Integrand size = 26, antiderivative size = 141 \[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{d+e \log \left (f x^m\right )} \, dx=\frac {b n x^3}{3 e m}-\frac {b e^{-\frac {3 d}{e m}} n x^3 \left (f x^m\right )^{-3/m} \operatorname {ExpIntegralEi}\left (\frac {3 \left (d+e \log \left (f x^m\right )\right )}{e m}\right ) \left (d+e \log \left (f x^m\right )\right )}{e^2 m^2}+\frac {e^{-\frac {3 d}{e m}} x^3 \left (f x^m\right )^{-3/m} \operatorname {ExpIntegralEi}\left (\frac {3 \left (d+e \log \left (f x^m\right )\right )}{e m}\right ) \left (a+b \log \left (c x^n\right )\right )}{e m} \]
1/3*b*n*x^3/e/m-b*n*x^3*Ei(3*(d+e*ln(f*x^m))/e/m)*(d+e*ln(f*x^m))/e^2/exp( 3*d/e/m)/m^2/((f*x^m)^(3/m))+x^3*Ei(3*(d+e*ln(f*x^m))/e/m)*(a+b*ln(c*x^n)) /e/exp(3*d/e/m)/m/((f*x^m)^(3/m))
Time = 0.13 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.66 \[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{d+e \log \left (f x^m\right )} \, dx=\frac {x^3 \left (b e m n+3 e^{-\frac {3 d}{e m}} \left (f x^m\right )^{-3/m} \operatorname {ExpIntegralEi}\left (\frac {3 \left (d+e \log \left (f x^m\right )\right )}{e m}\right ) \left (a e m-b d n-b e n \log \left (f x^m\right )+b e m \log \left (c x^n\right )\right )\right )}{3 e^2 m^2} \]
(x^3*(b*e*m*n + (3*ExpIntegralEi[(3*(d + e*Log[f*x^m]))/(e*m)]*(a*e*m - b* d*n - b*e*n*Log[f*x^m] + b*e*m*Log[c*x^n]))/(E^((3*d)/(e*m))*(f*x^m)^(3/m) )))/(3*e^2*m^2)
Time = 0.59 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.06, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2813, 27, 31, 3039, 7281, 7036}
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.
| \(\displaystyle \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{d+e \log \left (f x^m\right )} \, dx\) |
| \(\Big \downarrow \) 2813 |
| \(\displaystyle \frac {x^3 e^{-\frac {3 d}{e m}} \left (f x^m\right )^{-3/m} \left (a+b \log \left (c x^n\right )\right ) \operatorname {ExpIntegralEi}\left (\frac {3 \left (d+e \log \left (f x^m\right )\right )}{e m}\right )}{e m}-b n \int \frac {e^{-\frac {3 d}{e m}} x^2 \left (f x^m\right )^{-3/m} \operatorname {ExpIntegralEi}\left (\frac {3 \left (d+e \log \left (f x^m\right )\right )}{e m}\right )}{e m}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x^3 e^{-\frac {3 d}{e m}} \left (f x^m\right )^{-3/m} \left (a+b \log \left (c x^n\right )\right ) \operatorname {ExpIntegralEi}\left (\frac {3 \left (d+e \log \left (f x^m\right )\right )}{e m}\right )}{e m}-\frac {b n e^{-\frac {3 d}{e m}} \int x^2 \left (f x^m\right )^{-3/m} \operatorname {ExpIntegralEi}\left (\frac {3 \left (d+e \log \left (f x^m\right )\right )}{e m}\right )dx}{e m}\) |
