3.26.46 \(\int \frac {e^{2 x/3} (20+256 x-52 x^2)+e^{2 x/3} (540 x+18 x^2+450 x^3) \log (x)+e^{2 x/3} (972 x^3+162 x^4) \log ^2(x)}{-81+27 \log (4)} \, dx\)

Optimal. Leaf size=35 \[ 2-\frac {e^{2 x/3} \left (5-x+9 x^2 \log (x)\right )^2}{9 (3-\log (4))} \]

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Rubi [F]  time = 0.66, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {e^{2 x/3} \left (20+256 x-52 x^2\right )+e^{2 x/3} \left (540 x+18 x^2+450 x^3\right ) \log (x)+e^{2 x/3} \left (972 x^3+162 x^4\right ) \log ^2(x)}{-81+27 \log (4)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((2*x)/3)*(20 + 256*x - 52*x^2) + E^((2*x)/3)*(540*x + 18*x^2 + 450*x^3)*Log[x] + E^((2*x)/3)*(972*x^3
+ 162*x^4)*Log[x]^2)/(-81 + 27*Log[4]),x]

[Out]

(71971*E^((2*x)/3))/(72*(3 - Log[4])) - (10895*E^((2*x)/3)*x)/(36*(3 - Log[4])) + (727*E^((2*x)/3)*x^2)/(18*(3
- Log[4])) - (2187*ExpIntegralEi[(2*x)/3])/(4*(3 - Log[4])) + (2187*E^((2*x)/3)*Log[x])/(4*(3 - Log[4])) - (7
29*E^((2*x)/3)*x*Log[x])/(2*(3 - Log[4])) + (223*E^((2*x)/3)*x^2*Log[x])/(2*(3 - Log[4])) - (25*E^((2*x)/3)*x^
3*Log[x])/(3 - Log[4]) - (36*Defer[Int][E^((2*x)/3)*x^3*Log[x]^2, x])/(3 - Log[4]) - (6*Defer[Int][E^((2*x)/3)
*x^4*Log[x]^2, x])/(3 - Log[4])

