3.62.93 \(\int \frac {36+48 x+16 x^2+4 e^3 x^2+e^{3/2} (24 x+16 x^2)+(2 x^2+e^{3/2} x^2) \log (3)}{9 x^2+12 x^3+4 x^4+e^3 x^4+e^{3/2} (6 x^3+4 x^4)} \, dx\)
Optimal. Leaf size=25 \[ -2-\frac {4}{x}-\frac {\log (3)}{3+2 x+e^{3/2} x} \]
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Rubi [A] time = 0.24, antiderivative size = 33, normalized size of antiderivative = 1.32,
number of steps used = 6, number of rules used = 5, integrand size = 94, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used =
{6, 1680, 12, 1814, 8} \begin {gather*} -\frac {x \left (8+4 e^{3/2}+\log (3)\right )+12}{x \left (\left (2+e^{3/2}\right ) x+3\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(36 + 48*x + 16*x^2 + 4*E^3*x^2 + E^(3/2)*(24*x + 16*x^2) + (2*x^2 + E^(3/2)*x^2)*Log[3])/(9*x^2 + 12*x^3
+ 4*x^4 + E^3*x^4 + E^(3/2)*(6*x^3 + 4*x^4)),x]
[Out]
-((12 + x*(8 + 4*E^(3/2) + Log[3]))/(x*(3 + (2 + E^(3/2))*x)))
Rule 6
Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] && !FreeQ[v, x]
Rule 8
Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]
Rule 12
Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]
Rule 1680
Int[(Pq_)*(Q4_)^(p_), x_Symbol] :> With[{a = Coeff[Q4, x, 0], b = Coeff[Q4, x, 1], c = Coeff[Q4, x, 2], d = Co
eff[Q4, x, 3], e = Coeff[Q4, x, 4]}, Subst[Int[SimplifyIntegrand[(Pq /. x -> -(d/(4*e)) + x)*(a + d^4/(256*e^3
) - (b*d)/(8*e) + (c - (3*d^2)/(8*e))*x^2 + e*x^4)^p, x], x], x, d/(4*e) + x] /; EqQ[d^3 - 4*c*d*e + 8*b*e^2,
0] && NeQ[d, 0]] /; FreeQ[p, x] && PolyQ[Pq, x] && PolyQ[Q4, x, 4] && !IGtQ[p, 0]
Rule 1814
Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, a + b*x^2, x], f = Coeff[P
olynomialRemainder[Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 1]}, Simp[((a
*g - b*f*x)*(a + b*x^2)^(p + 1))/(2*a*b*(p + 1)), x] + Dist[1/(2*a*(p + 1)), Int[(a + b*x^2)^(p + 1)*ExpandToS
um[2*a*(p + 1)*Q + f*(2*p + 3), x], x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && LtQ[p, -1]
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {36+48 x+16 x^2+4 e^3 x^2+e^{3/2} \left (24 x+16 x^2\right )+\left (2 x^2+e^{3/2} x^2\right ) \log (3)}{9 x^2+12 x^3+\left (4+e^3\right ) x^4+e^{3/2} \left (6 x^3+4 x^4\right )} \, dx\\ &=\int \frac {36+48 x+\left (16+4 e^3\right ) x^2+e^{3/2} \left (24 x+16 x^2\right )+\left (2 x^2+e^{3/2} x^2\right ) \log (3)}{9 x^2+12 x^3+\left (4+e^3\right ) x^4+e^{3/2} \left (6 x^3+4 x^4\right )} \, dx\\ &=\operatorname {Subst}\left (\int \frac {4 \left (2+e^{3/2}\right ) \left (12 \left (2+e^{3/2}\right ) x \left (8+4 e^{3/2}-\log (3)\right )+9 \left (8+4 e^{3/2}+\log (3)\right )+4 \left (2+e^{3/2}\right )^2 x^2 \left (8+4 e^{3/2}+\log (3)\right )\right )}{\left (9-4 \left (2+e^{3/2}\right )^2 x^2\right )^2} \, dx,x,\frac {12+6 e^{3/2}}{4 \left (4+4 e^{3/2}+e^3\right )}+x\right )\\ &=\left (4 \left (2+e^{3/2}\right )\right ) \operatorname {Subst}\left (\int \frac {12 \left (2+e^{3/2}\right ) x \left (8+4 e^{3/2}-\log (3)\right )+9 \left (8+4 e^{3/2}+\log (3)\right )+4 \left (2+e^{3/2}\right )^2 x^2 \left (8+4 e^{3/2}+\log (3)\right )}{\left (9-4 \left (2+e^{3/2}\right )^2 x^2\right )^2} \, dx,x,\frac {12+6 e^{3/2}}{4 \left (4+4 e^{3/2}+e^3\right )}+x\right )\\ &=-\frac {12+x \left (8+4 e^{3/2}+\log (3)\right )}{x \left (3+\left (2+e^{3/2}\right ) x\right )}-\frac {1}{9} \left (2 \left (2+e^{3/2}\right )\right ) \operatorname {Subst}\left (\int 0 \, dx,x,\frac {12+6 e^{3/2}}{4 \left (4+4 e^{3/2}+e^3\right )}+x\right )\\ &=-\frac {12+x \left (8+4 e^{3/2}+\log (3)\right )}{x \left (3+\left (2+e^{3/2}\right ) x\right )}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.07, size = 41, normalized size = 1.64 \begin {gather*} -\frac {4}{x}-\frac {e^{3/2} \log (3)+\log (9)}{\left (2+e^{3/2}\right ) \left (3+\left (2+e^{3/2}\right ) x\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(36 + 48*x + 16*x^2 + 4*E^3*x^2 + E^(3/2)*(24*x + 16*x^2) + (2*x^2 + E^(3/2)*x^2)*Log[3])/(9*x^2 + 1
2*x^3 + 4*x^4 + E^3*x^4 + E^(3/2)*(6*x^3 + 4*x^4)),x]
[Out]
-4/x - (E^(3/2)*Log[3] + Log[9])/((2 + E^(3/2))*(3 + (2 + E^(3/2))*x))
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fricas [A] time = 0.71, size = 33, normalized size = 1.32 \begin {gather*} -\frac {4 \, x e^{\frac {3}{2}} + x \log \relax (3) + 8 \, x + 12}{x^{2} e^{\frac {3}{2}} + 2 \, x^{2} + 3 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(((x^2*exp(3/2)+2*x^2)*log(3)+4*x^2*exp(3/2)^2+(16*x^2+24*x)*exp(3/2)+16*x^2+48*x+36)/(x^4*exp(3/2)^2
+(4*x^4+6*x^3)*exp(3/2)+4*x^4+12*x^3+9*x^2),x, algorithm="fricas")
[Out]
-(4*x*e^(3/2) + x*log(3) + 8*x + 12)/(x^2*e^(3/2) + 2*x^2 + 3*x)
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(((x^2*exp(3/2)+2*x^2)*log(3)+4*x^2*exp(3/2)^2+(16*x^2+24*x)*exp(3/2)+16*x^2+48*x+36)/(x^4*exp(3/2)^2
+(4*x^4+6*x^3)*exp(3/2)+4*x^4+12*x^3+9*x^2),x, algorithm="giac")
[Out]
Exception raised: NotImplementedError >> Unable to parse Giac output: -4/sageVARx+(2*exp(3/2)*ln(3)+4*ln(3))*1
/6/sqrt(-exp(3/2)^2+exp(3))*atan((sageVARx*exp(3)+4*sageVARx*exp(3/2)+4*sageVARx+3*exp(3/2)+6)*1/3/sqrt(-exp(3
/2)^2+exp(3)))
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maple [A] time = 0.20, size = 30, normalized size = 1.20
|
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| method |
result |
size |
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| risch |
\(\frac {\left (-4 \,{\mathrm e}^{\frac {3}{2}}-\ln \relax (3)-8\right ) x -12}{x \left (3+x \,{\mathrm e}^{\frac {3}{2}}+2 x \right )}\) |
\(30\) |
| gosper |
\(-\frac {x \ln \relax (3)+4 x \,{\mathrm e}^{\frac {3}{2}}+8 x +12}{x \left (3+x \,{\mathrm e}^{\frac {3}{2}}+2 x \right )}\) |
\(31\) |
| norman |
\(\frac {-12+\left (\frac {\ln \relax (3) {\mathrm e}^{\frac {3}{2}}}{3}+\frac {4 \,{\mathrm e}^{3}}{3}+\frac {2 \ln \relax (3)}{3}+\frac {16 \,{\mathrm e}^{\frac {3}{2}}}{3}+\frac {16}{3}\right ) x^{2}}{x \left (3+x \,{\mathrm e}^{\frac {3}{2}}+2 x \right )}\) |
