3.6.71 \(\int \frac {-3 x+2 x^2}{(-2+2 x+x^3) \sqrt {-2 x+2 x^2+3 x^4}} \, dx\)
Optimal. Leaf size=44 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt {3 x^4+2 x^2-2 x}}{3 x^3+2 x-2}\right )}{\sqrt {2}} \]
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Rubi [F] time = 1.38, 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 {-3 x+2 x^2}{\left (-2+2 x+x^3\right ) \sqrt {-2 x+2 x^2+3 x^4}} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
Int[(-3*x + 2*x^2)/((-2 + 2*x + x^3)*Sqrt[-2*x + 2*x^2 + 3*x^4]),x]
[Out]
(-6*Sqrt[x]*Sqrt[-2 + 2*x + 3*x^3]*Defer[Subst][Defer[Int][x^2/((-2 + 2*x^2 + x^6)*Sqrt[-2 + 2*x^2 + 3*x^6]),
x], x, Sqrt[x]])/Sqrt[-2*x + 2*x^2 + 3*x^4] + (4*Sqrt[x]*Sqrt[-2 + 2*x + 3*x^3]*Defer[Subst][Defer[Int][x^4/((
-2 + 2*x^2 + x^6)*Sqrt[-2 + 2*x^2 + 3*x^6]), x], x, Sqrt[x]])/Sqrt[-2*x + 2*x^2 + 3*x^4]
Rubi steps
\begin {align*} \int \frac {-3 x+2 x^2}{\left (-2+2 x+x^3\right ) \sqrt {-2 x+2 x^2+3 x^4}} \, dx &=\int \frac {x (-3+2 x)}{\left (-2+2 x+x^3\right ) \sqrt {-2 x+2 x^2+3 x^4}} \, dx\\ &=\frac {\left (\sqrt {x} \sqrt {-2+2 x+3 x^3}\right ) \int \frac {\sqrt {x} (-3+2 x)}{\left (-2+2 x+x^3\right ) \sqrt {-2+2 x+3 x^3}} \, dx}{\sqrt {-2 x+2 x^2+3 x^4}}\\ &=\frac {\left (2 \sqrt {x} \sqrt {-2+2 x+3 x^3}\right ) \operatorname {Subst}\left (\int \frac {x^2 \left (-3+2 x^2\right )}{\left (-2+2 x^2+x^6\right ) \sqrt {-2+2 x^2+3 x^6}} \, dx,x,\sqrt {x}\right )}{\sqrt {-2 x+2 x^2+3 x^4}}\\ &=\frac {\left (2 \sqrt {x} \sqrt {-2+2 x+3 x^3}\right ) \operatorname {Subst}\left (\int \left (-\frac {3 x^2}{\left (-2+2 x^2+x^6\right ) \sqrt {-2+2 x^2+3 x^6}}+\frac {2 x^4}{\left (-2+2 x^2+x^6\right ) \sqrt {-2+2 x^2+3 x^6}}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {-2 x+2 x^2+3 x^4}}\\ &=\frac {\left (4 \sqrt {x} \sqrt {-2+2 x+3 x^3}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\left (-2+2 x^2+x^6\right ) \sqrt {-2+2 x^2+3 x^6}} \, dx,x,\sqrt {x}\right )}{\sqrt {-2 x+2 x^2+3 x^4}}-\frac {\left (6 \sqrt {x} \sqrt {-2+2 x+3 x^3}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (-2+2 x^2+x^6\right ) \sqrt {-2+2 x^2+3 x^6}} \, dx,x,\sqrt {x}\right )}{\sqrt {-2 x+2 x^2+3 x^4}}\\ \end {align*}
