∫ −72x+20x2−36x3+18x log(15e 1 ___81 (25−90x+81x2)) ______________________________________________________________________________________________________________________________ −576+432 log(15e 1 ___81 (25−90x+81x2))−108 log2(15e 1_81(25−90x+81x2))+9 log3(15e 1

Optimal. Leaf size=24 \[ \frac {x^2}{\left (4-\log \left (15 e^{\left (\frac {5}{9}-x\right )^2}\right )\right )^2} \]

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Rubi [F]
time = 0.38, 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 {-72 x+20 x^2-36 x^3+18 x \log \left (15 e^{\frac {1}{81} \left (25-90 x+81 x^2\right )}\right )}{-576+432 \log \left (15 e^{\frac {1}{81} \left (25-90 x+81 x^2\right )}\right )-108 \log ^2\left (15 e^{\frac {1}{81} \left (25-90 x+81 x^2\right )}\right )+9 \log ^3\left (15 e^{\frac {1}{81} \left (25-90 x+81 x^2\right )}\right )} \, dx \end {gather*}

Int[(-72*x + 20*x^2 - 36*x^3 + 18*x*Log[15*E^((25 - 90*x + 81*x^2)/81)])/(-576 + 432*Log[15*E^((25 - 90*x + 81

*x^2)/81)] - 108*Log[15*E^((25 - 90*x + 81*x^2)/81)]^2 + 9*Log[15*E^((25 - 90*x + 81*x^2)/81)]^3),x]

(20*Defer[Int][x^2/(-4 + Log[15*E^((5 - 9*x)^2/81)])^3, x])/9 - 4*Defer[Int][x^3/(-4 + Log[15*E^((5 - 9*x)^2/8

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 x \left (36-10 x+18 x^2-9 \log \left (15 e^{\frac {1}{81} (5-9 x)^2}\right )\right )}{9 \left (4-\log \left (15 e^{\frac {1}{81} (5-9 x)^2}\right )\right )^3} \, dx\\ &=\frac {2}{9} \int \frac {x \left (36-10 x+18 x^2-9 \log \left (15 e^{\frac {1}{81} (5-9 x)^2}\right )\right )}{\left (4-\log \left (15 e^{\frac {1}{81} (5-9 x)^2}\right )\right )^3} \, dx\\ &=\frac {2}{9} \int \left (-\frac {2 x^2 (-5+9 x)}{\left (-4+\log \left (15 e^{\frac {1}{81} (5-9 x)^2}\right )\right )^3}+\frac {9 x}{\left (-4+\log \left (15 e^{\frac {1}{81} (5-9 x)^2}\right )\right )^2}\right ) \, dx\\ &=-\left (\frac {4}{9} \int \frac {x^2 (-5+9 x)}{\left (-4+\log \left (15 e^{\frac {1}{81} (5-9 x)^2}\right )\right )^3} \, dx\right )+2 \int \frac {x}{\left (-4+\log \left (15 e^{\frac {1}{81} (5-9 x)^2}\right )\right )^2} \, dx\\ &=-\left (\frac {4}{9} \int \left (-\frac {5 x^2}{\left (-4+\log \left (15 e^{\frac {1}{81} (5-9 x)^2}\right )\right )^3}+\frac {9 x^3}{\left (-4+\log \left (15 e^{\frac {1}{81} (5-9 x)^2}\right )\right )^3}\right ) \, dx\right )+2 \int \frac {x}{\left (-4+\log \left (15 e^{\frac {1}{81} (5-9 x)^2}\right )\right )^2} \, dx\\ &=2 \int \frac {x}{\left (-4+\log \left (15 e^{\frac {1}{81} (5-9 x)^2}\right )\right )^2} \, dx+\frac {20}{9} \int \frac {x^2}{\left (-4+\log \left (15 e^{\frac {1}{81} (5-9 x)^2}\right )\right )^3} \, dx-4 \int \frac {x^3}{\left (-4+\log \left (15 e^{\frac {1}{81} (5-9 x)^2}\right )\right )^3} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.05, size = 24, normalized size = 1.00 \begin {gather*} \frac {x^2}{\left (-4+\log \left (15 e^{\frac {1}{81} (5-9 x)^2}\right )\right )^2} \end {gather*}

Integrate[(-72*x + 20*x^2 - 36*x^3 + 18*x*Log[15*E^((25 - 90*x + 81*x^2)/81)])/(-576 + 432*Log[15*E^((25 - 90*

x + 81*x^2)/81)] - 108*Log[15*E^((25 - 90*x + 81*x^2)/81)]^2 + 9*Log[15*E^((25 - 90*x + 81*x^2)/81)]^3),x]

