3.54.51 \(\int \frac {-202500-202500 x-67500 x^2-7500 x^3+(50625+50625 x+16875 x^2+1875 x^3) \log (3 x)+(40500+35100 x+7200 x^2-900 x^3-300 x^4) \log ^2(3 x)+(-20250-14850 x+1800 x^3+300 x^4) \log ^3(3 x)+(2025+945 x-567 x^2-279 x^3+9 x^4+9 x^5) \log ^5(3 x)}{(27+27 x+9 x^2+x^3) \log ^5(3 x)} \, dx\)

Optimal. Leaf size=23 \[ 3 x \left (-6+x+\frac {3}{3+x}+\frac {25}{\log ^2(3 x)}\right )^2 \]

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Rubi [F]  time = 0.82, 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 {-202500-202500 x-67500 x^2-7500 x^3+\left (50625+50625 x+16875 x^2+1875 x^3\right ) \log (3 x)+\left (40500+35100 x+7200 x^2-900 x^3-300 x^4\right ) \log ^2(3 x)+\left (-20250-14850 x+1800 x^3+300 x^4\right ) \log ^3(3 x)+\left (2025+945 x-567 x^2-279 x^3+9 x^4+9 x^5\right ) \log ^5(3 x)}{\left (27+27 x+9 x^2+x^3\right ) \log ^5(3 x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-202500 - 202500*x - 67500*x^2 - 7500*x^3 + (50625 + 50625*x + 16875*x^2 + 1875*x^3)*Log[3*x] + (40500 +
35100*x + 7200*x^2 - 900*x^3 - 300*x^4)*Log[3*x]^2 + (-20250 - 14850*x + 1800*x^3 + 300*x^4)*Log[3*x]^3 + (202
5 + 945*x - 567*x^2 - 279*x^3 + 9*x^4 + 9*x^5)*Log[3*x]^5)/((27 + 27*x + 9*x^2 + x^3)*Log[3*x]^5),x]

[Out]

