3.9.4 \(\int \frac {2+5 x}{\sqrt {-12+20 x+5 x^2+2 x^3+x^4}} \, dx\)
Optimal. Leaf size=64 \[ -\log \left (-x^5-3 x^4-8 x^3-24 x^2+\left (x^3+2 x^2+4 x+8\right ) \sqrt {x^4+2 x^3+5 x^2+20 x-12}-16 x-16\right ) \]
________________________________________________________________________________________
Rubi [F] time = 0.14, 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 {2+5 x}{\sqrt {-12+20 x+5 x^2+2 x^3+x^4}} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
Int[(2 + 5*x)/Sqrt[-12 + 20*x + 5*x^2 + 2*x^3 + x^4],x]
[Out]
2*Defer[Int][1/Sqrt[-12 + 20*x + 5*x^2 + 2*x^3 + x^4], x] + 5*Defer[Int][x/Sqrt[-12 + 20*x + 5*x^2 + 2*x^3 + x
^4], x]
Rubi steps
\begin {align*} \int \frac {2+5 x}{\sqrt {-12+20 x+5 x^2+2 x^3+x^4}} \, dx &=\int \left (\frac {2}{\sqrt {-12+20 x+5 x^2+2 x^3+x^4}}+\frac {5 x}{\sqrt {-12+20 x+5 x^2+2 x^3+x^4}}\right ) \, dx\\ &=2 \int \frac {1}{\sqrt {-12+20 x+5 x^2+2 x^3+x^4}} \, dx+5 \int \frac {x}{\sqrt {-12+20 x+5 x^2+2 x^3+x^4}} \, dx\\ \end {align*}
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Mathematica [C] time = 1.12, size = 1144, normalized size = 17.88
result too large to display
Antiderivative was successfully verified.
[In]
Integrate[(2 + 5*x)/Sqrt[-12 + 20*x + 5*x^2 + 2*x^3 + x^4],x]
[Out]
(-2*(3 + x)*(x - Root[-4 + 8*#1 - #1^2 + #1^3 & , 1, 0])*(2*EllipticF[ArcSin[Sqrt[((3 + x)*(-Root[-4 + 8*#1 -
#1^2 + #1^3 & , 1, 0] + Root[-4 + 8*#1 - #1^2 + #1^3 & , 3, 0]))/((x - Root[-4 + 8*#1 - #1^2 + #1^3 & , 1, 0])
*(3 + Root[-4 + 8*#1 - #1^2 + #1^3 & , 3, 0]))]], ((Root[-4 + 8*#1 - #1^2 + #1^3 & , 1, 0] - Root[-4 + 8*#1 -
#1^2 + #1^3 & , 2, 0])*(3 + Root[-4 + 8*#1 - #1^2 + #1^3 & , 3, 0]))/((3 + Root[-4 + 8*#1 - #1^2 + #1^3 & , 2,
0])*(Root[-4 + 8*#1 - #1^2 + #1^3 & , 1, 0] - Root[-4 + 8*#1 - #1^2 + #1^3 & , 3, 0]))] + 5*EllipticF[ArcSin[
Sqrt[-(((3 + x)*(Root[-4 + 8*#1 - #1^2 + #1^3 & , 1, 0] - Root[-4 + 8*#1 - #1^2 + #1^3 & , 3, 0]))/((x - Root[
-4 + 8*#1 - #1^2 + #1^3 & , 1, 0])*(3 + Root[-4 + 8*#1 - #1^2 + #1^3 & , 3, 0])))]], ((Root[-4 + 8*#1 - #1^2 +
#1^3 & , 1, 0] - Root[-4 + 8*#1 - #1^2 + #1^3 & , 2, 0])*(3 + Root[-4 + 8*#1 - #1^2 + #1^3 & , 3, 0]))/((3 +
