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

Optimal. Leaf size=130 \[ \frac{x e^{-\frac{d}{e m}} \left (f x^m\right )^{-1/m} \left (a+b \log \left (c x^n\right )\right ) \text{Ei}\left (\frac{d+e \log \left (f x^m\right )}{e m}\right )}{e m}-\frac{b n x e^{-\frac{d}{e m}} \left (f x^m\right )^{-1/m} \left (d+e \log \left (f x^m\right )\right ) \text{Ei}\left (\frac{d+e \log \left (f x^m\right )}{e m}\right )}{e^2 m^2}+\frac{b n x}{e m} \]

[Out]

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

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Rubi [A]  time = 0.122035, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {2300, 2178, 2361, 12, 15, 6482} \[ \frac{x e^{-\frac{d}{e m}} \left (f x^m\right )^{-1/m} \left (a+b \log \left (c x^n\right )\right ) \text{Ei}\left (\frac{d+e \log \left (f x^m\right )}{e m}\right )}{e m}-\frac{b n x e^{-\frac{d}{e m}} \left (f x^m\right )^{-1/m} \left (d+e \log \left (f x^m\right )\right ) \text{Ei}\left (\frac{d+e \log \left (f x^m\right )}{e m}\right )}{e^2 m^2}+\frac{b n x}{e m} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2300

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_), x_Symbol] :> Dist[x/(n*(c*x^n)^(1/n)), Subst[Int[E^(x/n)*(a +
b*x)^p, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

Rule 2178

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - (c*f)/d))*ExpIntegral
Ei[(f*g*(c + d*x)*Log[F])/d])/d, x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !$UseGamma === True

Rule 2361

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.) + Log[(f_.)*(x_)^(r_.)]*(e_.)), x_Symbol] :> With[{u =
IntHide[(a + b*Log[c*x^n])^p, x]}, Dist[d + e*Log[f*x^r], u, x] - Dist[e*r, Int[SimplifyIntegrand[u/x, x], x],
x]] /; FreeQ[{a, b, c, d, e, f, n, p, r}, x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[(a^IntPart[m]*(a*x^n)^FracPart[m])/x^(n*FracPart[m]), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

Rule 6482

Int[ExpIntegralEi[(a_.) + (b_.)*(x_)], x_Symbol] :> Simp[((a + b*x)*ExpIntegralEi[a + b*x])/b, x] - Simp[E^(a
+ b*x)/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \frac{a+b \log \left (c x^n\right )}{d+e \log \left (f x^m\right )} \, dx &=\frac{e^{-\frac{d}{e m}} x \left (f x^m\right )^{-1/m} \text{Ei}\left (\frac{d+e \log \left (f x^m\right )}{e m}\right ) \left (a+b \log \left (c x^n\right )\right )}{e m}-(b n) \int \frac{e^{-\frac{d}{e m}} \left (f x^m\right )^{-1/m} \text{Ei}\left (\frac{d+e \log \left (f x^m\right )}{e m}\right )}{e m} \, dx\\ &=\frac{e^{-\frac{d}{e m}} x \left (f x^m\right )^{-1/m} \text{Ei}\left (\frac{d+e \log \left (f x^m\right )}{e m}\right ) \left (a+b \log \left (c x^n\right )\right )}{e m}-\frac{\left (b e^{-\frac{d}{e m}} n\right ) \int \left (f x^m\right )^{-1/m} \text{Ei}\left (\frac{d+e \log \left (f x^m\right )}{e m}\right ) \, dx}{e m}\\ &=\frac{e^{-\frac{d}{e m}} x \left (f x^m\right )^{-1/m} \text{Ei}\left (\frac{d+e \log \left (f x^m\right )}{e m}\right ) \left (a+b \log \left (c x^n\right )\right )}{e m}-\frac{\left (b e^{-\frac{d}{e m}} n x \left (f x^m\right )^{-1/m}\right ) \int \frac{\text{Ei}\left (\frac{d+e \log \left (f x^m\right )}{e m}\right )}{x} \, dx}{e m}\\ &=\frac{e^{-\frac{d}{e m}} x \left (f x^m\right )^{-1/m} \text{Ei}\left (\frac{d+e \log \left (f x^m\right )}{e m}\right ) \left (a+b \log \left (c x^n\right )\right )}{e m}-\frac{\left (b e^{-\frac{d}{e m}} n x \left (f x^m\right )^{-1/m}\right ) \operatorname{Subst}\left (\int \text{Ei}\left (\frac{d+e x}{e m}\right ) \, dx,x,\log \left (f x^m\right )\right )}{e m^2}\\ &=\frac{e^{-\frac{d}{e m}} x \left (f x^m\right )^{-1/m} \text{Ei}\left (\frac{d+e \log \left (f x^m\right )}{e m}\right ) \left (a+b \log \left (c x^n\right )\right )}{e m}-\frac{\left (b e^{-\frac{d}{e m}} n x \left (f x^m\right )^{-1/m}\right ) \operatorname{Subst}\left (\int \text{Ei}(x) \, dx,x,\frac{d}{e m}+\frac{\log \left (f x^m\right )}{m}\right )}{e m}\\ &=\frac{b n x}{e m}-\frac{b e^{-\frac{d}{e m}} n x \left (f x^m\right )^{-1/m} \text{Ei}\left (\frac{d}{e m}+\frac{\log \left (f x^m\right )}{m}\right ) \left (\frac{d}{e m}+\frac{\log \left (f x^m\right )}{m}\right )}{e m}+\frac{e^{-\frac{d}{e m}} x \left (f x^m\right )^{-1/m} \text{Ei}\left (\frac{d+e \log \left (f x^m\right )}{e m}\right ) \left (a+b \log \left (c x^n\right )\right )}{e m}\\ \end{align*}

