3.16.36 \(\int \frac {8+e^{\frac {3+x}{2}} (-4-36 x)+e^x (3 e^{\frac {3+x}{2}} x+6 x^2)+(-8-72 x+6 e^x x) \log (\frac {4+36 x-3 e^x x}{x})}{-8-72 x+6 e^x x} \, dx\)

Optimal. Leaf size=25 \[ e^{\frac {3+x}{2}}+x \log \left (36-3 e^x+\frac {4}{x}\right ) \]

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Rubi [A]  time = 1.07, antiderivative size = 28, normalized size of antiderivative = 1.12, number of steps used = 13, number of rules used = 5, integrand size = 81, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {6741, 12, 6742, 2194, 2548} \begin {gather*} e^{\frac {x}{2}+\frac {3}{2}}+x \log \left (4 \left (\frac {1}{x}+9\right )-3 e^x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(8 + E^((3 + x)/2)*(-4 - 36*x) + E^x*(3*E^((3 + x)/2)*x + 6*x^2) + (-8 - 72*x + 6*E^x*x)*Log[(4 + 36*x - 3
*E^x*x)/x])/(-8 - 72*x + 6*E^x*x),x]

[Out]

E^(3/2 + x/2) + x*Log[-3*E^x + 4*(9 + x^(-1))]

Rule 12

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

Rule 2194

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

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 6741

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

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 \frac {-8-e^{\frac {3+x}{2}} (-4-36 x)-e^x \left (3 e^{\frac {3+x}{2}} x+6 x^2\right )-\left (-8-72 x+6 e^x x\right ) \log \left (\frac {4+36 x-3 e^x x}{x}\right )}{2 \left (4+36 x-3 e^x x\right )} \, dx\\ &=\frac {1}{2} \int \frac {-8-e^{\frac {3+x}{2}} (-4-36 x)-e^x \left (3 e^{\frac {3+x}{2}} x+6 x^2\right )-\left (-8-72 x+6 e^x x\right ) \log \left (\frac {4+36 x-3 e^x x}{x}\right )}{4+36 x-3 e^x x} \, dx\\ &=\frac {1}{2} \int \left (e^{\frac {3}{2}+\frac {x}{2}}+\frac {8 \left (1+x+9 x^2\right )}{-4-36 x+3 e^x x}+2 \left (x+\log \left (-3 e^x+4 \left (9+\frac {1}{x}\right )\right )\right )\right ) \, dx\\ &=\frac {1}{2} \int e^{\frac {3}{2}+\frac {x}{2}} \, dx+4 \int \frac {1+x+9 x^2}{-4-36 x+3 e^x x} \, dx+\int \left (x+\log \left (-3 e^x+4 \left (9+\frac {1}{x}\right )\right )\right ) \, dx\\ &=e^{\frac {3}{2}+\frac {x}{2}}+\frac {x^2}{2}+4 \int \left (\frac {1}{-4-36 x+3 e^x x}+\frac {x}{-4-36 x+3 e^x x}+\frac {9 x^2}{-4-36 x+3 e^x x}\right ) \, dx+\int \log \left (-3 e^x+4 \left (9+\frac {1}{x}\right )\right ) \, dx\\ &=e^{\frac {3}{2}+\frac {x}{2}}+\frac {x^2}{2}+x \log \left (-3 e^x+4 \left (9+\frac {1}{x}\right )\right )+4 \int \frac {1}{-4-36 x+3 e^x x} \, dx+4 \int \frac {x}{-4-36 x+3 e^x x} \, dx+36 \int \frac {x^2}{-4-36 x+3 e^x x} \, dx-\int \frac {-4-3 e^x x^2}{4-3 \left (-12+e^x\right ) x} \, dx\\ &=e^{\frac {3}{2}+\frac {x}{2}}+\frac {x^2}{2}+x \log \left (-3 e^x+4 \left (9+\frac {1}{x}\right )\right )+4 \int \frac {1}{-4-36 x+3 e^x x} \, dx+4 \int \frac {x}{-4-36 x+3 e^x x} \, dx+36 \int \frac {x^2}{-4-36 x+3 e^x x} \, dx-\int \left (x+\frac {4 \left (1+x+9 x^2\right )}{-4-36 x+3 e^x x}\right ) \, dx\\ &=e^{\frac {3}{2}+\frac {x}{2}}+x \log \left (-3 e^x+4 \left (9+\frac {1}{x}\right )\right )+4 \int \frac {1}{-4-36 x+3 e^x x} \, dx+4 \int \frac {x}{-4-36 x+3 e^x x} \, dx-4 \int \frac {1+x+9 x^2}{-4-36 x+3 e^x x} \, dx+36 \int \frac {x^2}{-4-36 x+3 e^x x} \, dx\\ &=e^{\frac {3}{2}+\frac {x}{2}}+x \log \left (-3 e^x+4 \left (9+\frac {1}{x}\right )\right )+4 \int \frac {1}{-4-36 x+3 e^x x} \, dx+4 \int \frac {x}{-4-36 x+3 e^x x} \, dx-4 \int \left (\frac {1}{-4-36 x+3 e^x x}+\frac {x}{-4-36 x+3 e^x x}+\frac {9 x^2}{-4-36 x+3 e^x x}\right ) \, dx+36 \int \frac {x^2}{-4-36 x+3 e^x x} \, dx\\ &=e^{\frac {3}{2}+\frac {x}{2}}+x \log \left (-3 e^x+4 \left (9+\frac {1}{x}\right )\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.34, size = 28, normalized size = 1.12 \begin {gather*} e^{\frac {3}{2}+\frac {x}{2}}+x \log \left (-3 e^x+4 \left (9+\frac {1}{x}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(8 + E^((3 + x)/2)*(-4 - 36*x) + E^x*(3*E^((3 + x)/2)*x + 6*x^2) + (-8 - 72*x + 6*E^x*x)*Log[(4 + 36
*x - 3*E^x*x)/x])/(-8 - 72*x + 6*E^x*x),x]

