3.47.91 \(\int \frac {1}{125} (-21349+20690 x+9450 x^2-13000 x^3+3125 x^4+(34020-5400 x-25500 x^2+10000 x^3) \log (3)+(-14850-12000 x+11250 x^2) \log ^2(3)+(500+5000 x) \log ^3(3)+625 \log ^4(3)) \, dx\)

Optimal. Leaf size=19 \[ 5 \left (-3+(2+x) \left (x+\left (-\frac {9}{5}+x+\log (3)\right )^4\right )\right ) \]

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Rubi [B]  time = 0.04, antiderivative size = 110, normalized size of antiderivative = 5.79, number of steps used = 4, number of rules used = 1, integrand size = 73, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.014, Rules used = {12} \begin {gather*} 5 x^5-26 x^4+20 x^4 \log (3)+\frac {126 x^3}{5}+30 x^3 \log ^2(3)-68 x^3 \log (3)+\frac {2069 x^2}{25}-48 x^2 \log ^2(3)-\frac {108}{5} x^2 \log (3)-\frac {1}{125} x \left (21349-625 \log ^4(3)\right )+\frac {1}{5} (10 x+1)^2 \log ^3(3)-\frac {594}{5} x \log ^2(3)+\frac {6804}{25} x \log (3) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-21349 + 20690*x + 9450*x^2 - 13000*x^3 + 3125*x^4 + (34020 - 5400*x - 25500*x^2 + 10000*x^3)*Log[3] + (-
14850 - 12000*x + 11250*x^2)*Log[3]^2 + (500 + 5000*x)*Log[3]^3 + 625*Log[3]^4)/125,x]

[Out]

(2069*x^2)/25 + (126*x^3)/5 - 26*x^4 + 5*x^5 + (6804*x*Log[3])/25 - (108*x^2*Log[3])/5 - 68*x^3*Log[3] + 20*x^
4*Log[3] - (594*x*Log[3]^2)/5 - 48*x^2*Log[3]^2 + 30*x^3*Log[3]^2 + ((1 + 10*x)^2*Log[3]^3)/5 - (x*(21349 - 62
5*Log[3]^4))/125

Rule 12

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{125} \int \left (-21349+20690 x+9450 x^2-13000 x^3+3125 x^4+\left (34020-5400 x-25500 x^2+10000 x^3\right ) \log (3)+\left (-14850-12000 x+11250 x^2\right ) \log ^2(3)+(500+5000 x) \log ^3(3)+625 \log ^4(3)\right ) \, dx\\ &=\frac {2069 x^2}{25}+\frac {126 x^3}{5}-26 x^4+5 x^5+\frac {1}{5} (1+10 x)^2 \log ^3(3)-\frac {1}{125} x \left (21349-625 \log ^4(3)\right )+\frac {1}{125} \log (3) \int \left (34020-5400 x-25500 x^2+10000 x^3\right ) \, dx+\frac {1}{125} \log ^2(3) \int \left (-14850-12000 x+11250 x^2\right ) \, dx\\ &=\frac {2069 x^2}{25}+\frac {126 x^3}{5}-26 x^4+5 x^5+\frac {6804}{25} x \log (3)-\frac {108}{5} x^2 \log (3)-68 x^3 \log (3)+20 x^4 \log (3)-\frac {594}{5} x \log ^2(3)-48 x^2 \log ^2(3)+30 x^3 \log ^2(3)+\frac {1}{5} (1+10 x)^2 \log ^3(3)-\frac {1}{125} x \left (21349-625 \log ^4(3)\right )\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.03, size = 105, normalized size = 5.53 \begin {gather*} -\frac {21349 x}{125}+\frac {2069 x^2}{25}+5 x^5+\frac {6804}{25} x \log (3)-\frac {108}{5} x^2 \log (3)-\frac {594}{5} x \log ^2(3)-48 x^2 \log ^2(3)+4 x \log ^3(3)+20 x^2 \log ^3(3)+5 x \log ^4(3)+2 x^4 (-13+10 \log (3))+\frac {2}{5} x^3 \left (63-170 \log (3)+75 \log ^2(3)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-21349 + 20690*x + 9450*x^2 - 13000*x^3 + 3125*x^4 + (34020 - 5400*x - 25500*x^2 + 10000*x^3)*Log[3
] + (-14850 - 12000*x + 11250*x^2)*Log[3]^2 + (500 + 5000*x)*Log[3]^3 + 625*Log[3]^4)/125,x]

