Integrand size = 42, antiderivative size = 722 \[ \int \frac {(a g+b g x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(c i+d i x)^2} \, dx=\frac {2 A B (b c-a d)^2 g^3 (a+b x)}{d^3 i^2 (c+d x)}-\frac {2 B^2 (b c-a d)^2 g^3 (a+b x)}{d^3 i^2 (c+d x)}+\frac {2 B^2 (b c-a d)^2 g^3 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{d^3 i^2 (c+d x)}-\frac {b B (b c-a d) g^3 (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d^3 i^2}-\frac {6 b B (b c-a d)^2 g^3 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d^4 i^2}-\frac {3 b (b c-a d) g^3 (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^3 i^2}-\frac {(b c-a d)^2 g^3 (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^3 i^2 (c+d x)}+\frac {b^3 g^3 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 d^4 i^2}-\frac {3 b (b c-a d)^2 g^3 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^4 i^2}+\frac {b B^2 (b c-a d)^2 g^3 \log (c+d x)}{d^4 i^2}+\frac {b B (b c-a d)^2 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right )}{d^4 i^2}-\frac {6 b B^2 (b c-a d)^2 g^3 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{d^4 i^2}-\frac {6 b B (b c-a d)^2 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{d^4 i^2}-\frac {b B^2 (b c-a d)^2 g^3 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{d (a+b x)}\right )}{d^4 i^2}+\frac {6 b B^2 (b c-a d)^2 g^3 \operatorname {PolyLog}\left (3,\frac {d (a+b x)}{b (c+d x)}\right )}{d^4 i^2} \]
2*A*B*(-a*d+b*c)^2*g^3*(b*x+a)/d^3/i^2/(d*x+c)-2*B^2*(-a*d+b*c)^2*g^3*(b*x +a)/d^3/i^2/(d*x+c)+2*B^2*(-a*d+b*c)^2*g^3*(b*x+a)*ln(e*(b*x+a)/(d*x+c))/d ^3/i^2/(d*x+c)-b*B*(-a*d+b*c)*g^3*(b*x+a)*(A+B*ln(e*(b*x+a)/(d*x+c)))/d^3/ i^2-6*b*B*(-a*d+b*c)^2*g^3*ln((-a*d+b*c)/b/(d*x+c))*(A+B*ln(e*(b*x+a)/(d*x +c)))/d^4/i^2-3*b*(-a*d+b*c)*g^3*(b*x+a)*(A+B*ln(e*(b*x+a)/(d*x+c)))^2/d^3 /i^2-(-a*d+b*c)^2*g^3*(b*x+a)*(A+B*ln(e*(b*x+a)/(d*x+c)))^2/d^3/i^2/(d*x+c )+1/2*b^3*g^3*(d*x+c)^2*(A+B*ln(e*(b*x+a)/(d*x+c)))^2/d^4/i^2-3*b*(-a*d+b* c)^2*g^3*ln((-a*d+b*c)/b/(d*x+c))*(A+B*ln(e*(b*x+a)/(d*x+c)))^2/d^4/i^2+b* B^2*(-a*d+b*c)^2*g^3*ln(d*x+c)/d^4/i^2+b*B*(-a*d+b*c)^2*g^3*(A+B*ln(e*(b*x +a)/(d*x+c)))*ln(1-b*(d*x+c)/d/(b*x+a))/d^4/i^2-6*b*B^2*(-a*d+b*c)^2*g^3*p olylog(2,d*(b*x+a)/b/(d*x+c))/d^4/i^2-6*b*B*(-a*d+b*c)^2*g^3*(A+B*ln(e*(b* x+a)/(d*x+c)))*polylog(2,d*(b*x+a)/b/(d*x+c))/d^4/i^2-b*B^2*(-a*d+b*c)^2*g ^3*polylog(2,b*(d*x+c)/d/(b*x+a))/d^4/i^2+6*b*B^2*(-a*d+b*c)^2*g^3*polylog (3,d*(b*x+a)/b/(d*x+c))/d^4/i^2
Leaf count is larger than twice the leaf count of optimal. \(4443\) vs. \(2(722)=1444\).
