3.12.42 \(\int \frac {50625+54000 x+20990 x^2+3560 x^3+225 x^4+e^4 (81+54 x+12 x^2)+e^2 (-4050-3510 x-1106 x^2-120 x^3)+(-16875-18000 x-7050 x^2-1200 x^3-75 x^4+e^4 (-27-18 x-3 x^2)+e^2 (1350+1170 x+330 x^2+30 x^3)) \log (x)}{16875 x^2+18000 x^3+7050 x^4+1200 x^5+75 x^6+e^4 (27 x^2+18 x^3+3 x^4)+e^2 (-1350 x^2-1170 x^3-330 x^4-30 x^5)} \, dx\)
Optimal. Leaf size=35 \[ \frac {-2+\frac {\left (\frac {4}{3}+e^2\right ) x}{(3+x) \left (-e^2+5 (5+x)\right )}+\log (x)}{x} \]
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Rubi [B] time = 1.95, antiderivative size = 517, normalized size of antiderivative = 14.77,
number of steps used = 11, number of rules used = 5, integrand size = 183, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.027, Rules used
= {6688, 12, 6742, 1612, 2304} \begin {gather*} -\frac {3 e^4}{\left (25-e^2\right )^2 x}+\frac {150 e^2}{\left (25-e^2\right )^2 x}-\frac {1875}{\left (25-e^2\right )^2 x}+\frac {1}{x}-\frac {e^4}{\left (10-e^2\right )^2 (x+3)}+\frac {26 e^2}{3 \left (10-e^2\right )^2 (x+3)}+\frac {40}{3 \left (10-e^2\right )^2 (x+3)}-\frac {5 \left (25000-7000 e^2+1340 e^4-188 e^6+9 e^8\right )}{3 \left (250-35 e^2+e^4\right )^2 \left (5 x-e^2+25\right )}-\frac {10 e^2 \left (10625-3625 e^2+347 e^4-12 e^6\right )}{3 \left (250-35 e^2+e^4\right )^2 \left (5 x-e^2+25\right )}-\frac {5 e^4 \left (925-110 e^2+4 e^4\right )}{\left (250-35 e^2+e^4\right )^2 \left (5 x-e^2+25\right )}+\frac {\log (x)}{x}+\frac {30 e^2 \left (25+e^2\right ) \log (x)}{\left (25-e^2\right )^3}-\frac {30 e^4 \log (x)}{\left (25-e^2\right )^3}-\frac {750 e^2 \log (x)}{\left (25-e^2\right )^3}-\frac {10 e^2 \left (4-3 e^2\right ) \log (x+3)}{3 \left (10-e^2\right )^3}-\frac {10 e^4 \log (x+3)}{\left (10-e^2\right )^3}+\frac {40 e^2 \log (x+3)}{3 \left (10-e^2\right )^3}-\frac {10 e^2 \left (162500-4125 e^2-1875 e^4+274 e^6-12 e^8\right ) \log \left (5 x-e^2+25\right )}{3 \left (250-35 e^2+e^4\right )^3}+\frac {10 e^4 \left (18625-2775 e^2+165 e^4-4 e^6\right ) \log \left (5 x-e^2+25\right )}{\left (250-35 e^2+e^4\right )^3}+\frac {10 e^2 \left (162500-60000 e^2+6450 e^4-221 e^6\right ) \log \left (5 x-e^2+25\right )}{3 \left (250-35 e^2+e^4\right )^3} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(50625 + 54000*x + 20990*x^2 + 3560*x^3 + 225*x^4 + E^4*(81 + 54*x + 12*x^2) + E^2*(-4050 - 3510*x - 1106*
