3.76.49 \(\int \frac {-54+70 x+33 x^2+e^{2 x} (19 x-8 x^2-6 x^3)+e^x (26 x-2 x^2-6 x^3)+(-63 x-54 x^2) \log (2 x)}{108 x-252 x^2+39 x^3+126 x^4+27 x^5} \, dx\)

Optimal. Leaf size=33 \[ \frac {\frac {1}{9} \left (1+e^x\right )^2-\log (2 x)}{\left (\frac {2}{3}-x\right ) (3+x)} \]

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Rubi [B]  time = 1.57, antiderivative size = 113, normalized size of antiderivative = 3.42, number of steps used = 32, number of rules used = 11, integrand size = 89, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.124, Rules used = {6741, 12, 6742, 44, 88, 77, 2177, 2178, 2357, 2314, 31} \begin {gather*} \frac {2 e^x}{11 (2-3 x)}+\frac {e^{2 x}}{11 (2-3 x)}+\frac {1}{11 (2-3 x)}+\frac {2 e^x}{33 (x+3)}+\frac {e^{2 x}}{33 (x+3)}+\frac {1}{33 (x+3)}-\frac {\log (x)}{2}-\frac {27 x \log (2 x)}{22 (2-3 x)}+\frac {x \log (2 x)}{11 (x+3)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-54 + 70*x + 33*x^2 + E^(2*x)*(19*x - 8*x^2 - 6*x^3) + E^x*(26*x - 2*x^2 - 6*x^3) + (-63*x - 54*x^2)*Log[
2*x])/(108*x - 252*x^2 + 39*x^3 + 126*x^4 + 27*x^5),x]

[Out]

1/(11*(2 - 3*x)) + (2*E^x)/(11*(2 - 3*x)) + E^(2*x)/(11*(2 - 3*x)) + 1/(33*(3 + x)) + (2*E^x)/(33*(3 + x)) + E
^(2*x)/(33*(3 + x)) - Log[x]/2 - (27*x*Log[2*x])/(22*(2 - 3*x)) + (x*Log[2*x])/(11*(3 + x))

Rule 12

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

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

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 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rule 2177

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[((c + d*x)^(m
+ 1)*(b*F^(g*(e + f*x)))^n)/(d*(m + 1)), x] - Dist[(f*g*n*Log[F])/(d*(m + 1)), Int[(c + d*x)^(m + 1)*(b*F^(g*
(e + f*x)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && LtQ[m, -1] && IntegerQ[2*m] &&  !$UseGamma ===
True

Rule 2178

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - (c*f)/d))*ExpIntegral
Ei[(f*g*(c + d*x)*Log[F])/d])/d, x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !$UseGamma === True

Rule 2314

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

Rule 2357

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*x^
n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, n}, x] && RationalFunctionQ[RFx, x] && IGtQ[p, 0]

