3.5.23 \(\int \frac {1+x}{\sqrt {2-5 x-3 x^2+2 x^3+x^4}} \, dx\)

Optimal. Leaf size=34 \[ \frac {2}{3} \tanh ^{-1}\left (\frac {x^2+x-2}{\sqrt {x^4+2 x^3-3 x^2-5 x+2}}\right ) \]

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Rubi [F]  time = 0.15, 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 {1+x}{\sqrt {2-5 x-3 x^2+2 x^3+x^4}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(1 + x)/Sqrt[2 - 5*x - 3*x^2 + 2*x^3 + x^4],x]

[Out]

Defer[Int][1/Sqrt[2 - 5*x - 3*x^2 + 2*x^3 + x^4], x] + Defer[Int][x/Sqrt[2 - 5*x - 3*x^2 + 2*x^3 + x^4], x]

Rubi steps

\begin {align*} \int \frac {1+x}{\sqrt {2-5 x-3 x^2+2 x^3+x^4}} \, dx &=\int \left (\frac {1}{\sqrt {2-5 x-3 x^2+2 x^3+x^4}}+\frac {x}{\sqrt {2-5 x-3 x^2+2 x^3+x^4}}\right ) \, dx\\ &=\int \frac {1}{\sqrt {2-5 x-3 x^2+2 x^3+x^4}} \, dx+\int \frac {x}{\sqrt {2-5 x-3 x^2+2 x^3+x^4}} \, dx\\ \end {align*}

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Mathematica [C]  time = 0.54, size = 830, normalized size = 24.41

result too large to display

Antiderivative was successfully verified.

[In]

Integrate[(1 + x)/Sqrt[2 - 5*x - 3*x^2 + 2*x^3 + x^4],x]

[Out]

(-2*Sqrt[(2 + x)/(x - Root[1 - 3*#1 + #1^3 & , 1, 0])]*(x - Root[1 - 3*#1 + #1^3 & , 1, 0])^2*(EllipticF[ArcSi
n[Sqrt[((2 + x)*(-Root[1 - 3*#1 + #1^3 & , 1, 0] + Root[1 - 3*#1 + #1^3 & , 3, 0]))/((x - Root[1 - 3*#1 + #1^3
& , 1, 0])*(2 + Root[1 - 3*#1 + #1^3 & , 3, 0]))]], ((Root[1 - 3*#1 + #1^3 & , 1, 0] - Root[1 - 3*#1 + #1^3 &
, 2, 0])*(2 + Root[1 - 3*#1 + #1^3 & , 3, 0]))/((2 + Root[1 - 3*#1 + #1^3 & , 2, 0])*(Root[1 - 3*#1 + #1^3 &
, 1, 0] - Root[1 - 3*#1 + #1^3 & , 3, 0]))] + EllipticF[ArcSin[Sqrt[-(((2 + x)*(Root[1 - 3*#1 + #1^3 & , 1, 0]
- Root[1 - 3*#1 + #1^3 & , 3, 0]))/((x - Root[1 - 3*#1 + #1^3 & , 1, 0])*(2 + Root[1 - 3*#1 + #1^3 & , 3, 0])
))]], ((Root[1 - 3*#1 + #1^3 & , 1, 0] - Root[1 - 3*#1 + #1^3 & , 2, 0])*(2 + Root[1 - 3*#1 + #1^3 & , 3, 0]))
/((2 + Root[1 - 3*#1 + #1^3 & , 2, 0])*(Root[1 - 3*#1 + #1^3 & , 1, 0] - Root[1 - 3*#1 + #1^3 & , 3, 0]))]*Roo
t[1 - 3*#1 + #1^3 & , 1, 0] - EllipticPi[(2 + Root[1 - 3*#1 + #1^3 & , 3, 0])/(-Root[1 - 3*#1 + #1^3 & , 1, 0]
+ Root[1 - 3*#1 + #1^3 & , 3, 0]), ArcSin[Sqrt[-(((2 + x)*(Root[1 - 3*#1 + #1^3 & , 1, 0] - Root[1 - 3*#1 + #
1^3 & , 3, 0]))/((x - Root[1 - 3*#1 + #1^3 & , 1, 0])*(2 + Root[1 - 3*#1 + #1^3 & , 3, 0])))]], ((Root[1 - 3*#
1 + #1^3 & , 1, 0] - Root[1 - 3*#1 + #1^3 & , 2, 0])*(2 + Root[1 - 3*#1 + #1^3 & , 3, 0]))/((2 + Root[1 - 3*#1
+ #1^3 & , 2, 0])*(Root[1 - 3*#1 + #1^3 & , 1, 0] - Root[1 - 3*#1 + #1^3 & , 3, 0]))]*(2 + Root[1 - 3*#1 + #1
^3 & , 1, 0]))*Sqrt[(x - Root[1 - 3*#1 + #1^3 & , 2, 0])/(x - Root[1 - 3*#1 + #1^3 & , 1, 0])]*Sqrt[(x - Root[
1 - 3*#1 + #1^3 & , 3, 0])/(x - Root[1 - 3*#1 + #1^3 & , 1, 0])])/(Sqrt[2 - 5*x - 3*x^2 + 2*x^3 + x^4]*Sqrt[(2
+ Root[1 - 3*#1 + #1^3 & , 2, 0])*(-Root[1 - 3*#1 + #1^3 & , 1, 0] + Root[1 - 3*#1 + #1^3 & , 3, 0])])

