3.38.72 \(\int \frac {36 x+348 x^2+841 x^3+e^x (-92 x+116 x^2)+(36 x+348 x^2+841 x^3+e^x (24+116 x)) \log (\frac {4 e^x+6 x+29 x^2}{6+29 x})}{36 x+348 x^2+841 x^3+e^x (24+116 x)} \, dx\)

Optimal. Leaf size=26 \[ x \log \left (x+\frac {e^x}{2 x+5 \left (x+\frac {6+x}{20}\right )}\right ) \]

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Rubi [A]  time = 1.39, antiderivative size = 17, normalized size of antiderivative = 0.65, number of steps used = 15, number of rules used = 4, integrand size = 100, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {6688, 6742, 77, 2548} \begin {gather*} x \log \left (x+\frac {4 e^x}{29 x+6}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(36*x + 348*x^2 + 841*x^3 + E^x*(-92*x + 116*x^2) + (36*x + 348*x^2 + 841*x^3 + E^x*(24 + 116*x))*Log[(4*E
^x + 6*x + 29*x^2)/(6 + 29*x)])/(36*x + 348*x^2 + 841*x^3 + E^x*(24 + 116*x)),x]

[Out]

x*Log[x + (4*E^x)/(6 + 29*x)]

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rule 2548

Int[Log[u_], x_Symbol] :> Simp[x*Log[u], x] - Int[SimplifyIntegrand[(x*D[u, x])/u, x], x] /; InverseFunctionFr
eeQ[u, x]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {x \left (4 e^x (-23+29 x)+(6+29 x)^2\right )}{(6+29 x) \left (4 e^x+x (6+29 x)\right )}+\log \left (x+\frac {4 e^x}{6+29 x}\right )\right ) \, dx\\ &=\int \frac {x \left (4 e^x (-23+29 x)+(6+29 x)^2\right )}{(6+29 x) \left (4 e^x+x (6+29 x)\right )} \, dx+\int \log \left (x+\frac {4 e^x}{6+29 x}\right ) \, dx\\ &=x \log \left (x+\frac {4 e^x}{6+29 x}\right )-\int \frac {x \left (4 e^x (-23+29 x)+(6+29 x)^2\right )}{(6+29 x) \left (4 e^x+x (6+29 x)\right )} \, dx+\int \left (\frac {x (-23+29 x)}{6+29 x}-\frac {x \left (-6-52 x+29 x^2\right )}{4 e^x+6 x+29 x^2}\right ) \, dx\\ &=x \log \left (x+\frac {4 e^x}{6+29 x}\right )+\int \frac {x (-23+29 x)}{6+29 x} \, dx-\int \frac {x \left (-6-52 x+29 x^2\right )}{4 e^x+6 x+29 x^2} \, dx-\int \left (\frac {x (-23+29 x)}{6+29 x}-\frac {x \left (-6-52 x+29 x^2\right )}{4 e^x+6 x+29 x^2}\right ) \, dx\\ &=x \log \left (x+\frac {4 e^x}{6+29 x}\right )-\int \frac {x (-23+29 x)}{6+29 x} \, dx+\int \frac {x \left (-6-52 x+29 x^2\right )}{4 e^x+6 x+29 x^2} \, dx+\int \left (-1+x+\frac {6}{6+29 x}\right ) \, dx-\int \left (-\frac {6 x}{4 e^x+6 x+29 x^2}-\frac {52 x^2}{4 e^x+6 x+29 x^2}+\frac {29 x^3}{4 e^x+6 x+29 x^2}\right ) \, dx\\ &=-x+\frac {x^2}{2}+\frac {6}{29} \log (6+29 x)+x \log \left (x+\frac {4 e^x}{6+29 x}\right )+6 \int \frac {x}{4 e^x+6 x+29 x^2} \, dx-29 \int \frac {x^3}{4 e^x+6 x+29 x^2} \, dx+52 \int \frac {x^2}{4 e^x+6 x+29 x^2} \, dx-\int \left (-1+x+\frac {6}{6+29 x}\right ) \, dx+\int \left (-\frac {6 x}{4 e^x+6 x+29 x^2}-\frac {52 x^2}{4 e^x+6 x+29 x^2}+\frac {29 x^3}{4 e^x+6 x+29 x^2}\right ) \, dx\\ &=x \log \left (x+\frac {4 e^x}{6+29 x}\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.38, size = 17, normalized size = 0.65 \begin {gather*} x \log \left (x+\frac {4 e^x}{6+29 x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(36*x + 348*x^2 + 841*x^3 + E^x*(-92*x + 116*x^2) + (36*x + 348*x^2 + 841*x^3 + E^x*(24 + 116*x))*Lo
g[(4*E^x + 6*x + 29*x^2)/(6 + 29*x)])/(36*x + 348*x^2 + 841*x^3 + E^x*(24 + 116*x)),x]

[Out]

x*Log[x + (4*E^x)/(6 + 29*x)]

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fricas [A]  time = 0.59, size = 24, normalized size = 0.92 \begin {gather*} x \log \left (\frac {29 \, x^{2} + 6 \, x + 4 \, e^{x}}{29 \, x + 6}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((116*x+24)*exp(x)+841*x^3+348*x^2+36*x)*log((4*exp(x)+29*x^2+6*x)/(29*x+6))+(116*x^2-92*x)*exp(x)+
841*x^3+348*x^2+36*x)/((116*x+24)*exp(x)+841*x^3+348*x^2+36*x),x, algorithm="fricas")

[Out]

x*log((29*x^2 + 6*x + 4*e^x)/(29*x + 6))