Mathematica [A]  time = 0.128079, size = 86, normalized size = 0.66 \[ \frac{x \left (e^{-\frac{d}{e m}} \left (f x^m\right )^{-1/m} \text{Ei}\left (\frac{d+e \log \left (f x^m\right )}{e m}\right ) \left (a e m+b e m \log \left (c x^n\right )-b d n-b e n \log \left (f x^m\right )\right )+b e m n\right )}{e^2 m^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(b*e*m*n + (ExpIntegralEi[(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^(d/(e*m))*(f*x^m)^m^(-1))))/(e^2*m^2)

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Maple [C]  time = 0.281, size = 2356, normalized size = 18.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \log \left (c x^{n}\right ) + a}{e \log \left (f x^{m}\right ) + d}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 0.784735, size = 220, normalized size = 1.69 \begin{align*} \frac{{\left (b e m n x e^{\left (\frac{e \log \left (f\right ) + d}{e m}\right )} +{\left (b e m \log \left (c\right ) - b e n \log \left (f\right ) + a e m - b d n\right )} \logintegral \left (x e^{\left (\frac{e \log \left (f\right ) + d}{e m}\right )}\right )\right )} e^{\left (-\frac{e \log \left (f\right ) + d}{e m}\right )}}{e^{2} m^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{a + b \log{\left (c x^{n} \right )}}{d + e \log{\left (f x^{m} \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.44772, size = 230, normalized size = 1.77 \begin{align*} -\frac{b d n{\rm Ei}\left (\frac{d e^{\left (-1\right )}}{m} + \frac{\log \left (f\right )}{m} + \log \left (x\right )\right ) e^{\left (-\frac{d e^{\left (-1\right )}}{m} - 2\right )}}{f^{\left (\frac{1}{m}\right )} m^{2}} + \frac{b{\rm Ei}\left (\frac{d e^{\left (-1\right )}}{m} + \frac{\log \left (f\right )}{m} + \log \left (x\right )\right ) e^{\left (-\frac{d e^{\left (-1\right )}}{m} - 1\right )} \log \left (c\right )}{f^{\left (\frac{1}{m}\right )} m} - \frac{b n{\rm Ei}\left (\frac{d e^{\left (-1\right )}}{m} + \frac{\log \left (f\right )}{m} + \log \left (x\right )\right ) e^{\left (-\frac{d e^{\left (-1\right )}}{m} - 1\right )} \log \left (f\right )}{f^{\left (\frac{1}{m}\right )} m^{2}} + \frac{a{\rm Ei}\left (\frac{d e^{\left (-1\right )}}{m} + \frac{\log \left (f\right )}{m} + \log \left (x\right )\right ) e^{\left (-\frac{d e^{\left (-1\right )}}{m} - 1\right )}}{f^{\left (\frac{1}{m}\right )} m} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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