Integrand size = 49, antiderivative size = 306 \[ \int \frac {(a g+b g x)^{-2-m} (c i+d i x)^m}{\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^3} \, dx=\frac {e^{\frac {A (1+m)}{B n}} (1+m)^2 (a+b x) (g (a+b x))^{-2-m} \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )^{\frac {1+m}{n}} (i (c+d x))^{2+m} \operatorname {ExpIntegralEi}\left (-\frac {(1+m) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{B n}\right )}{2 B^3 (b c-a d) i^2 n^3 (c+d x)}-\frac {(a+b x) (g (a+b x))^{-2-m} (i (c+d x))^{2+m}}{2 B (b c-a d) i^2 n (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}+\frac {(1+m) (a+b x) (g (a+b x))^{-2-m} (i (c+d x))^{2+m}}{2 B^2 (b c-a d) i^2 n^2 (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )} \]
1/2*exp(A*(1+m)/B/n)*(1+m)^2*(b*x+a)*(g*(b*x+a))^(-2-m)*(e*((b*x+a)/(d*x+c ))^n)^((1+m)/n)*(i*(d*x+c))^(2+m)*Ei(-(1+m)*(A+B*ln(e*((b*x+a)/(d*x+c))^n) )/B/n)/B^3/(-a*d+b*c)/i^2/n^3/(d*x+c)-1/2*(b*x+a)*(g*(b*x+a))^(-2-m)*(i*(d *x+c))^(2+m)/B/(-a*d+b*c)/i^2/n/(d*x+c)/(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2+ 1/2*(1+m)*(b*x+a)*(g*(b*x+a))^(-2-m)*(i*(d*x+c))^(2+m)/B^2/(-a*d+b*c)/i^2/ n^2/(d*x+c)/(A+B*ln(e*((b*x+a)/(d*x+c))^n))
\[ \int \frac {(a g+b g x)^{-2-m} (c i+d i x)^m}{\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^3} \, dx=\int \frac {(a g+b g x)^{-2-m} (c i+d i x)^m}{\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^3} \, dx \]
Integrate[((a*g + b*g*x)^(-2 - m)*(c*i + d*i*x)^m)/(A + B*Log[e*((a + b*x) /(c + d*x))^n])^3,x]
Integrate[((a*g + b*g*x)^(-2 - m)*(c*i + d*i*x)^m)/(A + B*Log[e*((a + b*x) /(c + d*x))^n])^3, x]
Time = 0.71 (sec) , antiderivative size = 276, normalized size of antiderivative = 0.90, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.102, Rules used = {2963, 2743, 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.
\(\displaystyle \int \frac {(a g+b g x)^{-m-2} (c i+d i x)^m}{\left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^3} \, dx\) |
\(\Big \downarrow \) 2963 |
\(\displaystyle \frac {(g (a+b x))^{-m-2} (i (c+d x))^{m+2} \left (\frac {a+b x}{c+d x}\right )^{m+2} \int \frac {\left (\frac {a+b x}{c+d x}\right )^{-m-2}}{\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^3}d\frac {a+b x}{c+d x}}{i^2 (b c-a d)}\) |
\(\Big \downarrow \) 2743 |
\(\displaystyle \frac {(g (a+b x))^{-m-2} (i (c+d x))^{m+2} \left (\frac {a+b x}{c+d x}\right )^{m+2} \left (-\frac {(m+1) \int \frac {\left (\frac {a+b x}{c+d x}\right )^{-m-2}}{\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}d\frac {a+b x}{c+d x}}{2 B n}-\frac {\left (\frac {a+b x}{c+d x}\right )^{-m-1}}{2 B n \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}\right )}{i^2 (b c-a d)}\) |
\(\Big \downarrow \) 2743 |
\(\displaystyle \frac {(g (a+b x))^{-m-2} (i (c+d x))^{m+2} \left (\frac {a+b x}{c+d x}\right )^{m+2} \left (-\frac {(m+1) \left (-\frac {(m+1) \int \frac {\left (\frac {a+b x}{c+d x}\right )^{-m-2}}{A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}d\frac {a+b x}{c+d x}}{B n}-\frac {\left (\frac {a+b x}{c+d x}\right )^{-m-1}}{B n \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}\right )}{2 B n}-\frac {\left (\frac {a+b x}{c+d x}\right )^{-m-1}}{2 B n \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}\right )}{i^2 (b c-a d)}\) |
