n\right )}{48 e^4} \]
(e*x*(12*a*e^3*(4*g^3 + 6*f*g^2*x + 4*f^2*g*x^2 + f^3*x^3) + b*n*(12*d^3*f ^3 - 6*d^2*e*f^2*(8*g + f*x) + 4*d*e^2*f*(18*g^2 + 6*f*g*x + f^2*x^2) - e^ 3*(48*g^3 + 36*f*g^2*x + 16*f^2*g*x^2 + 3*f^3*x^3))) - 12*b*d^2*f*(d^2*f^2 - 4*d*e*f*g + 6*e^2*g^2)*n*Log[d + e*x] + 12*b*e^3*(4*d*g^3 + e*x*(4*g^3 + 6*f*g^2*x + 4*f^2*g*x^2 + f^3*x^3))*Log[c*(d + e*x)^n])/(48*e^4)
Time = 0.29 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.92, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {2005, 2842, 49, 2009}
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.
| \(\displaystyle \int x^3 \left (f+\frac {g}{x}\right )^3 \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx\) |
| \(\Big \downarrow \) 2005 |
| \(\displaystyle \int (f x+g)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )dx\) |
| \(\Big \downarrow \) 2842 |
| \(\displaystyle \frac {(f x+g)^4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 f}-\frac {b e n \int \frac {(g+f x)^4}{d+e x}dx}{4 f}\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle \frac {(f x+g)^4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 f}-\frac {b e n \int \left (\frac {(e g-d f)^4}{e^4 (d+e x)}+\frac {f (e g-d f)^3}{e^4}+\frac {f (g+f x) (e g-d f)^2}{e^3}+\frac {f (g+f x)^2 (e g-d f)}{e^2}+\frac {f (g+f x)^3}{e}\right )dx}{4 f}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {(f x+g)^4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 f}-\frac {b e n \left (\frac {(d f-e g)^4 \log (d+e x)}{e^5}-\frac {f x (d f-e g)^3}{e^4}+\frac {(f x+g)^2 (d f-e g)^2}{2 e^3}-\frac {(f x+g)^3 (d f-e g)}{3 e^2}+\frac {(f x+g)^4}{4 e}\right )}{4 f}\) |
-1/4*(b*e*n*(-((f*(d*f - e*g)^3*x)/e^4) + ((d*f - e*g)^2*(g + f*x)^2)/(2*e ^3) - ((d*f - e*g)*(g + f*x)^3)/(3*e^2) + (g + f*x)^4/(4*e) + ((d*f - e*g) ^4*Log[d + e*x])/e^5))/f + ((g + f*x)^4*(a + b*Log[c*(d + e*x)^n]))/(4*f)
3.4.5.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Int[(Fx_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[x^(m + n*p)*(b + a/x^n)^p*Fx, x] /; FreeQ[{a, b, m, n}, x] && IntegerQ[p] && Neg Q[n]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*((f_.) + (g_.)*(x_ ))^(q_.), x_Symbol] :> Simp[(f + g*x)^(q + 1)*((a + b*Log[c*(d + e*x)^n])/( g*(q + 1))), x] - Simp[b*e*(n/(g*(q + 1))) Int[(f + g*x)^(q + 1)/(d + e*x ), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*f - d*g, 0] && NeQ[q, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(427\) vs. \(2(137)=274\).