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Mathematica [C] time = 4.60, size = 4253, normalized size = 96.66 \begin {gather*} \text {Result too large to show} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
Integrate[(-3*x + 2*x^2)/((-2 + 2*x + x^3)*Sqrt[-2*x + 2*x^2 + 3*x^4]),x]
[Out]
(2*(x - Root[-2 + 2*#1 + 3*#1^3 & , 1, 0])^2*(3*(Root[-2 + 2*#1 + #1^3 & , 2, 0] - Root[-2 + 2*#1 + #1^3 & , 3
, 0])*(Root[-2 + 2*#1 + #1^3 & , 2, 0] - Root[-2 + 2*#1 + 3*#1^3 & , 1, 0])*(Root[-2 + 2*#1 + #1^3 & , 3, 0] -
Root[-2 + 2*#1 + 3*#1^3 & , 1, 0])*(EllipticF[ArcSin[Sqrt[(x*(-Root[-2 + 2*#1 + 3*#1^3 & , 1, 0] + Root[-2 +
2*#1 + 3*#1^3 & , 3, 0]))/((x - Root[-2 + 2*#1 + 3*#1^3 & , 1, 0])*Root[-2 + 2*#1 + 3*#1^3 & , 3, 0])]], ((Roo
t[-2 + 2*#1 + 3*#1^3 & , 1, 0] - Root[-2 + 2*#1 + 3*#1^3 & , 2, 0])*Root[-2 + 2*#1 + 3*#1^3 & , 3, 0])/(Root[-
2 + 2*#1 + 3*#1^3 & , 2, 0]*(Root[-2 + 2*#1 + 3*#1^3 & , 1, 0] - Root[-2 + 2*#1 + 3*#1^3 & , 3, 0]))]*Root[-2
+ 2*#1 + #1^3 & , 1, 0] - EllipticPi[-(((Root[-2 + 2*#1 + #1^3 & , 1, 0] - Root[-2 + 2*#1 + 3*#1^3 & , 1, 0])*
Root[-2 + 2*#1 + 3*#1^3 & , 3, 0])/(Root[-2 + 2*#1 + #1^3 & , 1, 0]*(Root[-2 + 2*#1 + 3*#1^3 & , 1, 0] - Root[
-2 + 2*#1 + 3*#1^3 & , 3, 0]))), ArcSin[Sqrt[(x*(-Root[-2 + 2*#1 + 3*#1^3 & , 1, 0] + Root[-2 + 2*#1 + 3*#1^3
& , 3, 0]))/((x - Root[-2 + 2*#1 + 3*#1^3 & , 1, 0])*Root[-2 + 2*#1 + 3*#1^3 & , 3, 0])]], ((Root[-2 + 2*#1 +
3*#1^3 & , 1, 0] - Root[-2 + 2*#1 + 3*#1^3 & , 2, 0])*Root[-2 + 2*#1 + 3*#1^3 & , 3, 0])/(Root[-2 + 2*#1 + 3*#
1^3 & , 2, 0]*(Root[-2 + 2*#1 + 3*#1^3 & , 1, 0] - Root[-2 + 2*#1 + 3*#1^3 & , 3, 0]))]*Root[-2 + 2*#1 + 3*#1^
3 & , 1, 0]) - 2*Root[-2 + 2*#1 + #1^3 & , 1, 0]*(Root[-2 + 2*#1 + #1^3 & , 2, 0] - Root[-2 + 2*#1 + #1^3 & ,
3, 0])*(Root[-2 + 2*#1 + #1^3 & , 2, 0] - Root[-2 + 2*#1 + 3*#1^3 & , 1, 0])*(Root[-2 + 2*#1 + #1^3 & , 3, 0]
- Root[-2 + 2*#1 + 3*#1^3 & , 1, 0])*(EllipticF[ArcSin[Sqrt[(x*(-Root[-2 + 2*#1 + 3*#1^3 & , 1, 0] + Root[-2 +
2*#1 + 3*#1^3 & , 3, 0]))/((x - Root[-2 + 2*#1 + 3*#1^3 & , 1, 0])*Root[-2 + 2*#1 + 3*#1^3 & , 3, 0])]], ((Ro
