3.23.35 \(\int \frac {-2268-486 x+(270 x+162 x^2) \log (x)+(-756 x+162 x^2) \log ^2(x)+(576 x+306 x^2) \log ^3(x)+(252+306 x-54 x^2) \log ^4(x)-54 x \log ^5(x)}{(2744 x+1764 x^2+378 x^3+27 x^4) \log ^3(x)} \, dx\)

Optimal. Leaf size=24 \[ 5+\frac {\left (x-\frac {3}{\log (x)}+\log (x)\right )^2}{\left (-\frac {14}{3}-x\right )^2} \]

________________________________________________________________________________________

Rubi [F]  time = 1.03, 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 {-2268-486 x+\left (270 x+162 x^2\right ) \log (x)+\left (-756 x+162 x^2\right ) \log ^2(x)+\left (576 x+306 x^2\right ) \log ^3(x)+\left (252+306 x-54 x^2\right ) \log ^4(x)-54 x \log ^5(x)}{\left (2744 x+1764 x^2+378 x^3+27 x^4\right ) \log ^3(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-2268 - 486*x + (270*x + 162*x^2)*Log[x] + (-756*x + 162*x^2)*Log[x]^2 + (576*x + 306*x^2)*Log[x]^3 + (25
2 + 306*x - 54*x^2)*Log[x]^4 - 54*x*Log[x]^5)/((2744*x + 1764*x^2 + 378*x^3 + 27*x^4)*Log[x]^3),x]

[Out]