Time = 6.29 (sec) , antiderivative size = 4443, normalized size of antiderivative = 6.15 \[ \int \frac {(a g+b g x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(c i+d i x)^2} \, dx=\text {Result too large to show} \]
(g^3*(-4*A^2*b^2*d*(2*b*c - 3*a*d)*x + 2*A^2*b^3*d^2*x^2 + (4*A^2*(b*c - a *d)^3)/(c + d*x) + 12*A^2*b*(b*c - a*d)^2*Log[c + d*x] + (8*a^3*A*B*d^3*(b *c - a*d + b*(c + d*x)*Log[a/b + x] + (-(b*c) + a*d)*Log[(e*(a + b*x))/(c + d*x)] - b*c*Log[(b*(c + d*x))/(b*c - a*d)] - b*d*x*Log[(b*(c + d*x))/(b* c - a*d)]))/((b*c - a*d)*(c + d*x)) + 12*a^2*A*b*B*d^2*(-Log[c/d + x]^2 + 2*Log[c/d + x]*Log[c + d*x] + 2*(-(c/(c + d*x)) + (b*c*Log[a + b*x])/(-(b* c) + a*d) + (b*c*Log[c + d*x])/(b*c - a*d) - Log[a/b + x]*Log[c + d*x] + L og[(e*(a + b*x))/(c + d*x)]*(c/(c + d*x) + Log[c + d*x]) + Log[a/b + x]*Lo g[(b*(c + d*x))/(b*c - a*d)]) + 2*PolyLog[2, (d*(a + b*x))/(-(b*c) + a*d)] ) + 4*A*b^3*B*(-4*c^2 + (4*a*c*d)/b - c*d*x + (a*d^2*x)/b - (2*c^3)/(c + d *x) + 4*c^2*Log[c/d + x] - 3*c^2*Log[c/d + x]^2 - (a^2*d^2*Log[a + b*x])/b ^2 + (2*b*c^3*Log[a + b*x])/(-(b*c) + a*d) - 4*c*d*x*Log[(e*(a + b*x))/(c + d*x)] + d^2*x^2*Log[(e*(a + b*x))/(c + d*x)] + (2*c^3*Log[(e*(a + b*x))/ (c + d*x)])/(c + d*x) + c^2*Log[c + d*x] + (2*b*c^3*Log[c + d*x])/(b*c - a *d) + 6*c^2*Log[c/d + x]*Log[c + d*x] + 6*c^2*Log[(e*(a + b*x))/(c + d*x)] *Log[c + d*x] - (2*c*Log[a/b + x]*(2*a*d + 3*b*c*Log[c + d*x] - 3*b*c*Log[ (b*(c + d*x))/(b*c - a*d)]))/b + 6*c^2*PolyLog[2, (d*(a + b*x))/(-(b*c) + a*d)]) + 24*a*A*b^2*B*d*(d*(a/b + x)*(-1 + Log[a/b + x]) - (c^2*Log[a/b + x])/(c + d*x) - (c + d*x)*(-1 + Log[c/d + x]) + c*Log[c/d + x]^2 + (c^2*(1 + Log[c/d + x]))/(c + d*x) + (b*c^2*(Log[a + b*x] - Log[c + d*x]))/(b*...
Time = 0.84 (sec) , antiderivative size = 592, normalized size of antiderivative = 0.82, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {2962, 2795, 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.