126*x - 36*x^2 + 3*x^3 - 81/(3 + x)^2 + 513/(3 + x) + (1875*x)/Log[3*x]^4 - 300*Defer[Int][(-15 - 3*x + x^2)/(
(3 + x)*Log[3*x]^3), x] + 150*Defer[Int][(-45 - 18*x + 6*x^2 + 2*x^3)/((3 + x)^2*Log[3*x]^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {9 \left (225+105 x-63 x^2-31 x^3+x^4+x^5\right )}{(3+x)^3}-\frac {7500}{\log ^5(3 x)}+\frac {1875}{\log ^4(3 x)}-\frac {300 \left (-15-3 x+x^2\right )}{(3+x) \log ^3(3 x)}+\frac {150 \left (-45-18 x+6 x^2+2 x^3\right )}{(3+x)^2 \log ^2(3 x)}\right ) \, dx\\ &=9 \int \frac {225+105 x-63 x^2-31 x^3+x^4+x^5}{(3+x)^3} \, dx+150 \int \frac {-45-18 x+6 x^2+2 x^3}{(3+x)^2 \log ^2(3 x)} \, dx-300 \int \frac {-15-3 x+x^2}{(3+x) \log ^3(3 x)} \, dx+1875 \int \frac {1}{\log ^4(3 x)} \, dx-7500 \int \frac {1}{\log ^5(3 x)} \, dx\\ &=\frac {1875 x}{\log ^4(3 x)}-\frac {625 x}{\log ^3(3 x)}+9 \int \left (14-8 x+x^2+\frac {18}{(3+x)^3}-\frac {57}{(3+x)^2}\right ) \, dx+150 \int \frac {-45-18 x+6 x^2+2 x^3}{(3+x)^2 \log ^2(3 x)} \, dx-300 \int \frac {-15-3 x+x^2}{(3+x) \log ^3(3 x)} \, dx+625 \int \frac {1}{\log ^3(3 x)} \, dx-1875 \int \frac {1}{\log ^4(3 x)} \, dx\\ &=126 x-36 x^2+3 x^3-\frac {81}{(3+x)^2}+\frac {513}{3+x}+\frac {1875 x}{\log ^4(3 x)}-\frac {625 x}{2 \log ^2(3 x)}+150 \int \frac {-45-18 x+6 x^2+2 x^3}{(3+x)^2 \log ^2(3 x)} \, dx-300 \int \frac {-15-3 x+x^2}{(3+x) \log ^3(3 x)} \, dx+\frac {625}{2} \int \frac {1}{\log ^2(3 x)} \, dx-625 \int \frac {1}{\log ^3(3 x)} \, dx\\ &=126 x-36 x^2+3 x^3-\frac {81}{(3+x)^2}+\frac {513}{3+x}+\frac {1875 x}{\log ^4(3 x)}-\frac {625 x}{2 \log (3 x)}+150 \int \frac {-45-18 x+6 x^2+2 x^3}{(3+x)^2 \log ^2(3 x)} \, dx-300 \int \frac {-15-3 x+x^2}{(3+x) \log ^3(3 x)} \, dx-\frac {625}{2} \int \frac {1}{\log ^2(3 x)} \, dx+\frac {625}{2} \int \frac {1}{\log (3 x)} \, dx\\ &=126 x-36 x^2+3 x^3-\frac {81}{(3+x)^2}+\frac {513}{3+x}+\frac {1875 x}{\log ^4(3 x)}+\frac {625 \text {li}(3 x)}{6}+150 \int \frac {-45-18 x+6 x^2+2 x^3}{(3+x)^2 \log ^2(3 x)} \, dx-300 \int \frac {-15-3 x+x^2}{(3+x) \log ^3(3 x)} \, dx-\frac {625}{2} \int \frac {1}{\log (3 x)} \, dx\\ &=126 x-36 x^2+3 x^3-\frac {81}{(3+x)^2}+\frac {513}{3+x}+\frac {1875 x}{\log ^4(3 x)}+150 \int \frac {-45-18 x+6 x^2+2 x^3}{(3+x)^2 \log ^2(3 x)} \, dx-300 \int \frac {-15-3 x+x^2}{(3+x) \log ^3(3 x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 1.45, size = 59, normalized size = 2.57 \begin {gather*} 126 x-36 x^2+3 x^3-\frac {81}{(3+x)^2}+\frac {513}{3+x}+\frac {1875 x}{\log ^4(3 x)}+\frac {150 x \left (-15-3 x+x^2\right )}{(3+x) \log ^2(3 x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-202500 - 202500*x - 67500*x^2 - 7500*x^3 + (50625 + 50625*x + 16875*x^2 + 1875*x^3)*Log[3*x] + (40
500 + 35100*x + 7200*x^2 - 900*x^3 - 300*x^4)*Log[3*x]^2 + (-20250 - 14850*x + 1800*x^3 + 300*x^4)*Log[3*x]^3
+ (2025 + 945*x - 567*x^2 - 279*x^3 + 9*x^4 + 9*x^5)*Log[3*x]^5)/((27 + 27*x + 9*x^2 + x^3)*Log[3*x]^5),x]

[Out]

126*x - 36*x^2 + 3*x^3 - 81/(3 + x)^2 + 513/(3 + x) + (1875*x)/Log[3*x]^4 + (150*x*(-15 - 3*x + x^2))/((3 + x)
*Log[3*x]^2)