| \(\Big \downarrow \) 31 |
| \(\displaystyle \frac {x^3 e^{-\frac {3 d}{e m}} \left (f x^m\right )^{-3/m} \left (a+b \log \left (c x^n\right )\right ) \operatorname {ExpIntegralEi}\left (\frac {3 \left (d+e \log \left (f x^m\right )\right )}{e m}\right )}{e m}-\frac {b n x^3 e^{-\frac {3 d}{e m}} \left (f x^m\right )^{-3/m} \int \frac {\operatorname {ExpIntegralEi}\left (\frac {3 \left (d+e \log \left (f x^m\right )\right )}{e m}\right )}{x}dx}{e m}\) |
| \(\Big \downarrow \) 3039 |
| \(\displaystyle \frac {x^3 e^{-\frac {3 d}{e m}} \left (f x^m\right )^{-3/m} \left (a+b \log \left (c x^n\right )\right ) \operatorname {ExpIntegralEi}\left (\frac {3 \left (d+e \log \left (f x^m\right )\right )}{e m}\right )}{e m}-\frac {b n x^3 e^{-\frac {3 d}{e m}} \left (f x^m\right )^{-3/m} \int \operatorname {ExpIntegralEi}\left (\frac {3 \left (d+e \log \left (f x^m\right )\right )}{e m}\right )d\log \left (f x^m\right )}{e m^2}\) |
| \(\Big \downarrow \) 7281 |
| \(\displaystyle \frac {x^3 e^{-\frac {3 d}{e m}} \left (f x^m\right )^{-3/m} \left (a+b \log \left (c x^n\right )\right ) \operatorname {ExpIntegralEi}\left (\frac {3 \left (d+e \log \left (f x^m\right )\right )}{e m}\right )}{e m}-\frac {b n x^3 e^{-\frac {3 d}{e m}} \left (f x^m\right )^{-3/m} \int \operatorname {ExpIntegralEi}\left (\frac {3 d}{e m}+\frac {3 \log \left (f x^m\right )}{m}\right )d\left (\frac {3 d}{e m}+\frac {3 \log \left (f x^m\right )}{m}\right )}{3 e m}\) |
| \(\Big \downarrow \) 7036 |
| \(\displaystyle \frac {x^3 e^{-\frac {3 d}{e m}} \left (f x^m\right )^{-3/m} \left (a+b \log \left (c x^n\right )\right ) \operatorname {ExpIntegralEi}\left (\frac {3 \left (d+e \log \left (f x^m\right )\right )}{e m}\right )}{e m}-\frac {b n x^3 e^{-\frac {3 d}{e m}} \left (f x^m\right )^{-3/m} \left (\left (\frac {3 d}{e m}+\frac {3 \log \left (f x^m\right )}{m}\right ) \operatorname {ExpIntegralEi}\left (\frac {3 d}{e m}+\frac {3 \log \left (f x^m\right )}{m}\right )-f x^m\right )}{3 e m}\) |
-1/3*(b*n*x^3*(-(f*x^m) + ExpIntegralEi[(3*d)/(e*m) + (3*Log[f*x^m])/m]*(( 3*d)/(e*m) + (3*Log[f*x^m])/m)))/(e*E^((3*d)/(e*m))*m*(f*x^m)^(3/m)) + (x^ 3*ExpIntegralEi[(3*(d + e*Log[f*x^m]))/(e*m)]*(a + b*Log[c*x^n]))/(e*E^((3 *d)/(e*m))*m*(f*x^m)^(3/m))