Root[-4 + 8*#1 - #1^2 + #1^3 & , 2, 0])*(Root[-4 + 8*#1 - #1^2 + #1^3 & , 1, 0] - Root[-4 + 8*#1 - #1^2 + #1^3
& , 3, 0]))]*Root[-4 + 8*#1 - #1^2 + #1^3 & , 1, 0] - 5*EllipticPi[(3 + Root[-4 + 8*#1 - #1^2 + #1^3 & , 3, 0
])/(-Root[-4 + 8*#1 - #1^2 + #1^3 & , 1, 0] + Root[-4 + 8*#1 - #1^2 + #1^3 & , 3, 0]), ArcSin[Sqrt[-(((3 + x)*
(Root[-4 + 8*#1 - #1^2 + #1^3 & , 1, 0] - Root[-4 + 8*#1 - #1^2 + #1^3 & , 3, 0]))/((x - Root[-4 + 8*#1 - #1^2
+ #1^3 & , 1, 0])*(3 + Root[-4 + 8*#1 - #1^2 + #1^3 & , 3, 0])))]], ((Root[-4 + 8*#1 - #1^2 + #1^3 & , 1, 0]
- Root[-4 + 8*#1 - #1^2 + #1^3 & , 2, 0])*(3 + Root[-4 + 8*#1 - #1^2 + #1^3 & , 3, 0]))/((3 + Root[-4 + 8*#1 -
#1^2 + #1^3 & , 2, 0])*(Root[-4 + 8*#1 - #1^2 + #1^3 & , 1, 0] - Root[-4 + 8*#1 - #1^2 + #1^3 & , 3, 0]))]*(3
+ Root[-4 + 8*#1 - #1^2 + #1^3 & , 1, 0]))*Sqrt[(x - Root[-4 + 8*#1 - #1^2 + #1^3 & , 2, 0])/((x - Root[-4 +
8*#1 - #1^2 + #1^3 & , 1, 0])*(3 + Root[-4 + 8*#1 - #1^2 + #1^3 & , 2, 0]))]*Sqrt[(x - Root[-4 + 8*#1 - #1^2 +
#1^3 & , 3, 0])/((x - Root[-4 + 8*#1 - #1^2 + #1^3 & , 1, 0])*(3 + Root[-4 + 8*#1 - #1^2 + #1^3 & , 3, 0]))])
/(Sqrt[-12 + 20*x + 5*x^2 + 2*x^3 + x^4]*Sqrt[-(((3 + x)*(Root[-4 + 8*#1 - #1^2 + #1^3 & , 1, 0] - Root[-4 + 8
*#1 - #1^2 + #1^3 & , 3, 0]))/((x - Root[-4 + 8*#1 - #1^2 + #1^3 & , 1, 0])*(3 + Root[-4 + 8*#1 - #1^2 + #1^3
& , 3, 0])))])
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IntegrateAlgebraic [A] time = 5.06, size = 64, normalized size = 1.00 \begin {gather*} -\log \left (-x^5-3 x^4-8 x^3-24 x^2+\left (x^3+2 x^2+4 x+8\right ) \sqrt {x^4+2 x^3+5 x^2+20 x-12}-16 x-16\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[(2 + 5*x)/Sqrt[-12 + 20*x + 5*x^2 + 2*x^3 + x^4],x]
[Out]
-Log[-16 - 16*x - 24*x^2 - 8*x^3 - 3*x^4 - x^5 + (8 + 4*x + 2*x^2 + x^3)*Sqrt[-12 + 20*x + 5*x^2 + 2*x^3 + x^4
]]
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fricas [A] time = 0.50, size = 58, normalized size = 0.91 \begin {gather*} \log \left (x^{5} + 3 \, x^{4} + 8 \, x^{3} + 24 \, x^{2} + \sqrt {x^{4} + 2 \, x^{3} + 5 \, x^{2} + 20 \, x - 12} {\left (x^{3} + 2 \, x^{2} + 4 \, x + 8\right )} + 16 \, x + 16\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((2+5*x)/(x^4+2*x^3+5*x^2+20*x-12)^(1/2),x, algorithm="fricas")