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fricas [B]  time = 0.95, size = 82, normalized size = 3.57 \begin {gather*} \frac {3 \, {\left ({\left (x^{5} - 6 \, x^{4} - 21 \, x^{3} + 144 \, x^{2} + 549 \, x + 486\right )} \log \left (3 \, x\right )^{4} + 625 \, x^{3} + 50 \, {\left (x^{4} - 24 \, x^{2} - 45 \, x\right )} \log \left (3 \, x\right )^{2} + 3750 \, x^{2} + 5625 \, x\right )}}{{\left (x^{2} + 6 \, x + 9\right )} \log \left (3 \, x\right )^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((9*x^5+9*x^4-279*x^3-567*x^2+945*x+2025)*log(3*x)^5+(300*x^4+1800*x^3-14850*x-20250)*log(3*x)^3+(-3
00*x^4-900*x^3+7200*x^2+35100*x+40500)*log(3*x)^2+(1875*x^3+16875*x^2+50625*x+50625)*log(3*x)-7500*x^3-67500*x
^2-202500*x-202500)/(x^3+9*x^2+27*x+27)/log(3*x)^5,x, algorithm="fricas")

[Out]

3*((x^5 - 6*x^4 - 21*x^3 + 144*x^2 + 549*x + 486)*log(3*x)^4 + 625*x^3 + 50*(x^4 - 24*x^2 - 45*x)*log(3*x)^2 +
3750*x^2 + 5625*x)/((x^2 + 6*x + 9)*log(3*x)^4)

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giac [B]  time = 0.22, size = 92, normalized size = 4.00 \begin {gather*} 3 \, x^{3} - 36 \, x^{2} + 126 \, x + \frac {75 \, {\left (2 \, x^{3} \log \left (3 \, x\right )^{2} - 6 \, x^{2} \log \left (3 \, x\right )^{2} - 30 \, x \log \left (3 \, x\right )^{2} + 25 \, x^{2} + 75 \, x\right )}}{x \log \left (3 \, x\right )^{4} + 3 \, \log \left (3 \, x\right )^{4}} + \frac {27 \, {\left (19 \, x + 54\right )}}{x^{2} + 6 \, x + 9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((9*x^5+9*x^4-279*x^3-567*x^2+945*x+2025)*log(3*x)^5+(300*x^4+1800*x^3-14850*x-20250)*log(3*x)^3+(-3
00*x^4-900*x^3+7200*x^2+35100*x+40500)*log(3*x)^2+(1875*x^3+16875*x^2+50625*x+50625)*log(3*x)-7500*x^3-67500*x
^2-202500*x-202500)/(x^3+9*x^2+27*x+27)/log(3*x)^5,x, algorithm="giac")

[Out]

3*x^3 - 36*x^2 + 126*x + 75*(2*x^3*log(3*x)^2 - 6*x^2*log(3*x)^2 - 30*x*log(3*x)^2 + 25*x^2 + 75*x)/(x*log(3*x
)^4 + 3*log(3*x)^4) + 27*(19*x + 54)/(x^2 + 6*x + 9)

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maple [B]  time = 0.11, size = 84, normalized size = 3.65

method	result	size
risch	\(\frac {3 x^{5}-18 x^{4}-63 x^{3}+432 x^{2}+1647 x +1458}{x^{2}+6 x +9}+\frac {75 x \left (2 x^{2} \ln \left (3 x \right )^{2}-6 x \ln \left (3 x \right )^{2}-30 \ln \left (3 x \right )^{2}+25 x +75\right )}{\left (3+x \right ) \ln \left (3 x \right )^{4}}\)	\(84\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((9*x^5+9*x^4-279*x^3-567*x^2+945*x+2025)*ln(3*x)^5+(300*x^4+1800*x^3-14850*x-20250)*ln(3*x)^3+(-300*x^4-9
00*x^3+7200*x^2+35100*x+40500)*ln(3*x)^2+(1875*x^3+16875*x^2+50625*x+50625)*ln(3*x)-7500*x^3-67500*x^2-202500*
x-202500)/(x^3+9*x^2+27*x+27)/ln(3*x)^5,x,method=_RETURNVERBOSE)

[Out]

3*(x^5-6*x^4-21*x^3+144*x^2+549*x+486)/(x^2+6*x+9)+75*x*(2*x^2*ln(3*x)^2-6*x*ln(3*x)^2-30*ln(3*x)^2+25*x+75)/(
3+x)/ln(3*x)^4