3.2.70.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(u_.)*((a_.)*(x_))^(m_.)*((b_.)*(x_)^(i_.))^(p_), x_Symbol] :> Simp[(b* x^i)^p/(a*x)^(i*p) Int[u*(a*x)^(m + i*p), x], x] /; FreeQ[{a, b, i, m, p} , x] && !IntegerQ[p]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.) + Log[(f_.)*(x_)^(r_ .)]*(e_.))*((g_.)*(x_))^(m_.), x_Symbol] :> With[{u = IntHide[(g*x)^m*(a + b*Log[c*x^n])^p, x]}, Simp[(d + e*Log[f*x^r]) u, x] - Simp[e*r Int[Simp lifyIntegrand[u/x, x], x], x]] /; FreeQ[{a, b, c, d, e, f, g, m, n, p, r}, x] && !(EqQ[p, 1] && EqQ[a, 0] && NeQ[d, 0])
Int[u_, x_Symbol] :> With[{lst = FunctionOfLog[Cancel[x*u], x]}, Simp[1/lst [[3]] Subst[Int[lst[[1]], x], x, Log[lst[[2]]]], x] /; !FalseQ[lst]] /; NonsumQ[u]
Int[ExpIntegralEi[(a_.) + (b_.)*(x_)], x_Symbol] :> Simp[(a + b*x)*(ExpInte gralEi[a + b*x]/b), x] - Simp[E^(a + b*x)/b, x] /; FreeQ[{a, b}, x]
Int[u_, x_Symbol] :> With[{lst = FunctionOfLinear[u, x]}, Simp[1/lst[[3]] Subst[Int[lst[[1]], x], x, lst[[2]] + lst[[3]]*x], x] /; !FalseQ[lst]]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 3.38 (sec) , antiderivative size = 2350, normalized size of antiderivative = 16.67
-1/2*(-I*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+I*b*Pi*csgn(I*c)*csgn(I* c*x^n)^2+I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-I*b*Pi*csgn(I*c*x^n)^3+2*b*ln( c)+2*a)/e/m*x^3*f^(-3/m)*(x^m)^(-3/m)*exp(-3/2*(-I*Pi*csgn(I*f)*csgn(I*x^m )*csgn(I*f*x^m)*e+I*Pi*csgn(I*f)*csgn(I*f*x^m)^2*e+I*Pi*csgn(I*x^m)*csgn(I *f*x^m)^2*e-I*Pi*csgn(I*f*x^m)^3*e+2*d)/e/m)*Ei(1,-3*ln(x)+3/2*I*(e*Pi*csg n(I*f)*csgn(I*x^m)*csgn(I*f*x^m)-e*Pi*csgn(I*f)*csgn(I*f*x^m)^2-e*Pi*csgn( I*x^m)*csgn(I*f*x^m)^2+e*Pi*csgn(I*f*x^m)^3+2*I*e*ln(f)+2*I*e*(ln(x^m)-m*l n(x))+2*I*d)/e/m)-b/e/m*x^3*f^(-3/m)*(x^m)^(-3/m)*exp(-3/2*(-I*Pi*csgn(I*f )*csgn(I*x^m)*csgn(I*f*x^m)*e+I*Pi*csgn(I*f)*csgn(I*f*x^m)^2*e+I*Pi*csgn(I *x^m)*csgn(I*f*x^m)^2*e-I*Pi*csgn(I*f*x^m)^3*e+2*d)/e/m)*Ei(1,-3*ln(x)+3/2 *I*(e*Pi*csgn(I*f)*csgn(I*x^m)*csgn(I*f*x^m)-e*Pi*csgn(I*f)*csgn(I*f*x^m)^ 2-e*Pi*csgn(I*x^m)*csgn(I*f*x^m)^2+e*Pi*csgn(I*f*x^m)^3+2*I*e*ln(f)+2*I*e* (ln(x^m)-m*ln(x))+2*I*d)/e/m)*ln(x^n)+1/3*b*n*x^3/e/m-1/2*I*b*n/e/m^2*x^3* f^(-3/m)*(x^m)^(-3/m)*exp(-3/2*(-I*Pi*csgn(I*f)*csgn(I*x^m)*csgn(I*f*x^m)* e+I*Pi*csgn(I*f)*csgn(I*f*x^m)^2*e+I*Pi*csgn(I*x^m)*csgn(I*f*x^m)^2*e-I*Pi *csgn(I*f*x^m)^3*e+2*d)/e/m)*Ei(1,-3*ln(x)+3/2*I*(e*Pi*csgn(I*f)*csgn(I*x^ m)*csgn(I*f*x^m)-e*Pi*csgn(I*f)*csgn(I*f*x^m)^2-e*Pi*csgn(I*x^m)*csgn(I*f* x^m)^2+e*Pi*csgn(I*f*x^m)^3+2*I*e*ln(f)+2*I*e*(ln(x^m)-m*ln(x))+2*I*d)/e/m )*Pi*csgn(I*f)*csgn(I*x^m)*csgn(I*f*x^m)+1/2*I*b*n/e/m^2*x^3*f^(-3/m)*(x^m )^(-3/m)*exp(-3/2*(-I*Pi*csgn(I*f)*csgn(I*x^m)*csgn(I*f*x^m)*e+I*Pi*csg...