[Out]
log(x^5 + 3*x^4 + 8*x^3 + 24*x^2 + sqrt(x^4 + 2*x^3 + 5*x^2 + 20*x - 12)*(x^3 + 2*x^2 + 4*x + 8) + 16*x + 16)
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {5 \, x + 2}{\sqrt {x^{4} + 2 \, x^{3} + 5 \, x^{2} + 20 \, x - 12}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((2+5*x)/(x^4+2*x^3+5*x^2+20*x-12)^(1/2),x, algorithm="giac")
[Out]
integrate((5*x + 2)/sqrt(x^4 + 2*x^3 + 5*x^2 + 20*x - 12), x)
________________________________________________________________________________________
maple [C] time = 1.30, size = 2769, normalized size = 43.27 \begin {gather*} \text {Expression too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((2+5*x)/(x^4+2*x^3+5*x^2+20*x-12)^(1/2),x)
[Out]
4*(-10/3+1/6*(19+12*87^(1/2))^(1/3)-23/6/(19+12*87^(1/2))^(1/3)+1/2*I*3^(1/2)*(1/3*(19+12*87^(1/2))^(1/3)+23/3
/(19+12*87^(1/2))^(1/3)))*((-1/2*(19+12*87^(1/2))^(1/3)+23/2/(19+12*87^(1/2))^(1/3)-1/2*I*3^(1/2)*(1/3*(19+12*
87^(1/2))^(1/3)+23/3/(19+12*87^(1/2))^(1/3)))*(3+x)/(-1/6*(19+12*87^(1/2))^(1/3)+23/6/(19+12*87^(1/2))^(1/3)+1
0/3-1/2*I*3^(1/2)*(1/3*(19+12*87^(1/2))^(1/3)+23/3/(19+12*87^(1/2))^(1/3)))/(x-1/3*(19+12*87^(1/2))^(1/3)+23/3
/(19+12*87^(1/2))^(1/3)-1/3))^(1/2)*(x-1/3*(19+12*87^(1/2))^(1/3)+23/3/(19+12*87^(1/2))^(1/3)-1/3)^2*((1/3*(19
+12*87^(1/2))^(1/3)-23/3/(19+12*87^(1/2))^(1/3)+10/3)*(x+1/6*(19+12*87^(1/2))^(1/3)-23/6/(19+12*87^(1/2))^(1/3
)-1/3-1/2*I*3^(1/2)*(1/3*(19+12*87^(1/2))^(1/3)+23/3/(19+12*87^(1/2))^(1/3)))/(-1/6*(19+12*87^(1/2))^(1/3)+23/
6/(19+12*87^(1/2))^(1/3)+10/3+1/2*I*3^(1/2)*(1/3*(19+12*87^(1/2))^(1/3)+23/3/(19+12*87^(1/2))^(1/3)))/(x-1/3*(
19+12*87^(1/2))^(1/3)+23/3/(19+12*87^(1/2))^(1/3)-1/3))^(1/2)*((1/3*(19+12*87^(1/2))^(1/3)-23/3/(19+12*87^(1/2
))^(1/3)+10/3)*(x+1/6*(19+12*87^(1/2))^(1/3)-23/6/(19+12*87^(1/2))^(1/3)-1/3+1/2*I*3^(1/2)*(1/3*(19+12*87^(1/2
))^(1/3)+23/3/(19+12*87^(1/2))^(1/3)))/(-1/6*(19+12*87^(1/2))^(1/3)+23/6/(19+12*87^(1/2))^(1/3)+10/3-1/2*I*3^(
1/2)*(1/3*(19+12*87^(1/2))^(1/3)+23/3/(19+12*87^(1/2))^(1/3)))/(x-1/3*(19+12*87^(1/2))^(1/3)+23/3/(19+12*87^(1
