3.9.47 \(\int \frac {2+5 x}{\sqrt {-12+20 x+5 x^2+2 x^3+x^4}} \, dx\)

Optimal. Leaf size=64 \[ -\log \left (-x^5-3 x^4-8 x^3-24 x^2+\left (x^3+2 x^2+4 x+8\right ) \sqrt {x^4+2 x^3+5 x^2+20 x-12}-16 x-16\right ) \]

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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 = 5.16, size = 64, normalized size = 1.00 \begin {gather*} -\log \left (-16-16 x-24 x^2-8 x^3-3 x^4-x^5+\left (8+4 x+2 x^2+x^3\right ) \sqrt {-12+20 x+5 x^2+2 x^3+x^4}\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 = 0.52, 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)

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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)

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maple [A]  time = 1.24, size = 123, normalized size = 1.92

method	result	size
trager	\(-\ln \left (-x^{5}+\sqrt {x^{4}+2 x^{3}+5 x^{2}+20 x -12}\, x^{3}-3 x^{4}+2 \sqrt {x^{4}+2 x^{3}+5 x^{2}+20 x -12}\, x^{2}-8 x^{3}+4 \sqrt {x^{4}+2 x^{3}+5 x^{2}+20 x -12}\, x -24 x^{2}+8 \sqrt {x^{4}+2 x^{3}+5 x^{2}+20 x -12}-16 x -16\right )\)	\(123\)
default	\(\text {Expression too large to display}\)	\(2769\)
elliptic	\(\text {Expression too large to display}\)	\(2769\)

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,method=_RETURNVERBOSE)

[Out]

-ln(-x^5+(x^4+2*x^3+5*x^2+20*x-12)^(1/2)*x^3-3*x^4+2*(x^4+2*x^3+5*x^2+20*x-12)^(1/2)*x^2-8*x^3+4*(x^4+2*x^3+5*
x^2+20*x-12)^(1/2)*x-24*x^2+8*(x^4+2*x^3+5*x^2+20*x-12)^(1/2)-16*x-16)

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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)

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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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