ot[-2 + 2*#1 + 3*#1^3 & , 1, 0] - Root[-2 + 2*#1 + 3*#1^3 & , 2, 0])*Root[-2 + 2*#1 + 3*#1^3 & , 3, 0])/(Root[
-2 + 2*#1 + 3*#1^3 & , 2, 0]*(Root[-2 + 2*#1 + 3*#1^3 & , 1, 0] - Root[-2 + 2*#1 + 3*#1^3 & , 3, 0]))]*Root[-2
+ 2*#1 + #1^3 & , 1, 0] - EllipticPi[-(((Root[-2 + 2*#1 + #1^3 & , 1, 0] - Root[-2 + 2*#1 + 3*#1^3 & , 1, 0])
*Root[-2 + 2*#1 + 3*#1^3 & , 3, 0])/(Root[-2 + 2*#1 + #1^3 & , 1, 0]*(Root[-2 + 2*#1 + 3*#1^3 & , 1, 0] - Root
[-2 + 2*#1 + 3*#1^3 & , 3, 0]))), ArcSin[Sqrt[(x*(-Root[-2 + 2*#1 + 3*#1^3 & , 1, 0] + Root[-2 + 2*#1 + 3*#1^3
& , 3, 0]))/((x - Root[-2 + 2*#1 + 3*#1^3 & , 1, 0])*Root[-2 + 2*#1 + 3*#1^3 & , 3, 0])]], ((Root[-2 + 2*#1 +
3*#1^3 & , 1, 0] - Root[-2 + 2*#1 + 3*#1^3 & , 2, 0])*Root[-2 + 2*#1 + 3*#1^3 & , 3, 0])/(Root[-2 + 2*#1 + 3*
#1^3 & , 2, 0]*(Root[-2 + 2*#1 + 3*#1^3 & , 1, 0] - Root[-2 + 2*#1 + 3*#1^3 & , 3, 0]))]*Root[-2 + 2*#1 + 3*#1
^3 & , 1, 0]) + 2*Root[-2 + 2*#1 + #1^3 & , 2, 0]*(Root[-2 + 2*#1 + #1^3 & , 1, 0] - Root[-2 + 2*#1 + #1^3 & ,
3, 0])*(Root[-2 + 2*#1 + #1^3 & , 1, 0] - Root[-2 + 2*#1 + 3*#1^3 & , 1, 0])*(Root[-2 + 2*#1 + #1^3 & , 3, 0]
- Root[-2 + 2*#1 + 3*#1^3 & , 1, 0])*(EllipticF[ArcSin[Sqrt[(x*(-Root[-2 + 2*#1 + 3*#1^3 & , 1, 0] + Root[-2
+ 2*#1 + 3*#1^3 & , 3, 0]))/((x - Root[-2 + 2*#1 + 3*#1^3 & , 1, 0])*Root[-2 + 2*#1 + 3*#1^3 & , 3, 0])]], ((R
oot[-2 + 2*#1 + 3*#1^3 & , 1, 0] - Root[-2 + 2*#1 + 3*#1^3 & , 2, 0])*Root[-2 + 2*#1 + 3*#1^3 & , 3, 0])/(Root
[-2 + 2*#1 + 3*#1^3 & , 2, 0]*(Root[-2 + 2*#1 + 3*#1^3 & , 1, 0] - Root[-2 + 2*#1 + 3*#1^3 & , 3, 0]))]*Root[-
2 + 2*#1 + #1^3 & , 2, 0] - EllipticPi[-(((Root[-2 + 2*#1 + #1^3 & , 2, 0] - Root[-2 + 2*#1 + 3*#1^3 & , 1, 0]
)*Root[-2 + 2*#1 + 3*#1^3 & , 3, 0])/(Root[-2 + 2*#1 + #1^3 & , 2, 0]*(Root[-2 + 2*#1 + 3*#1^3 & , 1, 0] - Roo
t[-2 + 2*#1 + 3*#1^3 & , 3, 0]))), ArcSin[Sqrt[(x*(-Root[-2 + 2*#1 + 3*#1^3 & , 1, 0] + Root[-2 + 2*#1 + 3*#1^
3 & , 3, 0]))/((x - Root[-2 + 2*#1 + 3*#1^3 & , 1, 0])*Root[-2 + 2*#1 + 3*#1^3 & , 3, 0])]], ((Root[-2 + 2*#1
+ 3*#1^3 & , 1, 0] - Root[-2 + 2*#1 + 3*#1^3 & , 2, 0])*Root[-2 + 2*#1 + 3*#1^3 & , 3, 0])/(Root[-2 + 2*#1 + 3