x^2 - 120*x^3) + (-16875 - 18000*x - 7050*x^2 - 1200*x^3 - 75*x^4 + E^4*(-27 - 18*x - 3*x^2) + E^2*(1350 + 117
0*x + 330*x^2 + 30*x^3))*Log[x])/(16875*x^2 + 18000*x^3 + 7050*x^4 + 1200*x^5 + 75*x^6 + E^4*(27*x^2 + 18*x^3
+ 3*x^4) + E^2*(-1350*x^2 - 1170*x^3 - 330*x^4 - 30*x^5)),x]
[Out]
x^(-1) - 1875/((25 - E^2)^2*x) + (150*E^2)/((25 - E^2)^2*x) - (3*E^4)/((25 - E^2)^2*x) + 40/(3*(10 - E^2)^2*(3
+ x)) + (26*E^2)/(3*(10 - E^2)^2*(3 + x)) - E^4/((10 - E^2)^2*(3 + x)) - (5*E^4*(925 - 110*E^2 + 4*E^4))/((25
0 - 35*E^2 + E^4)^2*(25 - E^2 + 5*x)) - (10*E^2*(10625 - 3625*E^2 + 347*E^4 - 12*E^6))/(3*(250 - 35*E^2 + E^4)
^2*(25 - E^2 + 5*x)) - (5*(25000 - 7000*E^2 + 1340*E^4 - 188*E^6 + 9*E^8))/(3*(250 - 35*E^2 + E^4)^2*(25 - E^2
+ 5*x)) - (750*E^2*Log[x])/(25 - E^2)^3 - (30*E^4*Log[x])/(25 - E^2)^3 + (30*E^2*(25 + E^2)*Log[x])/(25 - E^2
)^3 + Log[x]/x + (40*E^2*Log[3 + x])/(3*(10 - E^2)^3) - (10*E^4*Log[3 + x])/(10 - E^2)^3 - (10*E^2*(4 - 3*E^2)
*Log[3 + x])/(3*(10 - E^2)^3) + (10*E^2*(162500 - 60000*E^2 + 6450*E^4 - 221*E^6)*Log[25 - E^2 + 5*x])/(3*(250
- 35*E^2 + E^4)^3) + (10*E^4*(18625 - 2775*E^2 + 165*E^4 - 4*E^6)*Log[25 - E^2 + 5*x])/(250 - 35*E^2 + E^4)^3
- (10*E^2*(162500 - 4125*E^2 - 1875*E^4 + 274*E^6 - 12*E^8)*Log[25 - E^2 + 5*x])/(3*(250 - 35*E^2 + E^4)^3)
Rule 12
Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]
Rule 1612
Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[E
xpandIntegrand[Px*(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && Poly
Q[Px, x] && IntegersQ[m, n]
Rule 2304
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Log[c*x^
n]))/(d*(m + 1)), x] - Simp[(b*n*(d*x)^(m + 1))/(d*(m + 1)^2), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]
Rule 6688
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]
Rule 6742
Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {3 e^4 \left (27+18 x+4 x^2\right )-2 e^2 \left (2025+1755 x+553 x^2+60 x^3\right )+5 \left (10125+10800 x+4198 x^2+712 x^3+45 x^4\right )-3 (3+x)^2 \left (e^2-5 (5+x)\right )^2 \log (x)}{3 x^2 (3+x)^2 \left (25-e^2+5 x\right )^2} \, dx\\ &=\frac {1}{3} \int \frac {3 e^4 \left (27+18 x+4 x^2\right )-2 e^2 \left (2025+1755 x+553 x^2+60 x^3\right )+5 \left (10125+10800 x+4198 x^2+712 x^3+45 x^4\right )-3 (3+x)^2 \left (e^2-5 (5+x)\right )^2 \log (x)}{x^2 (3+x)^2 \left (25-e^2+5 