6/(14 + 3*x) + (9*(32 + 17*x)^2)/(142*(14 + 3*x)^2) + (3*Log[x])/7 - (84*Log[x])/(14 + 3*x)^2 - (9*x*Log[x])/(
7*(14 + 3*x)) + (9*Log[x]^2)/(14 + 3*x)^2 - 162*Defer[Int][1/(x*(14 + 3*x)^2*Log[x]^3), x] + 54*Defer[Int][(5
+ 3*x)/((14 + 3*x)^3*Log[x]^2), x] + 54*Defer[Int][(-14 + 3*x)/((14 + 3*x)^3*Log[x]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-2268-486 x+\left (270 x+162 x^2\right ) \log (x)+\left (-756 x+162 x^2\right ) \log ^2(x)+\left (576 x+306 x^2\right ) \log ^3(x)+\left (252+306 x-54 x^2\right ) \log ^4(x)-54 x \log ^5(x)}{x \left (2744+1764 x+378 x^2+27 x^3\right ) \log ^3(x)} \, dx\\ &=\int \left (\frac {18 (32+17 x)}{(14+3 x)^3}-\frac {162}{x (14+3 x)^2 \log ^3(x)}+\frac {54 (5+3 x)}{(14+3 x)^3 \log ^2(x)}+\frac {54 (-14+3 x)}{(14+3 x)^3 \log (x)}-\frac {18 \left (-14-17 x+3 x^2\right ) \log (x)}{x (14+3 x)^3}-\frac {54 \log ^2(x)}{(14+3 x)^3}\right ) \, dx\\ &=18 \int \frac {32+17 x}{(14+3 x)^3} \, dx-18 \int \frac {\left (-14-17 x+3 x^2\right ) \log (x)}{x (14+3 x)^3} \, dx+54 \int \frac {5+3 x}{(14+3 x)^3 \log ^2(x)} \, dx+54 \int \frac {-14+3 x}{(14+3 x)^3 \log (x)} \, dx-54 \int \frac {\log ^2(x)}{(14+3 x)^3} \, dx-162 \int \frac {1}{x (14+3 x)^2 \log ^3(x)} \, dx\\ &=\frac {9 (32+17 x)^2}{142 (14+3 x)^2}+\frac {9 \log ^2(x)}{(14+3 x)^2}-18 \int \frac {\log (x)}{x (14+3 x)^2} \, dx-18 \int \left (-\frac {\log (x)}{196 x}-\frac {28 \log (x)}{(14+3 x)^3}+\frac {17 \log (x)}{14 (14+3 x)^2}+\frac {3 \log (x)}{196 (14+3 x)}\right ) \, dx+54 \int \frac {5+3 x}{(14+3 x)^3 \log ^2(x)} \, dx+54 \int \frac {-14+3 x}{(14+3 x)^3 \log (x)} \, dx-162 \int \frac {1}{x (14+3 x)^2 \log ^3(x)} \, dx\\ &=\frac {9 (32+17 x)^2}{142 (14+3 x)^2}+\frac {9 \log ^2(x)}{(14+3 x)^2}+\frac {9}{98} \int \frac {\log (x)}{x} \, dx-\frac {27}{98} \int \frac {\log (x)}{14+3 x} \, dx-\frac {9}{7} \int \frac {\log (x)}{x (14+3 x)} \, dx+\frac {27}{7} \int \frac {\log (x)}{(14+3 x)^2} \, dx-\frac {153}{7} \int \frac {\log (x)}{(14+3 x)^2} \, dx+54 \int \frac {5+3 x}{(14+3 x)^3 \log ^2(x)} \, dx+54 \int \frac {-14+3 x}{(14+3 x)^3 \log (x)} \, dx-162 \int \frac {1}{x (14+3 x)^2 \log ^3(x)} \, dx+504 \int \frac {\log (x)}{(14+3 x)^3} \, dx\\ &=\frac {9 (32+17 x)^2}{142 (14+3 x)^2}-\frac {84 \log (x)}{(14+3 x)^2}-\frac {9 x \log (x)}{7 (14+3 x)}-\frac {9}{98} \log \left (1+\frac {3 x}{14}\right ) \log (x)+\frac {9 \log ^2(x)}{196}+\frac {9 \log ^2(x)}{(14+3 x)^2}+\frac {9}{98} \int \frac {\log \left (1+\frac {3 x}{14}\right )}{x} \, dx-\frac {9}{98} \int \frac {\log (x)}{x} \, dx-\frac {27}{98} \int \frac {1}{14+3 x} \, dx+\frac {27}{98} \int \frac {\log (x)}{14+3 x} \, dx+\frac {153}{98} \int \frac {1}{14+3 x} \, dx+54 \int \frac {5+3 x}{(14+3 x)^3 \log ^2(x)} \, dx+54 \int \frac {-14+3 x}{(14+3 x)^3 \log (x)} \, dx+84 \int \frac {1}{x (14+3 x)^2} \, dx-162 \int \frac {1}{x (14+3 x)^2 \log ^3(x)} \, dx\\ &=\frac {9 (32+17 x)^2}{142 (14+3 x)^2}-\frac {84 \log (x)}{(14+3 x)^2}-\frac {9 x \log (x)}{7 (14+3 x)}+\frac {9 \log ^2(x)}{(14+3 x)^2}+\frac {3}{7} \log (14+3 x)-\frac {9}{98} \text {Li}_2\left (-\frac {3 x}{14}\right )-\frac {9}{98} \int \frac {\log \left (1+\frac {3 x}{14}\right )}{x} \, dx+54 \int \frac {5+3 x}{(14+3 x)^3 \log ^2(x)} \, dx+54 \int \frac {-14+3 x}{(14+3 x)^3 \log (x)} \, dx+84 \int \left (\frac {1}{196 x}-\frac {3}{14 (14+3 x)^2}-\frac {3}{196 (14+3 x)}\right ) \, dx-162 \int \frac {1}{x (14+3 x)^2 \log ^3(x)} \, dx\\ &=\frac {6}{14+3 x}+\frac {9 (32+17 x)^2}{142 (14+3 x)^2}+\frac {3 \log (x)}{7}-\frac {84 \log (x)}{(14+3 x)^2}-\frac {9 x \log (x)}{7 (14+3 x)}+\frac {9 \log ^2(x)}{(14+3 x)^2}+54 \int \frac {5+3 x}{(14+3 x)^3 \log ^2(x)} \, dx+54 \int \frac {-14+3 x}{(14+3 x)^3 \log (x)} \, dx-162 \int \frac {1}{x (14+3 x)^2 \log ^3(x)} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.31, size = 41, normalized size = 1.71 \begin {gather*} \frac {\left (-9-14 \log (x)+3 \log ^2(x)\right ) \left (-9+2 (7+3 x) \log (x)+3 \log ^2(x)\right )}{(14+3 x)^2 \log ^2(x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-2268 - 486*x + (270*x + 162*x^2)*Log[x] + (-756*x + 162*x^2)*Log[x]^2 + (576*x + 306*x^2)*Log[x]^3
+ (252 + 306*x - 54*x^2)*Log[x]^4 - 54*x*Log[x]^5)/((2744*x + 1764*x^2 + 378*x^3 + 27*x^4)*Log[x]^3),x]