[Out]

E^(3/2 + x/2) + x*Log[-3*E^x + 4*(9 + x^(-1))]

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fricas [A]  time = 0.56, size = 33, normalized size = 1.32 \begin {gather*} x \log \left (\frac {{\left (4 \, {\left (9 \, x + 1\right )} e^{3} - 3 \, x e^{\left (x + 3\right )}\right )} e^{\left (-3\right )}}{x}\right ) + e^{\left (\frac {1}{2} \, x + \frac {3}{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6*exp(x)*x-72*x-8)*log((-3*exp(x)*x+36*x+4)/x)+(3*x*exp(3/2+1/2*x)+6*x^2)*exp(x)+(-36*x-4)*exp(3/2
+1/2*x)+8)/(6*exp(x)*x-72*x-8),x, algorithm="fricas")

[Out]

x*log((4*(9*x + 1)*e^3 - 3*x*e^(x + 3))*e^(-3)/x) + e^(1/2*x + 3/2)

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giac [A]  time = 0.26, size = 25, normalized size = 1.00 \begin {gather*} x \log \left (-\frac {3 \, x e^{x} - 36 \, x - 4}{x}\right ) + e^{\left (\frac {1}{2} \, x + \frac {3}{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6*exp(x)*x-72*x-8)*log((-3*exp(x)*x+36*x+4)/x)+(3*x*exp(3/2+1/2*x)+6*x^2)*exp(x)+(-36*x-4)*exp(3/2
+1/2*x)+8)/(6*exp(x)*x-72*x-8),x, algorithm="giac")

[Out]

x*log(-(3*x*e^x - 36*x - 4)/x) + e^(1/2*x + 3/2)

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maple [C]  time = 0.16, size = 179, normalized size = 7.16