/2))^(1/3)-1/3))^(1/2)/(-1/2*(19+12*87^(1/2))^(1/3)+23/2/(19+12*87^(1/2))^(1/3)-1/2*I*3^(1/2)*(1/3*(19+12*87^(
1/2))^(1/3)+23/3/(19+12*87^(1/2))^(1/3)))/(1/3*(19+12*87^(1/2))^(1/3)-23/3/(19+12*87^(1/2))^(1/3)+10/3)/((3+x)
*(x-1/3*(19+12*87^(1/2))^(1/3)+23/3/(19+12*87^(1/2))^(1/3)-1/3)*(x+1/6*(19+12*87^(1/2))^(1/3)-23/6/(19+12*87^(
1/2))^(1/3)-1/3-1/2*I*3^(1/2)*(1/3*(19+12*87^(1/2))^(1/3)+23/3/(19+12*87^(1/2))^(1/3)))*(x+1/6*(19+12*87^(1/2)
)^(1/3)-23/6/(19+12*87^(1/2))^(1/3)-1/3+1/2*I*3^(1/2)*(1/3*(19+12*87^(1/2))^(1/3)+23/3/(19+12*87^(1/2))^(1/3))
))^(1/2)*EllipticF(((-1/2*(19+12*87^(1/2))^(1/3)+23/2/(19+12*87^(1/2))^(1/3)-1/2*I*3^(1/2)*(1/3*(19+12*87^(1/2
))^(1/3)+23/3/(19+12*87^(1/2))^(1/3)))*(3+x)/(-1/6*(19+12*87^(1/2))^(1/3)+23/6/(19+12*87^(1/2))^(1/3)+10/3-1/2
*I*3^(1/2)*(1/3*(19+12*87^(1/2))^(1/3)+23/3/(19+12*87^(1/2))^(1/3)))/(x-1/3*(19+12*87^(1/2))^(1/3)+23/3/(19+12
*87^(1/2))^(1/3)-1/3))^(1/2),((1/2*(19+12*87^(1/2))^(1/3)-23/2/(19+12*87^(1/2))^(1/3)-1/2*I*3^(1/2)*(1/3*(19+1
2*87^(1/2))^(1/3)+23/3/(19+12*87^(1/2))^(1/3)))*(-10/3+1/6*(19+12*87^(1/2))^(1/3)-23/6/(19+12*87^(1/2))^(1/3)+
1/2*I*3^(1/2)*(1/3*(19+12*87^(1/2))^(1/3)+23/3/(19+12*87^(1/2))^(1/3)))/(-10/3+1/6*(19+12*87^(1/2))^(1/3)-23/6
/(19+12*87^(1/2))^(1/3)-1/2*I*3^(1/2)*(1/3*(19+12*87^(1/2))^(1/3)+23/3/(19+12*87^(1/2))^(1/3)))/(1/2*(19+12*87
^(1/2))^(1/3)-23/2/(19+12*87^(1/2))^(1/3)+1/2*I*3^(1/2)*(1/3*(19+12*87^(1/2))^(1/3)+23/3/(19+12*87^(1/2))^(1/3
))))^(1/2))+10*(-10/3+1/6*(19+12*87^(1/2))^(1/3)-23/6/(19+12*87^(1/2))^(1/3)+1/2*I*3^(1/2)*(1/3*(19+12*87^(1/2
))^(1/3)+23/3/(19+12*87^(1/2))^(1/3)))*((-1/2*(19+12*87^(1/2))^(1/3)+23/2/(19+12*87^(1/2))^(1/3)-1/2*I*3^(1/2)
*(1/3*(19+12*87^(1/2))^(1/3)+23/3/(19+12*87^(1/2))^(1/3)))*(3+x)/(-1/6*(19+12*87^(1/2))^(1/3)+23/6/(19+12*87^(
1/2))^(1/3)+10/3-1/2*I*3^(1/2)*(1/3*(19+12*87^(1/2))^(1/3)+23/3/(19+12*87^(1/2))^(1/3)))/(x-1/3*(19+12*87^(1/2
))^(1/3)+23/3/(19+12*87^(1/2))^(1/3)-1/3))^(1/2)*(x-1/3*(19+12*87^(1/2))^(1/3)+23/3/(19+12*87^(1/2))^(1/3)-1/3
)^2*((1/3*(19+12*87^(1/2))^(1/3)-23/3/(19+12*87^(1/2))^(1/3)+10/3)*(x+1/6*(19+12*87^(1/2))^(1/3)-23/6/(19+12*8