Time = 0.72 (sec) , antiderivative size = 428, normalized size of antiderivative = 2.87
| method | result | size |
| parallelrisch | \(-\frac {-48 b d \,e^{3} g^{3} n +12 b \,d^{4} f^{3} n +36 x^{2} b \,e^{4} f \,g^{2} n -48 x^{3} \ln \left (c \left (e x +d \right )^{n}\right ) b \,e^{4} f^{2} g -4 x^{3} b d \,e^{3} f^{3} n +16 x^{3} b \,e^{4} f^{2} g n +6 x^{2} b \,d^{2} e^{2} f^{3} n -12 x b \,d^{3} e \,f^{3} n -96 \ln \left (e x +d \right ) b d \,e^{3} g^{3} n +72 b \,d^{2} e^{2} f \,g^{2} n -72 x^{2} a \,e^{4} f \,g^{2}-12 x^{4} a \,e^{4} f^{3}-48 x a \,e^{4} g^{3}-48 b \,d^{3} e \,f^{2} g n +48 a d \,g^{3} e^{3}-24 x^{2} b d \,e^{3} f^{2} g n +48 x b \,d^{2} e^{2} f^{2} g n -48 \ln \left (e x +d \right ) b \,d^{3} e \,f^{2} g n +72 \ln \left (e x +d \right ) b \,d^{2} e^{2} f \,g^{2} n -72 x^{2} \ln \left (c \left (e x +d \right )^{n}\right ) b \,e^{4} f \,g^{2}-72 x b d \,e^{3} f \,g^{2} n -48 x^{3} a \,e^{4} f^{2} g +12 \ln \left (e x +d \right ) b \,d^{4} f^{3} n -12 x^{4} \ln \left (c \left (e x +d \right )^{n}\right ) b \,e^{4} f^{3}+48 \ln \left (c \left (e x +d \right )^{n}\right ) b d \,e^{3} g^{3}+48 x b \,e^{4} g^{3} n +3 x^{4} b \,e^{4} f^{3} n -48 x \ln \left (c \left (e x +d \right )^{n}\right ) b \,e^{4} g^{3}}{48 e^{4}}\) | \(428\) |
| risch | \(\text {Expression too large to display}\) | \(836\) |
-1/48*(-48*b*d*e^3*g^3*n+12*b*d^4*f^3*n+36*x^2*b*e^4*f*g^2*n-48*x^3*ln(c*( e*x+d)^n)*b*e^4*f^2*g-4*x^3*b*d*e^3*f^3*n+16*x^3*b*e^4*f^2*g*n+6*x^2*b*d^2 *e^2*f^3*n-12*x*b*d^3*e*f^3*n-96*ln(e*x+d)*b*d*e^3*g^3*n+72*b*d^2*e^2*f*g^ 2*n-72*x^2*a*e^4*f*g^2-12*x^4*a*e^4*f^3-48*x*a*e^4*g^3-48*b*d^3*e*f^2*g*n+ 48*a*d*g^3*e^3-24*x^2*b*d*e^3*f^2*g*n+48*x*b*d^2*e^2*f^2*g*n-48*ln(e*x+d)* b*d^3*e*f^2*g*n+72*ln(e*x+d)*b*d^2*e^2*f*g^2*n-72*x^2*ln(c*(e*x+d)^n)*b*e^ 4*f*g^2-72*x*b*d*e^3*f*g^2*n-48*x^3*a*e^4*f^2*g+12*ln(e*x+d)*b*d^4*f^3*n-1 2*x^4*ln(c*(e*x+d)^n)*b*e^4*f^3+48*ln(c*(e*x+d)^n)*b*d*e^3*g^3+48*x*b*e^4* g^3*n+3*x^4*b*e^4*f^3*n-48*x*ln(c*(e*x+d)^n)*b*e^4*g^3)/e^4
Leaf count of result is larger than twice the leaf count of optimal. 336 vs. \(2 (137) = 274\).