x\right )^2} \, dx\\ &=\frac {1}{3} \int \left (\frac {3 e^4 \left (27+18 x+4 x^2\right )}{\left (-25+e^2-5 x\right )^2 x^2 (3+x)^2}-\frac {2 e^2 \left (2025+1755 x+553 x^2+60 x^3\right )}{\left (-25+e^2-5 x\right )^2 x^2 (3+x)^2}+\frac {5 \left (10125+10800 x+4198 x^2+712 x^3+45 x^4\right )}{x^2 (3+x)^2 \left (25-e^2+5 x\right )^2}-\frac {3 \log (x)}{x^2}\right ) \, dx\\ &=\frac {5}{3} \int \frac {10125+10800 x+4198 x^2+712 x^3+45 x^4}{x^2 (3+x)^2 \left (25-e^2+5 x\right )^2} \, dx-\frac {1}{3} \left (2 e^2\right ) \int \frac {2025+1755 x+553 x^2+60 x^3}{\left (-25+e^2-5 x\right )^2 x^2 (3+x)^2} \, dx+e^4 \int \frac {27+18 x+4 x^2}{\left (-25+e^2-5 x\right )^2 x^2 (3+x)^2} \, dx-\int \frac {\log (x)}{x^2} \, dx\\ &=\frac {1}{x}+\frac {\log (x)}{x}+\frac {5}{3} \int \left (\frac {5 \left (25000-7000 e^2+1340 e^4-188 e^6+9 e^8\right )}{\left (-25+e^2\right )^2 \left (-10+e^2\right )^2 \left (-25+e^2-5 x\right )^2}+\frac {10 e^2 \left (-162500+60000 e^2-6450 e^4+221 e^6\right )}{\left (250-35 e^2+e^4\right )^3 \left (-25+e^2-5 x\right )}+\frac {1125}{\left (-25+e^2\right )^2 x^2}+\frac {450 e^2}{\left (-25+e^2\right )^3 x}-\frac {8}{\left (-10+e^2\right )^2 (3+x)^2}-\frac {8 e^2}{\left (-10+e^2\right )^3 (3+x)}\right ) \, dx-\frac {1}{3} \left (2 e^2\right ) \int \left (\frac {25 \left (-10625+3625 e^2-347 e^4+12 e^6\right )}{\left (-25+e^2\right )^2 \left (-10+e^2\right )^2 \left (-25+e^2-5 x\right )^2}+\frac {25 \left (-162500+4125 e^2+1875 e^4-274 e^6+12 e^8\right )}{\left (250-35 e^2+e^4\right )^3 \left (-25+e^2-5 x\right )}+\frac {225}{\left (-25+e^2\right )^2 x^2}+\frac {45 \left (25+e^2\right )}{\left (-25+e^2\right )^3 x}+\frac {13}{\left (-10+e^2\right )^2 (3+x)^2}+\frac {5 \left (-4+3 e^2\right )}{\left (-10+e^2\right )^3 (3+x)}\right ) \, dx+e^4 \int \left (\frac {25 \left (925-110 e^2+4 e^4\right )}{\left (-25+e^2\right )^2 \left (-10+e^2\right )^2 \left (-25+e^2-5 x\right )^2}+\frac {50 \left (-18625+2775 e^2-165 e^4+4 e^6\right )}{\left (250-35 e^2+e^4\right )^3 \left (-25+e^2-5 x\right )}+\frac {3}{\left (-25+e^2\right )^2 x^2}+\frac {30}{\left (-25+e^2\right )^3 x}+\frac {1}{\left (-10+e^2\right )^2 (3+x)^2}+\frac {10}{\left (-10+e^2\right )^3 (3+x)}\right ) \, dx\\ &=\frac {1}{x}-\frac {1875}{\left (25-e^2\right )^2 x}+\frac {150 e^2}{\left (25-e^2\right )^2 x}-\frac {3 