[Out]

(-21349*x)/125 + (2069*x^2)/25 + 5*x^5 + (6804*x*Log[3])/25 - (108*x^2*Log[3])/5 - (594*x*Log[3]^2)/5 - 48*x^2
*Log[3]^2 + 4*x*Log[3]^3 + 20*x^2*Log[3]^3 + 5*x*Log[3]^4 + 2*x^4*(-13 + 10*Log[3]) + (2*x^3*(63 - 170*Log[3]
+ 75*Log[3]^2))/5

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fricas [B]  time = 1.06, size = 87, normalized size = 4.58 \begin {gather*} 5 \, x^{5} + 5 \, x \log \relax (3)^{4} - 26 \, x^{4} + 4 \, {\left (5 \, x^{2} + x\right )} \log \relax (3)^{3} + \frac {126}{5} \, x^{3} + \frac {6}{5} \, {\left (25 \, x^{3} - 40 \, x^{2} - 99 \, x\right )} \log \relax (3)^{2} + \frac {2069}{25} \, x^{2} + \frac {4}{25} \, {\left (125 \, x^{4} - 425 \, x^{3} - 135 \, x^{2} + 1701 \, x\right )} \log \relax (3) - \frac {21349}{125} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(5*log(3)^4+1/125*(5000*x+500)*log(3)^3+1/125*(11250*x^2-12000*x-14850)*log(3)^2+1/125*(10000*x^3-255
00*x^2-5400*x+34020)*log(3)+25*x^4-104*x^3+378/5*x^2+4138/25*x-21349/125,x, algorithm="fricas")

[Out]

5*x^5 + 5*x*log(3)^4 - 26*x^4 + 4*(5*x^2 + x)*log(3)^3 + 126/5*x^3 + 6/5*(25*x^3 - 40*x^2 - 99*x)*log(3)^2 + 2
069/25*x^2 + 4/25*(125*x^4 - 425*x^3 - 135*x^2 + 1701*x)*log(3) - 21349/125*x

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giac [B]  time = 0.15, size = 87, normalized size = 4.58 \begin {gather*} 5 \, x^{5} + 5 \, x \log \relax (3)^{4} - 26 \, x^{4} + 4 \, {\left (5 \, x^{2} + x\right )} \log \relax (3)^{3} + \frac {126}{5} \, x^{3} + \frac {6}{5} \, {\left (25 \, x^{3} - 40 \, x^{2} - 99 \, x\right )} \log \relax (3)^{2} + \frac {2069}{25} \, x^{2} + \frac {4}{25} \, {\left (125 \, x^{4} - 425 \, x^{3} - 135 \, x^{2} + 1701 \, x\right )} \log \relax (3) - \frac {21349}{125} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(5*log(3)^4+1/125*(5000*x+500)*log(3)^3+1/125*(11250*x^2-12000*x-14850)*log(3)^2+1/125*(10000*x^3-255
00*x^2-5400*x+34020)*log(3)+25*x^4-104*x^3+378/5*x^2+4138/25*x-21349/125,x, algorithm="giac")

[Out]

5*x^5 + 5*x*log(3)^4 - 26*x^4 + 4*(5*x^2 + x)*log(3)^3 + 126/5*x^3 + 6/5*(25*x^3 - 40*x^2 - 99*x)*log(3)^2 + 2
069/25*x^2 + 4/25*(125*x^4 - 425*x^3 - 135*x^2 + 1701*x)*log(3) - 21349/125*x