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giac [A]  time = 0.20, size = 24, normalized size = 0.92 \begin {gather*} x \log \left (\frac {29 \, x^{2} + 6 \, x + 4 \, e^{x}}{29 \, x + 6}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((116*x+24)*exp(x)+841*x^3+348*x^2+36*x)*log((4*exp(x)+29*x^2+6*x)/(29*x+6))+(116*x^2-92*x)*exp(x)+
841*x^3+348*x^2+36*x)/((116*x+24)*exp(x)+841*x^3+348*x^2+36*x),x, algorithm="giac")

[Out]

x*log((29*x^2 + 6*x + 4*e^x)/(29*x + 6))

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maple [A]  time = 0.14, size = 25, normalized size = 0.96

method	result	size
norman	\(\ln \left (\frac {4 \,{\mathrm e}^{x}+29 x^{2}+6 x}{29 x +6}\right ) x\)	\(25\)
risch	\(x \ln \left (x^{2}+\frac {6 x}{29}+\frac {4 \,{\mathrm e}^{x}}{29}\right )-x \ln \left (x +\frac {6}{29}\right )-\frac {i \pi x \,\mathrm {csgn}\left (\frac {i}{x +\frac {6}{29}}\right ) \mathrm {csgn}\left (i \left (x^{2}+\frac {6 x}{29}+\frac {4 \,{\mathrm e}^{x}}{29}\right )\right ) \mathrm {csgn}\left (\frac {i \left (x^{2}+\frac {6 x}{29}+\frac {4 \,{\mathrm e}^{x}}{29}\right )}{x +\frac {6}{29}}\right )}{2}+\frac {i \pi x \,\mathrm {csgn}\left (\frac {i}{x +\frac {6}{29}}\right ) \mathrm {csgn}\left (\frac {i \left (x^{2}+\frac {6 x}{29}+\frac {4 \,{\mathrm e}^{x}}{29}\right )}{x +\frac {6}{29}}\right )^{2}}{2}+\frac {i \pi x \,\mathrm {csgn}\left (i \left (x^{2}+\frac {6 x}{29}+\frac {4 \,{\mathrm e}^{x}}{29}\right )\right ) \mathrm {csgn}\left (\frac {i \left (x^{2}+\frac {6 x}{29}+\frac {4 \,{\mathrm e}^{x}}{29}\right )}{x +\frac {6}{29}}\right )^{2}}{2}-\frac {i \pi x \mathrm {csgn}\left (\frac {i \left (x^{2}+\frac {6 x}{29}+\frac {4 \,{\mathrm e}^{x}}{29}\right )}{x +\frac {6}{29}}\right )^{3}}{2}\)	\(177\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((116*x+24)*exp(x)+841*x^3+348*x^2+36*x)*ln((4*exp(x)+29*x^2+6*x)/(29*x+6))+(116*x^2-92*x)*exp(x)+841*x^3
+348*x^2+36*x)/((116*x+24)*exp(x)+841*x^3+348*x^2+36*x),x,method=_RETURNVERBOSE)

[Out]

ln((4*exp(x)+29*x^2+6*x)/(29*x+6))*x

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maxima [A]  time = 0.56, size = 26, normalized size = 1.00 \begin {gather*} x \log \left (29 \, x^{2} + 6 \, x + 4 \, e^{x}\right ) - x \log \left (29 \, x + 6\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((116*x+24)*exp(x)+841*x^3+348*x^2+36*x)*log((4*exp(x)+29*x^2+6*x)/(29*x+6))+(116*x^2-92*x)*exp(x)+
841*x^3+348*x^2+36*x)/((116*x+24)*exp(x)+841*x^3+348*x^2+36*x),x, algorithm="maxima")

[Out]

x*log(29*x^2 + 6*x + 4*e^x) - x*log(29*x + 6)

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mupad [B]  time = 2.73, size = 24, normalized size = 0.92 \begin {gather*} x\,\ln \left (\frac {6\,x+4\,{\mathrm {e}}^x+29\,x^2}{29\,x+6}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((36*x + log((6*x + 4*exp(x) + 29*x^2)/(29*x + 6))*(36*x + exp(x)*(116*x + 24) + 348*x^2 + 841*x^3) - exp(x
)*(92*x - 116*x^2) + 348*x^2 + 841*x^3)/(36*x + exp(x)*(116*x + 24) + 348*x^2 + 841*x^3),x)

[Out]

x*log((6*x + 4*exp(x) + 29*x^2)/(29*x + 6))

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sympy [B]  time = 0.69, size = 54, normalized size = 2.08 \begin {gather*} \left (x + \frac {3}{29}\right ) \log {\left (\frac {29 x^{2} + 6 x + 4 e^{x}}{29 x + 6} \right )} + \frac {3 \log {\left (29 x + 6 \right )}}{29} - \frac {3 \log {\left (\frac {29 x^{2}}{4} + \frac {3 x}{2} + e^{x} \right )}}{29} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((116*x+24)*exp(x)+841*x**3+348*x**2+36*x)*ln((4*exp(x)+29*x**2+6*x)/(29*x+6))+(116*x**2-92*x)*exp(
x)+841*x**3+348*x**2+36*x)/((116*x+24)*exp(x)+841*x**3+348*x**2+36*x),x)

[Out]

(x + 3/29)*log((29*x**2 + 6*x + 4*exp(x))/(29*x + 6)) + 3*log(29*x + 6)/29 - 3*log(29*x**2/4 + 3*x/2 + exp(x))
/29

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