e^4}{\left (25-e^2\right )^2 x}+\frac {40}{3 \left (10-e^2\right )^2 (3+x)}+\frac {26 e^2}{3 \left (10-e^2\right )^2 (3+x)}-\frac {e^4}{\left (10-e^2\right )^2 (3+x)}-\frac {5 e^4 \left (925-110 e^2+4 e^4\right )}{\left (250-35 e^2+e^4\right )^2 \left (25-e^2+5 x\right )}-\frac {10 e^2 \left (10625-3625 e^2+347 e^4-12 e^6\right )}{3 \left (250-35 e^2+e^4\right )^2 \left (25-e^2+5 x\right )}-\frac {5 \left (25000-7000 e^2+1340 e^4-188 e^6+9 e^8\right )}{3 \left (250-35 e^2+e^4\right )^2 \left (25-e^2+5 x\right )}-\frac {750 e^2 \log (x)}{\left (25-e^2\right )^3}-\frac {30 e^4 \log (x)}{\left (25-e^2\right )^3}+\frac {30 e^2 \left (25+e^2\right ) \log (x)}{\left (25-e^2\right )^3}+\frac {\log (x)}{x}+\frac {40 e^2 \log (3+x)}{3 \left (10-e^2\right )^3}-\frac {10 e^4 \log (3+x)}{\left (10-e^2\right )^3}-\frac {10 e^2 \left (4-3 e^2\right ) \log (3+x)}{3 \left (10-e^2\right )^3}+\frac {10 e^2 \left (162500-60000 e^2+6450 e^4-221 e^6\right ) \log \left (25-e^2+5 x\right )}{3 \left (250-35 e^2+e^4\right )^3}+\frac {10 e^4 \left (18625-2775 e^2+165 e^4-4 e^6\right ) \log \left (25-e^2+5 x\right )}{\left (250-35 e^2+e^4\right )^3}-\frac {10 e^2 \left (162500-4125 e^2-1875 e^4+274 e^6-12 e^8\right ) \log \left (25-e^2+5 x\right )}{3 \left (250-35 e^2+e^4\right )^3}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.13, size = 65, normalized size = 1.86 \begin {gather*} \frac {1}{3} \left (-\frac {6}{x}+\frac {-4-3 e^2}{\left (-10+e^2\right ) (3+x)}+\frac {5 \left (4+3 e^2\right )}{\left (-10+e^2\right ) \left (25-e^2+5 x\right )}+\frac {3 \log (x)}{x}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(50625 + 54000*x + 20990*x^2 + 3560*x^3 + 225*x^4 + E^4*(81 + 54*x + 12*x^2) + E^2*(-4050 - 3510*x -
1106*x^2 - 120*x^3) + (-16875 - 18000*x - 7050*x^2 - 1200*x^3 - 75*x^4 + E^4*(-27 - 18*x - 3*x^2) + E^2*(1350
+ 1170*x + 330*x^2 + 30*x^3))*Log[x])/(16875*x^2 + 18000*x^3 + 7050*x^4 + 1200*x^5 + 75*x^6 + E^4*(27*x^2 + 1
8*x^3 + 3*x^4) + E^2*(-1350*x^2 - 1170*x^3 - 330*x^4 - 30*x^5)),x]
[Out]
(-6/x + (-4 - 3*E^2)/((-10 + E^2)*(3 + x)) + (5*(4 + 3*E^2))/((-10 + E^2)*(25 - E^2 + 5*x)) + (3*Log[x])/x)/3
