Integrand size = 34, antiderivative size = 266 \[ \int \frac {1}{(a g+b g x)^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2} \, dx=\frac {d e^{\frac {A}{2 B}} \sqrt {\frac {e (a+b x)^2}{(c+d x)^2}} (c+d x) \operatorname {ExpIntegralEi}\left (\frac {-A-B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{2 B}\right )}{4 B^2 (b c-a d)^2 g^3 (a+b x)}-\frac {b e e^{A/B} \operatorname {ExpIntegralEi}\left (-\frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{B}\right )}{2 B^2 (b c-a d)^2 g^3}+\frac {d (c+d x)}{2 B (b c-a d)^2 g^3 (a+b x) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}-\frac {b (c+d x)^2}{2 B (b c-a d)^2 g^3 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )} \]
-1/2*b*e*exp(A/B)*Ei((-A-B*ln(e*(b*x+a)^2/(d*x+c)^2))/B)/B^2/(-a*d+b*c)^2/ g^3+1/2*d*(d*x+c)/B/(-a*d+b*c)^2/g^3/(b*x+a)/(A+B*ln(e*(b*x+a)^2/(d*x+c)^2 ))-1/2*b*(d*x+c)^2/B/(-a*d+b*c)^2/g^3/(b*x+a)^2/(A+B*ln(e*(b*x+a)^2/(d*x+c )^2))+1/4*d*exp(1/2*A/B)*(d*x+c)*Ei(1/2*(-A-B*ln(e*(b*x+a)^2/(d*x+c)^2))/B )*(e*(b*x+a)^2/(d*x+c)^2)^(1/2)/B^2/(-a*d+b*c)^2/g^3/(b*x+a)
\[ \int \frac {1}{(a g+b g x)^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2} \, dx=\int \frac {1}{(a g+b g x)^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2} \, dx \]
Time = 0.48 (sec) , antiderivative size = 225, normalized size of antiderivative = 0.85, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.088, Rules used = {2950, 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.
| \(\displaystyle \int \frac {1}{(a g+b g x)^3 \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )^2} \, dx\) |
| \(\Big \downarrow \) 2950 |
| \(\displaystyle \frac {\int \frac {(c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )}{(a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}d\frac {a+b x}{c+d x}}{g^3 (b c-a d)^2}\) |
| \(\Big \downarrow \) 2795 |
| \(\displaystyle \frac {\int \left (\frac {b (c+d x)^3}{(a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}-\frac {d (c+d x)^2}{(a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}\right )d\frac {a+b x}{c+d x}}{g^3 (b c-a d)^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {d e^{\frac {A}{2 B}} (c+d x) \sqrt {\frac {e (a+b x)^2}{(c+d x)^2}} \operatorname {ExpIntegralEi}\left (-\frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{2 B}\right )}{4 B^2 (a+b x)}-\frac {b e e^{A/B} \operatorname {ExpIntegralEi}\left (-\frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{B}\right )}{2 B^2}-\frac {b (c+d x)^2}{2 B (a+b x)^2 \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )}+\frac {d (c+d x)}{2 B (a+b x) \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )}}{g^3 (b c-a d)^2}\) |
(-1/2*(b*e*E^(A/B)*ExpIntegralEi[-((A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2] )/B)])/B^2 + (d*E^(A/(2*B))*Sqrt[(e*(a + b*x)^2)/(c + d*x)^2]*(c + d*x)*Ex pIntegralEi[-1/2*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])/B])/(4*B^2*(a + b*x)) + (d*(c + d*x))/(2*B*(a + b*x)*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^ 2])) - (b*(c + d*x)^2)/(2*B*(a + b*x)^2*(A + B*Log[(e*(a + b*x)^2)/(c + d* x)^2])))/((b*c - a*d)^2*g^3)