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IntegrateAlgebraic [A]  time = 0.19, size = 34, normalized size = 1.00 \begin {gather*} \frac {2}{3} \tanh ^{-1}\left (\frac {x^2+x-2}{\sqrt {x^4+2 x^3-3 x^2-5 x+2}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(1 + x)/Sqrt[2 - 5*x - 3*x^2 + 2*x^3 + x^4],x]

[Out]

(2*ArcTanh[(-2 + x + x^2)/Sqrt[2 - 5*x - 3*x^2 + 2*x^3 + x^4]])/3

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fricas [A]  time = 0.43, size = 38, normalized size = 1.12 \begin {gather*} \frac {1}{3} \, \log \left (2 \, x^{3} + 2 \, \sqrt {x^{4} + 2 \, x^{3} - 3 \, x^{2} - 5 \, x + 2} {\left (x - 1\right )} - 6 \, x + 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)/(x^4+2*x^3-3*x^2-5*x+2)^(1/2),x, algorithm="fricas")

[Out]

1/3*log(2*x^3 + 2*sqrt(x^4 + 2*x^3 - 3*x^2 - 5*x + 2)*(x - 1) - 6*x + 3)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x + 1}{\sqrt {x^{4} + 2 \, x^{3} - 3 \, x^{2} - 5 \, x + 2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)/(x^4+2*x^3-3*x^2-5*x+2)^(1/2),x, algorithm="giac")

[Out]

integrate((x + 1)/sqrt(x^4 + 2*x^3 - 3*x^2 - 5*x + 2), x)

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maple [C]  time = 0.79, size = 2934, normalized size = 86.29 \begin {gather*} \text {output too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1+x)/(x^4+2*x^3-3*x^2-5*x+2)^(1/2),x)

[Out]