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=-\frac {\int \left (e^{2 x/3} \left (20+256 x-52 x^2\right )+e^{2 x/3} \left (540 x+18 x^2+450 x^3\right ) \log (x)+e^{2 x/3} \left (972 x^3+162 x^4\right ) \log ^2(x)\right ) \, dx}{27 (3-\log (4))}\\ &=-\frac {\int e^{2 x/3} \left (20+256 x-52 x^2\right ) \, dx}{27 (3-\log (4))}-\frac {\int e^{2 x/3} \left (540 x+18 x^2+450 x^3\right ) \log (x) \, dx}{27 (3-\log (4))}-\frac {\int e^{2 x/3} \left (972 x^3+162 x^4\right ) \log ^2(x) \, dx}{27 (3-\log (4))}\\ &=-\frac {\int \left (20 e^{2 x/3}+256 e^{2 x/3} x-52 e^{2 x/3} x^2\right ) \, dx}{27 (3-\log (4))}-\frac {\int e^{2 x/3} x \left (540+18 x+450 x^2\right ) \log (x) \, dx}{27 (3-\log (4))}-\frac {\int e^{2 x/3} x^3 (972+162 x) \log ^2(x) \, dx}{27 (3-\log (4))}\\ &=\frac {2187 e^{2 x/3} \log (x)}{4 (3-\log (4))}-\frac {729 e^{2 x/3} x \log (x)}{2 (3-\log (4))}+\frac {223 e^{2 x/3} x^2 \log (x)}{2 (3-\log (4))}-\frac {25 e^{2 x/3} x^3 \log (x)}{3-\log (4)}+\frac {\int \frac {27 e^{2 x/3} \left (-2187+1458 x-446 x^2+100 x^3\right )}{4 x} \, dx}{27 (3-\log (4))}-\frac {\int \left (972 e^{2 x/3} x^3 \log ^2(x)+162 e^{2 x/3} x^4 \log ^2(x)\right ) \, dx}{27 (3-\log (4))}-\frac {20 \int e^{2 x/3} \, dx}{27 (3-\log (4))}+\frac {52 \int e^{2 x/3} x^2 \, dx}{27 (3-\log (4))}-\frac {256 \int e^{2 x/3} x \, dx}{27 (3-\log (4))}\\ &=-\frac {10 e^{2 x/3}}{9 (3-\log (4))}-\frac {128 e^{2 x/3} x}{9 (3-\log (4))}+\frac {26 e^{2 x/3} x^2}{9 (3-\log (4))}+\frac {2187 e^{2 x/3} \log (x)}{4 (3-\log (4))}-\frac {729 e^{2 x/3} x \log (x)}{2 (3-\log (4))}+\frac {223 e^{2 x/3} x^2 \log (x)}{2 (3-\log (4))}-\frac {25 e^{2 x/3} x^3 \log (x)}{3-\log (4)}+\frac {\int \frac {e^{2 x/3} \left (-2187+1458 x-446 x^2+100 x^3\right )}{x} \, dx}{4 (3-\log (4))}-\frac {52 \int e^{2 x/3} x \, dx}{9 (3-\log (4))}-\frac {6 \int e^{2 x/3} x^4 \log ^2(x) \, dx}{3-\log (4)}+\frac {128 \int e^{2 x/3} \, dx}{9 (3-\log (4))}-\frac {36 \int e^{2 x/3} x^3 \log ^2(x) \, dx}{3-\log (4)}\\ &=\frac {182 e^{2 x/3}}{9 (3-\log (4))}-\frac {206 e^{2 x/3} x}{9 (3-\log (4))}+\frac {26 e^{2 x/3} x^2}{9 (3-\log (4))}+\frac {2187 e^{2 x/3} \log (x)}{4 (3-\log (4))}-\frac {729 e^{2 x/3} x \log (x)}{2 (3-\log (4))}+\frac {223 e^{2 x/3} x^2 \log (x)}{2 (3-\log (4))}-\frac {25 e^{2 x/3} x^3 \log (x)}{3-\log (4)}+\frac {\int \left (1458 e^{2 x/3}-\frac {2187 e^{2 x/3}}{x}-446 e^{2 x/3} x+100 e^{2 x/3} x^2\right ) \, dx}{4 (3-\log (4))}-\frac {6 \int e^{2 x/3} x^4 \log ^2(x) \, dx}{3-\log (4)}+\frac {26 \int e^{2 x/3} \, dx}{3 (3-\log (4))}-\frac {36 \int e^{2 x/3} x^3 \log ^2(x) \, dx}{3-\log (4)}\\ &=\frac {299 e^{2 x/3}}{9 (3-\log (4))}-\frac {206 e^{2 x/3} x}{9 (3-\log (4))}+\frac {26 e^{2 x/3} x^2}{9 (3-\log (4))}+\frac {2187 e^{2 x/3} \log (x)}{4 (3-\log (4))}-\frac {729 e^{2 x/3} x \log (x)}{2 (3-\log (4))}+\frac {223 e^{2 x/3} x^2 \log (x)}{2 (3-\log (4))}-\frac {25 e^{2 x/3} x^3 \log (x)}{3-\log (4)}-\frac {6 \int e^{2 x/3} x^4 \log ^2(x) \, dx}{3-\log (4)}+\frac {25 \int e^{2 x/3} x^2 \, dx}{3-\log (4)}-\frac {36 \int e^{2 x/3} x^3 \log ^2(x) \, dx}{3-\log (4)}-\frac {223 \int e^{2 x/3} x \, dx}{2 (3-\log (4))}+\frac {729 \int e^{2 x/3} \, dx}{2 (3-\log (4))}-\frac {2187 \int \frac {e^{2 x/3}}{x} \, dx}{4 (3-\log (4))}\\ &=\frac {20879 e^{2 x/3}}{36 (3-\log (4))}-\frac {6845 e^{2 x/3} x}{36 (3-\log (4))}+\frac {727 e^{2 x/3} x^2}{18 (3-\log (4))}-\frac {2187 \text {Ei}\left (\frac {2 x}{3}\right )}{4 (3-\log (4))}+\frac {2187 e^{2 x/3} \log (x)}{4 (3-\log (4))}-\frac {729 e^{2 x/3} x \log (x)}{2 (3-\log (4))}+\frac {223 e^{2 x/3} x^2 \log (x)}{2 (3-\log (4))}-\frac {25 e^{2 x/3} x^3 \log (x)}{3-\log (4)}-\frac {6 \int e^{2 x/3} x^4 \log ^2(x) \, dx}{3-\log (4)}-\frac {36 \int e^{2 x/3} x^3 \log ^2(x) \, dx}{3-\log (4)}-\frac {75 \int e^{2 x/3} x \, dx}{3-\log (4)}+\frac {669 \int e^{2 x/3} \, dx}{4 (3-\log (4))}\\ &=\frac {59821 e^{2 x/3}}{72 (3-\log (4))}-\frac {10895 e^{2 x/3} x}{36 (3-\log (4))}+\frac {727 e^{2 x/3} x^2}{18 (3-\log (4))}-\frac {2187 \text {Ei}\left (\frac {2 x}{3}\right )}{4 (3-\log (4))}+\frac {2187 e^{2 x/3} \log (x)}{4 (3-\log (4))}-\frac {729 e^{2 x/3} x \log (x)}{2 (3-\log (4))}+\frac {223 e^{2 x/3} x^2 \log (x)}{2 (3-\log (4))}-\frac {25 e^{2 x/3} x^3 \log (x)}{3-\log (4)}-\frac {6 \int e^{2 x/3} x^4 \log ^2(x) \, dx}{3-\log (4)}-\frac {36 \int e^{2 x/3} x^3 \log ^2(x) \, dx}{3-\log (4)}+\frac {225 \int e^{2 x/3} \, dx}{2 (3-\log (4))}\\ &=\frac {71971 e^{2 x/3}}{72 (3-\log (4))}-\frac {10895 e^{2 x/3} x}{36 (3-\log (4))}+\frac {727 e^{2 x/3} x^2}{18 (3-\log (4))}-\frac {2187 \text {Ei}\left (\frac {2 x}{3}\right )}{4 (3-\log (4))}+\frac {2187 e^{2 x/3} \log (x)}{4 (3-\log (4))}-\frac {729 e^{2 x/3} x \log (x)}{2 (3-\log (4))}+\frac {223 e^{2 x/3} x^2 \log (x)}{2 (3-\log (4))}-\frac {25 e^{2 x/3} x^3 \log (x)}{3-\log (4)}-\frac {6 \int e^{2 x/3} x^4 \log ^2(x) \, dx}{3-\log (4)}-\frac {36 \int e^{2 x/3} x^3 \log ^2(x) \, dx}{3-\log (4)}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.18, size = 29, normalized size = 0.83 \begin {gather*} \frac {e^{2 x/3} \left (-5+x-9 x^2 \log (x)\right )^2}{9 (-3+\log (4))} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((2*x)/3)*(20 + 256*x - 52*x^2) + E^((2*x)/3)*(540*x + 18*x^2 + 450*x^3)*Log[x] + E^((2*x)/3)*(97
2*x^3 + 162*x^4)*Log[x]^2)/(-81 + 27*Log[4]),x]

