3.2.79 \(\int \frac {a+b \log (c (e+f x))}{d e+d f x} \, dx\) [179]

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

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {2437, 12, 2338} \begin {gather*} \frac {(a+b \log (c (e+f x)))^2}{2 b d f} \end {gather*}

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 12

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

Rule 2338

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]

Rule 2437

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

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Mathematica [A]
time = 0.00, size = 27, normalized size = 1.00 \begin {gather*} \frac {(a+b \log (c (e+f x)))^2}{2 b d f} \end {gather*}

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 = 0.29, size = 42, normalized size = 1.56

method	result	size
risch	\(\frac {b \ln \left (c \left (f x +e \right )\right )^{2}}{2 d f}+\frac {a \ln \left (f x +e \right )}{d f}\)	\(35\)
norman	\(\frac {a \ln \left (c \left (f x +e \right )\right )}{d f}+\frac {b \ln \left (c \left (f x +e \right )\right )^{2}}{2 d f}\)	\(37\)
derivativedivides	\(\frac {\frac {c a \ln \left (c f x +c e \right )}{d}+\frac {c b \ln \left (c f x +c e \right )^{2}}{2 d}}{c f}\)	\(42\)
default	\(\frac {\frac {c a \ln \left (c f x +c e \right )}{d}+\frac {c b \ln \left (c f x +c e \right )^{2}}{2 d}}{c f}\)	\(42\)

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,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 108 vs. \(2 (26) = 52\).
time = 0.31, size = 108, normalized size = 4.00 \begin {gather*} -\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 {gather*}

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 = 0.37, size = 36, normalized size = 1.33 \begin {gather*} \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 {gather*}

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 = 0.09, size = 31, normalized size = 1.15 \begin {gather*} \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 {gather*}

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 = 3.63, size = 33, normalized size = 1.22 \begin {gather*} \frac {b \log \left (c f x + c e\right )^{2} + 2 \, a \log \left (f x + e\right )}{2 \, d f} \end {gather*}

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

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Mupad [B]
time = 0.35, size = 31, normalized size = 1.15 \begin {gather*} \frac {b\,{\ln \left (c\,e+c\,f\,x\right )}^2+2\,a\,\ln \left (e+f\,x\right )}{2\,d\,f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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