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 {-54+70 x+33 x^2+e^{2 x} \left (19 x-8 x^2-6 x^3\right )+e^x \left (26 x-2 x^2-6 x^3\right )+\left (-63 x-54 x^2\right ) \log (2 x)}{3 x \left (6-7 x-3 x^2\right )^2} \, dx\\ &=\frac {1}{3} \int \frac {-54+70 x+33 x^2+e^{2 x} \left (19 x-8 x^2-6 x^3\right )+e^x \left (26 x-2 x^2-6 x^3\right )+\left (-63 x-54 x^2\right ) \log (2 x)}{x \left (6-7 x-3 x^2\right )^2} \, dx\\ &=\frac {1}{3} \int \left (\frac {70}{(3+x)^2 (-2+3 x)^2}-\frac {54}{x (3+x)^2 (-2+3 x)^2}+\frac {33 x}{(3+x)^2 (-2+3 x)^2}-\frac {2 e^x \left (-13+x+3 x^2\right )}{(3+x)^2 (-2+3 x)^2}-\frac {e^{2 x} \left (-19+8 x+6 x^2\right )}{(3+x)^2 (-2+3 x)^2}-\frac {9 (7+6 x) \log (2 x)}{(3+x)^2 (-2+3 x)^2}\right ) \, dx\\ &=-\left (\frac {1}{3} \int \frac {e^{2 x} \left (-19+8 x+6 x^2\right )}{(3+x)^2 (-2+3 x)^2} \, dx\right )-\frac {2}{3} \int \frac {e^x \left (-13+x+3 x^2\right )}{(3+x)^2 (-2+3 x)^2} \, dx-3 \int \frac {(7+6 x) \log (2 x)}{(3+x)^2 (-2+3 x)^2} \, dx+11 \int \frac {x}{(3+x)^2 (-2+3 x)^2} \, dx-18 \int \frac {1}{x (3+x)^2 (-2+3 x)^2} \, dx+\frac {70}{3} \int \frac {1}{(3+x)^2 (-2+3 x)^2} \, dx\\ &=-\left (\frac {1}{3} \int \left (\frac {e^{2 x}}{11 (3+x)^2}-\frac {2 e^{2 x}}{11 (3+x)}-\frac {9 e^{2 x}}{11 (-2+3 x)^2}+\frac {6 e^{2 x}}{11 (-2+3 x)}\right ) \, dx\right )-\frac {2}{3} \int \left (\frac {e^x}{11 (3+x)^2}-\frac {e^x}{11 (3+x)}-\frac {9 e^x}{11 (-2+3 x)^2}+\frac {3 e^x}{11 (-2+3 x)}\right ) \, dx-3 \int \left (-\frac {\log (2 x)}{11 (3+x)^2}+\frac {9 \log (2 x)}{11 (-2+3 x)^2}\right ) \, dx+11 \int \left (-\frac {3}{121 (3+x)^2}-\frac {7}{1331 (3+x)}+\frac {6}{121 (-2+3 x)^2}+\frac {21}{1331 (-2+3 x)}\right ) \, dx-18 \int \left (\frac {1}{36 x}-\frac {1}{363 (3+x)^2}-\frac {29}{11979 (3+x)}+\frac {27}{242 (-2+3 x)^2}-\frac {405}{5324 (-2+3 x)}\right ) \, dx+\frac {70}{3} \int \left (\frac {1}{121 (3+x)^2}+\frac {6}{1331 (3+x)}+\frac {9}{121 (-2+3 x)^2}-\frac {18}{1331 (-2+3 x)}\right ) \, dx\\ &=\frac {1}{11 (2-3 x)}+\frac {1}{33 (3+x)}+\frac {9}{22} \log (2-3 x)-\frac {\log (x)}{2}+\frac {1}{11} \log (3+x)-\frac {1}{33} \int \frac {e^{2 x}}{(3+x)^2} \, dx-\frac {2}{33} \int \frac {e^x}{(3+x)^2} \, dx+\frac {2}{33} \int \frac {e^x}{3+x} \, dx+\frac {2}{33} \int \frac {e^{2 x}}{3+x} \, dx-\frac {2}{11} \int \frac {e^x}{-2+3 x} \, dx-\frac {2}{11} \int \frac {e^{2 x}}{-2+3 x} \, dx+\frac {3}{11} \int \frac {e^{2 x}}{(-2+3 x)^2} \, dx+\frac {3}{11} \int \frac {\log (2 x)}{(3+x)^2} \, dx+\frac {6}{11} \int \frac {e^x}{(-2+3 x)^2} \, dx-\frac {27}{11} \int \frac {\log (2 x)}{(-2+3 x)^2} \, dx\\ &=\frac {1}{11 (2-3 x)}+\frac {2 e^x}{11 (2-3 x)}+\frac {e^{2 x}}{11 (2-3 x)}+\frac {1}{33 (3+x)}+\frac {2 e^x}{33 (3+x)}+\frac {e^{2 x}}{33 (3+x)}-\frac {2}{33} e^{4/3} \text {Ei}\left (-\frac {2}{3} (2-3 x)\right )+\frac {2 \text {Ei}(3+x)}{33 e^3}+\frac {2 \text {Ei}(2 (3+x))}{33 e^6}-\frac {2}{33} e^{2/3} \text {Ei}\left (\frac {1}{3} (-2+3 x)\right )+\frac {9}{22} \log (2-3 x)-\frac {\log (x)}{2}-\frac {27 x \log (2 x)}{22 (2-3 x)}+\frac {x \log (2 x)}{11 (3+x)}+\frac {1}{11} \log (3+x)-\frac {2}{33} \int \frac {e^x}{3+x} \, dx-\frac {2}{33} \int \frac {e^{2 x}}{3+x} \, dx-\frac {1}{11} \int \frac {1}{3+x} \, dx+\frac {2}{11} \int \frac {e^x}{-2+3 x} \, dx+\frac {2}{11} \int \frac {e^{2 x}}{-2+3 x} \, dx-\frac {27}{22} \int \frac {1}{-2+3 x} \, dx\\ &=\frac {1}{11 (2-3 x)}+\frac {2 e^x}{11 (2-3 x)}+\frac {e^{2 x}}{11 (2-3 x)}+\frac {1}{33 (3+x)}+\frac {2 e^x}{33 (3+x)}+\frac {e^{2 x}}{33 (3+x)}-\frac {\log (x)}{2}-\frac {27 x \log (2 x)}{22 (2-3 x)}+\frac {x \log (2 x)}{11 (3+x)}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.42, size = 32, normalized size = 0.97 \begin {gather*} \frac {-\left (1+e^x\right )^2+9 \log (2 x)}{3 (3+x) (-2+3 x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-54 + 70*x + 33*x^2 + E^(2*x)*(19*x - 8*x^2 - 6*x^3) + E^x*(26*x - 2*x^2 - 6*x^3) + (-63*x - 54*x^2
)*Log[2*x])/(108*x - 252*x^2 + 39*x^3 + 126*x^4 + 27*x^5),x]