*#1^3 & , 2, 0]*(Root[-2 + 2*#1 + 3*#1^3 & , 1, 0] - Root[-2 + 2*#1 + 3*#1^3 & , 3, 0]))]*Root[-2 + 2*#1 + 3*#
1^3 & , 1, 0]) + 3*(Root[-2 + 2*#1 + #1^3 & , 1, 0] - Root[-2 + 2*#1 + #1^3 & , 3, 0])*(Root[-2 + 2*#1 + #1^3
& , 1, 0] - Root[-2 + 2*#1 + 3*#1^3 & , 1, 0])*(Root[-2 + 2*#1 + #1^3 & , 3, 0] - Root[-2 + 2*#1 + 3*#1^3 & ,
1, 0])*(-(EllipticF[ArcSin[Sqrt[(x*(-Root[-2 + 2*#1 + 3*#1^3 & , 1, 0] + Root[-2 + 2*#1 + 3*#1^3 & , 3, 0]))/(
(x - Root[-2 + 2*#1 + 3*#1^3 & , 1, 0])*Root[-2 + 2*#1 + 3*#1^3 & , 3, 0])]], ((Root[-2 + 2*#1 + 3*#1^3 & , 1,
0] - Root[-2 + 2*#1 + 3*#1^3 & , 2, 0])*Root[-2 + 2*#1 + 3*#1^3 & , 3, 0])/(Root[-2 + 2*#1 + 3*#1^3 & , 2, 0]
*(Root[-2 + 2*#1 + 3*#1^3 & , 1, 0] - Root[-2 + 2*#1 + 3*#1^3 & , 3, 0]))]*Root[-2 + 2*#1 + #1^3 & , 2, 0]) +
EllipticPi[-(((Root[-2 + 2*#1 + #1^3 & , 2, 0] - Root[-2 + 2*#1 + 3*#1^3 & , 1, 0])*Root[-2 + 2*#1 + 3*#1^3 &
, 3, 0])/(Root[-2 + 2*#1 + #1^3 & , 2, 0]*(Root[-2 + 2*#1 + 3*#1^3 & , 1, 0] - Root[-2 + 2*#1 + 3*#1^3 & , 3,
0]))), ArcSin[Sqrt[(x*(-Root[-2 + 2*#1 + 3*#1^3 & , 1, 0] + Root[-2 + 2*#1 + 3*#1^3 & , 3, 0]))/((x - Root[-2
+ 2*#1 + 3*#1^3 & , 1, 0])*Root[-2 + 2*#1 + 3*#1^3 & , 3, 0])]], ((Root[-2 + 2*#1 + 3*#1^3 & , 1, 0] - Root[-2
+ 2*#1 + 3*#1^3 & , 2, 0])*Root[-2 + 2*#1 + 3*#1^3 & , 3, 0])/(Root[-2 + 2*#1 + 3*#1^3 & , 2, 0]*(Root[-2 + 2
*#1 + 3*#1^3 & , 1, 0] - Root[-2 + 2*#1 + 3*#1^3 & , 3, 0]))]*Root[-2 + 2*#1 + 3*#1^3 & , 1, 0]) + 3*(Root[-2
+ 2*#1 + #1^3 & , 1, 0] - Root[-2 + 2*#1 + #1^3 & , 2, 0])*(Root[-2 + 2*#1 + #1^3 & , 1, 0] - Root[-2 + 2*#1 +
3*#1^3 & , 1, 0])*(Root[-2 + 2*#1 + #1^3 & , 2, 0] - Root[-2 + 2*#1 + 3*#1^3 & , 1, 0])*(EllipticF[ArcSin[Sqr
t[(x*(-Root[-2 + 2*#1 + 3*#1^3 & , 1, 0] + Root[-2 + 2*#1 + 3*#1^3 & , 3, 0]))/((x - Root[-2 + 2*#1 + 3*#1^3 &
, 1, 0])*Root[-2 + 2*#1 + 3*#1^3 & , 3, 0])]], ((Root[-2 + 2*#1 + 3*#1^3 & , 1, 0] - Root[-2 + 2*#1 + 3*#1^3
& , 2, 0])*Root[-2 + 2*#1 + 3*#1^3 & , 3, 0])/(Root[-2 + 2*#1 + 3*#1^3 & , 2, 0]*(Root[-2 + 2*#1 + 3*#1^3 & ,
1, 0] - Root[-2 + 2*#1 + 3*#1^3 & , 3, 0]))]*Root[-2 + 2*#1 + #1^3 & , 3, 0] - EllipticPi[-(((Root[-2 + 2*#1 +