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fricas [A] time = 0.93, size = 67, normalized size = 1.91 \begin {gather*} -\frac {30 \, x^{2} - 9 \, {\left (x + 2\right )} e^{2} - 3 \, {\left (5 \, x^{2} - {\left (x + 3\right )} e^{2} + 40 \, x + 75\right )} \log \relax (x) + 236 \, x + 450}{3 \, {\left (5 \, x^{3} + 40 \, x^{2} - {\left (x^{2} + 3 \, x\right )} e^{2} + 75 \, x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((((-3*x^2-18*x-27)*exp(2)^2+(30*x^3+330*x^2+1170*x+1350)*exp(2)-75*x^4-1200*x^3-7050*x^2-18000*x-168
75)*log(x)+(12*x^2+54*x+81)*exp(2)^2+(-120*x^3-1106*x^2-3510*x-4050)*exp(2)+225*x^4+3560*x^3+20990*x^2+54000*x
+50625)/((3*x^4+18*x^3+27*x^2)*exp(2)^2+(-30*x^5-330*x^4-1170*x^3-1350*x^2)*exp(2)+75*x^6+1200*x^5+7050*x^4+18
000*x^3+16875*x^2),x, algorithm="fricas")
[Out]
-1/3*(30*x^2 - 9*(x + 2)*e^2 - 3*(5*x^2 - (x + 3)*e^2 + 40*x + 75)*log(x) + 236*x + 450)/(5*x^3 + 40*x^2 - (x^
2 + 3*x)*e^2 + 75*x)
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giac [B] time = 0.60, size = 78, normalized size = 2.23 \begin {gather*} \frac {15 \, x^{2} \log \relax (x) - 3 \, x e^{2} \log \relax (x) - 30 \, x^{2} + 9 \, x e^{2} + 120 \, x \log \relax (x) - 9 \, e^{2} \log \relax (x) - 236 \, x + 18 \, e^{2} + 225 \, \log \relax (x) - 450}{3 \, {\left (5 \, x^{3} - x^{2} e^{2} + 40 \, x^{2} - 3 \, x e^{2} + 75 \, x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((((-3*x^2-18*x-27)*exp(2)^2+(30*x^3+330*x^2+1170*x+1350)*exp(2)-75*x^4-1200*x^3-7050*x^2-18000*x-168
75)*log(x)+(12*x^2+54*x+81)*exp(2)^2+(-120*x^3-1106*x^2-3510*x-4050)*exp(2)+225*x^4+3560*x^3+20990*x^2+54000*x
+50625)/((3*x^4+18*x^3+27*x^2)*exp(2)^2+(-30*x^5-330*x^4-1170*x^3-1350*x^2)*exp(2)+75*x^6+1200*x^5+7050*x^4+18
000*x^3+16875*x^2),x, algorithm="giac")
[Out]
1/3*(15*x^2*log(x) - 3*x*e^2*log(x) - 30*x^2 + 9*x*e^2 + 120*x*log(x) - 9*e^2*log(x) - 236*x + 18*e^2 + 225*lo
g(x) - 450)/(5*x^3 - x^2*e^2 + 40*x^2 - 3*x*e^2 + 75*x)
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maple [F(-1)] time = 180.00, size = 0, normalized size = 0.00 hanged
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((((-3*x^2-18*x-27)*exp(2)^2+(30*x^3+330*x^2+1170*x+1350)*exp(2)-75*x^4-1200*x^3-7050*x^2-18000*x-16875)*ln
(x)+(12*x^2+54*x+81)*exp(2)^2+(-120*x^3-1106*x^2-3510*x-4050)*exp(2)+225*x^4+3560*x^3+20990*x^2+54000*x+50625)
/((3*x^4+18*x^3+27*x^2)*exp(2)^2+(-30*x^5-330*x^4-1170*x^3-1350*x^2)*exp(2)+75*x^6+1200*x^5+7050*x^4+18000*x^3
+16875*x^2),x)
[Out]
int((((-3*x^2-18*x-27)*exp(2)^2+(30*x^3+330*x^2+1170*x+1350)*exp(2)-75*x^4-1200*x^3-7050*x^2-18000*x-16875)*ln
