Integrand size = 49, antiderivative size = 206 \[ \int \frac {(a g+b g x)^m (c i+d i x)^{-2-m}}{\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2} \, dx=\frac {e^{-\frac {A (1+m)}{B n}} (1+m) (a+b x) (g (a+b x))^m \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )^{-\frac {1+m}{n}} (i (c+d x))^{-m} \operatorname {ExpIntegralEi}\left (\frac {(1+m) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{B n}\right )}{B^2 (b c-a d) i^2 n^2 (c+d x)}-\frac {(a+b x) (g (a+b x))^m (i (c+d x))^{-m}}{B (b c-a d) i^2 n (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )} \]
(1+m)*(b*x+a)*(g*(b*x+a))^m*Ei((1+m)*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/B/n)/ B^2/(-a*d+b*c)/exp(A*(1+m)/B/n)/i^2/n^2/((e*((b*x+a)/(d*x+c))^n)^((1+m)/n) )/(d*x+c)/((i*(d*x+c))^m)-(b*x+a)*(g*(b*x+a))^m/B/(-a*d+b*c)/i^2/n/(d*x+c) /((i*(d*x+c))^m)/(A+B*ln(e*((b*x+a)/(d*x+c))^n))
\[ \int \frac {(a g+b g x)^m (c i+d i x)^{-2-m}}{\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2} \, dx=\int \frac {(a g+b g x)^m (c i+d i x)^{-2-m}}{\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2} \, dx \]
Integrate[((a*g + b*g*x)^m*(c*i + d*i*x)^(-2 - m))/(A + B*Log[e*((a + b*x) /(c + d*x))^n])^2,x]
Integrate[((a*g + b*g*x)^m*(c*i + d*i*x)^(-2 - m))/(A + B*Log[e*((a + b*x) /(c + d*x))^n])^2, x]
Time = 0.59 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.98, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.082, Rules used = {2963, 2743, 2747, 2609}
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.
| \(\displaystyle \int \frac {(a g+b g x)^m (c i+d i x)^{-m-2}}{\left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2} \, dx\) |
| \(\Big \downarrow \) 2963 |
| \(\displaystyle \frac {(g (a+b x))^m (i (c+d x))^{-m} \left (\frac {a+b x}{c+d x}\right )^{-m} \int \frac {\left (\frac {a+b x}{c+d x}\right )^m}{\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}d\frac {a+b x}{c+d x}}{i^2 (b c-a d)}\) |
| \(\Big \downarrow \) 2743 |
| \(\displaystyle \frac {(g (a+b x))^m (i (c+d x))^{-m} \left (\frac {a+b x}{c+d x}\right )^{-m} \left (\frac {(m+1) \int \frac {\left (\frac {a+b x}{c+d x}\right )^m}{A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}d\frac {a+b x}{c+d x}}{B n}-\frac {\left (\frac {a+b x}{c+d x}\right )^{m+1}}{B n \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}\right )}{i^2 (b c-a d)}\) |
| \(\Big \downarrow \) 2747 |
| \(\displaystyle \frac {(g (a+b x))^m (i (c+d x))^{-m} \left (\frac {a+b x}{c+d x}\right )^{-m} \left (\frac {(m+1) \left (\frac {a+b x}{c+d x}\right )^{m+1} \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )^{-\frac {m+1}{n}} \int \frac {\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )^{\frac {m+1}{n}}}{A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}d\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{B n^2}-\frac {\left (\frac {a+b x}{c+d x}\right )^{m+1}}{B n \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}\right )}{i^2 (b c-a d)}\) |
| \(\Big \downarrow \) 2609 |