\(\displaystyle \int \frac {(a g+b g x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{(c i+d i x)^2} \, dx\) |
\(\Big \downarrow \) 2962 |
\(\displaystyle \frac {g^3 (b c-a d)^2 \int \frac {(a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}d\frac {a+b x}{c+d x}}{i^2}\) |
\(\Big \downarrow \) 2795 |
\(\displaystyle \frac {g^3 (b c-a d)^2 \int \left (\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 b^3}{d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 b^2}{d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 b}{d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^3}\right )d\frac {a+b x}{c+d x}}{i^2}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {g^3 (b c-a d)^2 \left (\frac {b^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{2 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {6 b B \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{d^4}-\frac {3 b \log \left (1-\frac {d (a+b x)}{b (c+d x)}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{d^4}-\frac {6 b B \log \left (1-\frac {d (a+b x)}{b (c+d x)}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{d^4}+\frac {b B \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{d^4}-\frac {3 b (a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{d^3 (c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {b B (a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{d^3 (c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {(a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{d^3 (c+d x)}+\frac {2 A B (a+b x)}{d^3 (c+d x)}-\frac {6 b B^2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{d^4}-\frac {b B^2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{d (a+b x)}\right )}{d^4}+\frac {6 b B^2 \operatorname {PolyLog}\left (3,\frac {d (a+b x)}{b (c+d x)}\right )}{d^4}-\frac {b B^2 \log \left (b-\frac {d (a+b x)}{c+d x}\right )}{d^4}+\frac {2 B^2 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{d^3 (c+d x)}-\frac {2 B^2 (a+b x)}{d^3 (c+d x)}\right )}{i^2}\) |
((b*c - a*d)^2*g^3*((2*A*B*(a + b*x))/(d^3*(c + d*x)) - (2*B^2*(a + b*x))/ (d^3*(c + d*x)) + (2*B^2*(a + b*x)*Log[(e*(a + b*x))/(c + d*x)])/(d^3*(c + d*x)) - (b*B*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(d^3*(c + d* x)*(b - (d*(a + b*x))/(c + d*x))) - ((a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(d^3*(c + d*x)) + (b^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2 )/(2*d^4*(b - (d*(a + b*x))/(c + d*x))^2) - (3*b*(a + b*x)*(A + B*Log[(e*( a + b*x))/(c + d*x)])^2)/(d^3*(c + d*x)*(b - (d*(a + b*x))/(c + d*x))) - ( b*B^2*Log[b - (d*(a + b*x))/(c + d*x)])/d^4 - (6*b*B*(A + B*Log[(e*(a + b* x))/(c + d*x)])*Log[1 - (d*(a + b*x))/(b*(c + d*x))])/d^4 - (3*b*(A + B*Lo g[(e*(a + b*x))/(c + d*x)])^2*Log[1 - (d*(a + b*x))/(b*(c + d*x))])/d^4 + (b*B*(A + B*Log[(e*(a + b*x))/(c + d*x)])*Log[1 - (b*(c + d*x))/(d*(a + b* x))])/d^4 - (6*b*B^2*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/d^4 - (6*b*B *(A + B*Log[(e*(a + b*x))/(c + d*x)])*PolyLog[2, (d*(a + b*x))/(b*(c + d*x ))])/d^4 - (b*B^2*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))])/d^4 + (6*b*B^2* PolyLog[3, (d*(a + b*x))/(b*(c + d*x))])/d^4))/i^2
3.1.92.3.1 Defintions of rubi rules used
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[ c*x^n])^p, (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b , c, d, e, f, m, n, p, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IGtQ[p, 0 ] && IntegerQ[m] && IntegerQ[r]))