2*(-2+1/4*(-4+4*I*3^(1/2))^(1/3)+1/(-4+4*I*3^(1/2))^(1/3)+1/2*I*3^(1/2)*(1/2*(-4+4*I*3^(1/2))^(1/3)-2/(-4+4*I*
3^(1/2))^(1/3)))*(-(3/4*(-4+4*I*3^(1/2))^(1/3)+3/(-4+4*I*3^(1/2))^(1/3)+1/2*I*3^(1/2)*(1/2*(-4+4*I*3^(1/2))^(1
/3)-2/(-4+4*I*3^(1/2))^(1/3)))*(2+x)/(-1/4*(-4+4*I*3^(1/2))^(1/3)-1/(-4+4*I*3^(1/2))^(1/3)-1/2*I*3^(1/2)*(1/2*
(-4+4*I*3^(1/2))^(1/3)-2/(-4+4*I*3^(1/2))^(1/3))+2)/(x-1/2*(-4+4*I*3^(1/2))^(1/3)-2/(-4+4*I*3^(1/2))^(1/3)))^(
1/2)*(x-1/2*(-4+4*I*3^(1/2))^(1/3)-2/(-4+4*I*3^(1/2))^(1/3))^2*((1/2*(-4+4*I*3^(1/2))^(1/3)+2/(-4+4*I*3^(1/2))
^(1/3)+2)*(x+1/4*(-4+4*I*3^(1/2))^(1/3)+1/(-4+4*I*3^(1/2))^(1/3)-1/2*I*3^(1/2)*(1/2*(-4+4*I*3^(1/2))^(1/3)-2/(
-4+4*I*3^(1/2))^(1/3)))/(-1/4*(-4+4*I*3^(1/2))^(1/3)-1/(-4+4*I*3^(1/2))^(1/3)+1/2*I*3^(1/2)*(1/2*(-4+4*I*3^(1/
2))^(1/3)-2/(-4+4*I*3^(1/2))^(1/3))+2)/(x-1/2*(-4+4*I*3^(1/2))^(1/3)-2/(-4+4*I*3^(1/2))^(1/3)))^(1/2)*((1/2*(-
4+4*I*3^(1/2))^(1/3)+2/(-4+4*I*3^(1/2))^(1/3)+2)*(x+1/4*(-4+4*I*3^(1/2))^(1/3)+1/(-4+4*I*3^(1/2))^(1/3)+1/2*I*
3^(1/2)*(1/2*(-4+4*I*3^(1/2))^(1/3)-2/(-4+4*I*3^(1/2))^(1/3)))/(-1/4*(-4+4*I*3^(1/2))^(1/3)-1/(-4+4*I*3^(1/2))
^(1/3)-1/2*I*3^(1/2)*(1/2*(-4+4*I*3^(1/2))^(1/3)-2/(-4+4*I*3^(1/2))^(1/3))+2)/(x-1/2*(-4+4*I*3^(1/2))^(1/3)-2/
(-4+4*I*3^(1/2))^(1/3)))^(1/2)/(-3/4*(-4+4*I*3^(1/2))^(1/3)-3/(-4+4*I*3^(1/2))^(1/3)-1/2*I*3^(1/2)*(1/2*(-4+4*
I*3^(1/2))^(1/3)-2/(-4+4*I*3^(1/2))^(1/3)))/(1/2*(-4+4*I*3^(1/2))^(1/3)+2/(-4+4*I*3^(1/2))^(1/3)+2)/((2+x)*(x-
1/2*(-4+4*I*3^(1/2))^(1/3)-2/(-4+4*I*3^(1/2))^(1/3))*(x+1/4*(-4+4*I*3^(1/2))^(1/3)+1/(-4+4*I*3^(1/2))^(1/3)-1/
2*I*3^(1/2)*(1/2*(-4+4*I*3^(1/2))^(1/3)-2/(-4+4*I*3^(1/2))^(1/3)))*(x+1/4*(-4+4*I*3^(1/2))^(1/3)+1/(-4+4*I*3^(
1/2))^(1/3)+1/2*I*3^(1/2)*(1/2*(-4+4*I*3^(1/2))^(1/3)-2/(-4+4*I*3^(1/2))^(1/3))))^(1/2)*((1/2*(-4+4*I*3^(1/2))