Time = 0.31 (sec) , antiderivative size = 336, normalized size of antiderivative = 2.26 \[ \int \left (f+\frac {g}{x}\right )^3 x^3 \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx=-\frac {3 \, {\left (b e^{4} f^{3} n - 4 \, a e^{4} f^{3}\right )} x^{4} - 4 \, {\left (12 \, a e^{4} f^{2} g + {\left (b d e^{3} f^{3} - 4 \, b e^{4} f^{2} g\right )} n\right )} x^{3} - 6 \, {\left (12 \, a e^{4} f g^{2} - {\left (b d^{2} e^{2} f^{3} - 4 \, b d e^{3} f^{2} g + 6 \, b e^{4} f g^{2}\right )} n\right )} x^{2} - 12 \, {\left (4 \, a e^{4} g^{3} + {\left (b d^{3} e f^{3} - 4 \, b d^{2} e^{2} f^{2} g + 6 \, b d e^{3} f g^{2} - 4 \, b e^{4} g^{3}\right )} n\right )} x - 12 \, {\left (b e^{4} f^{3} n x^{4} + 4 \, b e^{4} f^{2} g n x^{3} + 6 \, b e^{4} f g^{2} n x^{2} + 4 \, b e^{4} g^{3} n x - {\left (b d^{4} f^{3} - 4 \, b d^{3} e f^{2} g + 6 \, b d^{2} e^{2} f g^{2} - 4 \, b d e^{3} g^{3}\right )} n\right )} \log \left (e x + d\right ) - 12 \, {\left (b e^{4} f^{3} x^{4} + 4 \, b e^{4} f^{2} g x^{3} + 6 \, b e^{4} f g^{2} x^{2} + 4 \, b e^{4} g^{3} x\right )} \log \left (c\right )}{48 \, e^{4}} \]
-1/48*(3*(b*e^4*f^3*n - 4*a*e^4*f^3)*x^4 - 4*(12*a*e^4*f^2*g + (b*d*e^3*f^ 3 - 4*b*e^4*f^2*g)*n)*x^3 - 6*(12*a*e^4*f*g^2 - (b*d^2*e^2*f^3 - 4*b*d*e^3 *f^2*g + 6*b*e^4*f*g^2)*n)*x^2 - 12*(4*a*e^4*g^3 + (b*d^3*e*f^3 - 4*b*d^2* e^2*f^2*g + 6*b*d*e^3*f*g^2 - 4*b*e^4*g^3)*n)*x - 12*(b*e^4*f^3*n*x^4 + 4* b*e^4*f^2*g*n*x^3 + 6*b*e^4*f*g^2*n*x^2 + 4*b*e^4*g^3*n*x - (b*d^4*f^3 - 4 *b*d^3*e*f^2*g + 6*b*d^2*e^2*f*g^2 - 4*b*d*e^3*g^3)*n)*log(e*x + d) - 12*( b*e^4*f^3*x^4 + 4*b*e^4*f^2*g*x^3 + 6*b*e^4*f*g^2*x^2 + 4*b*e^4*g^3*x)*log (c))/e^4
Leaf count of result is larger than twice the leaf count of optimal. 410 vs. \(2 (128) = 256\).
Time = 19.60 (sec) , antiderivative size = 410, normalized size of antiderivative = 2.75 \[ \int \left (f+\frac {g}{x}\right )^3 x^3 \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx=\begin {cases} \frac {a f^{3} x^{4}}{4} + a f^{2} g x^{3} + \frac {3 a f g^{2} x^{2}}{2} + a g^{3} x - \frac {b d^{4} f^{3} \log {\left (c \left (d + e x\right )^{n} \right )}}{4 e^{4}} + \frac {b d^{3} f^{3} n x}{4 e^{3}} + \frac {b d^{3} f^{2} g \log {\left (c \left (d + e x\right )^{n} \right )}}{e^{3}} - \frac {b d^{2} f^{3} n x^{2}}{8 e^{2}} - \frac {b d^{2} f^{2} g n x}{e^{2}} - \frac {3 b d^{2} f g^{2} \log {\left (c \left (d + e x\right )^{n} \right )}}{2 e^{2}} + \frac {b d f^{3} n x^{3}}{12 e} + \frac {b d f^{2} g n x^{2}}{2 e} + \frac {3 b d f g^{2} n x}{2 e} + \frac {b d g^{3} \log {\left (c \left (d + e x\right )^{n} \right )}}{e} - \frac {b f^{3} n x^{4}}{16} + \frac {b f^{3} x^{4} \log {\left (c \left (d + e x\right )^{n} \right )}}{4} - \frac {b f^{2} g n x^{3}}{3} + b f^{2} g x^{3} \log {\left (c \left (d + e x\right )^{n} \right )} - \frac {3 b f g^{2} n x^{2}}{4} + \frac {3 b f g^{2} x^{2} \log {\left (c \left (d + e x\right )^{n} \right )}}{2} - b g^{3} n x + b g^{3} x \log {\left (c \left (d + e x\right )^{n} \right )} & \text {for}\: e \neq 0 \\\left (a + b \log {\left (c d^{n} \right )}\right ) \left (\frac {f^{3} x^{4}}{4} + f^{2} g x^{3} + \frac {3 f g^{2} x^{2}}{2} + g^{3} x\right ) & \text {otherwise} \end {cases} \]