[Out]

(E^((2*x)/3)*(-5 + x - 9*x^2*Log[x])^2)/(9*(-3 + Log[4]))

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fricas [A]  time = 0.92, size = 54, normalized size = 1.54 \begin {gather*} \frac {81 \, x^{4} e^{\left (\frac {2}{3} \, x\right )} \log \relax (x)^{2} - 18 \, {\left (x^{3} - 5 \, x^{2}\right )} e^{\left (\frac {2}{3} \, x\right )} \log \relax (x) + {\left (x^{2} - 10 \, x + 25\right )} e^{\left (\frac {2}{3} \, x\right )}}{9 \, {\left (2 \, \log \relax (2) - 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((162*x^4+972*x^3)*exp(1/3*x)^2*log(x)^2+(450*x^3+18*x^2+540*x)*exp(1/3*x)^2*log(x)+(-52*x^2+256*x+2
0)*exp(1/3*x)^2)/(54*log(2)-81),x, algorithm="fricas")

[Out]

1/9*(81*x^4*e^(2/3*x)*log(x)^2 - 18*(x^3 - 5*x^2)*e^(2/3*x)*log(x) + (x^2 - 10*x + 25)*e^(2/3*x))/(2*log(2) -
3)

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giac [B]  time = 0.29, size = 108, normalized size = 3.09 \begin {gather*} \frac {324 \, x^{4} e^{\left (\frac {2}{3} \, x\right )} \log \relax (x)^{2} + 108 \, x^{2} e^{\left (\frac {2}{3} \, x\right )} + 9 \, {\left (100 \, x^{3} - 446 \, x^{2} + 1458 \, x - 2187\right )} e^{\left (\frac {2}{3} \, x\right )} \log \relax (x) - 243 \, {\left (4 \, x^{3} - 18 \, x^{2} + 54 \, x - 81\right )} e^{\left (\frac {2}{3} \, x\right )} \log \relax (x) - 4 \, {\left (26 \, x^{2} - 206 \, x + 299\right )} e^{\left (\frac {2}{3} \, x\right )} - 864 \, x e^{\left (\frac {2}{3} \, x\right )} + 1296 \, e^{\left (\frac {2}{3} \, x\right )}}{36 \, {\left (2 \, \log \relax (2) - 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((162*x^4+972*x^3)*exp(1/3*x)^2*log(x)^2+(450*x^3+18*x^2+540*x)*exp(1/3*x)^2*log(x)+(-52*x^2+256*x+2
0)*exp(1/3*x)^2)/(54*log(2)-81),x, algorithm="giac")

[Out]

1/36*(324*x^4*e^(2/3*x)*log(x)^2 + 108*x^2*e^(2/3*x) + 9*(100*x^3 - 446*x^2 + 1458*x - 2187)*e^(2/3*x)*log(x)
- 243*(4*x^3 - 18*x^2 + 54*x - 81)*e^(2/3*x)*log(x) - 4*(26*x^2 - 206*x + 299)*e^(2/3*x) - 864*x*e^(2/3*x) + 1
296*e^(2/3*x))/(2*log(2) - 3)