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maple [B]  time = 0.05, size = 81, normalized size = 4.26

method	result	size
norman	\(\left (20 \ln \relax (3)-26\right ) x^{4}+\left (30 \ln \relax (3)^{2}-68 \ln \relax (3)+\frac {126}{5}\right ) x^{3}+\left (20 \ln \relax (3)^{3}-48 \ln \relax (3)^{2}-\frac {108 \ln \relax (3)}{5}+\frac {2069}{25}\right ) x^{2}+\left (5 \ln \relax (3)^{4}+4 \ln \relax (3)^{3}-\frac {594 \ln \relax (3)^{2}}{5}+\frac {6804 \ln \relax (3)}{25}-\frac {21349}{125}\right ) x +5 x^{5}\)	\(81\)
gosper	\(\frac {x \left (625 \ln \relax (3)^{4}+2500 x \ln \relax (3)^{3}+3750 x^{2} \ln \relax (3)^{2}+2500 x^{3} \ln \relax (3)+625 x^{4}+500 \ln \relax (3)^{3}-6000 x \ln \relax (3)^{2}-8500 x^{2} \ln \relax (3)-3250 x^{3}-14850 \ln \relax (3)^{2}-2700 x \ln \relax (3)+3150 x^{2}+34020 \ln \relax (3)+10345 x -21349\right )}{125}\)	\(88\)
default	\(5 x \ln \relax (3)^{4}+\frac {\ln \relax (3)^{3} \left (2500 x^{2}+500 x \right )}{125}+\frac {\ln \relax (3)^{2} \left (3750 x^{3}-6000 x^{2}-14850 x \right )}{125}+\frac {\ln \relax (3) \left (2500 x^{4}-8500 x^{3}-2700 x^{2}+34020 x \right )}{125}+5 x^{5}-26 x^{4}+\frac {126 x^{3}}{5}+\frac {2069 x^{2}}{25}-\frac {21349 x}{125}\)	\(90\)
risch	\(5 x \ln \relax (3)^{4}+20 \ln \relax (3)^{3} x^{2}+4 x \ln \relax (3)^{3}+30 x^{3} \ln \relax (3)^{2}-48 x^{2} \ln \relax (3)^{2}-\frac {594 x \ln \relax (3)^{2}}{5}+20 x^{4} \ln \relax (3)-68 x^{3} \ln \relax (3)-\frac {108 x^{2} \ln \relax (3)}{5}+\frac {6804 x \ln \relax (3)}{25}+5 x^{5}-26 x^{4}+\frac {126 x^{3}}{5}+\frac {2069 x^{2}}{25}-\frac {21349 x}{125}\)	\(99\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(5*ln(3)^4+1/125*(5000*x+500)*ln(3)^3+1/125*(11250*x^2-12000*x-14850)*ln(3)^2+1/125*(10000*x^3-25500*x^2-54
00*x+34020)*ln(3)+25*x^4-104*x^3+378/5*x^2+4138/25*x-21349/125,x,method=_RETURNVERBOSE)

[Out]

(20*ln(3)-26)*x^4+(30*ln(3)^2-68*ln(3)+126/5)*x^3+(20*ln(3)^3-48*ln(3)^2-108/5*ln(3)+2069/25)*x^2+(5*ln(3)^4+4
*ln(3)^3-594/5*ln(3)^2+6804/25*ln(3)-21349/125)*x+5*x^5

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maxima [B]  time = 0.37, size = 87, normalized size = 4.58 \begin {gather*} 5 \, x^{5} + 5 \, x \log \relax (3)^{4} - 26 \, x^{4} + 4 \, {\left (5 \, x^{2} + x\right )} \log \relax (3)^{3} + \frac {126}{5} \, x^{3} + \frac {6}{5} \, {\left (25 \, x^{3} - 40 \, x^{2} - 99 \, x\right )} \log \relax (3)^{2} + \frac {2069}{25} \, x^{2} + \frac {4}{25} \, {\left (125 \, x^{4} - 425 \, x^{3} - 135 \, x^{2} + 1701 \, x\right )} \log \relax (3) - \frac {21349}{125} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(5*log(3)^4+1/125*(5000*x+500)*log(3)^3+1/125*(11250*x^2-12000*x-14850)*log(3)^2+1/125*(10000*x^3-255
00*x^2-5400*x+34020)*log(3)+25*x^4-104*x^3+378/5*x^2+4138/25*x-21349/125,x, algorithm="maxima")