7^(1/2))^(1/3)-1/3-1/2*I*3^(1/2)*(1/3*(19+12*87^(1/2))^(1/3)+23/3/(19+12*87^(1/2))^(1/3)))/(-1/6*(19+12*87^(1/
2))^(1/3)+23/6/(19+12*87^(1/2))^(1/3)+10/3+1/2*I*3^(1/2)*(1/3*(19+12*87^(1/2))^(1/3)+23/3/(19+12*87^(1/2))^(1/
3)))/(x-1/3*(19+12*87^(1/2))^(1/3)+23/3/(19+12*87^(1/2))^(1/3)-1/3))^(1/2)*((1/3*(19+12*87^(1/2))^(1/3)-23/3/(
19+12*87^(1/2))^(1/3)+10/3)*(x+1/6*(19+12*87^(1/2))^(1/3)-23/6/(19+12*87^(1/2))^(1/3)-1/3+1/2*I*3^(1/2)*(1/3*(
19+12*87^(1/2))^(1/3)+23/3/(19+12*87^(1/2))^(1/3)))/(-1/6*(19+12*87^(1/2))^(1/3)+23/6/(19+12*87^(1/2))^(1/3)+1
0/3-1/2*I*3^(1/2)*(1/3*(19+12*87^(1/2))^(1/3)+23/3/(19+12*87^(1/2))^(1/3)))/(x-1/3*(19+12*87^(1/2))^(1/3)+23/3
/(19+12*87^(1/2))^(1/3)-1/3))^(1/2)/(-1/2*(19+12*87^(1/2))^(1/3)+23/2/(19+12*87^(1/2))^(1/3)-1/2*I*3^(1/2)*(1/
3*(19+12*87^(1/2))^(1/3)+23/3/(19+12*87^(1/2))^(1/3)))/(1/3*(19+12*87^(1/2))^(1/3)-23/3/(19+12*87^(1/2))^(1/3)
+10/3)/((3+x)*(x-1/3*(19+12*87^(1/2))^(1/3)+23/3/(19+12*87^(1/2))^(1/3)-1/3)*(x+1/6*(19+12*87^(1/2))^(1/3)-23/
6/(19+12*87^(1/2))^(1/3)-1/3-1/2*I*3^(1/2)*(1/3*(19+12*87^(1/2))^(1/3)+23/3/(19+12*87^(1/2))^(1/3)))*(x+1/6*(1
9+12*87^(1/2))^(1/3)-23/6/(19+12*87^(1/2))^(1/3)-1/3+1/2*I*3^(1/2)*(1/3*(19+12*87^(1/2))^(1/3)+23/3/(19+12*87^
(1/2))^(1/3))))^(1/2)*((1/3*(19+12*87^(1/2))^(1/3)-23/3/(19+12*87^(1/2))^(1/3)+1/3)*EllipticF(((-1/2*(19+12*87
^(1/2))^(1/3)+23/2/(19+12*87^(1/2))^(1/3)-1/2*I*3^(1/2)*(1/3*(19+12*87^(1/2))^(1/3)+23/3/(19+12*87^(1/2))^(1/3
)))*(3+x)/(-1/6*(19+12*87^(1/2))^(1/3)+23/6/(19+12*87^(1/2))^(1/3)+10/3-1/2*I*3^(1/2)*(1/3*(19+12*87^(1/2))^(1
/3)+23/3/(19+12*87^(1/2))^(1/3)))/(x-1/3*(19+12*87^(1/2))^(1/3)+23/3/(19+12*87^(1/2))^(1/3)-1/3))^(1/2),((1/2*
(19+12*87^(1/2))^(1/3)-23/2/(19+12*87^(1/2))^(1/3)-1/2*I*3^(1/2)*(1/3*(19+12*87^(1/2))^(1/3)+23/3/(19+12*87^(1
/2))^(1/3)))*(-10/3+1/6*(19+12*87^(1/2))^(1/3)-23/6/(19+12*87^(1/2))^(1/3)+1/2*I*3^(1/2)*(1/3*(19+12*87^(1/2))
^(1/3)+23/3/(19+12*87^(1/2))^(1/3)))/(-10/3+1/6*(19+12*87^(1/2))^(1/3)-23/6/(19+12*87^(1/2))^(1/3)-1/2*I*3^(1/
2)*(1/3*(19+12*87^(1/2))^(1/3)+23/3/(19+12*87^(1/2))^(1/3)))/(1/2*(19+12*87^(1/2))^(1/3)-23/2/(19+12*87^(1/2))
^(1/3)+1/2*I*3^(1/2)*(1/3*(19+12*87^(1/2))^(1/3)+23/3/(19+12*87^(1/2))^(1/3))))^(1/2))+(-10/3-1/3*(19+12*87^(1
