\(\int \frac {x (a+b \log (c x^n))}{d+e \log (f x^m)} \, dx\) [177]

Optimal result

Integrand size = 24, antiderivative size = 141 \[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{d+e \log \left (f x^m\right )} \, dx=\frac {b n x^2}{2 e m}-\frac {b e^{-\frac {2 d}{e m}} n x^2 \left (f x^m\right )^{-2/m} \operatorname {ExpIntegralEi}\left (\frac {2 \left (d+e \log \left (f x^m\right )\right )}{e m}\right ) \left (d+e \log \left (f x^m\right )\right )}{e^2 m^2}+\frac {e^{-\frac {2 d}{e m}} x^2 \left (f x^m\right )^{-2/m} \operatorname {ExpIntegralEi}\left (\frac {2 \left (d+e \log \left (f x^m\right )\right )}{e m}\right ) \left (a+b \log \left (c x^n\right )\right )}{e m} \] Output:

1/2*b*n*x^2/e/m-b*n*x^2*Ei(2*(d+e*ln(f*x^m))/e/m)*(d+e*ln(f*x^m))/e^2/exp( 
2*d/e/m)/m^2/((f*x^m)^(2/m))+x^2*Ei(2*(d+e*ln(f*x^m))/e/m)*(a+b*ln(c*x^n)) 
/e/exp(2*d/e/m)/m/((f*x^m)^(2/m))
 

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.66 \[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{d+e \log \left (f x^m\right )} \, dx=\frac {x^2 \left (b e m n+2 e^{-\frac {2 d}{e m}} \left (f x^m\right )^{-2/m} \operatorname {ExpIntegralEi}\left (\frac {2 \left (d+e \log \left (f x^m\right )\right )}{e m}\right ) \left (a e m-b d n-b e n \log \left (f x^m\right )+b e m \log \left (c x^n\right )\right )\right )}{2 e^2 m^2} \] Input:

Integrate[(x*(a + b*Log[c*x^n]))/(d + e*Log[f*x^m]),x]
 

Output:

(x^2*(b*e*m*n + (2*ExpIntegralEi[(2*(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^((2*d)/(e*m))*(f*x^m)^(2/m) 
)))/(2*e^2*m^2)
 

Rubi [A] (warning: unable to verify)

Time = 0.61 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.06, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, 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.

\(\displaystyle \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{d+e \log \left (f x^m\right )} \, dx\)

\(\Big \downarrow \) 2813

\(\displaystyle \frac {x^2 e^{-\frac {2 d}{e m}} \left (f x^m\right )^{-2/m} \left (a+b \log \left (c x^n\right )\right ) \operatorname {ExpIntegralEi}\left (\frac {2 \left (d+e \log \left (f x^m\right )\right )}{e m}\right )}{e m}-b n \int \frac {e^{-\frac {2 d}{e m}} x \left (f x^m\right )^{-2/m} \operatorname {ExpIntegralEi}\left (\frac {2 \left (d+e \log \left (f x^m\right )\right )}{e m}\right )}{e m}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {x^2 e^{-\frac {2 d}{e m}} \left (f x^m\right )^{-2/m} \left (a+b \log \left (c x^n\right )\right ) \operatorname {ExpIntegralEi}\left (\frac {2 \left (d+e \log \left (f x^m\right )\right )}{e m}\right )}{e m}-\frac {b n e^{-\frac {2 d}{e m}} \int x \left (f x^m\right )^{-2/m} \operatorname {ExpIntegralEi}\left (\frac {2 \left (d+e \log \left (f x^m\right )\right )}{e m}\right )dx}{e m}\)

\(\Big \downarrow \) 31

\(\displaystyle \frac {x^2 e^{-\frac {2 d}{e m}} \left (f x^m\right )^{-2/m} \left (a+b \log \left (c x^n\right )\right ) \operatorname {ExpIntegralEi}\left (\frac {2 \left (d+e \log \left (f x^m\right )\right )}{e m}\right )}{e m}-\frac {b n x^2 e^{-\frac {2 d}{e m}} \left (f x^m\right )^{-2/m} \int \frac {\operatorname {ExpIntegralEi}\left (\frac {2 \left (d+e \log \left (f x^m\right )\right )}{e m}\right )}{x}dx}{e m}\)

\(\Big \downarrow \) 3039

\(\displaystyle \frac {x^2 e^{-\frac {2 d}{e m}} \left (f x^m\right )^{-2/m} \left (a+b \log \left (c x^n\right )\right ) \operatorname {ExpIntegralEi}\left (\frac {2 \left (d+e \log \left (f x^m\right )\right )}{e m}\right )}{e m}-\frac {b n x^2 e^{-\frac {2 d}{e m}} \left (f x^m\right )^{-2/m} \int \operatorname {ExpIntegralEi}\left (\frac {2 \left (d+e \log \left (f x^m\right )\right )}{e m}\right )d\log \left (f x^m\right )}{e m^2}\)

