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

Optimal. Leaf size=101 \[ \frac {19}{96} \sqrt {2 x^2-x+3} x^2-\frac {409}{768} \sqrt {2 x^2-x+3} x-\frac {505 \sqrt {2 x^2-x+3}}{1024}+\frac {5}{8} \sqrt {2 x^2-x+3} x^3-\frac {6863 \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{2048 \sqrt {2}} \]

[Out]

-6863/4096*arcsinh(1/23*(1-4*x)*23^(1/2))*2^(1/2)-505/1024*(2*x^2-x+3)^(1/2)-409/768*x*(2*x^2-x+3)^(1/2)+19/96
*x^2*(2*x^2-x+3)^(1/2)+5/8*x^3*(2*x^2-x+3)^(1/2)

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Rubi [A]  time = 0.08, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {1661, 640, 619, 215} \[ \frac {5}{8} \sqrt {2 x^2-x+3} x^3+\frac {19}{96} \sqrt {2 x^2-x+3} x^2-\frac {409}{768} \sqrt {2 x^2-x+3} x-\frac {505 \sqrt {2 x^2-x+3}}{1024}-\frac {6863 \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{2048 \sqrt {2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-505*Sqrt[3 - x + 2*x^2])/1024 - (409*x*Sqrt[3 - x + 2*x^2])/768 + (19*x^2*Sqrt[3 - x + 2*x^2])/96 + (5*x^3*S
qrt[3 - x + 2*x^2])/8 - (6863*ArcSinh[(1 - 4*x)/Sqrt[23]])/(2048*Sqrt[2])

Rule 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rule 619

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[1/(2*c*((-4*c)/(b^2 - 4*a*c))^p), Subst[Int[Si
mp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/c, 0]

Rule 640

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(a + b*x + c*x^2)^(p +
1))/(2*c*(p + 1)), x] + Dist[(2*c*d - b*e)/(2*c), Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}
, x] && NeQ[2*c*d - b*e, 0] && NeQ[p, -1]

Rule 1661

Int[(Pq_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], e = Coeff[Pq, x, Expo
n[Pq, x]]}, Simp[(e*x^(q - 1)*(a + b*x + c*x^2)^(p + 1))/(c*(q + 2*p + 1)), x] + Dist[1/(c*(q + 2*p + 1)), Int
[(a + b*x + c*x^2)^p*ExpandToSum[c*(q + 2*p + 1)*Pq - a*e*(q - 1)*x^(q - 2) - b*e*(q + p)*x^(q - 1) - c*e*(q +
2*p + 1)*x^q, x], x], x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] &&  !LeQ[p, -1]

Rubi steps

\begin {align*} \int \frac {2+x+3 x^2-x^3+5 x^4}{\sqrt {3-x+2 x^2}} \, dx &=\frac {5}{8} x^3 \sqrt {3-x+2 x^2}+\frac {1}{8} \int \frac {16+8 x-21 x^2+\frac {19 x^3}{2}}{\sqrt {3-x+2 x^2}} \, dx\\ &=\frac {19}{96} x^2 \sqrt {3-x+2 x^2}+\frac {5}{8} x^3 \sqrt {3-x+2 x^2}+\frac {1}{48} \int \frac {96-9 x-\frac {409 x^2}{4}}{\sqrt {3-x+2 x^2}} \, dx\\ &=-\frac {409}{768} x \sqrt {3-x+2 x^2}+\frac {19}{96} x^2 \sqrt {3-x+2 x^2}+\frac {5}{8} x^3 \sqrt {3-x+2 x^2}+\frac {1}{192} \int \frac {\frac {2763}{4}-\frac {1515 x}{8}}{\sqrt {3-x+2 x^2}} \, dx\\ &=-\frac {505 \sqrt {3-x+2 x^2}}{1024}-\frac {409}{768} x \sqrt {3-x+2 x^2}+\frac {19}{96} x^2 \sqrt {3-x+2 x^2}+\frac {5}{8} x^3 \sqrt {3-x+2 x^2}+\frac {6863 \int \frac {1}{\sqrt {3-x+2 x^2}} \, dx}{2048}\\ &=-\frac {505 \sqrt {3-x+2 x^2}}{1024}-\frac {409}{768} x \sqrt {3-x+2 x^2}+\frac {19}{96} x^2 \sqrt {3-x+2 x^2}+\frac {5}{8} x^3 \sqrt {3-x+2 x^2}+\frac {6863 \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{23}}} \, dx,x,-1+4 x\right )}{2048 \sqrt {46}}\\ &=-\frac {505 \sqrt {3-x+2 x^2}}{1024}-\frac {409}{768} x \sqrt {3-x+2 x^2}+\frac {19}{96} x^2 \sqrt {3-x+2 x^2}+\frac {5}{8} x^3 \sqrt {3-x+2 x^2}-\frac {6863 \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{2048 \sqrt {2}}\\ \end {align*}