\(\Big \downarrow \) 2747 |
\(\displaystyle \frac {(g (a+b x))^{-m-2} (i (c+d x))^{m+2} \left (\frac {a+b x}{c+d x}\right )^{m+2} \left (-\frac {(m+1) \left (-\frac {(m+1) \left (\frac {a+b x}{c+d x}\right )^{-m-1} \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )^{\frac {m+1}{n}} \int \frac {\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )^{-\frac {m+1}{n}}}{A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}d\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{B n^2}-\frac {\left (\frac {a+b x}{c+d x}\right )^{-m-1}}{B n \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}\right )}{2 B n}-\frac {\left (\frac {a+b x}{c+d x}\right )^{-m-1}}{2 B n \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}\right )}{i^2 (b c-a d)}\) |
\(\Big \downarrow \) 2609 |
\(\displaystyle \frac {(g (a+b x))^{-m-2} (i (c+d x))^{m+2} \left (\frac {a+b x}{c+d x}\right )^{m+2} \left (-\frac {(m+1) \left (-\frac {(m+1) e^{\frac {A (m+1)}{B n}} \left (\frac {a+b x}{c+d x}\right )^{-m-1} \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )^{\frac {m+1}{n}} \operatorname {ExpIntegralEi}\left (-\frac {(m+1) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{B n}\right )}{B^2 n^2}-\frac {\left (\frac {a+b x}{c+d x}\right )^{-m-1}}{B n \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}\right )}{2 B n}-\frac {\left (\frac {a+b x}{c+d x}\right )^{-m-1}}{2 B n \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}\right )}{i^2 (b c-a d)}\) |
((g*(a + b*x))^(-2 - m)*((a + b*x)/(c + d*x))^(2 + m)*(i*(c + d*x))^(2 + m )*(-1/2*((a + b*x)/(c + d*x))^(-1 - m)/(B*n*(A + B*Log[e*((a + b*x)/(c + d *x))^n])^2) - ((1 + m)*(-((E^((A*(1 + m))/(B*n))*(1 + m)*(e*((a + b*x)/(c + d*x))^n)^((1 + m)/n)*((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*n^2)) - ((a + b *x)/(c + d*x))^(-1 - m)/(B*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))))/(2* B*n)))/((b*c - a*d)*i^2)
3.3.23.3.1 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]
\[\int \frac {\left (b g x +a g \right )^{-2-m} \left (d i x +c i \right )^{m}}{{\left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right )}^{3}}d x\]
Leaf count of result is larger than twice the leaf count of optimal. 815 vs. \(2 (299) = 598\).
Time = 0.39 (sec) , antiderivative size = 815, normalized size of antiderivative = 2.66 \[ \int \frac {(a g+b g x)^{-2-m} (c i+d i x)^m}{\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^3} \, dx=-\frac {{\left (B^{2} a c g^{2} n^{2} + {\left (B^{2} b d g^{2} n^{2} - {\left (A B b d g^{2} m + A B b d g^{2}\right )} n\right )} x^{2} - {\left (A B a c g^{2} m + A B a c g^{2}\right )} n + {\left ({\left (B^{2} b c + B^{2} a d\right )} g^{2} n^{2} - {\left ({\left (A B b c + A B a d\right )} g^{2} m + {\left (A B b c + A B a d\right )} g^{2}\right )} n\right )} x - {\left ({\left (B^{2} b d g^{2} m + B^{2} b d g^{2}\right )} n x^{2} + {\left ({\left (B^{2} b