Integrand size = 35, antiderivative size = 742 \[ \int \frac {(i+j x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{g+h x} \, dx=\frac {6 a b^2 j (f i-e j) p^2 q^2 x}{f h}+\frac {6 a b^2 j (h i-g j) p^2 q^2 x}{h^2}-\frac {6 b^3 j (f i-e j) p^3 q^3 x}{f h}-\frac {6 b^3 j (h i-g j) p^3 q^3 x}{h^2}-\frac {3 b^3 j^2 p^3 q^3 (e+f x)^2}{8 f^2 h}+\frac {6 b^3 j (f i-e j) p^2 q^2 (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )}{f^2 h}+\frac {6 b^3 j (h i-g j) p^2 q^2 (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )}{f h^2}+\frac {3 b^2 j^2 p^2 q^2 (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{4 f^2 h}-\frac {3 b j (f i-e j) p q (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{f^2 h}-\frac {3 b j (h i-g j) p q (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{f h^2}-\frac {3 b j^2 p q (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{4 f^2 h}+\frac {j (f i-e j) (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{f^2 h}+\frac {j (h i-g j) (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{f h^2}+\frac {j^2 (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{2 f^2 h}+\frac {(h i-g j)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3 \log \left (\frac {f (g+h x)}{f g-e h}\right )}{h^3}+\frac {3 b (h i-g j)^2 p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \operatorname {PolyLog}\left (2,-\frac {h (e+f x)}{f g-e h}\right )}{h^3}-\frac {6 b^2 (h i-g j)^2 p^2 q^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \operatorname {PolyLog}\left (3,-\frac {h (e+f x)}{f g-e h}\right )}{h^3}+\frac {6 b^3 (h i-g j)^2 p^3 q^3 \operatorname {PolyLog}\left (4,-\frac {h (e+f x)}{f g-e h}\right )}{h^3} \]
6*a*b^2*j*(-e*j+f*i)*p^2*q^2*x/f/h+6*a*b^2*j*(-g*j+h*i)*p^2*q^2*x/h^2-6*b^ 3*j*(-e*j+f*i)*p^3*q^3*x/f/h-6*b^3*j*(-g*j+h*i)*p^3*q^3*x/h^2-3/8*b^3*j^2* p^3*q^3*(f*x+e)^2/f^2/h+6*b^3*j*(-e*j+f*i)*p^2*q^2*(f*x+e)*ln(c*(d*(f*x+e) ^p)^q)/f^2/h+6*b^3*j*(-g*j+h*i)*p^2*q^2*(f*x+e)*ln(c*(d*(f*x+e)^p)^q)/f/h^ 2+3/4*b^2*j^2*p^2*q^2*(f*x+e)^2*(a+b*ln(c*(d*(f*x+e)^p)^q))/f^2/h-3*b*j*(- e*j+f*i)*p*q*(f*x+e)*(a+b*ln(c*(d*(f*x+e)^p)^q))^2/f^2/h-3*b*j*(-g*j+h*i)* p*q*(f*x+e)*(a+b*ln(c*(d*(f*x+e)^p)^q))^2/f/h^2-3/4*b*j^2*p*q*(f*x+e)^2*(a +b*ln(c*(d*(f*x+e)^p)^q))^2/f^2/h+j*(-e*j+f*i)*(f*x+e)*(a+b*ln(c*(d*(f*x+e )^p)^q))^3/f^2/h+j*(-g*j+h*i)*(f*x+e)*(a+b*ln(c*(d*(f*x+e)^p)^q))^3/f/h^2+ 1/2*j^2*(f*x+e)^2*(a+b*ln(c*(d*(f*x+e)^p)^q))^3/f^2/h+(-g*j+h*i)^2*(a+b*ln (c*(d*(f*x+e)^p)^q))^3*ln(f*(h*x+g)/(-e*h+f*g))/h^3+3*b*(-g*j+h*i)^2*p*q*( a+b*ln(c*(d*(f*x+e)^p)^q))^2*polylog(2,-h*(f*x+e)/(-e*h+f*g))/h^3-6*b^2*(- g*j+h*i)^2*p^2*q^2*(a+b*ln(c*(d*(f*x+e)^p)^q))*polylog(3,-h*(f*x+e)/(-e*h+ f*g))/h^3+6*b^3*(-g*j+h*i)^2*p^3*q^3*polylog(4,-h*(f*x+e)/(-e*h+f*g))/h^3
Leaf count is larger than twice the leaf count of optimal. \(4056\) vs. \(2(742)=1484\).