Time = 0.27 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.65 \[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{d+e \log \left (f x^m\right )} \, dx=\frac {{\left (b e m n x^{3} e^{\left (\frac {3 \, {\left (e \log \left (f\right ) + d\right )}}{e m}\right )} + 3 \, {\left (b e m \log \left (c\right ) - b e n \log \left (f\right ) + a e m - b d n\right )} \operatorname {log\_integral}\left (x^{3} e^{\left (\frac {3 \, {\left (e \log \left (f\right ) + d\right )}}{e m}\right )}\right )\right )} e^{\left (-\frac {3 \, {\left (e \log \left (f\right ) + d\right )}}{e m}\right )}}{3 \, e^{2} m^{2}} \]
1/3*(b*e*m*n*x^3*e^(3*(e*log(f) + d)/(e*m)) + 3*(b*e*m*log(c) - b*e*n*log( f) + a*e*m - b*d*n)*log_integral(x^3*e^(3*(e*log(f) + d)/(e*m))))*e^(-3*(e *log(f) + d)/(e*m))/(e^2*m^2)
\[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{d+e \log \left (f x^m\right )} \, dx=\int \frac {x^{2} \left (a + b \log {\left (c x^{n} \right )}\right )}{d + e \log {\left (f x^{m} \right )}}\, dx \]
\[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{d+e \log \left (f x^m\right )} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x^{2}}{e \log \left (f x^{m}\right ) + d} \,d x } \]
Time = 0.34 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.55 \[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{d+e \log \left (f x^m\right )} \, dx=\frac {b n x^{3}}{3 \, e m} + \frac {b {\rm Ei}\left (\frac {3 \, \log \left (f\right )}{m} + \frac {3 \, d}{e m} + 3 \, \log \left (x\right )\right ) e^{\left (-\frac {3 \, d}{e m}\right )} \log \left (c\right )}{e f^{\frac {3}{m}} m} - \frac {b n {\rm Ei}\left (\frac {3 \, \log \left (f\right )}{m} + \frac {3 \, d}{e m} + 3 \, \log \left (x\right )\right ) e^{\left (-\frac {3 \, d}{e m}\right )} \log \left (f\right )}{e f^{\frac {3}{m}} m^{2}} + \frac {a {\rm Ei}\left (\frac {3 \, \log \left (f\right )}{m} + \frac {3 \, d}{e m} + 3 \, \log \left (x\right )\right ) e^{\left (-\frac {3 \, d}{e m}\right )}}{e f^{\frac {3}{m}} m} - \frac {b d n {\rm Ei}\left (\frac {3 \, \log \left (f\right )}{m} + \frac {3 \, d}{e m} + 3 \, \log \left (x\right )\right ) e^{\left (-\frac {3 \, d}{e m}\right )}}{e^{2} f^{\frac {3}{m}} m^{2}} \]
1/3*b*n*x^3/(e*m) + b*Ei(3*log(f)/m + 3*d/(e*m) + 3*log(x))*e^(-3*d/(e*m)) *log(c)/(e*f^(3/m)*m) - b*n*Ei(3*log(f)/m + 3*d/(e*m) + 3*log(x))*e^(-3*d/ (e*m))*log(f)/(e*f^(3/m)*m^2) + a*Ei(3*log(f)/m + 3*d/(e*m) + 3*log(x))*e^ (-3*d/(e*m))/(e*f^(3/m)*m) - b*d*n*Ei(3*log(f)/m + 3*d/(e*m) + 3*log(x))*e ^(-3*d/(e*m))/(e^2*f^(3/m)*m^2)
Timed out. \[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{d+e \log \left (f x^m\right )} \, dx=\int \frac {x^2\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{d+e\,\ln \left (f\,x^m\right )} \,d x \]