[Out]

((-9 - 14*Log[x] + 3*Log[x]^2)*(-9 + 2*(7 + 3*x)*Log[x] + 3*Log[x]^2))/((14 + 3*x)^2*Log[x]^2)

________________________________________________________________________________________

fricas [B]  time = 0.72, size = 48, normalized size = 2.00 \begin {gather*} \frac {18 \, x \log \relax (x)^{3} + 9 \, \log \relax (x)^{4} - 2 \, {\left (42 \, x + 125\right )} \log \relax (x)^{2} - 54 \, x \log \relax (x) + 81}{{\left (9 \, x^{2} + 84 \, x + 196\right )} \log \relax (x)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-54*x*log(x)^5+(-54*x^2+306*x+252)*log(x)^4+(306*x^2+576*x)*log(x)^3+(162*x^2-756*x)*log(x)^2+(162*
x^2+270*x)*log(x)-486*x-2268)/(27*x^4+378*x^3+1764*x^2+2744*x)/log(x)^3,x, algorithm="fricas")

[Out]

(18*x*log(x)^3 + 9*log(x)^4 - 2*(42*x + 125)*log(x)^2 - 54*x*log(x) + 81)/((9*x^2 + 84*x + 196)*log(x)^2)

________________________________________________________________________________________

giac [B]  time = 0.38, size = 89, normalized size = 3.71 \begin {gather*} \frac {18 \, x \log \relax (x)}{9 \, x^{2} + 84 \, x + 196} + \frac {9 \, \log \relax (x)^{2}}{9 \, x^{2} + 84 \, x + 196} - \frac {27 \, {\left (2 \, x \log \relax (x) - 3\right )}}{9 \, x^{2} \log \relax (x)^{2} + 84 \, x \log \relax (x)^{2} + 196 \, \log \relax (x)^{2}} - \frac {2 \, {\left (42 \, x + 125\right )}}{9 \, x^{2} + 84 \, x + 196} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-54*x*log(x)^5+(-54*x^2+306*x+252)*log(x)^4+(306*x^2+576*x)*log(x)^3+(162*x^2-756*x)*log(x)^2+(162*
x^2+270*x)*log(x)-486*x-2268)/(27*x^4+378*x^3+1764*x^2+2744*x)/log(x)^3,x, algorithm="giac")

[Out]

18*x*log(x)/(9*x^2 + 84*x + 196) + 9*log(x)^2/(9*x^2 + 84*x + 196) - 27*(2*x*log(x) - 3)/(9*x^2*log(x)^2 + 84*
x*log(x)^2 + 196*log(x)^2) - 2*(42*x + 125)/(9*x^2 + 84*x + 196)

________________________________________________________________________________________

maple [B]  time = 0.07, size = 81, normalized size = 3.38

method	result	size
risch	\(\frac {9 \ln \relax (x )^{2}}{9 x^{2}+84 x +196}+\frac {18 x \ln \relax (x )}{9 x^{2}+84 x +196}-\frac {2 \left (42 x +125\right )}{9 x^{2}+84 x +196}-\frac {27 \left (2 x \ln \relax (x )-3\right )}{\left (9 x^{2}+84 x +196\right ) \ln \relax (x )^{2}}\)	\(81\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-54*x*ln(x)^5+(-54*x^2+306*x+252)*ln(x)^4+(306*x^2+576*x)*ln(x)^3+(162*x^2-756*x)*ln(x)^2+(162*x^2+270*x)
*ln(x)-486*x-2268)/(27*x^4+378*x^3+1764*x^2+2744*x)/ln(x)^3,x,method=_RETURNVERBOSE)