#1^3 & , 3, 0] - Root[-2 + 2*#1 + 3*#1^3 & , 1, 0])*Root[-2 + 2*#1 + 3*#1^3 & , 3, 0])/(Root[-2 + 2*#1 + #1^3
& , 3, 0]*(Root[-2 + 2*#1 + 3*#1^3 & , 1, 0] - Root[-2 + 2*#1 + 3*#1^3 & , 3, 0]))), ArcSin[Sqrt[(x*(-Root[-2
+ 2*#1 + 3*#1^3 & , 1, 0] + Root[-2 + 2*#1 + 3*#1^3 & , 3, 0]))/((x - Root[-2 + 2*#1 + 3*#1^3 & , 1, 0])*Root
[-2 + 2*#1 + 3*#1^3 & , 3, 0])]], ((Root[-2 + 2*#1 + 3*#1^3 & , 1, 0] - Root[-2 + 2*#1 + 3*#1^3 & , 2, 0])*Roo
t[-2 + 2*#1 + 3*#1^3 & , 3, 0])/(Root[-2 + 2*#1 + 3*#1^3 & , 2, 0]*(Root[-2 + 2*#1 + 3*#1^3 & , 1, 0] - Root[-
2 + 2*#1 + 3*#1^3 & , 3, 0]))]*Root[-2 + 2*#1 + 3*#1^3 & , 1, 0]) - 2*(Root[-2 + 2*#1 + #1^3 & , 1, 0] - Root[
-2 + 2*#1 + #1^3 & , 2, 0])*Root[-2 + 2*#1 + #1^3 & , 3, 0]*(Root[-2 + 2*#1 + #1^3 & , 1, 0] - Root[-2 + 2*#1
+ 3*#1^3 & , 1, 0])*(Root[-2 + 2*#1 + #1^3 & , 2, 0] - Root[-2 + 2*#1 + 3*#1^3 & , 1, 0])*(EllipticF[ArcSin[Sq
rt[(x*(-Root[-2 + 2*#1 + 3*#1^3 & , 1, 0] + Root[-2 + 2*#1 + 3*#1^3 & , 3, 0]))/((x - Root[-2 + 2*#1 + 3*#1^3
& , 1, 0])*Root[-2 + 2*#1 + 3*#1^3 & , 3, 0])]], ((Root[-2 + 2*#1 + 3*#1^3 & , 1, 0] - Root[-2 + 2*#1 + 3*#1^3
& , 2, 0])*Root[-2 + 2*#1 + 3*#1^3 & , 3, 0])/(Root[-2 + 2*#1 + 3*#1^3 & , 2, 0]*(Root[-2 + 2*#1 + 3*#1^3 & ,
1, 0] - Root[-2 + 2*#1 + 3*#1^3 & , 3, 0]))]*Root[-2 + 2*#1 + #1^3 & , 3, 0] - EllipticPi[-(((Root[-2 + 2*#1
+ #1^3 & , 3, 0] - Root[-2 + 2*#1 + 3*#1^3 & , 1, 0])*Root[-2 + 2*#1 + 3*#1^3 & , 3, 0])/(Root[-2 + 2*#1 + #1^
3 & , 3, 0]*(Root[-2 + 2*#1 + 3*#1^3 & , 1, 0] - Root[-2 + 2*#1 + 3*#1^3 & , 3, 0]))), ArcSin[Sqrt[(x*(-Root[-
2 + 2*#1 + 3*#1^3 & , 1, 0] + Root[-2 + 2*#1 + 3*#1^3 & , 3, 0]))/((x - Root[-2 + 2*#1 + 3*#1^3 & , 1, 0])*Roo
t[-2 + 2*#1 + 3*#1^3 & , 3, 0])]], ((Root[-2 + 2*#1 + 3*#1^3 & , 1, 0] - Root[-2 + 2*#1 + 3*#1^3 & , 2, 0])*Ro
ot[-2 + 2*#1 + 3*#1^3 & , 3, 0])/(Root[-2 + 2*#1 + 3*#1^3 & , 2, 0]*(Root[-2 + 2*#1 + 3*#1^3 & , 1, 0] - Root[
-2 + 2*#1 + 3*#1^3 & , 3, 0]))]*Root[-2 + 2*#1 + 3*#1^3 & , 1, 0]))*Sqrt[(x - Root[-2 + 2*#1 + 3*#1^3 & , 2, 0
])/((x - Root[-2 + 2*#1 + 3*#1^3 & , 1, 0])*Root[-2 + 2*#1 + 3*#1^3 & , 2, 0])]*Sqrt[(x - Root[-2 + 2*#1 + 3*#