[Out]

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

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fricas [A]  time = 0.58, size = 30, normalized size = 0.91 \begin {gather*} -\frac {e^{\left (2 \, x\right )} + 2 \, e^{x} - 9 \, \log \left (2 \, x\right ) + 1}{3 \, {\left (3 \, x^{2} + 7 \, x - 6\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-54*x^2-63*x)*log(2*x)+(-6*x^3-8*x^2+19*x)*exp(x)^2+(-6*x^3-2*x^2+26*x)*exp(x)+33*x^2+70*x-54)/(27
*x^5+126*x^4+39*x^3-252*x^2+108*x),x, algorithm="fricas")

[Out]

-1/3*(e^(2*x) + 2*e^x - 9*log(2*x) + 1)/(3*x^2 + 7*x - 6)

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giac [A]  time = 0.29, size = 32, normalized size = 0.97 \begin {gather*} -\frac {e^{\left (2 \, x\right )} + 2 \, e^{x} - 9 \, \log \relax (2) - 9 \, \log \relax (x) + 1}{3 \, {\left (3 \, x^{2} + 7 \, x - 6\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-54*x^2-63*x)*log(2*x)+(-6*x^3-8*x^2+19*x)*exp(x)^2+(-6*x^3-2*x^2+26*x)*exp(x)+33*x^2+70*x-54)/(27
*x^5+126*x^4+39*x^3-252*x^2+108*x),x, algorithm="giac")

[Out]

-1/3*(e^(2*x) + 2*e^x - 9*log(2) - 9*log(x) + 1)/(3*x^2 + 7*x - 6)

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maple [A]  time = 0.12, size = 44, normalized size = 1.33

method	result	size
risch	\(\frac {3 \ln \left (2 x \right )}{3 x^{2}+7 x -6}-\frac {{\mathrm e}^{2 x}+2 \,{\mathrm e}^{x}+1}{3 \left (3 x^{2}+7 x -6\right )}\)	\(44\)
default	\(\frac {1}{99+33 x}+\frac {\ln \left (3+x \right )}{11}-\frac {\ln \relax (x )}{2}-\frac {1}{11 \left (3 x -2\right )}+\frac {9 \ln \left (3 x -2\right )}{22}-\frac {9 \ln \left (6 x -4\right )}{22}+\frac {27 \ln \left (2 x \right ) x}{11 \left (6 x -4\right )}-\frac {\ln \left (2 x +6\right )}{11}+\frac {2 \ln \left (2 x \right ) x}{11 \left (2 x +6\right )}-\frac {2 \,{\mathrm e}^{x}}{33 \left (x -\frac {2}{3}\right )}+\frac {2 \,{\mathrm e}^{x}}{33 \left (3+x \right )}-\frac {{\mathrm e}^{2 x}}{33 \left (x -\frac {2}{3}\right )}+\frac {{\mathrm e}^{2 x}}{99+33 x}\)	\(120\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-54*x^2-63*x)*ln(2*x)+(-6*x^3-8*x^2+19*x)*exp(x)^2+(-6*x^3-2*x^2+26*x)*exp(x)+33*x^2+70*x-54)/(27*x^5+12
6*x^4+39*x^3-252*x^2+108*x),x,method=_RETURNVERBOSE)