^(1/3)+2/(-4+4*I*3^(1/2))^(1/3))*EllipticF((-(3/4*(-4+4*I*3^(1/2))^(1/3)+3/(-4+4*I*3^(1/2))^(1/3)+1/2*I*3^(1/2
)*(1/2*(-4+4*I*3^(1/2))^(1/3)-2/(-4+4*I*3^(1/2))^(1/3)))*(2+x)/(-1/4*(-4+4*I*3^(1/2))^(1/3)-1/(-4+4*I*3^(1/2))
^(1/3)-1/2*I*3^(1/2)*(1/2*(-4+4*I*3^(1/2))^(1/3)-2/(-4+4*I*3^(1/2))^(1/3))+2)/(x-1/2*(-4+4*I*3^(1/2))^(1/3)-2/
(-4+4*I*3^(1/2))^(1/3)))^(1/2),((3/4*(-4+4*I*3^(1/2))^(1/3)+3/(-4+4*I*3^(1/2))^(1/3)-1/2*I*3^(1/2)*(1/2*(-4+4*
I*3^(1/2))^(1/3)-2/(-4+4*I*3^(1/2))^(1/3)))*(-1/4*(-4+4*I*3^(1/2))^(1/3)-1/(-4+4*I*3^(1/2))^(1/3)-1/2*I*3^(1/2
)*(1/2*(-4+4*I*3^(1/2))^(1/3)-2/(-4+4*I*3^(1/2))^(1/3))+2)/(-1/4*(-4+4*I*3^(1/2))^(1/3)-1/(-4+4*I*3^(1/2))^(1/
3)+1/2*I*3^(1/2)*(1/2*(-4+4*I*3^(1/2))^(1/3)-2/(-4+4*I*3^(1/2))^(1/3))+2)/(3/4*(-4+4*I*3^(1/2))^(1/3)+3/(-4+4*
I*3^(1/2))^(1/3)+1/2*I*3^(1/2)*(1/2*(-4+4*I*3^(1/2))^(1/3)-2/(-4+4*I*3^(1/2))^(1/3))))^(1/2))+(-2-1/2*(-4+4*I*
3^(1/2))^(1/3)-2/(-4+4*I*3^(1/2))^(1/3))*EllipticPi((-(3/4*(-4+4*I*3^(1/2))^(1/3)+3/(-4+4*I*3^(1/2))^(1/3)+1/2
*I*3^(1/2)*(1/2*(-4+4*I*3^(1/2))^(1/3)-2/(-4+4*I*3^(1/2))^(1/3)))*(2+x)/(-1/4*(-4+4*I*3^(1/2))^(1/3)-1/(-4+4*I
*3^(1/2))^(1/3)-1/2*I*3^(1/2)*(1/2*(-4+4*I*3^(1/2))^(1/3)-2/(-4+4*I*3^(1/2))^(1/3))+2)/(x-1/2*(-4+4*I*3^(1/2))
^(1/3)-2/(-4+4*I*3^(1/2))^(1/3)))^(1/2),(-1/4*(-4+4*I*3^(1/2))^(1/3)-1/(-4+4*I*3^(1/2))^(1/3)-1/2*I*3^(1/2)*(1
/2*(-4+4*I*3^(1/2))^(1/3)-2/(-4+4*I*3^(1/2))^(1/3))+2)/(-3/4*(-4+4*I*3^(1/2))^(1/3)-3/(-4+4*I*3^(1/2))^(1/3)-1
/2*I*3^(1/2)*(1/2*(-4+4*I*3^(1/2))^(1/3)-2/(-4+4*I*3^(1/2))^(1/3))),((3/4*(-4+4*I*3^(1/2))^(1/3)+3/(-4+4*I*3^(
1/2))^(1/3)-1/2*I*3^(1/2)*(1/2*(-4+4*I*3^(1/2))^(1/3)-2/(-4+4*I*3^(1/2))^(1/3)))*(-1/4*(-4+4*I*3^(1/2))^(1/3)-
1/(-4+4*I*3^(1/2))^(1/3)-1/2*I*3^(1/2)*(1/2*(-4+4*I*3^(1/2))^(1/3)-2/(-4+4*I*3^(1/2))^(1/3))+2)/(-1/4*(-4+4*I*