\(\Big \downarrow \) 7281

\(\displaystyle \frac {x^2 e^{-\frac {2 d}{e m}} \left (f x^m\right )^{-2/m} \left (a+b \log \left (c x^n\right )\right ) \operatorname {ExpIntegralEi}\left (\frac {2 \left (d+e \log \left (f x^m\right )\right )}{e m}\right )}{e m}-\frac {b n x^2 e^{-\frac {2 d}{e m}} \left (f x^m\right )^{-2/m} \int \operatorname {ExpIntegralEi}\left (\frac {2 d}{e m}+\frac {2 \log \left (f x^m\right )}{m}\right )d\left (\frac {2 d}{e m}+\frac {2 \log \left (f x^m\right )}{m}\right )}{2 e m}\)

\(\Big \downarrow \) 7036

\(\displaystyle \frac {x^2 e^{-\frac {2 d}{e m}} \left (f x^m\right )^{-2/m} \left (a+b \log \left (c x^n\right )\right ) \operatorname {ExpIntegralEi}\left (\frac {2 \left (d+e \log \left (f x^m\right )\right )}{e m}\right )}{e m}-\frac {b n x^2 e^{-\frac {2 d}{e m}} \left (f x^m\right )^{-2/m} \left (\left (\frac {2 d}{e m}+\frac {2 \log \left (f x^m\right )}{m}\right ) \operatorname {ExpIntegralEi}\left (\frac {2 d}{e m}+\frac {2 \log \left (f x^m\right )}{m}\right )-f x^m\right )}{2 e m}\)

Input:

Int[(x*(a + b*Log[c*x^n]))/(d + e*Log[f*x^m]),x]
 

Output:

-1/2*(b*n*x^2*(-(f*x^m) + ExpIntegralEi[(2*d)/(e*m) + (2*Log[f*x^m])/m]*(( 
2*d)/(e*m) + (2*Log[f*x^m])/m)))/(e*E^((2*d)/(e*m))*m*(f*x^m)^(2/m)) + (x^ 
2*ExpIntegralEi[(2*(d + e*Log[f*x^m]))/(e*m)]*(a + b*Log[c*x^n]))/(e*E^((2 
*d)/(e*m))*m*(f*x^m)^(2/m))
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 31

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]
 

rule 2813

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])
 

rule 3039

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]
 

rule 7036

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]
 

rule 7281

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]]
 

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 1.48 (sec) , antiderivative size = 2350, normalized size of antiderivative = 16.67

method	result	size
risch	\(\text {Expression too large to display}\)	\(2350\)

Input:

int(x*(a+b*ln(c*x^n))/(d+e*ln(f*x^m)),x,method=_RETURNVERBOSE)
 

Output:

-1/2*(I*Pi*b*csgn(I*x^n)*csgn(I*c*x^n)^2-I*Pi*b*csgn(I*x^n)*csgn(I*c*x^n)* 
csgn(I*c)-I*Pi*b*csgn(I*c*x^n)^3+I*Pi*b*csgn(I*c*x^n)^2*csgn(I*c)+2*b*ln(c 
)+2*a)/m/e*x^2*f^(-2/m)*(x^m)^(-2/m)*exp(-(-I*Pi*csgn(I*f)*csgn(I*x^m)*csg 
n(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)/m/e)*Ei(1,-2*ln(x)+I*(e*Pi*csgn(I*f)*cs 
gn(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)*cs 
gn(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)/m/e)-b/m/e*x^2*f^(-2/m)*(x^m)^(-2/m)*exp(-(-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)/m/e)*Ei(1,-2*ln(x)+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)/m/e)*ln(x^n)+1/2*b*n*x^2/e/m-1/2*I*b*n/m^2/e*x^2*f^(-2/m)*(x^m)^(- 
2/m)*exp(-(-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)/m/e)*Ei(1,-2*ln(x)+I*(e*Pi*csgn(I*f)*csgn(I*x^m)*csgn(I*f*x^m)-e*Pi*csg 
n(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)/m/e)*Pi*csgn(I*f)*csgn(I*x^m 
)*csgn(I*f*x^m)+1/2*I*b*n/m^2/e*x^2*f^(-2/m)*(x^m)^(-2/m)*exp(-(-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...
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.65 \[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{d+e \log \left (f x^m\right )} \, dx=\frac {{\left (b e m n x^{2} e^{\left (\frac {2 \, {\left (e \log \left (f\right ) + d\right )}}{e m}\right )} + 2 \, {\left (b e m \log \left (c\right ) - b e n \log \left (f\right ) + a e m - b d n\right )} \operatorname {log\_integral}\left (x^{2} e^{\left (\frac {2 \, {\left (e \log \left (f\right ) + d\right )}}{e m}\right )}\right )\right )} e^{\left (-\frac {2 \, {\left (e \log \left (f\right ) + d\right )}}{e m}\right )}}{2 \, e^{2} m^{2}} \] Input:

integrate(x*(a+b*log(c*x^n))/(d+e*log(f*x^m)),x, algorithm="fricas")
 