Piecewise((a*f**3*x**4/4 + a*f**2*g*x**3 + 3*a*f*g**2*x**2/2 + a*g**3*x - b*d**4*f**3*log(c*(d + e*x)**n)/(4*e**4) + b*d**3*f**3*n*x/(4*e**3) + b*d* *3*f**2*g*log(c*(d + e*x)**n)/e**3 - b*d**2*f**3*n*x**2/(8*e**2) - b*d**2* f**2*g*n*x/e**2 - 3*b*d**2*f*g**2*log(c*(d + e*x)**n)/(2*e**2) + b*d*f**3* n*x**3/(12*e) + b*d*f**2*g*n*x**2/(2*e) + 3*b*d*f*g**2*n*x/(2*e) + b*d*g** 3*log(c*(d + e*x)**n)/e - b*f**3*n*x**4/16 + b*f**3*x**4*log(c*(d + e*x)** n)/4 - b*f**2*g*n*x**3/3 + b*f**2*g*x**3*log(c*(d + e*x)**n) - 3*b*f*g**2* n*x**2/4 + 3*b*f*g**2*x**2*log(c*(d + e*x)**n)/2 - b*g**3*n*x + b*g**3*x*l og(c*(d + e*x)**n), Ne(e, 0)), ((a + b*log(c*d**n))*(f**3*x**4/4 + f**2*g* x**3 + 3*f*g**2*x**2/2 + g**3*x), True))
Leaf count of result is larger than twice the leaf count of optimal. 284 vs. \(2 (137) = 274\).
Time = 0.21 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.91 \[ \int \left (f+\frac {g}{x}\right )^3 x^3 \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx=\frac {1}{4} \, b f^{3} x^{4} \log \left ({\left (e x + d\right )}^{n} c\right ) + \frac {1}{4} \, a f^{3} x^{4} + b f^{2} g x^{3} \log \left ({\left (e x + d\right )}^{n} c\right ) + a f^{2} g x^{3} - b e g^{3} n {\left (\frac {x}{e} - \frac {d \log \left (e x + d\right )}{e^{2}}\right )} - \frac {1}{48} \, b e f^{3} n {\left (\frac {12 \, d^{4} \log \left (e x + d\right )}{e^{5}} + \frac {3 \, e^{3} x^{4} - 4 \, d e^{2} x^{3} + 6 \, d^{2} e x^{2} - 12 \, d^{3} x}{e^{4}}\right )} + \frac {1}{6} \, b e f^{2} g n {\left (\frac {6 \, d^{3} \log \left (e x + d\right )}{e^{4}} - \frac {2 \, e^{2} x^{3} - 3 \, d e x^{2} + 6 \, d^{2} x}{e^{3}}\right )} - \frac {3}{4} \, b e f g^{2} n {\left (\frac {2 \, d^{2} \log \left (e x + d\right )}{e^{3}} + \frac {e x^{2} - 2 \, d x}{e^{2}}\right )} + \frac {3}{2} \, b f g^{2} x^{2} \log \left ({\left (e x + d\right )}^{n} c\right ) + \frac {3}{2} \, a f g^{2} x^{2} + b g^{3} x \log \left ({\left (e x + d\right )}^{n} c\right ) + a g^{3} x \]
1/4*b*f^3*x^4*log((e*x + d)^n*c) + 1/4*a*f^3*x^4 + b*f^2*g*x^3*log((e*x + d)^n*c) + a*f^2*g*x^3 - b*e*g^3*n*(x/e - d*log(e*x + d)/e^2) - 1/48*b*e*f^ 3*n*(12*d^4*log(e*x + d)/e^5 + (3*e^3*x^4 - 4*d*e^2*x^3 + 6*d^2*e*x^2 - 12 *d^3*x)/e^4) + 1/6*b*e*f^2*g*n*(6*d^3*log(e*x + d)/e^4 - (2*e^2*x^3 - 3*d* e*x^2 + 6*d^2*x)/e^3) - 3/4*b*e*f*g^2*n*(2*d^2*log(e*x + d)/e^3 + (e*x^2 - 2*d*x)/e^2) + 3/2*b*f*g^2*x^2*log((e*x + d)^n*c) + 3/2*a*f*g^2*x^2 + b*g^ 3*x*log((e*x + d)^n*c) + a*g^3*x
Leaf count of result is larger than twice the leaf count of optimal. 770 vs. \(2 (137) = 274\).