3^(1/2))^(1/3)-1/(-4+4*I*3^(1/2))^(1/3)+1/2*I*3^(1/2)*(1/2*(-4+4*I*3^(1/2))^(1/3)-2/(-4+4*I*3^(1/2))^(1/3))+2)
/(3/4*(-4+4*I*3^(1/2))^(1/3)+3/(-4+4*I*3^(1/2))^(1/3)+1/2*I*3^(1/2)*(1/2*(-4+4*I*3^(1/2))^(1/3)-2/(-4+4*I*3^(1
/2))^(1/3))))^(1/2)))+2*(-2+1/4*(-4+4*I*3^(1/2))^(1/3)+1/(-4+4*I*3^(1/2))^(1/3)+1/2*I*3^(1/2)*(1/2*(-4+4*I*3^(
1/2))^(1/3)-2/(-4+4*I*3^(1/2))^(1/3)))*(-(3/4*(-4+4*I*3^(1/2))^(1/3)+3/(-4+4*I*3^(1/2))^(1/3)+1/2*I*3^(1/2)*(1
/2*(-4+4*I*3^(1/2))^(1/3)-2/(-4+4*I*3^(1/2))^(1/3)))*(2+x)/(-1/4*(-4+4*I*3^(1/2))^(1/3)-1/(-4+4*I*3^(1/2))^(1/
3)-1/2*I*3^(1/2)*(1/2*(-4+4*I*3^(1/2))^(1/3)-2/(-4+4*I*3^(1/2))^(1/3))+2)/(x-1/2*(-4+4*I*3^(1/2))^(1/3)-2/(-4+
4*I*3^(1/2))^(1/3)))^(1/2)*(x-1/2*(-4+4*I*3^(1/2))^(1/3)-2/(-4+4*I*3^(1/2))^(1/3))^2*((1/2*(-4+4*I*3^(1/2))^(1
/3)+2/(-4+4*I*3^(1/2))^(1/3)+2)*(x+1/4*(-4+4*I*3^(1/2))^(1/3)+1/(-4+4*I*3^(1/2))^(1/3)-1/2*I*3^(1/2)*(1/2*(-4+
4*I*3^(1/2))^(1/3)-2/(-4+4*I*3^(1/2))^(1/3)))/(-1/4*(-4+4*I*3^(1/2))^(1/3)-1/(-4+4*I*3^(1/2))^(1/3)+1/2*I*3^(1
/2)*(1/2*(-4+4*I*3^(1/2))^(1/3)-2/(-4+4*I*3^(1/2))^(1/3))+2)/(x-1/2*(-4+4*I*3^(1/2))^(1/3)-2/(-4+4*I*3^(1/2))^
(1/3)))^(1/2)*((1/2*(-4+4*I*3^(1/2))^(1/3)+2/(-4+4*I*3^(1/2))^(1/3)+2)*(x+1/4*(-4+4*I*3^(1/2))^(1/3)+1/(-4+4*I
*3^(1/2))^(1/3)+1/2*I*3^(1/2)*(1/2*(-4+4*I*3^(1/2))^(1/3)-2/(-4+4*I*3^(1/2))^(1/3)))/(-1/4*(-4+4*I*3^(1/2))^(1
/3)-1/(-4+4*I*3^(1/2))^(1/3)-1/2*I*3^(1/2)*(1/2*(-4+4*I*3^(1/2))^(1/3)-2/(-4+4*I*3^(1/2))^(1/3))+2)/(x-1/2*(-4
+4*I*3^(1/2))^(1/3)-2/(-4+4*I*3^(1/2))^(1/3)))^(1/2)/(-3/4*(-4+4*I*3^(1/2))^(1/3)-3/(-4+4*I*3^(1/2))^(1/3)-1/2
*I*3^(1/2)*(1/2*(-4+4*I*3^(1/2))^(1/3)-2/(-4+4*I*3^(1/2))^(1/3)))/(1/2*(-4+4*I*3^(1/2))^(1/3)+2/(-4+4*I*3^(1/2
))^(1/3)+2)/((2+x)*(x-1/2*(-4+4*I*3^(1/2))^(1/3)-2/(-4+4*I*3^(1/2))^(1/3))*(x+1/4*(-4+4*I*3^(1/2))^(1/3)+1/(-4
+4*I*3^(1/2))^(1/3)-1/2*I*3^(1/2)*(1/2*(-4+4*I*3^(1/2))^(1/3)-2/(-4+4*I*3^(1/2))^(1/3)))*(x+1/4*(-4+4*I*3^(1/2