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Mathematica [A]  time = 0.07, size = 55, normalized size = 0.54 \[ \frac {4 \sqrt {2 x^2-x+3} \left (1920 x^3+608 x^2-1636 x-1515\right )-20589 \sqrt {2} \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{12288} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*Sqrt[3 - x + 2*x^2]*(-1515 - 1636*x + 608*x^2 + 1920*x^3) - 20589*Sqrt[2]*ArcSinh[(1 - 4*x)/Sqrt[23]])/1228
8

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fricas [A]  time = 0.78, size = 68, normalized size = 0.67 \[ \frac {1}{3072} \, {\left (1920 \, x^{3} + 608 \, x^{2} - 1636 \, x - 1515\right )} \sqrt {2 \, x^{2} - x + 3} + \frac {6863}{8192} \, \sqrt {2} \log \left (-4 \, \sqrt {2} \sqrt {2 \, x^{2} - x + 3} {\left (4 \, x - 1\right )} - 32 \, x^{2} + 16 \, x - 25\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3072*(1920*x^3 + 608*x^2 - 1636*x - 1515)*sqrt(2*x^2 - x + 3) + 6863/8192*sqrt(2)*log(-4*sqrt(2)*sqrt(2*x^2
- x + 3)*(4*x - 1) - 32*x^2 + 16*x - 25)

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giac [A]  time = 0.20, size = 63, normalized size = 0.62 \[ \frac {1}{3072} \, {\left (4 \, {\left (8 \, {\left (60 \, x + 19\right )} x - 409\right )} x - 1515\right )} \sqrt {2 \, x^{2} - x + 3} - \frac {6863}{4096} \, \sqrt {2} \log \left (-2 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3072*(4*(8*(60*x + 19)*x - 409)*x - 1515)*sqrt(2*x^2 - x + 3) - 6863/4096*sqrt(2)*log(-2*sqrt(2)*(sqrt(2)*x
- sqrt(2*x^2 - x + 3)) + 1)

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maple [A]  time = 0.01, size = 79, normalized size = 0.78 \[ \frac {5 \sqrt {2 x^{2}-x +3}\, x^{3}}{8}+\frac {19 \sqrt {2 x^{2}-x +3}\, x^{2}}{96}-\frac {409 \sqrt {2 x^{2}-x +3}\, x}{768}+\frac {6863 \sqrt {2}\, \arcsinh \left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{4096}-\frac {505 \sqrt {2 x^{2}-x +3}}{1024} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5/8*(2*x^2-x+3)^(1/2)*x^3+19/96*(2*x^2-x+3)^(1/2)*x^2-409/768*(2*x^2-x+3)^(1/2)*x-505/1024*(2*x^2-x+3)^(1/2)+6
863/4096*2^(1/2)*arcsinh(4/23*23^(1/2)*(x-1/4))

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maxima [A]  time = 0.96, size = 80, normalized size = 0.79 \[ \frac {5}{8} \, \sqrt {2 \, x^{2} - x + 3} x^{3} + \frac {19}{96} \, \sqrt {2 \, x^{2} - x + 3} x^{2} - \frac {409}{768} \, \sqrt {2 \, x^{2} - x + 3} x + \frac {6863}{4096} \, \sqrt {2} \operatorname {arsinh}\left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) - \frac {505}{1024} \, \sqrt {2 \, x^{2} - x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5/8*sqrt(2*x^2 - x + 3)*x^3 + 19/96*sqrt(2*x^2 - x + 3)*x^2 - 409/768*sqrt(2*x^2 - x + 3)*x + 6863/4096*sqrt(2
)*arcsinh(1/23*sqrt(23)*(4*x - 1)) - 505/1024*sqrt(2*x^2 - x + 3)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {5\,x^4-x^3+3\,x^2+x+2}{\sqrt {2\,x^2-x+3}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {5 x^{4} - x^{3} + 3 x^{2} + x + 2}{\sqrt {2 x^{2} - x + 3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((5*x**4 - x**3 + 3*x**2 + x + 2)/sqrt(2*x**2 - x + 3), x)

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