3.174 \(\int \frac{a+b \log (c x^n)}{x^2 (d+e \log (f x^m))} \, dx\)

Optimal. Leaf size=133 \[ \frac{e^{\frac{d}{e m}} \left (f x^m\right )^{\frac{1}{m}} \left (a+b \log \left (c x^n\right )\right ) \text{Ei}\left (-\frac{d+e \log \left (f x^m\right )}{e m}\right )}{e m x}-\frac{b n e^{\frac{d}{e m}} \left (f x^m\right )^{\frac{1}{m}} \left (d+e \log \left (f x^m\right )\right ) \text{Ei}\left (-\frac{d+e \log \left (f x^m\right )}{e m}\right )}{e^2 m^2 x}-\frac{b n}{e m x} \]

[Out]

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

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Rubi [A]  time = 0.171626, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {2310, 2178, 2366, 12, 15, 6482} \[ \frac{e^{\frac{d}{e m}} \left (f x^m\right )^{\frac{1}{m}} \left (a+b \log \left (c x^n\right )\right ) \text{Ei}\left (-\frac{d+e \log \left (f x^m\right )}{e m}\right )}{e m x}-\frac{b n e^{\frac{d}{e m}} \left (f x^m\right )^{\frac{1}{m}} \left (d+e \log \left (f x^m\right )\right ) \text{Ei}\left (-\frac{d+e \log \left (f x^m\right )}{e m}\right )}{e^2 m^2 x}-\frac{b n}{e m x} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[c*x^n])/(x^2*(d + e*Log[f*x^m])),x]

[Out]

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

Rule 2310

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_)*((d_.)*(x_))^(m_.), x_Symbol] :> Dist[(d*x)^(m + 1)/(d*n*(c*x^n
)^((m + 1)/n)), Subst[Int[E^(((m + 1)*x)/n)*(a + b*x)^p, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, d, m, 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 2366

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.) + Log[(f_.)*(x_)^(r_.)]*(e_.))*((g_.)*(x_))^(m_.), x_Sy
mbol] :> With[{u = IntHide[(g*x)^m*(a + b*Log[c*x^n])^p, x]}, Dist[d + e*Log[f*x^r], u, x] - Dist[e*r, Int[Sim
plifyIntegrand[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 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 )}{x^2 \left (d+e \log \left (f x^m\right )\right )} \, dx &=\frac{e^{\frac{d}{e m}} \left (f x^m\right )^{\frac{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 x}-(b n) \int \frac{e^{\frac{d}{e m}} \left (f x^m\right )^{\frac{1}{m}} \text{Ei}\left (-\frac{d+e \log \left (f x^m\right )}{e m}\right )}{e m x^2} \, dx\\ &=\frac{e^{\frac{d}{e m}} \left (f x^m\right )^{\frac{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 x}-\frac{\left (b e^{\frac{d}{e m}} n\right ) \int \frac{\left (f x^m\right )^{\frac{1}{m}} \text{Ei}\left (-\frac{d+e \log \left (f x^m\right )}{e m}\right )}{x^2} \, dx}{e m}\\ &=\frac{e^{\frac{d}{e m}} \left (f x^m\right )^{\frac{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 x}-\frac{\left (b e^{\frac{d}{e m}} n \left (f x^m\right )^{\frac{1}{m}}\right ) \int \frac{\text{Ei}\left (-\frac{d+e \log \left (f x^m\right )}{e m}\right )}{x} \, dx}{e m x}\\ &=\frac{e^{\frac{d}{e m}} \left (f x^m\right )^{\frac{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 x}-\frac{\left (b e^{\frac{d}{e m}} n \left (f x^m\right )^{\frac{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 x}\\ &=\frac{e^{\frac{d}{e m}} \left (f x^m\right )^{\frac{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 x}+\frac{\left (b e^{\frac{d}{e m}} n \left (f x^m\right )^{\frac{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 x}\\ &=-\frac{b n}{e m x}-\frac{b e^{\frac{d}{e m}} n \left (f x^m\right )^{\frac{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 x}+\frac{e^{\frac{d}{e m}} \left (f x^m\right )^{\frac{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 x}\\ \end{align*}

Mathematica [A]  time = 0.124643, size = 87, normalized size = 0.65 \[ \frac{e^{\frac{d}{e m}} \left (f x^m\right )^{\frac{1}{m}} \text{Ei}\left (-\frac{d+e \log \left (f x^m\right )}{e m}\right ) \left (a e m+b e m \log \left (c x^n\right )-b d n-b e n \log \left (f x^m\right )\right )-b e m n}{e^2 m^2 x} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Log[c*x^n])/(x^2*(d + e*Log[f*x^m])),x]

[Out]

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

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Maple [F]  time = 0.312, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b\ln \left ( c{x}^{n} \right ) }{{x}^{2} \left ( d+e\ln \left ( f{x}^{m} \right ) \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*ln(c*x^n))/x^2/(d+e*ln(f*x^m)),x)

[Out]

int((a+b*ln(c*x^n))/x^2/(d+e*ln(f*x^m)),x)

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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}{{\left (e \log \left (f x^{m}\right ) + d\right )} x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))/x^2/(d+e*log(f*x^m)),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.825772, size = 198, normalized size = 1.49 \begin{align*} -\frac{b e m n -{\left (b e m x \log \left (c\right ) - b e n x \log \left (f\right ) +{\left (a e m - b d n\right )} x\right )} e^{\left (\frac{e \log \left (f\right ) + d}{e m}\right )} \logintegral \left (\frac{e^{\left (-\frac{e \log \left (f\right ) + d}{e m}\right )}}{x}\right )}{e^{2} m^{2} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))/x^2/(d+e*log(f*x^m)),x, algorithm="fricas")

[Out]

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

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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 )}}{x^{2} \left (d + e \log{\left (f x^{m} \right )}\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*ln(c*x**n))/x**2/(d+e*ln(f*x**m)),x)

[Out]

Integral((a + b*log(c*x**n))/(x**2*(d + e*log(f*x**m))), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \log \left (c x^{n}\right ) + a}{{\left (e \log \left (f x^{m}\right ) + d\right )} x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))/x^2/(d+e*log(f*x^m)),x, algorithm="giac")

[Out]

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