[Out]

9/(9*x^2+84*x+196)*ln(x)^2+18*x/(9*x^2+84*x+196)*ln(x)-2*(42*x+125)/(9*x^2+84*x+196)-27*(2*x*ln(x)-3)/(9*x^2+8
4*x+196)/ln(x)^2

________________________________________________________________________________________

maxima [B]  time = 0.60, size = 48, normalized size = 2.00 \begin {gather*} \frac {18 \, x \log \relax (x)^{3} + 9 \, \log \relax (x)^{4} - 2 \, {\left (42 \, x + 125\right )} \log \relax (x)^{2} - 54 \, x \log \relax (x) + 81}{{\left (9 \, x^{2} + 84 \, x + 196\right )} \log \relax (x)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-54*x*log(x)^5+(-54*x^2+306*x+252)*log(x)^4+(306*x^2+576*x)*log(x)^3+(162*x^2-756*x)*log(x)^2+(162*
x^2+270*x)*log(x)-486*x-2268)/(27*x^4+378*x^3+1764*x^2+2744*x)/log(x)^3,x, algorithm="maxima")

[Out]

(18*x*log(x)^3 + 9*log(x)^4 - 2*(42*x + 125)*log(x)^2 - 54*x*log(x) + 81)/((9*x^2 + 84*x + 196)*log(x)^2)

________________________________________________________________________________________

mupad [B]  time = 1.39, size = 49, normalized size = 2.04 \begin {gather*} \frac {9\,\left (125\,x^2\,{\ln \relax (x)}^2+196\,x\,{\ln \relax (x)}^3+252\,x\,{\ln \relax (x)}^2-588\,x\,\ln \relax (x)+98\,{\ln \relax (x)}^4+882\right )}{98\,{\ln \relax (x)}^2\,{\left (3\,x+14\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(486*x - log(x)^4*(306*x - 54*x^2 + 252) - log(x)^3*(576*x + 306*x^2) + log(x)^2*(756*x - 162*x^2) + 54*x
*log(x)^5 - log(x)*(270*x + 162*x^2) + 2268)/(log(x)^3*(2744*x + 1764*x^2 + 378*x^3 + 27*x^4)),x)

[Out]

(9*(252*x*log(x)^2 + 196*x*log(x)^3 + 98*log(x)^4 + 125*x^2*log(x)^2 - 588*x*log(x) + 882))/(98*log(x)^2*(3*x
+ 14)^2)

________________________________________________________________________________________

sympy [B]  time = 0.27, size = 73, normalized size = 3.04 \begin {gather*} \frac {18 x \log {\relax (x )}}{9 x^{2} + 84 x + 196} + \frac {- 84 x - 250}{9 x^{2} + 84 x + 196} + \frac {- 54 x \log {\relax (x )} + 81}{\left (9 x^{2} + 84 x + 196\right ) \log {\relax (x )}^{2}} + \frac {9 \log {\relax (x )}^{2}}{9 x^{2} + 84 x + 196} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-54*x*ln(x)**5+(-54*x**2+306*x+252)*ln(x)**4+(306*x**2+576*x)*ln(x)**3+(162*x**2-756*x)*ln(x)**2+(1
62*x**2+270*x)*ln(x)-486*x-2268)/(27*x**4+378*x**3+1764*x**2+2744*x)/ln(x)**3,x)

[Out]

18*x*log(x)/(9*x**2 + 84*x + 196) + (-84*x - 250)/(9*x**2 + 84*x + 196) + (-54*x*log(x) + 81)/((9*x**2 + 84*x
+ 196)*log(x)**2) + 9*log(x)**2/(9*x**2 + 84*x + 196)

________________________________________________________________________________________