3.45.63 \(\int \frac {19+52 x+73 x^2+58 x^3+28 x^4+8 x^5+x^6+e^x (25+60 x+76 x^2+58 x^3+28 x^4+8 x^5+x^6)}{25+60 x+76 x^2+58 x^3+28 x^4+8 x^5+x^6} \, dx\) [4463]
Optimal. Leaf size=18 \[ 4+e^x+x+\frac {x}{-1+x+(1+x)^4} \]
[Out]
4+x+exp(x)+x/((1+x)^4+x-1)
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Rubi [F]
time = 180.01, 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*} \text {\$Aborted} \end {gather*}
Verification is not applicable to the result.
[In]
Int[(19 + 52*x + 73*x^2 + 58*x^3 + 28*x^4 + 8*x^5 + x^6 + E^x*(25 + 60*x + 76*x^2 + 58*x^3 + 28*x^4 + 8*x^5 +
x^6))/(25 + 60*x + 76*x^2 + 58*x^3 + 28*x^4 + 8*x^5 + x^6),x]
[Out]
$Aborted
Rubi steps
Aborted
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Mathematica [A]
time = 1.64, size = 20, normalized size = 1.11 \begin {gather*} e^x+x+\frac {1}{5+6 x+4 x^2+x^3} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(19 + 52*x + 73*x^2 + 58*x^3 + 28*x^4 + 8*x^5 + x^6 + E^x*(25 + 60*x + 76*x^2 + 58*x^3 + 28*x^4 + 8*
x^5 + x^6))/(25 + 60*x + 76*x^2 + 58*x^3 + 28*x^4 + 8*x^5 + x^6),x]
[Out]
E^x + x + (5 + 6*x + 4*x^2 + x^3)^(-1)
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.36, size = 1065, normalized size = 59.17
| | |
| method |
result |
size |
| | |
| risch |
\(x +\frac {1}{x^{3}+4 x^{2}+6 x +5}+{\mathrm e}^{x}\) |
\(20\) |
| norman |
\(\frac {x^{4}-19 x -10 x^{2}+{\mathrm e}^{x} x^{3}+6 \,{\mathrm e}^{x} x +4 \,{\mathrm e}^{x} x^{2}+5 \,{\mathrm e}^{x}-19}{x^{3}+4 x^{2}+6 x +5}\) |
\(52\) |
| default |
\(\text {Expression too large to display}\) |
\(1065\) |
| | |
|
|
|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(((x^6+8*x^5+28*x^4+58*x^3+76*x^2+60*x+25)*exp(x)+x^6+8*x^5+28*x^4+58*x^3+73*x^2+52*x+19)/(x^6+8*x^5+28*x^4
+58*x^3+76*x^2+60*x+25),x,method=_RETURNVERBOSE)
[Out]
x+exp(x)-28/83*sum((161*_R1^2+198*_R1+190)/(3*_R1^2+8*_R1+6)*exp(_R1)*Ei(1,-x+_R1),_R1=RootOf(_Z^3+4*_Z^2+6*_Z
+5))+60/83*sum((21*_R1^2+15*_R1+32)/(3*_R1^2+8*_R1+6)*exp(_R1)*Ei(1,-x+_R1),_R1=RootOf(_Z^3+4*_Z^2+6*_Z+5))+58
/83*exp(x)*(86*x^2+183*x+240)/(x^3+4*x^2+6*x+5)+8/83*sum((83*_R^2+36*_R+430)/(3*_R^2+8*_R+6)*ln(x-_R),_R=RootO
f(_Z^3+4*_Z^2+6*_Z+5))+19*(-4/83*x^2+5/83*x+12/83)/(x^3+4*x^2+6*x+5)+168/83*sum((-13*_R-40)/(3*_R^2+8*_R+6)*ln
(x-_R),_R=RootOf(_Z^3+4*_Z^2+6*_Z+5))-1/83*exp(x)*(936*x^2+1403*x+1840)/(x^3+4*x^2+6*x+5)+73*(-48/83*x^2-106/8
3*x-105/83)/(x^3+4*x^2+6*x+5)-4/83*sum((166*_R^2+151*_R+305)/(3*_R^2+8*_R+6)*ln(x-_R),_R=RootOf(_Z^3+4*_Z^2+6*
_Z+5))+156/83*sum((7*_R-4)/(3*_R^2+8*_R+6)*ln(x-_R),_R=RootOf(_Z^3+4*_Z^2+6*_Z+5))-76/83*exp(x)*(48*x^2+106*x+
105)/(x^3+4*x^2+6*x+5)-(936/83*x^2+1403/83*x+1840/83)/(x^3+4*x^2+6*x+5)+38/83*sum((-2*_R+13)/(3*_R^2+8*_R+6)*l
n(x-_R),_R=RootOf(_Z^3+4*_Z^2+6*_Z+5))-28/83*exp(x)*(161*x^2+276*x+430)/(x^3+4*x^2+6*x+5)+60/83*exp(x)*(21*x^2
+36*x+20)/(x^3+4*x^2+6*x+5)+292/83*sum((-12*_R-5)/(3*_R^2+8*_R+6)*ln(x-_R),_R=RootOf(_Z^3+4*_Z^2+6*_Z+5))+52*(
21/83*x^2+36/83*x+20/83)/(x^3+4*x^2+6*x+5)-1/83*sum((272*_R1^2+799*_R1+620)/(3*_R1^2+8*_R1+6)*exp(_R1)*Ei(1,-x
+_R1),_R1=RootOf(_Z^3+4*_Z^2+6*_Z+5))-25/83*sum((4*_R1^2-9*_R1+14)/(3*_R1^2+8*_R1+6)*exp(_R1)*Ei(1,-x+_R1),_R1
=RootOf(_Z^3+4*_Z^2+6*_Z+5))-25/83*exp(x)*(4*x^2-5*x-12)/(x^3+4*x^2+6*x+5)+58/83*sum((86*_R1^2+97*_R1+135)/(3*
_R1^2+8*_R1+6)*exp(_R1)*Ei(1,-x+_R1),_R1=RootOf(_Z^3+4*_Z^2+6*_Z+5))+8/83*exp(x)*(368*x^2+536*x+805)/(x^3+4*x^
2+6*x+5)+58/83*sum((86*_R+105)/(3*_R^2+8*_R+6)*ln(x-_R),_R=RootOf(_Z^3+4*_Z^2+6*_Z+5))+40/83*sum((57*_R1^2+100
*_R1+75)/(3*_R1^2+8*_R1+6)*exp(_R1)*Ei(1,-x+_R1),_R1=RootOf(_Z^3+4*_Z^2+6*_Z+5))+8*(368/83*x^2+536/83*x+805/83
)/(x^3+4*x^2+6*x+5)+28*(-161/83*x^2-276/83*x-430/83)/(x^3+4*x^2+6*x+5)+58*(86/83*x^2+183/83*x+240/83)/(x^3+4*x
^2+6*x+5)-76/83*sum((48*_R1^2+58*_R1+85)/(3*_R1^2+8*_R1+6)*exp(_R1)*Ei(1,-x+_R1),_R1=RootOf(_Z^3+4*_Z^2+6*_Z+5
))
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 50 vs.