\(44\) |
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int(((x^2*exp(3/2)+2*x^2)*ln(3)+4*x^2*exp(3/2)^2+(16*x^2+24*x)*exp(3/2)+16*x^2+48*x+36)/(x^4*exp(3/2)^2+(4*x^4
+6*x^3)*exp(3/2)+4*x^4+12*x^3+9*x^2),x,method=_RETURNVERBOSE)
[Out]
((-4*exp(3/2)-ln(3)-8)*x-12)/x/(3+x*exp(3/2)+2*x)
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maxima [A] time = 0.52, size = 32, normalized size = 1.28 \begin {gather*} -\frac {{\left (e^{\frac {3}{2}} + 2\right )} \log \relax (3)}{x {\left (e^{3} + 4 \, e^{\frac {3}{2}} + 4\right )} + 3 \, e^{\frac {3}{2}} + 6} - \frac {4}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(((x^2*exp(3/2)+2*x^2)*log(3)+4*x^2*exp(3/2)^2+(16*x^2+24*x)*exp(3/2)+16*x^2+48*x+36)/(x^4*exp(3/2)^2
+(4*x^4+6*x^3)*exp(3/2)+4*x^4+12*x^3+9*x^2),x, algorithm="maxima")
[Out]
-(e^(3/2) + 2)*log(3)/(x*(e^3 + 4*e^(3/2) + 4) + 3*e^(3/2) + 6) - 4/x
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mupad [B] time = 0.21, size = 20, normalized size = 0.80 \begin {gather*} -\frac {\ln \relax (3)}{x\,\left ({\mathrm {e}}^{3/2}+2\right )+3}-\frac {4}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((48*x + exp(3/2)*(24*x + 16*x^2) + 4*x^2*exp(3) + 16*x^2 + log(3)*(x^2*exp(3/2) + 2*x^2) + 36)/(exp(3/2)*(
6*x^3 + 4*x^4) + x^4*exp(3) + 9*x^2 + 12*x^3 + 4*x^4),x)
[Out]
- log(3)/(x*(exp(3/2) + 2) + 3) - 4/x
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sympy [B] time = 6.69, size = 1023, normalized size = 40.92 result too large to
display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(((x**2*exp(3/2)+2*x**2)*ln(3)+4*x**2*exp(3/2)**2+(16*x**2+24*x)*exp(3/2)+16*x**2+48*x+36)/(x**4*exp(
3/2)**2+(4*x**4+6*x**3)*exp(3/2)+4*x**4+12*x**3+9*x**2),x)
[Out]
(x*(-41287833600*exp(33) - 181666467840*exp(63/2) - 8078054400*exp(69/2) - 693635604480*exp(30) - 1346342400*e
xp(36) - 2312118681600*exp(57/2) - 188487936*exp(75/2) - 6758500761600*exp(27) - 17379001958400*exp(51/2) - 21
748608*exp(39) - 39392404439040*exp(24) - 2013760*exp(81/2) - 78784808878080*exp(45/2) - 7325260800*exp(63/2)*
log(3) - 30766095360*exp(30)*log(3) - 1498348800*exp(33)*log(3) - 111876710400*exp(57/2)*log(3) - 260582400*ex
p(69/2)*log(3) - 143840*exp(42) - 354276249600*exp(27)*log(3) - 139032015667200*exp(21) - 38001600*exp(36)*log
(3) - 981072691200*exp(51/2)*log(3) - 4560192*exp(75/2)*log(3) - 2382605107200*exp(24)*log(3) - 21627202437120
0*exp(39/2) - 7440*exp(87/2) - 438480*exp(39)*log(3) - 5082890895360*exp(45/2)*log(3) - 295951191244800*exp(18
) - 32480*exp(81/2)*log(3) - 9530420428800*exp(21)*log(3) - 248*exp(45) - 355141429493760*exp(33/2) - 15697163
059200*exp(39/2)*log(3) - 1740*exp(42)*log(3) - 22673679974400*exp(18)*log(3) - 372052926136320*exp(15) - 4*ex