1^3 & , 3, 0])/((x - Root[-2 + 2*#1 + 3*#1^3 & , 1, 0])*Root[-2 + 2*#1 + 3*#1^3 & , 3, 0])]*Root[-2 + 2*#1 + 3
*#1^3 & , 3, 0]*Sqrt[(x*(-Root[-2 + 2*#1 + 3*#1^3 & , 1, 0] + Root[-2 + 2*#1 + 3*#1^3 & , 3, 0]))/((x - Root[-
2 + 2*#1 + 3*#1^3 & , 1, 0])*Root[-2 + 2*#1 + 3*#1^3 & , 3, 0])])/(Sqrt[x*(-2 + 2*x + 3*x^3)]*(Root[-2 + 2*#1
+ #1^3 & , 1, 0] - Root[-2 + 2*#1 + #1^3 & , 2, 0])*(Root[-2 + 2*#1 + #1^3 & , 1, 0] - Root[-2 + 2*#1 + #1^3 &
, 3, 0])*(Root[-2 + 2*#1 + #1^3 & , 2, 0] - Root[-2 + 2*#1 + #1^3 & , 3, 0])*(Root[-2 + 2*#1 + #1^3 & , 1, 0]
- Root[-2 + 2*#1 + 3*#1^3 & , 1, 0])*(Root[-2 + 2*#1 + #1^3 & , 2, 0] - Root[-2 + 2*#1 + 3*#1^3 & , 1, 0])*(R
oot[-2 + 2*#1 + #1^3 & , 3, 0] - Root[-2 + 2*#1 + 3*#1^3 & , 1, 0])*(Root[-2 + 2*#1 + 3*#1^3 & , 1, 0] - Root[
-2 + 2*#1 + 3*#1^3 & , 3, 0]))
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IntegrateAlgebraic [A] time = 0.40, size = 44, normalized size = 1.00 \begin {gather*} \frac {\tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt {-2 x+2 x^2+3 x^4}}{-2+2 x+3 x^3}\right )}{\sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[(-3*x + 2*x^2)/((-2 + 2*x + x^3)*Sqrt[-2*x + 2*x^2 + 3*x^4]),x]
[Out]
ArcTanh[(Sqrt[2]*x*Sqrt[-2*x + 2*x^2 + 3*x^4])/(-2 + 2*x + 3*x^3)]/Sqrt[2]
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fricas [B] time = 0.53, size = 93, normalized size = 2.11 \begin {gather*} \frac {1}{8} \, \sqrt {2} \log \left (-\frac {49 \, x^{6} + 36 \, x^{4} - 36 \, x^{3} + 4 \, \sqrt {2} {\left (5 \, x^{4} + 2 \, x^{2} - 2 \, x\right )} \sqrt {3 \, x^{4} + 2 \, x^{2} - 2 \, x} + 4 \, x^{2} - 8 \, x + 4}{x^{6} + 4 \, x^{4} - 4 \, x^{3} + 4 \, x^{2} - 8 \, x + 4}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((2*x^2-3*x)/(x^3+2*x-2)/(3*x^4+2*x^2-2*x)^(1/2),x, algorithm="fricas")
[Out]
1/8*sqrt(2)*log(-(49*x^6 + 36*x^4 - 36*x^3 + 4*sqrt(2)*(5*x^4 + 2*x^2 - 2*x)*sqrt(3*x^4 + 2*x^2 - 2*x) + 4*x^2
- 8*x + 4)/(x^6 + 4*x^4 - 4*x^3 + 4*x^2 - 8*x + 4))
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 \, x^{2} - 3 \, x}{\sqrt {3 \, x^{4} + 2 \, x^{2} - 2 \, x} {\left (x^{3} + 2 \, x - 2\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((2*x^2-3*x)/(x^3+2*x-2)/(3*x^4+2*x^2-2*x)^(1/2),x, algorithm="giac")