[Out]

5*x^5 + 5*x*log(3)^4 - 26*x^4 + 4*(5*x^2 + x)*log(3)^3 + 126/5*x^3 + 6/5*(25*x^3 - 40*x^2 - 99*x)*log(3)^2 + 2
069/25*x^2 + 4/25*(125*x^4 - 425*x^3 - 135*x^2 + 1701*x)*log(3) - 21349/125*x

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mupad [B]  time = 3.19, size = 81, normalized size = 4.26 \begin {gather*} 5\,x^5+\left (20\,\ln \relax (3)-26\right )\,x^4+\left (30\,{\ln \relax (3)}^2-68\,\ln \relax (3)+\frac {126}{5}\right )\,x^3+\left (20\,{\ln \relax (3)}^3-48\,{\ln \relax (3)}^2-\frac {108\,\ln \relax (3)}{5}+\frac {2069}{25}\right )\,x^2+\left (\frac {6804\,\ln \relax (3)}{25}-\frac {594\,{\ln \relax (3)}^2}{5}+4\,{\ln \relax (3)}^3+5\,{\ln \relax (3)}^4-\frac {21349}{125}\right )\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4138*x)/25 + (log(3)^3*(5000*x + 500))/125 - (log(3)*(5400*x + 25500*x^2 - 10000*x^3 - 34020))/125 - (log
(3)^2*(12000*x - 11250*x^2 + 14850))/125 + 5*log(3)^4 + (378*x^2)/5 - 104*x^3 + 25*x^4 - 21349/125,x)

[Out]

x^4*(20*log(3) - 26) + x*((6804*log(3))/25 - (594*log(3)^2)/5 + 4*log(3)^3 + 5*log(3)^4 - 21349/125) + x^3*(30
*log(3)^2 - 68*log(3) + 126/5) - x^2*((108*log(3))/5 + 48*log(3)^2 - 20*log(3)^3 - 2069/25) + 5*x^5

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sympy [B]  time = 0.08, size = 94, normalized size = 4.95 \begin {gather*} 5 x^{5} + x^{4} \left (-26 + 20 \log {\relax (3 )}\right ) + x^{3} \left (- 68 \log {\relax (3 )} + \frac {126}{5} + 30 \log {\relax (3 )}^{2}\right ) + x^{2} \left (- 48 \log {\relax (3 )}^{2} - \frac {108 \log {\relax (3 )}}{5} + 20 \log {\relax (3 )}^{3} + \frac {2069}{25}\right ) + x \left (- \frac {21349}{125} - \frac {594 \log {\relax (3 )}^{2}}{5} + 4 \log {\relax (3 )}^{3} + 5 \log {\relax (3 )}^{4} + \frac {6804 \log {\relax (3 )}}{25}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(5*ln(3)**4+1/125*(5000*x+500)*ln(3)**3+1/125*(11250*x**2-12000*x-14850)*ln(3)**2+1/125*(10000*x**3-2
5500*x**2-5400*x+34020)*ln(3)+25*x**4-104*x**3+378/5*x**2+4138/25*x-21349/125,x)

[Out]

5*x**5 + x**4*(-26 + 20*log(3)) + x**3*(-68*log(3) + 126/5 + 30*log(3)**2) + x**2*(-48*log(3)**2 - 108*log(3)/
5 + 20*log(3)**3 + 2069/25) + x*(-21349/125 - 594*log(3)**2/5 + 4*log(3)**3 + 5*log(3)**4 + 6804*log(3)/25)

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