\(2 (17) = 34\).
time = 0.29, size = 50, normalized size = 2.78 \begin {gather*} \frac {x^{4} + 4 \, x^{3} + 6 \, x^{2} + {\left (x^{3} + 4 \, x^{2} + 6 \, x + 5\right )} e^{x} + 5 \, x + 1}{x^{3} + 4 \, x^{2} + 6 \, x + 5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(((x^6+8*x^5+28*x^4+58*x^3+76*x^2+60*x+25)*exp(x)+x^6+8*x^5+28*x^4+58*x^3+73*x^2+52*x+19)/(x^6+8*x^5+
28*x^4+58*x^3+76*x^2+60*x+25),x, algorithm="maxima")
[Out]
(x^4 + 4*x^3 + 6*x^2 + (x^3 + 4*x^2 + 6*x + 5)*e^x + 5*x + 1)/(x^3 + 4*x^2 + 6*x + 5)
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 50 vs.
\(2 (17) = 34\).
time = 0.38, size = 50, normalized size = 2.78 \begin {gather*} \frac {x^{4} + 4 \, x^{3} + 6 \, x^{2} + {\left (x^{3} + 4 \, x^{2} + 6 \, x + 5\right )} e^{x} + 5 \, x + 1}{x^{3} + 4 \, x^{2} + 6 \, x + 5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(((x^6+8*x^5+28*x^4+58*x^3+76*x^2+60*x+25)*exp(x)+x^6+8*x^5+28*x^4+58*x^3+73*x^2+52*x+19)/(x^6+8*x^5+
28*x^4+58*x^3+76*x^2+60*x+25),x, algorithm="fricas")
[Out]
(x^4 + 4*x^3 + 6*x^2 + (x^3 + 4*x^2 + 6*x + 5)*e^x + 5*x + 1)/(x^3 + 4*x^2 + 6*x + 5)
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Sympy [A]
time = 0.07, size = 19, normalized size = 1.06 \begin {gather*} x + e^{x} + \frac {1}{x^{3} + 4 x^{2} + 6 x + 5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(((x**6+8*x**5+28*x**4+58*x**3+76*x**2+60*x+25)*exp(x)+x**6+8*x**5+28*x**4+58*x**3+73*x**2+52*x+19)/(
x**6+8*x**5+28*x**4+58*x**3+76*x**2+60*x+25),x)
[Out]
x + exp(x) + 1/(x**3 + 4*x**2 + 6*x + 5)
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 56 vs.
\(2 (17) = 34\).
time = 0.41, size = 56, normalized size = 3.11 \begin {gather*} \frac {x^{4} + x^{3} e^{x} + 4 \, x^{3} + 4 \, x^{2} e^{x} + 6 \, x^{2} + 6 \, x e^{x} + 5 \, x + 5 \, e^{x} + 1}{x^{3} + 4 \, x^{2} + 6 \, x + 5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(((x^6+8*x^5+28*x^4+58*x^3+76*x^2+60*x+25)*exp(x)+x^6+8*x^5+28*x^4+58*x^3+73*x^2+52*x+19)/(x^6+8*x^5+
28*x^4+58*x^3+76*x^2+60*x+25),x, algorithm="giac")
[Out]
(x^4 + x^3*e^x + 4*x^3 + 4*x^2*e^x + 6*x^2 + 6*x*e^x + 5*x + 5*e^x + 1)/(x^3 + 4*x^2 + 6*x + 5)
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Mupad [B]
time = 0.16, size = 19, normalized size = 1.06 \begin {gather*} x+{\mathrm {e}}^x+\frac {1}{x^3+4\,x^2+6\,x+5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((52*x + exp(x)*(60*x + 76*x^2 + 58*x^3 + 28*x^4 + 8*x^5 + x^6 + 25) + 73*x^2 + 58*x^3 + 28*x^4 + 8*x^5 + x
^6 + 19)/(60*x + 76*x^2 + 58*x^3 + 28*x^4 + 8*x^5 + x^6 + 25),x)
[Out]
x + exp(x) + 1/(6*x + 4*x^2 + x^3 + 5)
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