[Out]
integrate((2*x^2 - 3*x)/(sqrt(3*x^4 + 2*x^2 - 2*x)*(x^3 + 2*x - 2)), x)
________________________________________________________________________________________
maple [C] time = 1.55, size = 69, normalized size = 1.57
| | |
| method |
result |
size |
| | |
| trager |
\(-\frac {\RootOf \left (\textit {\_Z}^{2}-2\right ) \ln \left (\frac {5 \RootOf \left (\textit {\_Z}^{2}-2\right ) x^{3}+2 \RootOf \left (\textit {\_Z}^{2}-2\right ) x -4 x \sqrt {3 x^{4}+2 x^{2}-2 x}-2 \RootOf \left (\textit {\_Z}^{2}-2\right )}{x^{3}+2 x -2}\right )}{4}\) |
\(69\) |
| default |
\(\text {Expression too large to display}\) |
\(1659\) |
| elliptic |
\(\text {Expression too large to display}\) |
\(1659\) |
| | |
|
|
|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((2*x^2-3*x)/(x^3+2*x-2)/(3*x^4+2*x^2-2*x)^(1/2),x,method=_RETURNVERBOSE)
[Out]
-1/4*RootOf(_Z^2-2)*ln((5*RootOf(_Z^2-2)*x^3+2*RootOf(_Z^2-2)*x-4*x*(3*x^4+2*x^2-2*x)^(1/2)-2*RootOf(_Z^2-2))/
(x^3+2*x-2))
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 \, x^{2} - 3 \, x}{\sqrt {3 \, x^{4} + 2 \, x^{2} - 2 \, x} {\left (x^{3} + 2 \, x - 2\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((2*x^2-3*x)/(x^3+2*x-2)/(3*x^4+2*x^2-2*x)^(1/2),x, algorithm="maxima")
[Out]
integrate((2*x^2 - 3*x)/(sqrt(3*x^4 + 2*x^2 - 2*x)*(x^3 + 2*x - 2)), x)
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} -\int \frac {3\,x-2\,x^2}{\left (x^3+2\,x-2\right )\,\sqrt {3\,x^4+2\,x^2-2\,x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(-(3*x - 2*x^2)/((2*x + x^3 - 2)*(2*x^2 - 2*x + 3*x^4)^(1/2)),x)
[Out]
-int((3*x - 2*x^2)/((2*x + x^3 - 2)*(2*x^2 - 2*x + 3*x^4)^(1/2)), x)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x \left (2 x - 3\right )}{\sqrt {x \left (3 x^{3} + 2 x - 2\right )} \left (x^{3} + 2 x - 2\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((2*x**2-3*x)/(x**3+2*x-2)/(3*x**4+2*x**2-2*x)**(1/2),x)
[Out]
Integral(x*(2*x - 3)/(sqrt(x*(3*x**3 + 2*x - 2))*(x**3 + 2*x - 2)), x)
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