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

Optimal. Leaf size=27 \[ \frac{(a+b \log (c (e+f x)))^2}{2 b d f} \]

[Out]

(a + b*Log[c*(e + f*x)])^2/(2*b*d*f)

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Rubi [A]  time = 0.0343237, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {2390, 12, 2301} \[ \frac{(a+b \log (c (e+f x)))^2}{2 b d f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a + b*Log[c*(e + f*x)])^2/(2*b*d*f)

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]

Rule 12

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

Rule 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rubi steps

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

Mathematica [A]  time = 0.0036606, size = 27, normalized size = 1. \[ \frac{(a+b \log (c (e+f x)))^2}{2 b d f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a + b*Log[c*(e + f*x)])^2/(2*b*d*f)

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Maple [A]  time = 0.062, size = 39, normalized size = 1.4 \begin{align*}{\frac{a\ln \left ( cfx+ce \right ) }{df}}+{\frac{b \left ( \ln \left ( cfx+ce \right ) \right ) ^{2}}{2\,df}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/f/d*a*ln(c*f*x+c*e)+1/2/f/d*b*ln(c*f*x+c*e)^2

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Maxima [B]  time = 1.13009, size = 136, normalized size = 5.04 \begin{align*} -\frac{1}{2} \, b{\left (\frac{2 \, \log \left (c f x + c e\right ) \log \left (d f x + d e\right )}{d f} - \frac{\log \left (f x + e\right )^{2} + 2 \, \log \left (f x + e\right ) \log \left (c\right )}{d f}\right )} + \frac{b \log \left (c f x + c e\right ) \log \left (d f x + d e\right )}{d f} + \frac{a \log \left (d f x + d e\right )}{d f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.65599, size = 77, normalized size = 2.85 \begin{align*} \frac{b \log \left (c f x + c e\right )^{2} + 2 \, a \log \left (c f x + c e\right )}{2 \, d f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.366937, size = 31, normalized size = 1.15 \begin{align*} \frac{a \log{\left (d e + d f x \right )}}{d f} + \frac{b \log{\left (c \left (e + f x\right ) \right )}^{2}}{2 d f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*log(d*e + d*f*x)/(d*f) + b*log(c*(e + f*x))**2/(2*d*f)

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Giac [A]  time = 1.19508, size = 46, normalized size = 1.7 \begin{align*} \frac{b \log \left ({\left (f x + e\right )} c\right )^{2} + 2 \, a \log \left ({\left (f x + e\right )} c\right )}{2 \, d f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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