(x)+(12*x^2+54*x+81)*exp(2)^2+(-120*x^3-1106*x^2-3510*x-4050)*exp(2)+225*x^4+3560*x^3+20990*x^2+54000*x+50625)
/((3*x^4+18*x^3+27*x^2)*exp(2)^2+(-30*x^5-330*x^4-1170*x^3-1350*x^2)*exp(2)+75*x^6+1200*x^5+7050*x^4+18000*x^3
+16875*x^2),x)
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maxima [B] time = 0.83, size = 1762, normalized size = 50.34 result too large to
display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((((-3*x^2-18*x-27)*exp(2)^2+(30*x^3+330*x^2+1170*x+1350)*exp(2)-75*x^4-1200*x^3-7050*x^2-18000*x-168
75)*log(x)+(12*x^2+54*x+81)*exp(2)^2+(-120*x^3-1106*x^2-3510*x-4050)*exp(2)+225*x^4+3560*x^3+20990*x^2+54000*x
+50625)/((3*x^4+18*x^3+27*x^2)*exp(2)^2+(-30*x^5-330*x^4-1170*x^3-1350*x^2)*exp(2)+75*x^6+1200*x^5+7050*x^4+18
000*x^3+16875*x^2),x, algorithm="maxima")
[Out]
-(6750*(2*e^2 - 35)*log(5*x - e^2 + 25)/(e^12 - 105*e^10 + 4425*e^8 - 95375*e^6 + 1106250*e^4 - 6562500*e^2 +
15625000) - 2*(e^2 + 5)*log(x + 3)/(e^6 - 30*e^4 + 300*e^2 - 1000) + 2*(e^2 - 40)*log(x)/(e^6 - 75*e^4 + 1875*
e^2 - 15625) + 3*(10*x^2*(e^4 - 35*e^2 + 475) - x*(2*e^6 - 135*e^4 + 2775*e^2 - 23000) - 3*e^6 + 135*e^4 - 180
0*e^2 + 7500)/(5*x^3*(e^8 - 70*e^6 + 1725*e^4 - 17500*e^2 + 62500) - x^2*(e^10 - 110*e^8 + 4525*e^6 - 86500*e^
4 + 762500*e^2 - 2500000) - 3*x*(e^10 - 95*e^8 + 3475*e^6 - 60625*e^4 + 500000*e^2 - 1562500)))*e^4 - 2*(675*(
e^2 - 20)*log(5*x - e^2 + 25)/(e^10 - 80*e^8 + 2425*e^6 - 34750*e^4 + 237500*e^2 - 625000) + (e^2 + 20)*log(x
+ 3)/(e^6 - 30*e^4 + 300*e^2 - 1000) - 3*(5*x*(e^2 - 40) - e^4 + 50*e^2 - 850)/(5*x^2*(e^6 - 45*e^4 + 600*e^2
- 2500) - x*(e^8 - 85*e^6 + 2400*e^4 - 26500*e^2 + 100000) - 3*e^8 + 210*e^6 - 5175*e^4 + 52500*e^2 - 187500)
- log(x)/(e^4 - 50*e^2 + 625))*e^4 - 4*((10*x - e^2 + 40)/(5*x^2*(e^4 - 20*e^2 + 100) - x*(e^6 - 60*e^4 + 900*
e^2 - 4000) - 3*e^6 + 135*e^4 - 1800*e^2 + 7500) + 10*log(5*x - e^2 + 25)/(e^6 - 30*e^4 + 300*e^2 - 1000) - 10
*log(x + 3)/(e^6 - 30*e^4 + 300*e^2 - 1000))*e^4 + 50*(6750*(2*e^2 - 35)*log(5*x - e^2 + 25)/(e^12 - 105*e^10
+ 4425*e^8 - 95375*e^6 + 1106250*e^4 - 6562500*e^2 + 15625000) - 2*(e^2 + 5)*log(x + 3)/(e^6 - 30*e^4 + 300*e^
2 - 1000) + 2*(e^2 - 40)*log(x)/(e^6 - 75*e^4 + 1875*e^2 - 15625) + 3*(10*x^2*(e^4 - 35*e^2 + 475) - x*(2*e^6