| \(\displaystyle \frac {(g (a+b x))^m (i (c+d x))^{-m} \left (\frac {a+b x}{c+d x}\right )^{-m} \left (\frac {(m+1) e^{-\frac {A (m+1)}{B n}} \left (\frac {a+b x}{c+d x}\right )^{m+1} \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )^{-\frac {m+1}{n}} \operatorname {ExpIntegralEi}\left (\frac {(m+1) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{B n}\right )}{B^2 n^2}-\frac {\left (\frac {a+b x}{c+d x}\right )^{m+1}}{B n \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}\right )}{i^2 (b c-a d)}\) |
((g*(a + b*x))^m*(((1 + m)*((a + b*x)/(c + d*x))^(1 + m)*ExpIntegralEi[((1 + m)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(B*n)])/(B^2*E^((A*(1 + m))/ (B*n))*n^2*(e*((a + b*x)/(c + d*x))^n)^((1 + m)/n)) - ((a + b*x)/(c + d*x) )^(1 + m)/(B*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))))/((b*c - a*d)*i^2* ((a + b*x)/(c + d*x))^m*(i*(c + d*x))^m)
3.3.16.3.1 Defintions of rubi rules used
Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Si mp[(F^(g*(e - c*(f/d)))/d)*ExpIntegralEi[f*g*(c + d*x)*(Log[F]/d)], x] /; F reeQ[{F, c, d, e, f, g}, x] && !TrueQ[$UseGamma]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_)*((d_.)*(x_))^(m_.), x_Symbol ] :> Simp[(d*x)^(m + 1)*((a + b*Log[c*x^n])^(p + 1)/(b*d*n*(p + 1))), x] - Simp[(m + 1)/(b*n*(p + 1)) Int[(d*x)^m*(a + b*Log[c*x^n])^(p + 1), x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1] && LtQ[p, -1]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_)*((d_.)*(x_))^(m_.), x_Symbol ] :> Simp[(d*x)^(m + 1)/(d*n*(c*x^n)^((m + 1)/n)) Subst[Int[E^(((m + 1)/n )*x)*(a + b*x)^p, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, d, m, n, p}, x]
Int[((A_.) + Log[(e_.)*(((a_.) + (b_.)*(x_))/((c_.) + (d_.)*(x_)))^(n_.)]*( B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m_.)*((h_.) + (i_.)*(x_))^(q_.), x_Symbol ] :> Simp[d^2*((g*((a + b*x)/b))^m/(i^2*(b*c - a*d)*(i*((c + d*x)/d))^m*((a + b*x)/(c + d*x))^m)) Subst[Int[x^m*(A + B*Log[e*x^n])^p, x], x, (a + b* x)/(c + d*x)], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, A, B, m, n, p, q}, x ] && NeQ[b*c - a*d, 0] && EqQ[b*f - a*g, 0] && EqQ[d*h - c*i, 0] && EqQ[m + q + 2, 0]
\[\int \frac {\left (b g x +a g \right )^{m} \left (d i x +c i \right )^{-2-m}}{{\left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right )}^{2}}d x\]
Time = 0.31 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.42 \[ \int \frac {(a g+b g x)^m (c i+d i x)^{-2-m}}{\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2} \, dx=-\frac {{\left (B b d g^{2} n x^{2} + B a c g^{2} n + {\left (B b c + B a d\right )} g^{2} n x\right )} {\left (b g x + a g\right )}^{m} e^{\left (-{\left (m + 2\right )} \log \left (b g x + a g\right ) + {\left (m + 2\right )} \log \left (\frac {b x + a}{d x + c}\right ) - {\left (m + 2\right )} \log \left (\frac {i}{g}\right )\right )} - {\left ({\left (B m + B\right )} n \log \left (\frac {b x + a}{d x + c}\right ) + A m + {\left (B m + B\right )} \log \left (e\right ) + A\right )} {\rm Ei}\left (\frac {{\left (B m + B\right )} n \log \left (\frac {b x + a}{d x + c}\right ) + A m + {\left (B m + B\right )} \log \left (e\right ) + A}{B n}\right ) e^{\left (-\frac {{\left (B m + 2 \, B\right )} n \log \left (\frac {i}{g}\right ) + A m + {\left (B m + B\right )} \log \left (e\right ) + A}{B n}\right )}}{{\left (B^{3} b c - B^{3} a d\right )} g^{2} n^{3} \log \left (\frac {b x + a}{d x + c}\right ) + {\left (B^{3} b c - B^{3} a d\right )} g^{2} n^{2} \log \left (e\right ) + {\left (A B^{2} b c - A B^{2} a d\right )} g^{2} n^{2}} \]