Time = 0.89 (sec) , antiderivative size = 4056, normalized size of antiderivative = 5.47 \[ \int \frac {(i+j x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{g+h x} \, dx=\text {Result too large to show} \]
(-48*a^2*b*e*f*h^2*i*j*p*q + 24*a^2*b*e*f*g*h*j^2*p*q + 16*a^3*f^2*h^2*i*j *x - 8*a^3*f^2*g*h*j^2*x - 48*a^2*b*f^2*h^2*i*j*p*q*x + 24*a^2*b*f^2*g*h*j ^2*p*q*x + 12*a^2*b*e*f*h^2*j^2*p*q*x + 96*a*b^2*f^2*h^2*i*j*p^2*q^2*x - 4 8*a*b^2*f^2*g*h*j^2*p^2*q^2*x - 36*a*b^2*e*f*h^2*j^2*p^2*q^2*x - 96*b^3*f^ 2*h^2*i*j*p^3*q^3*x + 48*b^3*f^2*g*h*j^2*p^3*q^3*x + 42*b^3*e*f*h^2*j^2*p^ 3*q^3*x + 4*a^3*f^2*h^2*j^2*x^2 - 6*a^2*b*f^2*h^2*j^2*p*q*x^2 + 6*a*b^2*f^ 2*h^2*j^2*p^2*q^2*x^2 - 3*b^3*f^2*h^2*j^2*p^3*q^3*x^2 + 48*a^2*b*e*f*h^2*i *j*p*q*Log[e + f*x] - 24*a^2*b*e*f*g*h*j^2*p*q*Log[e + f*x] - 12*a^2*b*e^2 *h^2*j^2*p*q*Log[e + f*x] + 36*a*b^2*e^2*h^2*j^2*p^2*q^2*Log[e + f*x] + 96 *b^3*e*f*h^2*i*j*p^3*q^3*Log[e + f*x] - 48*b^3*e*f*g*h*j^2*p^3*q^3*Log[e + f*x] - 42*b^3*e^2*h^2*j^2*p^3*q^3*Log[e + f*x] - 48*a*b^2*e*f*h^2*i*j*p^2 *q^2*Log[e + f*x]^2 + 24*a*b^2*e*f*g*h*j^2*p^2*q^2*Log[e + f*x]^2 + 12*a*b ^2*e^2*h^2*j^2*p^2*q^2*Log[e + f*x]^2 - 18*b^3*e^2*h^2*j^2*p^3*q^3*Log[e + f*x]^2 + 16*b^3*e*f*h^2*i*j*p^3*q^3*Log[e + f*x]^3 - 8*b^3*e*f*g*h*j^2*p^ 3*q^3*Log[e + f*x]^3 - 4*b^3*e^2*h^2*j^2*p^3*q^3*Log[e + f*x]^3 - 96*a*b^2 *e*f*h^2*i*j*p*q*Log[c*(d*(e + f*x)^p)^q] + 48*a*b^2*e*f*g*h*j^2*p*q*Log[c *(d*(e + f*x)^p)^q] + 48*a^2*b*f^2*h^2*i*j*x*Log[c*(d*(e + f*x)^p)^q] - 24 *a^2*b*f^2*g*h*j^2*x*Log[c*(d*(e + f*x)^p)^q] - 96*a*b^2*f^2*h^2*i*j*p*q*x *Log[c*(d*(e + f*x)^p)^q] + 48*a*b^2*f^2*g*h*j^2*p*q*x*Log[c*(d*(e + f*x)^ p)^q] + 24*a*b^2*e*f*h^2*j^2*p*q*x*Log[c*(d*(e + f*x)^p)^q] + 96*b^3*f^...
Time = 1.98 (sec) , antiderivative size = 742, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {2895, 2865, 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.