Output:

1/2*(b*e*m*n*x^2*e^(2*(e*log(f) + d)/(e*m)) + 2*(b*e*m*log(c) - b*e*n*log( 
f) + a*e*m - b*d*n)*log_integral(x^2*e^(2*(e*log(f) + d)/(e*m))))*e^(-2*(e 
*log(f) + d)/(e*m))/(e^2*m^2)
 

Sympy [F]

\[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{d+e \log \left (f x^m\right )} \, dx=\int \frac {x \left (a + b \log {\left (c x^{n} \right )}\right )}{d + e \log {\left (f x^{m} \right )}}\, dx \] Input:

integrate(x*(a+b*ln(c*x**n))/(d+e*ln(f*x**m)),x)
 

Output:

Integral(x*(a + b*log(c*x**n))/(d + e*log(f*x**m)), x)
 

Maxima [F]

\[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{d+e \log \left (f x^m\right )} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x}{e \log \left (f x^{m}\right ) + d} \,d x } \] Input:

integrate(x*(a+b*log(c*x^n))/(d+e*log(f*x^m)),x, algorithm="maxima")
 

Output:

integrate((b*log(c*x^n) + a)*x/(e*log(f*x^m) + d), x)
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.55 \[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{d+e \log \left (f x^m\right )} \, dx=\frac {b n x^{2}}{2 \, e m} + \frac {b {\rm Ei}\left (\frac {2 \, \log \left (f\right )}{m} + \frac {2 \, d}{e m} + 2 \, \log \left (x\right )\right ) e^{\left (-\frac {2 \, d}{e m}\right )} \log \left (c\right )}{e f^{\frac {2}{m}} m} - \frac {b n {\rm Ei}\left (\frac {2 \, \log \left (f\right )}{m} + \frac {2 \, d}{e m} + 2 \, \log \left (x\right )\right ) e^{\left (-\frac {2 \, d}{e m}\right )} \log \left (f\right )}{e f^{\frac {2}{m}} m^{2}} + \frac {a {\rm Ei}\left (\frac {2 \, \log \left (f\right )}{m} + \frac {2 \, d}{e m} + 2 \, \log \left (x\right )\right ) e^{\left (-\frac {2 \, d}{e m}\right )}}{e f^{\frac {2}{m}} m} - \frac {b d n {\rm Ei}\left (\frac {2 \, \log \left (f\right )}{m} + \frac {2 \, d}{e m} + 2 \, \log \left (x\right )\right ) e^{\left (-\frac {2 \, d}{e m}\right )}}{e^{2} f^{\frac {2}{m}} m^{2}} \] Input:

integrate(x*(a+b*log(c*x^n))/(d+e*log(f*x^m)),x, algorithm="giac")
 

Output:

1/2*b*n*x^2/(e*m) + b*Ei(2*log(f)/m + 2*d/(e*m) + 2*log(x))*e^(-2*d/(e*m)) 
*log(c)/(e*f^(2/m)*m) - b*n*Ei(2*log(f)/m + 2*d/(e*m) + 2*log(x))*e^(-2*d/ 
(e*m))*log(f)/(e*f^(2/m)*m^2) + a*Ei(2*log(f)/m + 2*d/(e*m) + 2*log(x))*e^ 
(-2*d/(e*m))/(e*f^(2/m)*m) - b*d*n*Ei(2*log(f)/m + 2*d/(e*m) + 2*log(x))*e 
^(-2*d/(e*m))/(e^2*f^(2/m)*m^2)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{d+e \log \left (f x^m\right )} \, dx=\int \frac {x\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{d+e\,\ln \left (f\,x^m\right )} \,d x \] Input:

int((x*(a + b*log(c*x^n)))/(d + e*log(f*x^m)),x)
 

Output:

int((x*(a + b*log(c*x^n)))/(d + e*log(f*x^m)), x)
 

Reduce [F]

\[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{d+e \log \left (f x^m\right )} \, dx=\left (\int \frac {\mathrm {log}\left (x^{n} c \right ) x}{\mathrm {log}\left (x^{m} f \right ) e +d}d x \right ) b +\left (\int \frac {x}{\mathrm {log}\left (x^{m} f \right ) e +d}d x \right ) a \] Input:

int(x*(a+b*log(c*x^n))/(d+e*log(f*x^m)),x)
 

Output:

int((log(x**n*c)*x)/(log(x**m*f)*e + d),x)*b + int(x/(log(x**m*f)*e + d),x 
)*a