Int[((A_.) + Log[(e_.)*((a_.) + (b_.)*(x_))^(n_.)*((c_.) + (d_.)*(x_))^(mn_ )]*(B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m_.)*((h_.) + (i_.)*(x_))^(q_.), x_Sy mbol] :> Simp[(b*c - a*d)^(m + q + 1)*(g/b)^m*(i/d)^q Subst[Int[x^m*((A + B*Log[e*x^n])^p/(b - d*x)^(m + q + 2)), x], x, (a + b*x)/(c + d*x)], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, A, B, n, p}, x] && EqQ[n + mn, 0] && IGt Q[n, 0] && NeQ[b*c - a*d, 0] && EqQ[b*f - a*g, 0] && EqQ[d*h - c*i, 0] && I ntegersQ[m, q]
\[\int \frac {\left (b g x +a g \right )^{3} \left (A +B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right )^{2}}{\left (d i x +c i \right )^{2}}d x\]
\[ \int \frac {(a g+b g x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(c i+d i x)^2} \, dx=\int { \frac {{\left (b g x + a g\right )}^{3} {\left (B \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right ) + A\right )}^{2}}{{\left (d i x + c i\right )}^{2}} \,d x } \]
integral((A^2*b^3*g^3*x^3 + 3*A^2*a*b^2*g^3*x^2 + 3*A^2*a^2*b*g^3*x + A^2* a^3*g^3 + (B^2*b^3*g^3*x^3 + 3*B^2*a*b^2*g^3*x^2 + 3*B^2*a^2*b*g^3*x + B^2 *a^3*g^3)*log((b*e*x + a*e)/(d*x + c))^2 + 2*(A*B*b^3*g^3*x^3 + 3*A*B*a*b^ 2*g^3*x^2 + 3*A*B*a^2*b*g^3*x + A*B*a^3*g^3)*log((b*e*x + a*e)/(d*x + c))) /(d^2*i^2*x^2 + 2*c*d*i^2*x + c^2*i^2), x)
Timed out. \[ \int \frac {(a g+b g x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(c i+d i x)^2} \, dx=\text {Timed out} \]
\[ \int \frac {(a g+b g x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(c i+d i x)^2} \, dx=\int { \frac {{\left (b g x + a g\right )}^{3} {\left (B \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right ) + A\right )}^{2}}{{\left (d i x + c i\right )}^{2}} \,d x } \]
1/2*(2*c^3/(d^5*i^2*x + c*d^4*i^2) + 6*c^2*log(d*x + c)/(d^4*i^2) + (d*x^2 - 4*c*x)/(d^3*i^2))*A^2*b^3*g^3 - 3*A^2*a*b^2*(c^2/(d^4*i^2*x + c*d^3*i^2 ) - x/(d^2*i^2) + 2*c*log(d*x + c)/(d^3*i^2))*g^3 + 3*A^2*a^2*b*g^3*(c/(d^ 3*i^2*x + c*d^2*i^2) + log(d*x + c)/(d^2*i^2)) - 2*A*B*a^3*g^3*(log(b*e*x/ (d*x + c) + a*e/(d*x + c))/(d^2*i^2*x + c*d*i^2) - 1/(d^2*i^2*x + c*d*i^2) - b*log(b*x + a)/((b*c*d - a*d^2)*i^2) + b*log(d*x + c)/((b*c*d - a*d^2)* i^2)) - A^2*a^3*g^3/(d^2*i^2*x + c*d*i^2) + 1/2*(2*((b^3*c^2*d*g^3 - 2*a*b ^2*c*d^2*g^3 + a^2*b*d^3*g^3)*B^2*x + (b^3*c^3*g^3 - 2*a*b^2*c^2*d*g^3 + a ^2*b*c*d^2*g^3)*B^2)*log(d*x + c)^3 + (B^2*b^3*d^3*g^3*x^3 - 3*(b^3*c*d^2* g^3 - 2*a*b^2*d^3*g^3)*B^2*x^2 - 2*(2*b^3*c^2*d*g^3 - 3*a*b^2*c*d^2*g^3)*B ^2*x + 2*(b^3*c^3*g^3 - 3*a*b^2*c^2*d*g^3 + 3*a^2*b*c*d^2*g^3 - a^3*d^3*g^ 3)*B^2)*log(d*x + c)^2)/(d^5*i^2*x + c*d^4*i^2) - integrate(-(B^2*a^3*d^3* g^3*log(e)^2 + (B^2*b^3*d^3*g^3*log(e)^2 + 2*A*B*b^3*d^3*g^3*log(e))*x^3 + 3*(B^2*a*b^2*d^3*g^3*log(e)^2 + 2*A*B*a*b^2*d^3*g^3*log(e))*x^2 + (B^2*b^ 3*d^3*g^3*x^3 + 3*B^2*a*b^2*d^3*g^3*x^2 + 3*B^2*a^2*b*d^3*g^3*x + B^2*a^3* d^3*g^3)*log(b*x + a)^2 + 3*(B^2*a^2*b*d^3*g^3*log(e)^2 + 2*A*B*a^2*b*d^3* g^3*log(e))*x + 2*(B^2*a^3*d^3*g^3*log(e) + (B^2*b^3*d^3*g^3*log(e) + A*B* b^3*d^3*g^3)*x^3 + 3*(B^2*a*b^2*d^3*g^3*log(e) + A*B*a*b^2*d^3*g^3)*x^2 + 3*(B^2*a^2*b*d^3*g^3*log(e) + A*B*a^2*b*d^3*g^3)*x)*log(b*x + a) - ((2*A*B *b^3*d^3*g^3 + (2*g^3*log(e) + g^3)*B^2*b^3*d^3)*x^3 + 2*(b^3*c^3*g^3 -...
\[ \int \frac {(a g+b g x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(c i+d i x)^2} \, dx=\int { \frac {{\left (b g x + a g\right )}^{3} {\left (B \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right ) + A\right )}^{2}}{{\left (d i x + c i\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {(a g+b g x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(c i+d i x)^2} \, dx=\int \frac {{\left (a\,g+b\,g\,x\right )}^3\,{\left (A+B\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\right )}^2}{{\left (c\,i+d\,i\,x\right )}^2} \,d x \]