))^(1/3)+1/(-4+4*I*3^(1/2))^(1/3)+1/2*I*3^(1/2)*(1/2*(-4+4*I*3^(1/2))^(1/3)-2/(-4+4*I*3^(1/2))^(1/3))))^(1/2)*
EllipticF((-(3/4*(-4+4*I*3^(1/2))^(1/3)+3/(-4+4*I*3^(1/2))^(1/3)+1/2*I*3^(1/2)*(1/2*(-4+4*I*3^(1/2))^(1/3)-2/(
-4+4*I*3^(1/2))^(1/3)))*(2+x)/(-1/4*(-4+4*I*3^(1/2))^(1/3)-1/(-4+4*I*3^(1/2))^(1/3)-1/2*I*3^(1/2)*(1/2*(-4+4*I
*3^(1/2))^(1/3)-2/(-4+4*I*3^(1/2))^(1/3))+2)/(x-1/2*(-4+4*I*3^(1/2))^(1/3)-2/(-4+4*I*3^(1/2))^(1/3)))^(1/2),((
3/4*(-4+4*I*3^(1/2))^(1/3)+3/(-4+4*I*3^(1/2))^(1/3)-1/2*I*3^(1/2)*(1/2*(-4+4*I*3^(1/2))^(1/3)-2/(-4+4*I*3^(1/2
))^(1/3)))*(-1/4*(-4+4*I*3^(1/2))^(1/3)-1/(-4+4*I*3^(1/2))^(1/3)-1/2*I*3^(1/2)*(1/2*(-4+4*I*3^(1/2))^(1/3)-2/(
-4+4*I*3^(1/2))^(1/3))+2)/(-1/4*(-4+4*I*3^(1/2))^(1/3)-1/(-4+4*I*3^(1/2))^(1/3)+1/2*I*3^(1/2)*(1/2*(-4+4*I*3^(
1/2))^(1/3)-2/(-4+4*I*3^(1/2))^(1/3))+2)/(3/4*(-4+4*I*3^(1/2))^(1/3)+3/(-4+4*I*3^(1/2))^(1/3)+1/2*I*3^(1/2)*(1
/2*(-4+4*I*3^(1/2))^(1/3)-2/(-4+4*I*3^(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 {x + 1}{\sqrt {x^{4} + 2 \, x^{3} - 3 \, x^{2} - 5 \, x + 2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)/(x^4+2*x^3-3*x^2-5*x+2)^(1/2),x, algorithm="maxima")

[Out]

integrate((x + 1)/sqrt(x^4 + 2*x^3 - 3*x^2 - 5*x + 2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {x+1}{\sqrt {x^4+2\,x^3-3\,x^2-5\,x+2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x + 1)/(2*x^3 - 3*x^2 - 5*x + x^4 + 2)^(1/2),x)

[Out]

int((x + 1)/(2*x^3 - 3*x^2 - 5*x + x^4 + 2)^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x + 1}{\sqrt {\left (x + 2\right ) \left (x^{3} - 3 x + 1\right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)/(x**4+2*x**3-3*x**2-5*x+2)**(1/2),x)

[Out]

Integral((x + 1)/sqrt((x + 2)*(x**3 - 3*x + 1)), x)

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