| \(\displaystyle \int \frac {(i+j x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{g+h x} \, dx\) |
| \(\Big \downarrow \) 2895 |
| \(\displaystyle \int \frac {(i+j x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{g+h x}dx\) |
| \(\Big \downarrow \) 2865 |
| \(\displaystyle \int \left (\frac {j (h i-g j) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{h^2}+\frac {(h i-g j)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{h^2 (g+h x)}+\frac {j (i+j x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{h}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3 b^2 j^2 p^2 q^2 (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{4 f^2 h}-\frac {6 b^2 p^2 q^2 (h i-g j)^2 \operatorname {PolyLog}\left (3,-\frac {h (e+f x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h^3}+\frac {6 a b^2 j p^2 q^2 x (f i-e j)}{f h}+\frac {6 a b^2 j p^2 q^2 x (h i-g j)}{h^2}-\frac {3 b j p q (e+f x) (f i-e j) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{f^2 h}+\frac {j (e+f x) (f i-e j) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{f^2 h}-\frac {3 b j^2 p q (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{4 f^2 h}+\frac {j^2 (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{2 f^2 h}+\frac {3 b p q (h i-g j)^2 \operatorname {PolyLog}\left (2,-\frac {h (e+f x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{h^3}+\frac {(h i-g j)^2 \log \left (\frac {f (g+h x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{h^3}-\frac {3 b j p q (e+f x) (h i-g j) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{f h^2}+\frac {j (e+f x) (h i-g j) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{f h^2}+\frac {6 b^3 j p^2 q^2 (e+f x) (f i-e j) \log \left (c \left (d (e+f x)^p\right )^q\right )}{f^2 h}+\frac {6 b^3 j p^2 q^2 (e+f x) (h i-g j) \log \left (c \left (d (e+f x)^p\right )^q\right )}{f h^2}-\frac {3 b^3 j^2 p^3 q^3 (e+f x)^2}{8 f^2 h}+\frac {6 b^3 p^3 q^3 (h i-g j)^2 \operatorname {PolyLog}\left (4,-\frac {h (e+f x)}{f g-e h}\right )}{h^3}-\frac {6 b^3 j p^3 q^3 x (f i-e j)}{f h}-\frac {6 b^3 j p^3 q^3 x (h i-g j)}{h^2}\) |
(6*a*b^2*j*(f*i - e*j)*p^2*q^2*x)/(f*h) + (6*a*b^2*j*(h*i - g*j)*p^2*q^2*x )/h^2 - (6*b^3*j*(f*i - e*j)*p^3*q^3*x)/(f*h) - (6*b^3*j*(h*i - g*j)*p^3*q ^3*x)/h^2 - (3*b^3*j^2*p^3*q^3*(e + f*x)^2)/(8*f^2*h) + (6*b^3*j*(f*i - e* j)*p^2*q^2*(e + f*x)*Log[c*(d*(e + f*x)^p)^q])/(f^2*h) + (6*b^3*j*(h*i - g *j)*p^2*q^2*(e + f*x)*Log[c*(d*(e + f*x)^p)^q])/(f*h^2) + (3*b^2*j^2*p^2*q ^2*(e + f*x)^2*(a + b*Log[c*(d*(e + f*x)^p)^q]))/(4*f^2*h) - (3*b*j*(f*i - e*j)*p*q*(e + f*x)*(a + b*Log[c*(d*(e + f*x)^p)^q])^2)/(f^2*h) - (3*b*j*( h*i - g*j)*p*q*(e + f*x)*(a + b*Log[c*(d*(e + f*x)^p)^q])^2)/(f*h^2) - (3* b*j^2*p*q*(e + f*x)^2*(a + b*Log[c*(d*(e + f*x)^p)^q])^2)/(4*f^2*h) + (j*( f*i - e*j)*(e + f*x)*(a + b*Log[c*(d*(e + f*x)^p)^q])^3)/(f^2*h) + (j*(h*i - g*j)*(e + f*x)*(a + b*Log[c*(d*(e + f*x)^p)^q])^3)/(f*h^2) + (j^2*(e + f*x)^2*(a + b*Log[c*(d*(e + f*x)^p)^q])^3)/(2*f^2*h) + ((h*i - g*j)^2*(a + b*Log[c*(d*(e + f*x)^p)^q])^3*Log[(f*(g + h*x))/(f*g - e*h)])/h^3 + (3*b* (h*i - g*j)^2*p*q*(a + b*Log[c*(d*(e + f*x)^p)^q])^2*PolyLog[2, -((h*(e + f*x))/(f*g - e*h))])/h^3 - (6*b^2*(h*i - g*j)^2*p^2*q^2*(a + b*Log[c*(d*(e + f*x)^p)^q])*PolyLog[3, -((h*(e + f*x))/(f*g - e*h))])/h^3 + (6*b^3*(h*i - g*j)^2*p^3*q^3*PolyLog[4, -((h*(e + f*x))/(f*g - e*h))])/h^3
3.6.35.3.1 Defintions of rubi rules used