[Out]

3/(3*x^2+7*x-6)*ln(2*x)-1/3*(exp(2*x)+2*exp(x)+1)/(3*x^2+7*x-6)

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maxima [B]  time = 0.51, size = 104, normalized size = 3.15 \begin {gather*} \frac {3 \, {\left (3 \, x^{2} + 7 \, x\right )} \log \relax (x) - 2 \, e^{\left (2 \, x\right )} - 4 \, e^{x} + 18 \, \log \relax (2)}{6 \, {\left (3 \, x^{2} + 7 \, x - 6\right )}} + \frac {3 \, {\left (21 \, x + 85\right )}}{121 \, {\left (3 \, x^{2} + 7 \, x - 6\right )}} + \frac {7 \, x - 12}{11 \, {\left (3 \, x^{2} + 7 \, x - 6\right )}} - \frac {70 \, {\left (6 \, x + 7\right )}}{363 \, {\left (3 \, x^{2} + 7 \, x - 6\right )}} - \frac {1}{2} \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-54*x^2-63*x)*log(2*x)+(-6*x^3-8*x^2+19*x)*exp(x)^2+(-6*x^3-2*x^2+26*x)*exp(x)+33*x^2+70*x-54)/(27
*x^5+126*x^4+39*x^3-252*x^2+108*x),x, algorithm="maxima")

[Out]

1/6*(3*(3*x^2 + 7*x)*log(x) - 2*e^(2*x) - 4*e^x + 18*log(2))/(3*x^2 + 7*x - 6) + 3/121*(21*x + 85)/(3*x^2 + 7*
x - 6) + 1/11*(7*x - 12)/(3*x^2 + 7*x - 6) - 70/363*(6*x + 7)/(3*x^2 + 7*x - 6) - 1/2*log(x)

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mupad [B]  time = 4.79, size = 64, normalized size = 1.94 \begin {gather*} \frac {\ln \left (2\,x\right )}{x^2+\frac {7\,x}{3}-2}-\frac {{\mathrm {e}}^{2\,x}}{9\,\left (x^2+\frac {7\,x}{3}-2\right )}-\frac {1}{9\,\left (x^2+\frac {7\,x}{3}-2\right )}-\frac {2\,{\mathrm {e}}^x}{9\,\left (x^2+\frac {7\,x}{3}-2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(2*x)*(63*x + 54*x^2) - 70*x + exp(2*x)*(8*x^2 - 19*x + 6*x^3) - 33*x^2 + exp(x)*(2*x^2 - 26*x + 6*x^
3) + 54)/(108*x - 252*x^2 + 39*x^3 + 126*x^4 + 27*x^5),x)

[Out]

log(2*x)/((7*x)/3 + x^2 - 2) - exp(2*x)/(9*((7*x)/3 + x^2 - 2)) - 1/(9*((7*x)/3 + x^2 - 2)) - (2*exp(x))/(9*((
7*x)/3 + x^2 - 2))

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sympy [B]  time = 0.46, size = 76, normalized size = 2.30 \begin {gather*} \frac {\left (- 18 x^{2} - 42 x + 36\right ) e^{x} + \left (- 9 x^{2} - 21 x + 18\right ) e^{2 x}}{81 x^{4} + 378 x^{3} + 117 x^{2} - 756 x + 324} - \frac {1}{9 x^{2} + 21 x - 18} + \frac {3 \log {\left (2 x \right )}}{3 x^{2} + 7 x - 6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-54*x**2-63*x)*ln(2*x)+(-6*x**3-8*x**2+19*x)*exp(x)**2+(-6*x**3-2*x**2+26*x)*exp(x)+33*x**2+70*x-5
4)/(27*x**5+126*x**4+39*x**3-252*x**2+108*x),x)

[Out]

((-18*x**2 - 42*x + 36)*exp(x) + (-9*x**2 - 21*x + 18)*exp(2*x))/(81*x**4 + 378*x**3 + 117*x**2 - 756*x + 324)
- 1/(9*x**2 + 21*x - 18) + 3*log(2*x)/(3*x**2 + 7*x - 6)

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