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method	result	size

default	\(\frac {6561 x^{2}}{\left (81 \ln \left (15 \,{\mathrm e}^{x^{2}-\frac {10}{9} x +\frac {25}{81}}\right )-324\right )^{2}}\)	\(24\)
risch	\(-\frac {4 x^{2}}{\left (2 i \ln \left (5\right )+2 i \ln \left (3\right )+2 i \ln \left ({\mathrm e}^{\frac {\left (9 x -5\right )^{2}}{81}}\right )-8 i\right )^{2}}\)	\(35\)

int((18*x*ln(15*exp(x^2-10/9*x+25/81))-36*x^3+20*x^2-72*x)/(9*ln(15*exp(x^2-10/9*x+25/81))^3-108*ln(15*exp(x^2

-10/9*x+25/81))^2+432*ln(15*exp(x^2-10/9*x+25/81))-576),x,method=_RETURNVERBOSE)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 2123 vs. \(2 (21) = 42\).
time = 0.88, size = 2123, normalized size = 88.46 \begin {gather*} \text {Too large to display} \end {gather*}

integrate((18*x*log(15*exp(x^2-10/9*x+25/81))-36*x^3+20*x^2-72*x)/(9*log(15*exp(x^2-10/9*x+25/81))^3-108*log(1

5*exp(x^2-10/9*x+25/81))^2+432*log(15*exp(x^2-10/9*x+25/81))-576),x, algorithm="maxima")

1/24*(18*(3645*x^3 - 6075*x^2 + 675*x*(9*log(5) + 9*log(3) - 31) - 4374*log(5)^2 - 27*(324*log(5) - 1171)*log(

3) - 4374*log(3)^2 + 31617*log(5) - 57109)/(6561*(log(5)^2 + 2*(log(5) - 4)*log(3) + log(3)^2 - 8*log(5) + 16)

*x^4 - 14580*(log(5)^2 + 2*(log(5) - 4)*log(3) + log(3)^2 - 8*log(5) + 16)*x^3 + 6561*log(5)^4 + 162*(162*log(

5) - 623)*log(3)^3 + 6561*log(3)^4 + 486*(27*log(5)^3 + (81*log(5) - 299)*log(3)^2 + 27*log(3)^3 - 299*log(5)^

2 + (81*log(5)^2 - 598*log(5) + 1096)*log(3) + 1096*log(5) - 1328)*x^2 - 100926*log(5)^3 + (39366*log(5)^2 - 3

02778*log(5) + 581881)*log(3)^2 - 180*(81*log(5)^3 + (243*log(5) - 947)*log(3)^2 + 81*log(3)^3 - 947*log(5)^2

+ (243*log(5)^2 - 1894*log(5) + 3688)*log(3) + 3688*log(5) - 4784)*x + 581881*log(5)^2 + 2*(13122*log(5)^3 - 1

51389*log(5)^2 + 581881*log(5) - 745108)*log(3) - 1490216*log(5) + 1430416) + 5*log((9*x - 9*sqrt(-log(5) - lo

g(3) + 4) - 5)/(9*x + 9*sqrt(-log(5) - log(3) + 4) - 5))/((log(5)^2 + 2*(log(5) - 4)*log(3) + log(3)^2 - 8*log

(5) + 16)*sqrt(-log(5) - log(3) + 4)))*log(15*e^(x^2 - 10/9*x + 25/81)) - 5/972*(81*log(5) + 81*log(3) - 299)*

log((9*x - 9*sqrt(-log(5) - log(3) + 4) - 5)/(9*x + 9*sqrt(-log(5) - log(3) + 4) - 5))/((log(5)^2 + 2*(log(5)

- 4)*log(3) + log(3)^2 - 8*log(5) + 16)*sqrt(-log(5) - log(3) + 4)) + 5/972*(27*log(5) + 27*log(3) - 83)*log((

9*x - 9*sqrt(-log(5) - log(3) + 4) - 5)/(9*x + 9*sqrt(-log(5) - log(3) + 4) - 5))/((log(5)^2 + 2*(log(5) - 4)*

log(3) + log(3)^2 - 8*log(5) + 16)*sqrt(-log(5) - log(3) + 4)) + 5/1944*(27*log(5) + 27*log(3) - 133)*log((9*x

- 9*sqrt(-log(5) - log(3) + 4) - 5)/(9*x + 9*sqrt(-log(5) - log(3) + 4) - 5))/((log(5)^2 + 2*(log(5) - 4)*log

(3) + log(3)^2 - 8*log(5) + 16)*sqrt(-log(5) - log(3) + 4)) - 1/54*(3645*x^3*(81*log(5) + 81*log(3) - 299) - 2