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*(RFx_), x_Sy mbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n}, x] && RationalFunctionQ[ RFx, x] && IntegerQ[p]
Int[((a_.) + Log[(c_.)*((d_.)*((e_.) + (f_.)*(x_))^(m_.))^(n_)]*(b_.))^(p_. )*(u_.), x_Symbol] :> Subst[Int[u*(a + b*Log[c*d^n*(e + f*x)^(m*n)])^p, x], c*d^n*(e + f*x)^(m*n), c*(d*(e + f*x)^m)^n] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && !IntegerQ[n] && !(EqQ[d, 1] && EqQ[m, 1]) && IntegralFreeQ[ IntHide[u*(a + b*Log[c*d^n*(e + f*x)^(m*n)])^p, x]]
\[\int \frac {\left (j x +i \right )^{2} {\left (a +b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )\right )}^{3}}{h x +g}d x\]
\[ \int \frac {(i+j x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{g+h x} \, dx=\int { \frac {{\left (j x + i\right )}^{2} {\left (b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}^{3}}{h x + g} \,d x } \]
integral((a^3*j^2*x^2 + 2*a^3*i*j*x + a^3*i^2 + (b^3*j^2*x^2 + 2*b^3*i*j*x + b^3*i^2)*log(((f*x + e)^p*d)^q*c)^3 + 3*(a*b^2*j^2*x^2 + 2*a*b^2*i*j*x + a*b^2*i^2)*log(((f*x + e)^p*d)^q*c)^2 + 3*(a^2*b*j^2*x^2 + 2*a^2*b*i*j*x + a^2*b*i^2)*log(((f*x + e)^p*d)^q*c))/(h*x + g), x)
\[ \int \frac {(i+j x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{g+h x} \, dx=\int \frac {\left (a + b \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}\right )^{3} \left (i + j x\right )^{2}}{g + h x}\, dx \]
\[ \int \frac {(i+j x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{g+h x} \, dx=\int { \frac {{\left (j x + i\right )}^{2} {\left (b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}^{3}}{h x + g} \,d x } \]
2*a^3*i*j*(x/h - g*log(h*x + g)/h^2) + 1/2*a^3*j^2*(2*g^2*log(h*x + g)/h^3 + (h*x^2 - 2*g*x)/h^2) + a^3*i^2*log(h*x + g)/h + integrate((3*(i^2*q*log (d) + i^2*log(c))*a^2*b + 3*(i^2*q^2*log(d)^2 + 2*i^2*q*log(c)*log(d) + i^ 2*log(c)^2)*a*b^2 + (i^2*q^3*log(d)^3 + 3*i^2*q^2*log(c)*log(d)^2 + 3*i^2* q*log(c)^2*log(d) + i^2*log(c)^3)*b^3 + (b^3*j^2*x^2 + 2*b^3*i*j*x + b^3*i ^2)*log(((f*x + e)^p)^q)^3 + (3*(j^2*q*log(d) + j^2*log(c))*a^2*b + 3*(j^2 *q^2*log(d)^2 + 2*j^2*q*log(c)*log(d) + j^2*log(c)^2)*a*b^2 + (j^2*q^3*log (d)^3 + 3*j^2*q^2*log(c)*log(d)^2 + 3*j^2*q*log(c)^2*log(d) + j^2*log(c)^3 )*b^3)*x^2 + 3*(a*b^2*i^2 + (i^2*q*log(d) + i^2*log(c))*b^3 + (a*b^2*j^2 + (j^2*q*log(d) + j^2*log(c))*b^3)*x^2 + 2*(a*b^2*i*j + (i*j*q*log(d) + i*j *log(c))*b^3)*x)*log(((f*x + e)^p)^q)^2 + 2*(3*(i*j*q*log(d) + i*j*log(c)) *a^2*b + 3*(i*j*q^2*log(d)^2 + 2*i*j*q*log(c)*log(d) + i*j*log(c)^2)*a*b^2 + (i*j*q^3*log(d)^3 + 3*i*j*q^2*log(c)*log(d)^2 + 3*i*j*q*log(c)^2*log(d) + i*j*log(c)^3)*b^3)*x + 3*(a^2*b*i^2 + 2*(i^2*q*log(d) + i^2*log(c))*a*b ^2 + (i^2*q^2*log(d)^2 + 2*i^2*q*log(c)*log(d) + i^2*log(c)^2)*b^3 + (a^2* b*j^2 + 2*(j^2*q*log(d) + j^2*log(c))*a*b^2 + (j^2*q^2*log(d)^2 + 2*j^2*q* log(c)*log(d) + j^2*log(c)^2)*b^3)*x^2 + 2*(a^2*b*i*j + 2*(i*j*q*log(d) + i*j*log(c))*a*b^2 + (i*j*q^2*log(d)^2 + 2*i*j*q*log(c)*log(d) + i*j*log(c) ^2)*b^3)*x)*log(((f*x + e)^p)^q))/(h*x + g), x)
\[ \int \frac {(i+j x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{g+h x} \, dx=\int { \frac {{\left (j x + i\right )}^{2} {\left (b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}^{3}}{h x + g} \,d x } \]
Timed out. \[ \int \frac {(i+j x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{g+h x} \, dx=\int \frac {{\left (i+j\,x\right )}^2\,{\left (a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )\right )}^3}{g+h\,x} \,d x \]