3.195 \(\int \frac {1}{(d e+d f x) (a+b \log (c (e+f x)))} \, dx\)

Optimal. Leaf size=23 \[ \frac {\log (a+b \log (c (e+f x)))}{b d f} \]

[Out]

ln(a+b*ln(c*(f*x+e)))/b/d/f

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Rubi [A]  time = 0.07, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {2390, 12, 2302, 29} \[ \frac {\log (a+b \log (c (e+f x)))}{b d f} \]

Antiderivative was successfully verified.

[In]

Int[1/((d*e + d*f*x)*(a + b*Log[c*(e + f*x)])),x]

[Out]

Log[a + b*Log[c*(e + f*x)]]/(b*d*f)

Rule 12

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

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 2302

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

Rule 2390

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_))^(q_.), x_Symbol] :> Dist[1/
e, Subst[Int[((f*x)/d)^q*(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p, q}, x]
&& EqQ[e*f - d*g, 0]

Rubi steps

\begin {align*} \int \frac {1}{(d e+d f x) (a+b \log (c (e+f x)))} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{d x (a+b \log (c x))} \, dx,x,e+f x\right )}{f}\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{x (a+b \log (c x))} \, dx,x,e+f x\right )}{d f}\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,a+b \log (c (e+f x))\right )}{b d f}\\ &=\frac {\log (a+b \log (c (e+f x)))}{b d f}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 23, normalized size = 1.00 \[ \frac {\log (a+b \log (c (e+f x)))}{b d f} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((d*e + d*f*x)*(a + b*Log[c*(e + f*x)])),x]

[Out]

Log[a + b*Log[c*(e + f*x)]]/(b*d*f)

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fricas [A]  time = 0.44, size = 24, normalized size = 1.04 \[ \frac {\log \left (b \log \left (c f x + c e\right ) + a\right )}{b d f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*f*x+d*e)/(a+b*log(c*(f*x+e))),x, algorithm="fricas")

[Out]

log(b*log(c*f*x + c*e) + a)/(b*d*f)

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giac [A]  time = 0.18, size = 25, normalized size = 1.09 \[ \frac {\log \left (b \log \left (c f x + c e\right ) + a\right )}{b d f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*f*x+d*e)/(a+b*log(c*(f*x+e))),x, algorithm="giac")

[Out]

log(b*log(c*f*x + c*e) + a)/(b*d*f)

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maple [A]  time = 0.05, size = 25, normalized size = 1.09 \[ \frac {\ln \left (b \ln \left (c f x +c e \right )+a \right )}{b d f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(d*f*x+d*e)/(b*ln((f*x+e)*c)+a),x)

[Out]

1/f/d*ln(a+b*ln(c*f*x+c*e))/b

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maxima [A]  time = 0.47, size = 29, normalized size = 1.26 \[ \frac {\log \left (\frac {b \log \left (f x + e\right ) + b \log \relax (c) + a}{b}\right )}{b d f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*f*x+d*e)/(a+b*log(c*(f*x+e))),x, algorithm="maxima")

[Out]

log((b*log(f*x + e) + b*log(c) + a)/b)/(b*d*f)

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mupad [B]  time = 0.85, size = 23, normalized size = 1.00 \[ \frac {\ln \left (a+b\,\ln \left (c\,\left (e+f\,x\right )\right )\right )}{b\,d\,f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((d*e + d*f*x)*(a + b*log(c*(e + f*x)))),x)

[Out]

log(a + b*log(c*(e + f*x)))/(b*d*f)

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sympy [A]  time = 0.22, size = 17, normalized size = 0.74 \[ \frac {\log {\left (\frac {a}{b} + \log {\left (c \left (e + f x\right ) \right )} \right )}}{b d f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*f*x+d*e)/(a+b*ln(c*(f*x+e))),x)

[Out]

log(a/b + log(c*(e + f*x)))/(b*d*f)

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