integrate((b*g*x+a*g)^m*(d*i*x+c*i)^(-2-m)/(A+B*log(e*((b*x+a)/(d*x+c))^n) )^2,x, algorithm="fricas")
-((B*b*d*g^2*n*x^2 + B*a*c*g^2*n + (B*b*c + B*a*d)*g^2*n*x)*(b*g*x + a*g)^ m*e^(-(m + 2)*log(b*g*x + a*g) + (m + 2)*log((b*x + a)/(d*x + c)) - (m + 2 )*log(i/g)) - ((B*m + B)*n*log((b*x + a)/(d*x + c)) + A*m + (B*m + B)*log( e) + A)*Ei(((B*m + B)*n*log((b*x + a)/(d*x + c)) + A*m + (B*m + B)*log(e) + A)/(B*n))*e^(-((B*m + 2*B)*n*log(i/g) + A*m + (B*m + B)*log(e) + A)/(B*n )))/((B^3*b*c - B^3*a*d)*g^2*n^3*log((b*x + a)/(d*x + c)) + (B^3*b*c - B^3 *a*d)*g^2*n^2*log(e) + (A*B^2*b*c - A*B^2*a*d)*g^2*n^2)
Exception generated. \[ \int \frac {(a g+b g x)^m (c i+d i x)^{-2-m}}{\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2} \, dx=\text {Exception raised: HeuristicGCDFailed} \]
\[ \int \frac {(a g+b g x)^m (c i+d i x)^{-2-m}}{\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2} \, dx=\int { \frac {{\left (b g x + a g\right )}^{m} {\left (d i x + c i\right )}^{-m - 2}}{{\left (B \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right ) + A\right )}^{2}} \,d x } \]
integrate((b*g*x+a*g)^m*(d*i*x+c*i)^(-2-m)/(A+B*log(e*((b*x+a)/(d*x+c))^n) )^2,x, algorithm="maxima")
-g^m*(m + 1)*integrate(-(b*x + a)^m/((B^2*d^2*i^(m + 2)*n*x^2 + 2*B^2*c*d* i^(m + 2)*n*x + B^2*c^2*i^(m + 2)*n)*(d*x + c)^m*log((b*x + a)^n) - (B^2*d ^2*i^(m + 2)*n*x^2 + 2*B^2*c*d*i^(m + 2)*n*x + B^2*c^2*i^(m + 2)*n)*(d*x + c)^m*log((d*x + c)^n) + (B^2*c^2*i^(m + 2)*n*log(e) + A*B*c^2*i^(m + 2)*n + (B^2*d^2*i^(m + 2)*n*log(e) + A*B*d^2*i^(m + 2)*n)*x^2 + 2*(B^2*c*d*i^( m + 2)*n*log(e) + A*B*c*d*i^(m + 2)*n)*x)*(d*x + c)^m), x) - (b*g^m*x + a* g^m)*(b*x + a)^m/(((b*c*d*i^(m + 2)*n - a*d^2*i^(m + 2)*n)*B^2*x + (b*c^2* i^(m + 2)*n - a*c*d*i^(m + 2)*n)*B^2)*(d*x + c)^m*log((b*x + a)^n) - ((b*c *d*i^(m + 2)*n - a*d^2*i^(m + 2)*n)*B^2*x + (b*c^2*i^(m + 2)*n - a*c*d*i^( m + 2)*n)*B^2)*(d*x + c)^m*log((d*x + c)^n) + ((b*c^2*i^(m + 2)*n - a*c*d* i^(m + 2)*n)*A*B + (b*c^2*i^(m + 2)*n*log(e) - a*c*d*i^(m + 2)*n*log(e))*B ^2 + ((b*c*d*i^(m + 2)*n - a*d^2*i^(m + 2)*n)*A*B + (b*c*d*i^(m + 2)*n*log (e) - a*d^2*i^(m + 2)*n*log(e))*B^2)*x)*(d*x + c)^m)
\[ \int \frac {(a g+b g x)^m (c i+d i x)^{-2-m}}{\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2} \, dx=\int { \frac {{\left (b g x + a g\right )}^{m} {\left (d i x + c i\right )}^{-m - 2}}{{\left (B \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right ) + A\right )}^{2}} \,d x } \]
integrate((b*g*x+a*g)^m*(d*i*x+c*i)^(-2-m)/(A+B*log(e*((b*x+a)/(d*x+c))^n) )^2,x, algorithm="giac")
integrate((b*g*x + a*g)^m*(d*i*x + c*i)^(-m - 2)/(B*log(e*((b*x + a)/(d*x + c))^n) + A)^2, x)
Timed out. \[ \int \frac {(a g+b g x)^m (c i+d i x)^{-2-m}}{\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2} \, dx=\int \frac {{\left (a\,g+b\,g\,x\right )}^m}{{\left (c\,i+d\,i\,x\right )}^{m+2}\,{\left (A+B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\right )}^2} \,d x \]