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

[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])

________________________________________________________________________________________

Rubi [A]  time = 0.0801967, antiderivative size = 101, normalized size of antiderivative = 1., 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 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]

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 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 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]

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*}

Mathematica [A]  time = 0.0750094, 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

________________________________________________________________________________________

Maple [A]  time = 0.053, size = 79, normalized size = 0.8 \begin{align*}{\frac{5\,{x}^{3}}{8}\sqrt{2\,{x}^{2}-x+3}}+{\frac{19\,{x}^{2}}{96}\sqrt{2\,{x}^{2}-x+3}}-{\frac{409\,x}{768}\sqrt{2\,{x}^{2}-x+3}}-{\frac{505}{1024}\sqrt{2\,{x}^{2}-x+3}}+{\frac{6863\,\sqrt{2}}{4096}{\it Arcsinh} \left ({\frac{4\,\sqrt{23}}{23} \left ( x-{\frac{1}{4}} \right ) } \right ) } \end{align*}

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

________________________________________________________________________________________

Maxima [A]  time = 1.47592, size = 108, normalized size = 1.07 \begin{align*} \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} \end{align*}

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)

________________________________________________________________________________________

Fricas [A]  time = 1.30573, size = 205, normalized size = 2.03 \begin{align*} \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 ) \end{align*}

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)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{5 x^{4} - x^{3} + 3 x^{2} + x + 2}{\sqrt{2 x^{2} - x + 3}}\, dx \end{align*}

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)

________________________________________________________________________________________

Giac [A]  time = 1.15413, size = 85, normalized size = 0.84 \begin{align*} \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 ) \end{align*}

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)