c + B^{2} a d\right )} g^{2} m + {\left (B^{2} b c + B^{2} a d\right )} g^{2}\right )} n x + {\left (B^{2} a c g^{2} m + B^{2} a c g^{2}\right )} n\right )} \log \left (e\right ) - {\left ({\left (B^{2} b d g^{2} m + B^{2} b d g^{2}\right )} n^{2} x^{2} + {\left ({\left (B^{2} b c + B^{2} a d\right )} g^{2} m + {\left (B^{2} b c + B^{2} a d\right )} g^{2}\right )} n^{2} x + {\left (B^{2} a c g^{2} m + B^{2} a c g^{2}\right )} n^{2}\right )} \log \left (\frac {b x + a}{d x + c}\right )\right )} {\left (b g x + a g\right )}^{-m - 2} e^{\left (m \log \left (b g x + a g\right ) - m \log \left (\frac {b x + a}{d x + c}\right ) + m \log \left (\frac {i}{g}\right )\right )} - {\left ({\left (B^{2} m^{2} + 2 \, B^{2} m + B^{2}\right )} n^{2} \log \left (\frac {b x + a}{d x + c}\right )^{2} + A^{2} m^{2} + 2 \, A^{2} m + {\left (B^{2} m^{2} + 2 \, B^{2} m + B^{2}\right )} \log \left (e\right )^{2} + 2 \, {\left (A B m^{2} + 2 \, A B m + A B\right )} n \log \left (\frac {b x + a}{d x + c}\right ) + A^{2} + 2 \, {\left (A B m^{2} + 2 \, A B m + {\left (B^{2} m^{2} + 2 \, B^{2} m + B^{2}\right )} n \log \left (\frac {b x + a}{d x + c}\right ) + A B\right )} \log \left (e\right )\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 {B m n \log \left (\frac {i}{g}\right ) + A m + {\left (B m + B\right )} \log \left (e\right ) + A}{B n}\right )}}{2 \, {\left ({\left (B^{5} b c - B^{5} a d\right )} g^{2} n^{5} \log \left (\frac {b x + a}{d x + c}\right )^{2} + {\left (B^{5} b c - B^{5} a d\right )} g^{2} n^{3} \log \left (e\right )^{2} + 2 \, {\left (A B^{4} b c - A B^{4} a d\right )} g^{2} n^{4} \log \left (\frac {b x + a}{d x + c}\right ) + {\left (A^{2} B^{3} b c - A^{2} B^{3} a d\right )} g^{2} n^{3} + 2 \, {\left ({\left (B^{5} b c - B^{5} a d\right )} g^{2} n^{4} \log \left (\frac {b x + a}{d x + c}\right ) + {\left (A B^{4} b c - A B^{4} a d\right )} g^{2} n^{3}\right )} \log \left (e\right )\right )}} \]
integrate((b*g*x+a*g)^(-2-m)*(d*i*x+c*i)^m/(A+B*log(e*((b*x+a)/(d*x+c))^n) )^3,x, algorithm="fricas")
-1/2*((B^2*a*c*g^2*n^2 + (B^2*b*d*g^2*n^2 - (A*B*b*d*g^2*m + A*B*b*d*g^2)* n)*x^2 - (A*B*a*c*g^2*m + A*B*a*c*g^2)*n + ((B^2*b*c + B^2*a*d)*g^2*n^2 - ((A*B*b*c + A*B*a*d)*g^2*m + (A*B*b*c + A*B*a*d)*g^2)*n)*x - ((B^2*b*d*g^2 *m + B^2*b*d*g^2)*n*x^2 + ((B^2*b*c + B^2*a*d)*g^2*m + (B^2*b*c + B^2*a*d) *g^2)*n*x + (B^2*a*c*g^2*m + B^2*a*c*g^2)*n)*log(e) - ((B^2*b*d*g^2*m + B^ 2*b*d*g^2)*n^2*x^2 + ((B^2*b*c + B^2*a*d)*g^2*m + (B^2*b*c + B^2*a*d)*g^2) *n^2*x + (B^2*a*c*g^2*m + B^2*a*c*g^2)*n^2)*log((b*x + a)/(d*x + c)))*(b*g *x + a*g)^(-m - 2)*e^(m*log(b*g*x + a*g) - m*log((b*x + a)/(d*x + c)) + m* log(i/g)) - ((B^2*m^2 + 2*B^2*m + B^2)*n^2*log((b*x + a)/(d*x + c))^2 + A^ 2*m^2 + 2*A^2*m + (B^2*m^2 + 2*B^2*m + B^2)*log(e)^2 + 2*(A*B*m^2 + 2*A*B* m + A*B)*n*log((b*x + a)/(d*x + c)) + A^2 + 2*(A*B*m^2 + 2*A*B*m + (B^2*m^ 2 + 2*B^2*m + B^2)*n*log((b*x + a)/(d*x + c)) + A*B)*log(e))*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* n*log(i/g) + A*m + (B*m + B)*log(e) + A)/(B*n)))/((B^5*b*c - B^5*a*d)*g^2* n^5*log((b*x + a)/(d*x + c))^2 + (B^5*b*c - B^5*a*d)*g^2*n^3*log(e)^2 + 2* (A*B^4*b*c - A*B^4*a*d)*g^2*n^4*log((b*x + a)/(d*x + c)) + (A^2*B^3*b*c - A^2*B^3*a*d)*g^2*n^3 + 2*((B^5*b*c - B^5*a*d)*g^2*n^4*log((b*x + a)/(d*x + c)) + (A*B^4*b*c - A*B^4*a*d)*g^2*n^3)*log(e))