Time = 0.33 (sec) , antiderivative size = 770, normalized size of antiderivative = 5.17 \[ \int \left (f+\frac {g}{x}\right )^3 x^3 \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx=\frac {{\left (e x + d\right )}^{4} b f^{3} n \log \left (e x + d\right )}{4 \, e^{4}} - \frac {{\left (e x + d\right )}^{3} b d f^{3} n \log \left (e x + d\right )}{e^{4}} + \frac {3 \, {\left (e x + d\right )}^{2} b d^{2} f^{3} n \log \left (e x + d\right )}{2 \, e^{4}} - \frac {{\left (e x + d\right )} b d^{3} f^{3} n \log \left (e x + d\right )}{e^{4}} + \frac {{\left (e x + d\right )}^{3} b f^{2} g n \log \left (e x + d\right )}{e^{3}} - \frac {3 \, {\left (e x + d\right )}^{2} b d f^{2} g n \log \left (e x + d\right )}{e^{3}} + \frac {3 \, {\left (e x + d\right )} b d^{2} f^{2} g n \log \left (e x + d\right )}{e^{3}} + \frac {3 \, {\left (e x + d\right )}^{2} b f g^{2} n \log \left (e x + d\right )}{2 \, e^{2}} - \frac {3 \, {\left (e x + d\right )} b d f g^{2} n \log \left (e x + d\right )}{e^{2}} + \frac {{\left (e x + d\right )} b g^{3} n \log \left (e x + d\right )}{e} - \frac {{\left (e x + d\right )}^{4} b f^{3} n}{16 \, e^{4}} + \frac {{\left (e x + d\right )}^{3} b d f^{3} n}{3 \, e^{4}} - \frac {3 \, {\left (e x + d\right )}^{2} b d^{2} f^{3} n}{4 \, e^{4}} + \frac {{\left (e x + d\right )} b d^{3} f^{3} n}{e^{4}} - \frac {{\left (e x + d\right )}^{3} b f^{2} g n}{3 \, e^{3}} + \frac {3 \, {\left (e x + d\right )}^{2} b d f^{2} g n}{2 \, e^{3}} - \frac {3 \, {\left (e x + d\right )} b d^{2} f^{2} g n}{e^{3}} - \frac {3 \, {\left (e x + d\right )}^{2} b f g^{2} n}{4 \, e^{2}} + \frac {3 \, {\left (e x + d\right )} b d f g^{2} n}{e^{2}} - \frac {{\left (e x + d\right )} b g^{3} n}{e} + \frac {{\left (e x + d\right )}^{4} b f^{3} \log \left (c\right )}{4 \, e^{4}} - \frac {{\left (e x + d\right )}^{3} b d f^{3} \log \left (c\right )}{e^{4}} + \frac {3 \, {\left (e x + d\right )}^{2} b d^{2} f^{3} \log \left (c\right )}{2 \, e^{4}} - \frac {{\left (e x + d\right )} b d^{3} f^{3} \log \left (c\right )}{e^{4}} + \frac {{\left (e x + d\right )}^{3} b f^{2} g \log \left (c\right )}{e^{3}} - \frac {3 \, {\left (e x + d\right )}^{2} b d f^{2} g \log \left (c\right )}{e^{3}} + \frac {3 \, {\left (e x + d\right )} b d^{2} f^{2} g \log \left (c\right )}{e^{3}} + \frac {3 \, {\left (e x + d\right )}^{2} b f g^{2} \log \left (c\right )}{2 \, e^{2}} - \frac {3 \, {\left (e x + d\right )} b d f g^{2} \log \left (c\right )}{e^{2}} + \frac {{\left (e x + d\right )} b g^{3} \log \left (c\right )}{e} + \frac {{\left (e x + d\right )}^{4} a f^{3}}{4 \, e^{4}} - \frac {{\left (e x + d\right )}^{3} a d f^{3}}{e^{4}} + \frac {3 \, {\left (e x + d\right )}^{2} a d^{2} f^{3}}{2 \, e^{4}} - \frac {{\left (e x + d\right )} a d^{3} f^{3}}{e^{4}} + \frac {{\left (e x + d\right )}^{3} a f^{2} g}{e^{3}} - \frac {3 \, {\left (e x + d\right )}^{2} a d f^{2} g}{e^{3}} + \frac {3 \, {\left (e x + d\right )} a d^{2} f^{2} g}{e^{3}} + \frac {3 \, {\left (e x + d\right )}^{2} a f g^{2}}{2 \, e^{2}} - \frac {3 \, {\left (e x + d\right )} a d f g^{2}}{e^{2}} + \frac {{\left (e x + d\right )} a g^{3}}{e} \]
1/4*(e*x + d)^4*b*f^3*n*log(e*x + d)/e^4 - (e*x + d)^3*b*d*f^3*n*log(e*x + d)/e^4 + 3/2*(e*x + d)^2*b*d^2*f^3*n*log(e*x + d)/e^4 - (e*x + d)*b*d^3*f ^3*n*log(e*x + d)/e^4 + (e*x + d)^3*b*f^2*g*n*log(e*x + d)/e^3 - 3*(e*x + d)^2*b*d*f^2*g*n*log(e*x + d)/e^3 + 3*(e*x + d)*b*d^2*f^2*g*n*log(e*x + d) /e^3 + 3/2*(e*x + d)^2*b*f*g^2*n*log(e*x + d)/e^2 - 3*(e*x + d)*b*d*f*g^2* n*log(e*x + d)/e^2 + (e*x + d)*b*g^3*n*log(e*x + d)/e - 1/16*(e*x + d)^4*b *f^3*n/e^4 + 1/3*(e*x + d)^3*b*d*f^3*n/e^4 - 3/4*(e*x + d)^2*b*d^2*f^3*n/e ^4 + (e*x + d)*b*d^3*f^3*n/e^4 - 1/3*(e*x + d)^3*b*f^2*g*n/e^3 + 3/2*(e*x + d)^2*b*d*f^2*g*n/e^3 - 3*(e*x + d)*b*d^2*f^2*g*n/e^3 - 3/4*(e*x + d)^2*b *f*g^2*n/e^2 + 3*(e*x + d)*b*d*f*g^2*n/e^2 - (e*x + d)*b*g^3*n/e + 1/4*(e* x + d)^4*b*f^3*log(c)/e^4 - (e*x + d)^3*b*d*f^3*log(c)/e^4 + 3/2*(e*x + d) ^2*b*d^2*f^3*log(c)/e^4 - (e*x + d)*b*d^3*f^3*log(c)/e^4 + (e*x + d)^3*b*f ^2*g*log(c)/e^3 - 3*(e*x + d)^2*b*d*f^2*g*log(c)/e^3 + 3*(e*x + d)*b*d^2*f ^2*g*log(c)/e^3 + 3/2*(e*x + d)^2*b*f*g^2*log(c)/e^2 - 3*(e*x + d)*b*d*f*g ^2*log(c)/e^2 + (e*x + d)*b*g^3*log(c)/e + 1/4*(e*x + d)^4*a*f^3/e^4 - (e* x + d)^3*a*d*f^3/e^4 + 3/2*(e*x + d)^2*a*d^2*f^3/e^4 - (e*x + d)*a*d^3*f^3 /e^4 + (e*x + d)^3*a*f^2*g/e^3 - 3*(e*x + d)^2*a*d*f^2*g/e^3 + 3*(e*x + d) *a*d^2*f^2*g/e^3 + 3/2*(e*x + d)^2*a*f*g^2/e^2 - 3*(e*x + d)*a*d*f*g^2/e^2 + (e*x + d)*a*g^3/e
