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3.5.23 \(\int \frac {1+x}{\sqrt {2-5 x-3 x^2+2 x^3+x^4}} \, dx\) [423]

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

[Out]

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

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Rubi [F]
time = 0.09, 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 [A]
time = 0.12, size = 34, normalized size = 1.00 \begin {gather*} \frac {2}{3} \tanh ^{-1}\left (\frac {-2+x+x^2}{\sqrt {2-5 x-3 x^2+2 x^3+x^4}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(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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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order 3.
time = 1.22, size = 2934, normalized size = 86.29

method result size
trager \(-\frac {\ln \left (-2 x^{3}+2 \sqrt {x^{4}+2 x^{3}-3 x^{2}-5 x +2}\, x -2 \sqrt {x^{4}+2 x^{3}-3 x^{2}-5 x +2}+6 x -3\right )}{3}\) \(59\)
default \(\text {Expression too large to display}\) \(2934\)
elliptic \(\text {Expression too large to display}\) \(2934\)

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,method=_RETURNVERBOSE)

[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)))*(x+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)/(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)/((x+2)*(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))
)*(x+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)/(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*(-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)))*(x+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)/(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)/((x+2)*(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)))*(x+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)/(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...

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \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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Fricas [A]
time = 0.38, 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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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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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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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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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