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maxima [B]  time = 0.50, size = 410, normalized size = 17.83 \begin {gather*} \frac {3 \, {\left (x^{5} \log \relax (3)^{4} - 2 \, {\left (3 \, \log \relax (3)^{4} - 25 \, \log \relax (3)^{2}\right )} x^{4} + {\left (x^{5} - 6 \, x^{4} - 21 \, x^{3} + 144 \, x^{2} + 549 \, x + 486\right )} \log \relax (x)^{4} - {\left (21 \, \log \relax (3)^{4} - 625\right )} x^{3} + 486 \, \log \relax (3)^{4} + 4 \, {\left (x^{5} \log \relax (3) - 6 \, x^{4} \log \relax (3) - 21 \, x^{3} \log \relax (3) + 144 \, x^{2} \log \relax (3) + 549 \, x \log \relax (3) + 486 \, \log \relax (3)\right )} \log \relax (x)^{3} + 6 \, {\left (24 \, \log \relax (3)^{4} - 200 \, \log \relax (3)^{2} + 625\right )} x^{2} + 2 \, {\left (3 \, x^{5} \log \relax (3)^{2} - {\left (18 \, \log \relax (3)^{2} - 25\right )} x^{4} - 63 \, x^{3} \log \relax (3)^{2} + 24 \, {\left (18 \, \log \relax (3)^{2} - 25\right )} x^{2} + 9 \, {\left (183 \, \log \relax (3)^{2} - 125\right )} x + 1458 \, \log \relax (3)^{2}\right )} \log \relax (x)^{2} + 9 \, {\left (61 \, \log \relax (3)^{4} - 250 \, \log \relax (3)^{2} + 625\right )} x + 4 \, {\left (x^{5} \log \relax (3)^{3} - 21 \, x^{3} \log \relax (3)^{3} - {\left (6 \, \log \relax (3)^{3} - 25 \, \log \relax (3)\right )} x^{4} + 24 \, {\left (6 \, \log \relax (3)^{3} - 25 \, \log \relax (3)\right )} x^{2} + 486 \, \log \relax (3)^{3} + 9 \, {\left (61 \, \log \relax (3)^{3} - 125 \, \log \relax (3)\right )} x\right )} \log \relax (x)\right )}}{x^{2} \log \relax (3)^{4} + 6 \, x \log \relax (3)^{4} + {\left (x^{2} + 6 \, x + 9\right )} \log \relax (x)^{4} + 9 \, \log \relax (3)^{4} + 4 \, {\left (x^{2} \log \relax (3) + 6 \, x \log \relax (3) + 9 \, \log \relax (3)\right )} \log \relax (x)^{3} + 6 \, {\left (x^{2} \log \relax (3)^{2} + 6 \, x \log \relax (3)^{2} + 9 \, \log \relax (3)^{2}\right )} \log \relax (x)^{2} + 4 \, {\left (x^{2} \log \relax (3)^{3} + 6 \, x \log \relax (3)^{3} + 9 \, \log \relax (3)^{3}\right )} \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((9*x^5+9*x^4-279*x^3-567*x^2+945*x+2025)*log(3*x)^5+(300*x^4+1800*x^3-14850*x-20250)*log(3*x)^3+(-3
00*x^4-900*x^3+7200*x^2+35100*x+40500)*log(3*x)^2+(1875*x^3+16875*x^2+50625*x+50625)*log(3*x)-7500*x^3-67500*x
^2-202500*x-202500)/(x^3+9*x^2+27*x+27)/log(3*x)^5,x, algorithm="maxima")

[Out]