Timed out. \[ \int \frac {(a g+b g x)^{-2-m} (c i+d i x)^m}{\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^3} \, dx=\text {Timed out} \]
\[ \int \frac {(a g+b g x)^{-2-m} (c i+d i x)^m}{\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^3} \, dx=\int { \frac {{\left (b g x + a g\right )}^{-m - 2} {\left (d i x + c i\right )}^{m}}{{\left (B \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right ) + A\right )}^{3}} \,d x } \]
integrate((b*g*x+a*g)^(-2-m)*(d*i*x+c*i)^m/(A+B*log(e*((b*x+a)/(d*x+c))^n) )^3,x, algorithm="maxima")
-(m^2 + 2*m + 1)*i^m*integrate(-1/2*(d*x + c)^m/((B^3*b^2*g^(m + 2)*n^2*x^ 2 + 2*B^3*a*b*g^(m + 2)*n^2*x + B^3*a^2*g^(m + 2)*n^2)*(b*x + a)^m*log((b* x + a)^n) - (B^3*b^2*g^(m + 2)*n^2*x^2 + 2*B^3*a*b*g^(m + 2)*n^2*x + B^3*a ^2*g^(m + 2)*n^2)*(b*x + a)^m*log((d*x + c)^n) + (B^3*a^2*g^(m + 2)*n^2*lo g(e) + A*B^2*a^2*g^(m + 2)*n^2 + (B^3*b^2*g^(m + 2)*n^2*log(e) + A*B^2*b^2 *g^(m + 2)*n^2)*x^2 + 2*(B^3*a*b*g^(m + 2)*n^2*log(e) + A*B^2*a*b*g^(m + 2 )*n^2)*x)*(b*x + a)^m), x) + 1/2*((B*d*i^m*(m + 1)*x + B*c*i^m*(m + 1))*(d *x + c)^m*log((b*x + a)^n) - (B*d*i^m*(m + 1)*x + B*c*i^m*(m + 1))*(d*x + c)^m*log((d*x + c)^n) + (A*c*i^m*(m + 1) + (i^m*(m + 1)*log(e) - i^m*n)*B* c + (A*d*i^m*(m + 1) + (i^m*(m + 1)*log(e) - i^m*n)*B*d)*x)*(d*x + c)^m)/( ((b^2*c*g^(m + 2)*n^2 - a*b*d*g^(m + 2)*n^2)*B^4*x + (a*b*c*g^(m + 2)*n^2 - a^2*d*g^(m + 2)*n^2)*B^4)*(b*x + a)^m*log((b*x + a)^n)^2 + ((b^2*c*g^(m + 2)*n^2 - a*b*d*g^(m + 2)*n^2)*B^4*x + (a*b*c*g^(m + 2)*n^2 - a^2*d*g^(m + 2)*n^2)*B^4)*(b*x + a)^m*log((d*x + c)^n)^2 + 2*((a*b*c*g^(m + 2)*n^2 - a^2*d*g^(m + 2)*n^2)*A*B^3 + (a*b*c*g^(m + 2)*n^2*log(e) - a^2*d*g^(m + 2) *n^2*log(e))*B^4 + ((b^2*c*g^(m + 2)*n^2 - a*b*d*g^(m + 2)*n^2)*A*B^3 + (b ^2*c*g^(m + 2)*n^2*log(e) - a*b*d*g^(m + 2)*n^2*log(e))*B^4)*x)*(b*x + a)^ m*log((b*x + a)^n) + ((a*b*c*g^(m + 2)*n^2 - a^2*d*g^(m + 2)*n^2)*A^2*B^2 + 2*(a*b*c*g^(m + 2)*n^2*log(e) - a^2*d*g^(m + 2)*n^2*log(e))*A*B^3 + (a*b *c*g^(m + 2)*n^2*log(e)^2 - a^2*d*g^(m + 2)*n^2*log(e)^2)*B^4 + ((b^2*c...
\[ \int \frac {(a g+b g x)^{-2-m} (c i+d i x)^m}{\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^3} \, dx=\int { \frac {{\left (b g x + a g\right )}^{-m - 2} {\left (d i x + c i\right )}^{m}}{{\left (B \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right ) + A\right )}^{3}} \,d x } \]
integrate((b*g*x+a*g)^(-2-m)*(d*i*x+c*i)^m/(A+B*log(e*((b*x+a)/(d*x+c))^n) )^3,x, algorithm="giac")
integrate((b*g*x + a*g)^(-m - 2)*(d*i*x + c*i)^m/(B*log(e*((b*x + a)/(d*x + c))^n) + A)^3, x)
Timed out. \[ \int \frac {(a g+b g x)^{-2-m} (c i+d i x)^m}{\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^3} \, dx=\int \frac {{\left (c\,i+d\,i\,x\right )}^m}{{\left (a\,g+b\,g\,x\right )}^{m+2}\,{\left (A+B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\right )}^3} \,d x \]