- 135*e^4 + 2775*e^2 - 23000) - 3*e^6 + 135*e^4 - 1800*e^2 + 7500)/(5*x^3*(e^8 - 70*e^6 + 1725*e^4 - 17500*e^2
+ 62500) - x^2*(e^10 - 110*e^8 + 4525*e^6 - 86500*e^4 + 762500*e^2 - 2500000) - 3*x*(e^10 - 95*e^8 + 3475*e^6
- 60625*e^4 + 500000*e^2 - 1562500)))*e^2 + 130*(675*(e^2 - 20)*log(5*x - e^2 + 25)/(e^10 - 80*e^8 + 2425*e^6
- 34750*e^4 + 237500*e^2 - 625000) + (e^2 + 20)*log(x + 3)/(e^6 - 30*e^4 + 300*e^2 - 1000) - 3*(5*x*(e^2 - 40
) - e^4 + 50*e^2 - 850)/(5*x^2*(e^6 - 45*e^4 + 600*e^2 - 2500) - x*(e^8 - 85*e^6 + 2400*e^4 - 26500*e^2 + 1000
00) - 3*e^8 + 210*e^6 - 5175*e^4 + 52500*e^2 - 187500) - log(x)/(e^4 - 50*e^2 + 625))*e^2 + 40*((e^2 - 40)*log
(5*x - e^2 + 25)/(e^6 - 30*e^4 + 300*e^2 - 1000) - (e^2 - 40)*log(x + 3)/(e^6 - 30*e^4 + 300*e^2 - 1000) + (x*
(e^2 - 40) + 6*e^2 - 150)/(5*x^2*(e^4 - 20*e^2 + 100) - x*(e^6 - 60*e^4 + 900*e^2 - 4000) - 3*e^6 + 135*e^4 -
1800*e^2 + 7500))*e^2 + 1106/3*((10*x - e^2 + 40)/(5*x^2*(e^4 - 20*e^2 + 100) - x*(e^6 - 60*e^4 + 900*e^2 - 40
00) - 3*e^6 + 135*e^4 - 1800*e^2 + 7500) + 10*log(5*x - e^2 + 25)/(e^6 - 30*e^4 + 300*e^2 - 1000) - 10*log(x +
3)/(e^6 - 30*e^4 + 300*e^2 - 1000))*e^2 - 4218750*(2*e^2 - 35)*log(5*x - e^2 + 25)/(e^12 - 105*e^10 + 4425*e^
8 - 95375*e^6 + 1106250*e^4 - 6562500*e^2 + 15625000) - 1350000*(e^2 - 20)*log(5*x - e^2 + 25)/(e^10 - 80*e^8
+ 2425*e^6 - 34750*e^4 + 237500*e^2 - 625000) + 450*(e^2 - 25)*log(5*x - e^2 + 25)/(e^6 - 30*e^4 + 300*e^2 - 1
000) - 3560/3*(e^2 - 40)*log(5*x - e^2 + 25)/(e^6 - 30*e^4 + 300*e^2 - 1000) - 2000*(e^2 + 20)*log(x + 3)/(e^6
- 30*e^4 + 300*e^2 - 1000) + 1250*(e^2 + 5)*log(x + 3)/(e^6 - 30*e^4 + 300*e^2 - 1000) - 450*(e^2 - 25)*log(x
+ 3)/(e^6 - 30*e^4 + 300*e^2 - 1000) + 3560/3*(e^2 - 40)*log(x + 3)/(e^6 - 30*e^4 + 300*e^2 - 1000) - 1250*(e
^2 - 40)*log(x)/(e^6 - 75*e^4 + 1875*e^2 - 15625) - 1875*(10*x^2*(e^4 - 35*e^2 + 475) - x*(2*e^6 - 135*e^4 + 2
775*e^2 - 23000) - 3*e^6 + 135*e^4 - 1800*e^2 + 7500)/(5*x^3*(e^8 - 70*e^6 + 1725*e^4 - 17500*e^2 + 62500) - x
^2*(e^10 - 110*e^8 + 4525*e^6 - 86500*e^4 + 762500*e^2 - 2500000) - 3*x*(e^10 - 95*e^8 + 3475*e^6 - 60625*e^4
+ 500000*e^2 - 1562500)) - 15*(x*(e^4 - 50*e^2 + 850) + 3*e^4 - 195*e^2 + 3000)/(5*x^2*(e^4 - 20*e^2 + 100) -
x*(e^6 - 60*e^4 + 900*e^2 - 4000) - 3*e^6 + 135*e^4 - 1800*e^2 + 7500) + 6000*(5*x*(e^2 - 40) - e^4 + 50*e^2 -
850)/(5*x^2*(e^6 - 45*e^4 + 600*e^2 - 2500) - x*(e^8 - 85*e^6 + 2400*e^4 - 26500*e^2 + 100000) - 3*e^8 + 210*