3*(x^5*log(3)^4 - 2*(3*log(3)^4 - 25*log(3)^2)*x^4 + (x^5 - 6*x^4 - 21*x^3 + 144*x^2 + 549*x + 486)*log(x)^4 -
(21*log(3)^4 - 625)*x^3 + 486*log(3)^4 + 4*(x^5*log(3) - 6*x^4*log(3) - 21*x^3*log(3) + 144*x^2*log(3) + 549*
x*log(3) + 486*log(3))*log(x)^3 + 6*(24*log(3)^4 - 200*log(3)^2 + 625)*x^2 + 2*(3*x^5*log(3)^2 - (18*log(3)^2
- 25)*x^4 - 63*x^3*log(3)^2 + 24*(18*log(3)^2 - 25)*x^2 + 9*(183*log(3)^2 - 125)*x + 1458*log(3)^2)*log(x)^2 +
9*(61*log(3)^4 - 250*log(3)^2 + 625)*x + 4*(x^5*log(3)^3 - 21*x^3*log(3)^3 - (6*log(3)^3 - 25*log(3))*x^4 + 2
4*(6*log(3)^3 - 25*log(3))*x^2 + 486*log(3)^3 + 9*(61*log(3)^3 - 125*log(3))*x)*log(x))/(x^2*log(3)^4 + 6*x*lo
g(3)^4 + (x^2 + 6*x + 9)*log(x)^4 + 9*log(3)^4 + 4*(x^2*log(3) + 6*x*log(3) + 9*log(3))*log(x)^3 + 6*(x^2*log(
3)^2 + 6*x*log(3)^2 + 9*log(3)^2)*log(x)^2 + 4*(x^2*log(3)^3 + 6*x*log(3)^3 + 9*log(3)^3)*log(x))

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mupad [B]  time = 3.79, size = 68, normalized size = 2.96 \begin {gather*} \frac {3\,x\,{\left (25\,x+75\right )}^2-6\,x\,{\ln \left (3\,x\right )}^2\,\left (25\,x+75\right )\,\left (-x^2+3\,x+15\right )}{{\ln \left (3\,x\right )}^4\,{\left (x+3\right )}^2}+\frac {3\,x\,{\left (-x^2+3\,x+15\right )}^2}{{\left (x+3\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(202500*x - log(3*x)^5*(945*x - 567*x^2 - 279*x^3 + 9*x^4 + 9*x^5 + 2025) - log(3*x)*(50625*x + 16875*x^2
+ 1875*x^3 + 50625) + log(3*x)^3*(14850*x - 1800*x^3 - 300*x^4 + 20250) + 67500*x^2 + 7500*x^3 - log(3*x)^2*(
35100*x + 7200*x^2 - 900*x^3 - 300*x^4 + 40500) + 202500)/(log(3*x)^5*(27*x + 9*x^2 + x^3 + 27)),x)

[Out]

(3*x*(25*x + 75)^2 - 6*x*log(3*x)^2*(25*x + 75)*(3*x - x^2 + 15))/(log(3*x)^4*(x + 3)^2) + (3*x*(3*x - x^2 + 1
5)^2)/(x + 3)^2

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sympy [B]  time = 0.44, size = 65, normalized size = 2.83 \begin {gather*} 3 x^{3} - 36 x^{2} + 126 x + \frac {513 x + 1458}{x^{2} + 6 x + 9} + \frac {1875 x^{2} + 5625 x + \left (150 x^{3} - 450 x^{2} - 2250 x\right ) \log {\left (3 x \right )}^{2}}{\left (x + 3\right ) \log {\left (3 x \right )}^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((9*x**5+9*x**4-279*x**3-567*x**2+945*x+2025)*ln(3*x)**5+(300*x**4+1800*x**3-14850*x-20250)*ln(3*x)*
*3+(-300*x**4-900*x**3+7200*x**2+35100*x+40500)*ln(3*x)**2+(1875*x**3+16875*x**2+50625*x+50625)*ln(3*x)-7500*x
**3-67500*x**2-202500*x-202500)/(x**3+9*x**2+27*x+27)/ln(3*x)**5,x)

[Out]

3*x**3 - 36*x**2 + 126*x + (513*x + 1458)/(x**2 + 6*x + 9) + (1875*x**2 + 5625*x + (150*x**3 - 450*x**2 - 2250
*x)*log(3*x)**2)/((x + 3)*log(3*x)**4)

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