Time = 1.39 (sec) , antiderivative size = 352, normalized size of antiderivative = 2.36 \[ \int \left (f+\frac {g}{x}\right )^3 x^3 \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx=x\,\left (\frac {4\,a\,e\,g^3+12\,a\,d\,f\,g^2-4\,b\,e\,g^3\,n}{4\,e}+\frac {d\,\left (\frac {d\,\left (\frac {f^2\,\left (a\,d\,f+3\,a\,e\,g-b\,e\,g\,n\right )}{e}-\frac {d\,f^3\,\left (4\,a-b\,n\right )}{4\,e}\right )}{e}-\frac {3\,f\,g\,\left (2\,a\,d\,f+2\,a\,e\,g-b\,e\,g\,n\right )}{2\,e}\right )}{e}\right )+x^3\,\left (\frac {f^2\,\left (a\,d\,f+3\,a\,e\,g-b\,e\,g\,n\right )}{3\,e}-\frac {d\,f^3\,\left (4\,a-b\,n\right )}{12\,e}\right )+\ln \left (c\,{\left (d+e\,x\right )}^n\right )\,\left (\frac {b\,f^3\,x^4}{4}+b\,f^2\,g\,x^3+\frac {3\,b\,f\,g^2\,x^2}{2}+b\,g^3\,x\right )-x^2\,\left (\frac {d\,\left (\frac {f^2\,\left (a\,d\,f+3\,a\,e\,g-b\,e\,g\,n\right )}{e}-\frac {d\,f^3\,\left (4\,a-b\,n\right )}{4\,e}\right )}{2\,e}-\frac {3\,f\,g\,\left (2\,a\,d\,f+2\,a\,e\,g-b\,e\,g\,n\right )}{4\,e}\right )-\frac {\ln \left (d+e\,x\right )\,\left (b\,n\,d^4\,f^3-4\,b\,n\,d^3\,e\,f^2\,g+6\,b\,n\,d^2\,e^2\,f\,g^2-4\,b\,n\,d\,e^3\,g^3\right )}{4\,e^4}+\frac {f^3\,x^4\,\left (4\,a-b\,n\right )}{16} \]
x*((4*a*e*g^3 + 12*a*d*f*g^2 - 4*b*e*g^3*n)/(4*e) + (d*((d*((f^2*(a*d*f + 3*a*e*g - b*e*g*n))/e - (d*f^3*(4*a - b*n))/(4*e)))/e - (3*f*g*(2*a*d*f + 2*a*e*g - b*e*g*n))/(2*e)))/e) + x^3*((f^2*(a*d*f + 3*a*e*g - b*e*g*n))/(3 *e) - (d*f^3*(4*a - b*n))/(12*e)) + log(c*(d + e*x)^n)*((b*f^3*x^4)/4 + b* g^3*x + (3*b*f*g^2*x^2)/2 + b*f^2*g*x^3) - x^2*((d*((f^2*(a*d*f + 3*a*e*g - b*e*g*n))/e - (d*f^3*(4*a - b*n))/(4*e)))/(2*e) - (3*f*g*(2*a*d*f + 2*a* e*g - b*e*g*n))/(4*e)) - (log(d + e*x)*(b*d^4*f^3*n - 4*b*d*e^3*g^3*n - 4* b*d^3*e*f^2*g*n + 6*b*d^2*e^2*f*g^2*n))/(4*e^4) + (f^3*x^4*(4*a - b*n))/16