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maple [B]  time = 0.07, size = 67, normalized size = 1.91

method	result	size
risch	\(\frac {243 \ln \relax (x )^{2} {\mathrm e}^{\frac {2 x}{3}} x^{4}}{54 \ln \relax (2)-81}-\frac {54 x^{2} \left (x -5\right ) {\mathrm e}^{\frac {2 x}{3}} \ln \relax (x )}{54 \ln \relax (2)-81}+\frac {3 \left (x^{2}-10 x +25\right ) {\mathrm e}^{\frac {2 x}{3}}}{54 \ln \relax (2)-81}\)	\(67\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((162*x^4+972*x^3)*exp(1/3*x)^2*ln(x)^2+(450*x^3+18*x^2+540*x)*exp(1/3*x)^2*ln(x)+(-52*x^2+256*x+20)*exp(1
/3*x)^2)/(54*ln(2)-81),x,method=_RETURNVERBOSE)

[Out]

243/(54*ln(2)-81)*ln(x)^2*exp(2/3*x)*x^4-54/(54*ln(2)-81)*x^2*(x-5)*exp(2/3*x)*ln(x)+3/(54*ln(2)-81)*(x^2-10*x
+25)*exp(2/3*x)

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maxima [B]  time = 0.86, size = 82, normalized size = 2.34 \begin {gather*} \frac {9 \, {\left (9 \, x^{4} \log \relax (x)^{2} + 3 \, x^{2} - 2 \, {\left (x^{3} - 5 \, x^{2}\right )} \log \relax (x) - 24 \, x + 36\right )} e^{\left (\frac {2}{3} \, x\right )} - 13 \, {\left (2 \, x^{2} - 6 \, x + 9\right )} e^{\left (\frac {2}{3} \, x\right )} + 64 \, {\left (2 \, x - 3\right )} e^{\left (\frac {2}{3} \, x\right )} + 10 \, e^{\left (\frac {2}{3} \, x\right )}}{9 \, {\left (2 \, \log \relax (2) - 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((162*x^4+972*x^3)*exp(1/3*x)^2*log(x)^2+(450*x^3+18*x^2+540*x)*exp(1/3*x)^2*log(x)+(-52*x^2+256*x+2
0)*exp(1/3*x)^2)/(54*log(2)-81),x, algorithm="maxima")

[Out]

1/9*(9*(9*x^4*log(x)^2 + 3*x^2 - 2*(x^3 - 5*x^2)*log(x) - 24*x + 36)*e^(2/3*x) - 13*(2*x^2 - 6*x + 9)*e^(2/3*x
) + 64*(2*x - 3)*e^(2/3*x) + 10*e^(2/3*x))/(2*log(2) - 3)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {{\mathrm {e}}^{\frac {2\,x}{3}}\,\left (162\,x^4+972\,x^3\right )\,{\ln \relax (x)}^2+{\mathrm {e}}^{\frac {2\,x}{3}}\,\left (450\,x^3+18\,x^2+540\,x\right )\,\ln \relax (x)+{\mathrm {e}}^{\frac {2\,x}{3}}\,\left (-52\,x^2+256\,x+20\right )}{54\,\ln \relax (2)-81} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp((2*x)/3)*(256*x - 52*x^2 + 20) + exp((2*x)/3)*log(x)*(540*x + 18*x^2 + 450*x^3) + exp((2*x)/3)*log(x)
^2*(972*x^3 + 162*x^4))/(54*log(2) - 81),x)

[Out]

int((exp((2*x)/3)*(256*x - 52*x^2 + 20) + exp((2*x)/3)*log(x)*(540*x + 18*x^2 + 450*x^3) + exp((2*x)/3)*log(x)
^2*(972*x^3 + 162*x^4))/(54*log(2) - 81), x)

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sympy [A]  time = 0.59, size = 48, normalized size = 1.37 \begin {gather*} \frac {\left (81 x^{4} \log {\relax (x )}^{2} - 18 x^{3} \log {\relax (x )} + 90 x^{2} \log {\relax (x )} + x^{2} - 10 x + 25\right ) e^{\frac {2 x}{3}}}{-27 + 18 \log {\relax (2 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((162*x**4+972*x**3)*exp(1/3*x)**2*ln(x)**2+(450*x**3+18*x**2+540*x)*exp(1/3*x)**2*ln(x)+(-52*x**2+2
56*x+20)*exp(1/3*x)**2)/(54*ln(2)-81),x)

[Out]

(81*x**4*log(x)**2 - 18*x**3*log(x) + 90*x**2*log(x) + x**2 - 10*x + 25)*exp(2*x/3)/(-27 + 18*log(2))

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