p(93/2) - 60*exp(87/2)*log(3) - 28640437862400*exp(33/2)*log(3) - 338229932851200*exp(27/2) - 31504481648640*e
xp(15)*log(3) - 264701686579200*exp(12) - exp(45)*log(3) - 30004268236800*exp(27/2)*log(3) - 176467791052800*e
xp(21/2) - 24548946739200*exp(12)*log(3) - 98821962989568*exp(9) - 17077528166400*exp(21/2)*log(3) - 996189143
0400*exp(9)*log(3) - 45610136764416*exp(15/2) - 4781707886592*exp(15/2)*log(3) - 16892643246080*exp(6) - 18391
18417920*exp(6)*log(3) - 4826469498880*exp(9/2) - 544923975680*exp(9/2)*log(3) - 998579896320*exp(3) - 1167694
23360*exp(3)*log(3) - 133143986176*exp(3/2) - 16106127360*exp(3/2)*log(3) - 8589934592 - 1073741824*log(3)) -
87903129600*exp(63/2) - 369193144320*exp(30) - 17980185600*exp(33) - 1342520524800*exp(57/2) - 3126988800*exp(
69/2) - 4251314995200*exp(27) - 456019200*exp(36) - 11772872294400*exp(51/2) - 54722304*exp(75/2) - 2859126128
6400*exp(24) - 5261760*exp(39) - 60994690744320*exp(45/2) - 389760*exp(81/2) - 114365045145600*exp(21) - 18836
5956710400*exp(39/2) - 20880*exp(42) - 272084159692800*exp(18) - 720*exp(87/2) - 343685254348800*exp(33/2) - 3
78053779783680*exp(15) - 12*exp(45) - 360051218841600*exp(27/2) - 294587360870400*exp(12) - 204930337996800*ex
p(21/2) - 119542697164800*exp(9) - 57380494639104*exp(15/2) - 22069421015040*exp(6) - 6539087708160*exp(9/2) -
1401233080320*exp(3) - 193273528320*exp(3/2) - 12884901888)/(x**2*(2147483648 + 33285996544*exp(3/2) + 249644
974080*exp(3) + 1206617374720*exp(9/2) + 4223160811520*exp(6) + 11402534191104*exp(15/2) + 24705490747392*exp(
9) + 44116947763200*exp(21/2) + 66175421644800*exp(12) + 84557483212800*exp(27/2) + exp(93/2) + 93013231534080
*exp(15) + 88785357373440*exp(33/2) + 62*exp(45) + 73987797811200*exp(18) + 1860*exp(87/2) + 54068006092800*ex
p(39/2) + 34758003916800*exp(21) + 35960*exp(42) + 19696202219520*exp(45/2) + 503440*exp(81/2) + 9848101109760
*exp(24) + 5437152*exp(39) + 4344750489600*exp(51/2) + 1689625190400*exp(27) + 47121984*exp(75/2) + 5780296704
00*exp(57/2) + 336585600*exp(36) + 173408901120*exp(30) + 2019513600*exp(69/2) + 45416616960*exp(63/2) + 10321
958400*exp(33)) + x*(3221225472 + 48318382080*exp(3/2) + 350308270080*exp(3) + 1634771927040*exp(9/2) + 551735
5253760*exp(6) + 14345123659776*exp(15/2) + 29885674291200*exp(9) + 51232584499200*exp(21/2) + 73646840217600*
exp(12) + 90012804710400*exp(27/2) + 3*exp(45) + 94513444945920*exp(15) + 85921313587200*exp(33/2) + 180*exp(8
7/2) + 68021039923200*exp(18) + 5220*exp(42) + 47091489177600*exp(39/2) + 28591261286400*exp(21) + 97440*exp(8
1/2) + 15248672686080*exp(45/2) + 1315440*exp(39) + 7147815321600*exp(24) + 13680576*exp(75/2) + 2943218073600
*exp(51/2) + 114004800*exp(36) + 1062828748800*exp(27) + 781747200*exp(69/2) + 335630131200*exp(57/2) + 449504
6400*exp(33) + 92298286080*exp(30) + 21975782400*exp(63/2)))
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