e^6 - 5175*e^4 + 52500*e^2 - 187500) - 3560/3*(x*(e^2 - 40) + 6*e^2 - 150)/(5*x^2*(e^4 - 20*e^2 + 100) - x*(e^
6 - 60*e^4 + 900*e^2 - 4000) - 3*e^6 + 135*e^4 - 1800*e^2 + 7500) - 20990/3*(10*x - e^2 + 40)/(5*x^2*(e^4 - 20
*e^2 + 100) - x*(e^6 - 60*e^4 + 900*e^2 - 4000) - 3*e^6 + 135*e^4 - 1800*e^2 + 7500) + (log(x) + 1)/x - 209900
/3*log(5*x - e^2 + 25)/(e^6 - 30*e^4 + 300*e^2 - 1000) + 209900/3*log(x + 3)/(e^6 - 30*e^4 + 300*e^2 - 1000) +
2000*log(x)/(e^4 - 50*e^2 + 625)
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mupad [B] time = 1.47, size = 54, normalized size = 1.54 \begin {gather*} \frac {\ln \relax (x)}{x}-\frac {-30\,x^2+\left (9\,{\mathrm {e}}^2-236\right )\,x+18\,{\mathrm {e}}^2-450}{-15\,x^3+\left (3\,{\mathrm {e}}^2-120\right )\,x^2+\left (9\,{\mathrm {e}}^2-225\right )\,x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((54000*x + exp(4)*(54*x + 12*x^2 + 81) - exp(2)*(3510*x + 1106*x^2 + 120*x^3 + 4050) - log(x)*(18000*x + e
xp(4)*(18*x + 3*x^2 + 27) - exp(2)*(1170*x + 330*x^2 + 30*x^3 + 1350) + 7050*x^2 + 1200*x^3 + 75*x^4 + 16875)
+ 20990*x^2 + 3560*x^3 + 225*x^4 + 50625)/(exp(4)*(27*x^2 + 18*x^3 + 3*x^4) + 16875*x^2 + 18000*x^3 + 7050*x^4
+ 1200*x^5 + 75*x^6 - exp(2)*(1350*x^2 + 1170*x^3 + 330*x^4 + 30*x^5)),x)
[Out]
log(x)/x - (18*exp(2) - 30*x^2 + x*(9*exp(2) - 236) - 450)/(x^2*(3*exp(2) - 120) - 15*x^3 + x*(9*exp(2) - 225)
)
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sympy [A] time = 2.81, size = 48, normalized size = 1.37 \begin {gather*} \frac {- 30 x^{2} + x \left (-236 + 9 e^{2}\right ) - 450 + 18 e^{2}}{15 x^{3} + x^{2} \left (120 - 3 e^{2}\right ) + x \left (225 - 9 e^{2}\right )} + \frac {\log {\relax (x )}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((((-3*x**2-18*x-27)*exp(2)**2+(30*x**3+330*x**2+1170*x+1350)*exp(2)-75*x**4-1200*x**3-7050*x**2-1800
0*x-16875)*ln(x)+(12*x**2+54*x+81)*exp(2)**2+(-120*x**3-1106*x**2-3510*x-4050)*exp(2)+225*x**4+3560*x**3+20990
*x**2+54000*x+50625)/((3*x**4+18*x**3+27*x**2)*exp(2)**2+(-30*x**5-330*x**4-1170*x**3-1350*x**2)*exp(2)+75*x**
6+1200*x**5+7050*x**4+18000*x**3+16875*x**2),x)
[Out]
(-30*x**2 + x*(-236 + 9*exp(2)) - 450 + 18*exp(2))/(15*x**3 + x**2*(120 - 3*exp(2)) + x*(225 - 9*exp(2))) + lo
g(x)/x
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