3.2.46.3.1 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_.), x_Symbol] :> Simp[(b*c - a*d)^( m + 1)*(g/b)^m Subst[Int[x^m*((A + B*Log[e*x^n])^p/(b - d*x)^(m + 2)), x] , x, (a + b*x)/(c + d*x)], x] /; FreeQ[{a, b, c, d, e, f, g, A, B, n}, x] & & EqQ[n + mn, 0] && IGtQ[n, 0] && NeQ[b*c - a*d, 0] && IntegersQ[m, p] && E qQ[b*f - a*g, 0] && (GtQ[p, 0] || LtQ[m, -1])
\[\int \frac {1}{\left (b g x +a g \right )^{3} {\left (A +B \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )\right )}^{2}}d x\]
\[ \int \frac {1}{(a g+b g x)^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2} \, dx=\int { \frac {1}{{\left (b g x + a g\right )}^{3} {\left (B \log \left (\frac {{\left (b x + a\right )}^{2} e}{{\left (d x + c\right )}^{2}}\right ) + A\right )}^{2}} \,d x } \]
integral(1/(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^2*e*x^2 + 2*a*b*e*x + a^2*e)/(d^2*x^2 + 2*c*d*x + c^2)) ^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^2*e*x^2 + 2*a*b*e*x + a^2*e)/(d^2*x^2 + 2*c*d*x + c^2))), x)
Timed out. \[ \int \frac {1}{(a g+b g x)^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2} \, dx=\text {Timed out} \]
\[ \int \frac {1}{(a g+b g x)^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2} \, dx=\int { \frac {1}{{\left (b g x + a g\right )}^{3} {\left (B \log \left (\frac {{\left (b x + a\right )}^{2} e}{{\left (d x + c\right )}^{2}}\right ) + A\right )}^{2}} \,d x } \]
-1/2*(d*x + c)/((a^2*b*c*g^3 - a^3*d*g^3)*A*B + (a^2*b*c*g^3*log(e) - a^3* d*g^3*log(e))*B^2 + ((b^3*c*g^3 - a*b^2*d*g^3)*A*B + (b^3*c*g^3*log(e) - a *b^2*d*g^3*log(e))*B^2)*x^2 + 2*((a*b^2*c*g^3 - a^2*b*d*g^3)*A*B + (a*b^2* c*g^3*log(e) - a^2*b*d*g^3*log(e))*B^2)*x + 2*((b^3*c*g^3 - a*b^2*d*g^3)*B ^2*x^2 + 2*(a*b^2*c*g^3 - a^2*b*d*g^3)*B^2*x + (a^2*b*c*g^3 - a^3*d*g^3)*B ^2)*log(b*x + a) - 2*((b^3*c*g^3 - a*b^2*d*g^3)*B^2*x^2 + 2*(a*b^2*c*g^3 - a^2*b*d*g^3)*B^2*x + (a^2*b*c*g^3 - a^3*d*g^3)*B^2)*log(d*x + c)) - integ rate(1/2*(b*d*x + 2*b*c - a*d)/(((b^4*c*g^3 - a*b^3*d*g^3)*A*B + (b^4*c*g^ 3*log(e) - a*b^3*d*g^3*log(e))*B^2)*x^3 + (a^3*b*c*g^3 - a^4*d*g^3)*A*B + (a^3*b*c*g^3*log(e) - a^4*d*g^3*log(e))*B^2 + 3*((a*b^3*c*g^3 - a^2*b^2*d* g^3)*A*B + (a*b^3*c*g^3*log(e) - a^2*b^2*d*g^3*log(e))*B^2)*x^2 + 3*((a^2* b^2*c*g^3 - a^3*b*d*g^3)*A*B + (a^2*b^2*c*g^3*log(e) - a^3*b*d*g^3*log(e)) *B^2)*x + 2*((b^4*c*g^3 - a*b^3*d*g^3)*B^2*x^3 + 3*(a*b^3*c*g^3 - a^2*b^2* d*g^3)*B^2*x^2 + 3*(a^2*b^2*c*g^3 - a^3*b*d*g^3)*B^2*x + (a^3*b*c*g^3 - a^ 4*d*g^3)*B^2)*log(b*x + a) - 2*((b^4*c*g^3 - a*b^3*d*g^3)*B^2*x^3 + 3*(a*b ^3*c*g^3 - a^2*b^2*d*g^3)*B^2*x^2 + 3*(a^2*b^2*c*g^3 - a^3*b*d*g^3)*B^2*x + (a^3*b*c*g^3 - a^4*d*g^3)*B^2)*log(d*x + c)), x)
\[ \int \frac {1}{(a g+b g x)^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2} \, dx=\int { \frac {1}{{\left (b g x + a g\right )}^{3} {\left (B \log \left (\frac {{\left (b x + a\right )}^{2} e}{{\left (d x + c\right )}^{2}}\right ) + A\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {1}{(a g+b g x)^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2} \, dx=\int \frac {1}{{\left (a\,g+b\,g\,x\right )}^3\,{\left (A+B\,\ln \left (\frac {e\,{\left (a+b\,x\right )}^2}{{\left (c+d\,x\right )}^2}\right )\right )}^2} \,d x \]