/2))^(1/3)+23/3/(19+12*87^(1/2))^(1/3))*EllipticPi(((-1/2*(19+12*87^(1/2))^(1/3)+23/2/(19+12*87^(1/2))^(1/3)-1
/2*I*3^(1/2)*(1/3*(19+12*87^(1/2))^(1/3)+23/3/(19+12*87^(1/2))^(1/3)))*(3+x)/(-1/6*(19+12*87^(1/2))^(1/3)+23/6
/(19+12*87^(1/2))^(1/3)+10/3-1/2*I*3^(1/2)*(1/3*(19+12*87^(1/2))^(1/3)+23/3/(19+12*87^(1/2))^(1/3)))/(x-1/3*(1
9+12*87^(1/2))^(1/3)+23/3/(19+12*87^(1/2))^(1/3)-1/3))^(1/2),(-1/6*(19+12*87^(1/2))^(1/3)+23/6/(19+12*87^(1/2)
)^(1/3)+10/3-1/2*I*3^(1/2)*(1/3*(19+12*87^(1/2))^(1/3)+23/3/(19+12*87^(1/2))^(1/3)))/(-1/2*(19+12*87^(1/2))^(1
/3)+23/2/(19+12*87^(1/2))^(1/3)-1/2*I*3^(1/2)*(1/3*(19+12*87^(1/2))^(1/3)+23/3/(19+12*87^(1/2))^(1/3))),((1/2*
(19+12*87^(1/2))^(1/3)-23/2/(19+12*87^(1/2))^(1/3)-1/2*I*3^(1/2)*(1/3*(19+12*87^(1/2))^(1/3)+23/3/(19+12*87^(1
/2))^(1/3)))*(-10/3+1/6*(19+12*87^(1/2))^(1/3)-23/6/(19+12*87^(1/2))^(1/3)+1/2*I*3^(1/2)*(1/3*(19+12*87^(1/2))
^(1/3)+23/3/(19+12*87^(1/2))^(1/3)))/(-10/3+1/6*(19+12*87^(1/2))^(1/3)-23/6/(19+12*87^(1/2))^(1/3)-1/2*I*3^(1/
2)*(1/3*(19+12*87^(1/2))^(1/3)+23/3/(19+12*87^(1/2))^(1/3)))/(1/2*(19+12*87^(1/2))^(1/3)-23/2/(19+12*87^(1/2))
^(1/3)+1/2*I*3^(1/2)*(1/3*(19+12*87^(1/2))^(1/3)+23/3/(19+12*87^(1/2))^(1/3))))^(1/2)))
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {5 \, x + 2}{\sqrt {x^{4} + 2 \, x^{3} + 5 \, x^{2} + 20 \, x - 12}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((2+5*x)/(x^4+2*x^3+5*x^2+20*x-12)^(1/2),x, algorithm="maxima")
[Out]
integrate((5*x + 2)/sqrt(x^4 + 2*x^3 + 5*x^2 + 20*x - 12), x)
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {5\,x+2}{\sqrt {x^4+2\,x^3+5\,x^2+20\,x-12}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((5*x + 2)/(20*x + 5*x^2 + 2*x^3 + x^4 - 12)^(1/2),x)
[Out]
int((5*x + 2)/(20*x + 5*x^2 + 2*x^3 + x^4 - 12)^(1/2), x)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {5 x + 2}{\sqrt {\left (x + 3\right ) \left (x^{3} - x^{2} + 8 x - 4\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((2+5*x)/(x**4+2*x**3+5*x**2+20*x-12)**(1/2),x)
[Out]
Integral((5*x + 2)/sqrt((x + 3)*(x**3 - x**2 + 8*x - 4)), x)
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