43*(2916*log(5)^2 + 81*(72*log(5) - 263)*log(3) + 2916*log(3)^2 - 21303*log(5) + 39181)*x^2 - 354294*log(5)^3

- 2187*(486*log(5) - 1769)*log(3)^2 - 354294*log(3)^3 + 675*(729*log(5)^2 + 18*(81*log(5) - 289)*log(3) + 729*

log(3)^2 - 5202*log(5) + 9269)*x + 3868803*log(5)^2 - 54*(19683*log(5)^2 - 143289*log(5) + 260728)*log(3) - 14

079312*log(5) + 17075591)/(6561*(log(5)^2 + 2*(log(5) - 4)*log(3) + log(3)^2 - 8*log(5) + 16)*x^4 - 14580*(log

(5)^2 + 2*(log(5) - 4)*log(3) + log(3)^2 - 8*log(5) + 16)*x^3 + 6561*log(5)^4 + 162*(162*log(5) - 623)*log(3)^

3 + 6561*log(3)^4 + 486*(27*log(5)^3 + (81*log(5) - 299)*log(3)^2 + 27*log(3)^3 - 299*log(5)^2 + (81*log(5)^2

- 598*log(5) + 1096)*log(3) + 1096*log(5) - 1328)*x^2 - 100926*log(5)^3 + (39366*log(5)^2 - 302778*log(5) + 58

1881)*log(3)^2 - 180*(81*log(5)^3 + (243*log(5) - 947)*log(3)^2 + 81*log(3)^3 - 947*log(5)^2 + (243*log(5)^2 -

1894*log(5) + 3688)*log(3) + 3688*log(5) - 4784)*x + 581881*log(5)^2 + 2*(13122*log(5)^3 - 151389*log(5)^2 +

581881*log(5) - 745108)*log(3) - 1490216*log(5) + 1430416) + 5/54*(729*x^3*(27*log(5) + 27*log(3) - 83) - 1215

*x^2*(27*log(5) + 27*log(3) - 83) - 27*(729*log(5)^2 + 18*(81*log(5) - 424)*log(3) + 729*log(3)^2 - 7632*log(5

) + 18239)*x - 32805*log(5)^2 - 810*(81*log(5) - 299)*log(3) - 32805*log(3)^2 + 242190*log(5) - 447005)/(6561*

(log(5)^2 + 2*(log(5) - 4)*log(3) + log(3)^2 - 8*log(5) + 16)*x^4 - 14580*(log(5)^2 + 2*(log(5) - 4)*log(3) +

log(3)^2 - 8*log(5) + 16)*x^3 + 6561*log(5)^4 + 162*(162*log(5) - 623)*log(3)^3 + 6561*log(3)^4 + 486*(27*log(

5)^3 + (81*log(5) - 299)*log(3)^2 + 27*log(3)^3 - 299*log(5)^2 + (81*log(5)^2 - 598*log(5) + 1096)*log(3) + 10

96*log(5) - 1328)*x^2 - 100926*log(5)^3 + (39366*log(5)^2 - 302778*log(5) + 581881)*log(3)^2 - 180*(81*log(5)^

3 + (243*log(5) - 947)*log(3)^2 + 81*log(3)^3 - 947*log(5)^2 + (243*log(5)^2 - 1894*log(5) + 3688)*log(3) + 36

88*log(5) - 4784)*x + 581881*log(5)^2 + 2*(13122*log(5)^3 - 151389*log(5)^2 + 581881*log(5) - 745108)*log(3) -

1490216*log(5) + 1430416) - 1/216*(7290*x^3*sqrt(-log(5) - log(3) + 4) - 8100*x^2*sqrt(-log(5) - log(3) + 4)

+ 90*x*(27*log(5) + 27*log(3) - 83)*sqrt(-log(5) - log(3) + 4) - 5*(729*x^4 - 1620*x^3 + 9*x^2*(81*log(5) + 81

*log(3) - 199) - 10*x*(81*log(5) + 81*log(3) - 299))*log(9*x + 9*sqrt(-log(5) - log(3) + 4) - 5) + 5*(729*x^4

- 1620*x^3 + 9*x^2*(81*log(5) + 81*log(3) - 199) - 10*x*(81*log(5) + 81*log(3) - 299))*log(9*x - 9*sqrt(-log(5

) - log(3) + 4) - 5) + 108*(81*log(5)^2 + (162*log(5) - 623)*log(3) + 81*log(3)^2 - 623*log(5) + 1196)*sqrt(-l

og(5) - log(3) + 4))/(81*(log(5)^2 + 2*(log(5) - 4)*log(3) + log(3)^2 - 8*log(5) + 16)*x^2*sqrt(-log(5) - log(