method	result	size
risch	\(x \ln \left (-\frac {4}{3}+\left ({\mathrm e}^{x}-12\right ) x \right )-x \ln \relax (x )-\frac {i \pi x \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (i \left (-\frac {4}{3}+\left ({\mathrm e}^{x}-12\right ) x \right )\right ) \mathrm {csgn}\left (\frac {i \left (-\frac {4}{3}+\left ({\mathrm e}^{x}-12\right ) x \right )}{x}\right )}{2}+\frac {i \pi x \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i \left (-\frac {4}{3}+\left ({\mathrm e}^{x}-12\right ) x \right )}{x}\right )^{2}}{2}-i \pi x \mathrm {csgn}\left (\frac {i \left (-\frac {4}{3}+\left ({\mathrm e}^{x}-12\right ) x \right )}{x}\right )^{2}+\frac {i \pi x \,\mathrm {csgn}\left (i \left (-\frac {4}{3}+\left ({\mathrm e}^{x}-12\right ) x \right )\right ) \mathrm {csgn}\left (\frac {i \left (-\frac {4}{3}+\left ({\mathrm e}^{x}-12\right ) x \right )}{x}\right )^{2}}{2}+\frac {i \pi x \mathrm {csgn}\left (\frac {i \left (-\frac {4}{3}+\left ({\mathrm e}^{x}-12\right ) x \right )}{x}\right )^{3}}{2}+i \pi x +x \ln \relax (3)+{\mathrm e}^{\frac {3}{2}+\frac {x}{2}}\)	\(179\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((6*exp(x)*x-72*x-8)*ln((-3*exp(x)*x+36*x+4)/x)+(3*x*exp(3/2+1/2*x)+6*x^2)*exp(x)+(-36*x-4)*exp(3/2+1/2*x)
+8)/(6*exp(x)*x-72*x-8),x,method=_RETURNVERBOSE)

[Out]

x*ln(-4/3+(exp(x)-12)*x)-x*ln(x)-1/2*I*Pi*x*csgn(I/x)*csgn(I*(-4/3+(exp(x)-12)*x))*csgn(I/x*(-4/3+(exp(x)-12)*
x))+1/2*I*Pi*x*csgn(I/x)*csgn(I/x*(-4/3+(exp(x)-12)*x))^2-I*Pi*x*csgn(I/x*(-4/3+(exp(x)-12)*x))^2+1/2*I*Pi*x*c
sgn(I*(-4/3+(exp(x)-12)*x))*csgn(I/x*(-4/3+(exp(x)-12)*x))^2+1/2*I*Pi*x*csgn(I/x*(-4/3+(exp(x)-12)*x))^3+I*Pi*
x+x*ln(3)+exp(3/2+1/2*x)

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maxima [A]  time = 0.92, size = 25, normalized size = 1.00 \begin {gather*} x \log \left (-3 \, x e^{x} + 36 \, x + 4\right ) - x \log \relax (x) + e^{\left (\frac {1}{2} \, x + \frac {3}{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6*exp(x)*x-72*x-8)*log((-3*exp(x)*x+36*x+4)/x)+(3*x*exp(3/2+1/2*x)+6*x^2)*exp(x)+(-36*x-4)*exp(3/2
+1/2*x)+8)/(6*exp(x)*x-72*x-8),x, algorithm="maxima")

[Out]

x*log(-3*x*e^x + 36*x + 4) - x*log(x) + e^(1/2*x + 3/2)

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mupad [B]  time = 1.37, size = 24, normalized size = 0.96 \begin {gather*} {\mathrm {e}}^{\frac {x}{2}+\frac {3}{2}}+x\,\ln \left (\frac {36\,x-3\,x\,{\mathrm {e}}^x+4}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x/2 + 3/2)*(36*x + 4) - exp(x)*(3*x*exp(x/2 + 3/2) + 6*x^2) + log((36*x - 3*x*exp(x) + 4)/x)*(72*x -
6*x*exp(x) + 8) - 8)/(72*x - 6*x*exp(x) + 8),x)

[Out]

exp(x/2 + 3/2) + x*log((36*x - 3*x*exp(x) + 4)/x)

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sympy [A]  time = 0.46, size = 27, normalized size = 1.08 \begin {gather*} x \log {\left (\frac {- 3 x e^{x} + 36 x + 4}{x} \right )} + e^{\frac {3}{2}} \sqrt {e^{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6*exp(x)*x-72*x-8)*ln((-3*exp(x)*x+36*x+4)/x)+(3*x*exp(3/2+1/2*x)+6*x**2)*exp(x)+(-36*x-4)*exp(3/2
+1/2*x)+8)/(6*exp(x)*x-72*x-8),x)

[Out]

x*log((-3*x*exp(x) + 36*x + 4)/x) + exp(3/2)*sqrt(exp(x))

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