3) + 4) - 90*(log(5)^2 + 2*(log(5) - 4)*log(3) + log(3)^2 - 8*log(5) + 16)*x*sqrt(-log(5) - log(3) + 4) + (81*

log(5)^3 + (243*log(5) - 947)*log(3)^2 + 81*log(3)^3 - 947*log(5)^2 + (243*log(5)^2 - 1894*log(5) + 3688)*log(

3) + 3688*log(5) - 4784)*sqrt(-log(5) - log(3) + 4)) - 3*(3645*x^3 - 6075*x^2 + 675*x*(9*log(5) + 9*log(3) - 3

1) - 4374*log(5)^2 - 27*(324*log(5) - 1171)*log(3) - 4374*log(3)^2 + 31617*log(5) - 57109)/(6561*(log(5)^2 + 2

*(log(5) - 4)*log(3) + log(3)^2 - 8*log(5) + 16)*x^4 - 14580*(log(5)^2 + 2*(log(5) - 4)*log(3) + log(3)^2 - 8*

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 47 vs. \(2 (21) = 42\).
time = 0.35, size = 47, normalized size = 1.96 \begin {gather*} \frac {6561 \, x^{2}}{6561 \, x^{4} - 14580 \, x^{3} - 40338 \, x^{2} + 162 \, {\left (81 \, x^{2} - 90 \, x - 299\right )} \log \left (15\right ) + 6561 \, \log \left (15\right )^{2} + 53820 \, x + 89401} \end {gather*}

integrate((18*x*log(15*exp(x^2-10/9*x+25/81))-36*x^3+20*x^2-72*x)/(9*log(15*exp(x^2-10/9*x+25/81))^3-108*log(1

5*exp(x^2-10/9*x+25/81))^2+432*log(15*exp(x^2-10/9*x+25/81))-576),x, algorithm="fricas")

6561*x^2/(6561*x^4 - 14580*x^3 - 40338*x^2 + 162*(81*x^2 - 90*x - 299)*log(15) + 6561*log(15)^2 + 53820*x + 89

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (17) = 34\).
time = 0.78, size = 46, normalized size = 1.92 \begin {gather*} \frac {6561 x^{2}}{6561 x^{4} - 14580 x^{3} + x^{2} \left (-40338 + 13122 \log {\left (15 \right )}\right ) + x \left (53820 - 14580 \log {\left (15 \right )}\right ) - 48438 \log {\left (15 \right )} + 6561 \log {\left (15 \right )}^{2} + 89401} \end {gather*}

integrate((18*x*ln(15*exp(x**2-10/9*x+25/81))-36*x**3+20*x**2-72*x)/(9*ln(15*exp(x**2-10/9*x+25/81))**3-108*ln

(15*exp(x**2-10/9*x+25/81))**2+432*ln(15*exp(x**2-10/9*x+25/81))-576),x)

6561*x**2/(6561*x**4 - 14580*x**3 + x**2*(-40338 + 13122*log(15)) + x*(53820 - 14580*log(15)) - 48438*log(15)

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Giac [A]
time = 0.39, size = 21, normalized size = 0.88 \begin {gather*} \frac {6561 \, x^{2}}{{\left (81 \, x^{2} - 90 \, x + 81 \, \log \left (15\right ) - 299\right )}^{2}} \end {gather*}

integrate((18*x*log(15*exp(x^2-10/9*x+25/81))-36*x^3+20*x^2-72*x)/(9*log(15*exp(x^2-10/9*x+25/81))^3-108*log(1

5*exp(x^2-10/9*x+25/81))^2+432*log(15*exp(x^2-10/9*x+25/81))-576),x, algorithm="giac")

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Mupad [B]
time = 0.34, size = 21, normalized size = 0.88 \begin {gather*} \frac {6561\,x^2}{{\left (-81\,x^2+90\,x-81\,\ln \left (15\right )+299\right )}^2} \end {gather*}

int(-(72*x - 18*x*log(15*exp(x^2 - (10*x)/9 + 25/81)) - 20*x^2 + 36*x^3)/(432*log(15*exp(x^2 - (10*x)/9 + 25/8

1)) - 108*log(15*exp(x^2 - (10*x)/9 + 25/81))^2 + 9*log(15*exp(x^2 - (10*x)/9 + 25/81))^3 - 576),x)

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3.21.7 \(\int \frac {-72 x+20 x^2-36 x^3+18 x \log (15 e^{\frac {1}{81} (25-90 x+81 x^2)})}{-576+432 \log (15 e^{\frac {1}{81} (25-90 x+81 x^2)})-108 \log ^2(15 e^{\frac {1}{81} (25-90 x+81 x^2)})+9 \log ^3(15 e^{\frac {1}{81} (25-90 x+81 x^2)})} \, dx\) [2007]