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

Optimal. Leaf size=101 \[ \frac{25}{8} \sqrt{2 x^2-x+3} x^3+\frac{655}{96} \sqrt{2 x^2-x+3} x^2+\frac{3443}{768} \sqrt{2 x^2-x+3} x-\frac{11373 \sqrt{2 x^2-x+3}}{1024}+\frac{30725 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{2048 \sqrt{2}} \]

[Out]

(-11373*Sqrt[3 - x + 2*x^2])/1024 + (3443*x*Sqrt[3 - x + 2*x^2])/768 + (655*x^2*Sqrt[3 - x + 2*x^2])/96 + (25*
x^3*Sqrt[3 - x + 2*x^2])/8 + (30725*ArcSinh[(1 - 4*x)/Sqrt[23]])/(2048*Sqrt[2])

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Rubi [A]  time = 0.0883339, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148, Rules used = {1661, 640, 619, 215} \[ \frac{25}{8} \sqrt{2 x^2-x+3} x^3+\frac{655}{96} \sqrt{2 x^2-x+3} x^2+\frac{3443}{768} \sqrt{2 x^2-x+3} x-\frac{11373 \sqrt{2 x^2-x+3}}{1024}+\frac{30725 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{2048 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-11373*Sqrt[3 - x + 2*x^2])/1024 + (3443*x*Sqrt[3 - x + 2*x^2])/768 + (655*x^2*Sqrt[3 - x + 2*x^2])/96 + (25*
x^3*Sqrt[3 - x + 2*x^2])/8 + (30725*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{\left (2+3 x+5 x^2\right )^2}{\sqrt{3-x+2 x^2}} \, dx &=\frac{25}{8} x^3 \sqrt{3-x+2 x^2}+\frac{1}{8} \int \frac{32+96 x+7 x^2+\frac{655 x^3}{2}}{\sqrt{3-x+2 x^2}} \, dx\\ &=\frac{655}{96} x^2 \sqrt{3-x+2 x^2}+\frac{25}{8} x^3 \sqrt{3-x+2 x^2}+\frac{1}{48} \int \frac{192-1389 x+\frac{3443 x^2}{4}}{\sqrt{3-x+2 x^2}} \, dx\\ &=\frac{3443}{768} x \sqrt{3-x+2 x^2}+\frac{655}{96} x^2 \sqrt{3-x+2 x^2}+\frac{25}{8} x^3 \sqrt{3-x+2 x^2}+\frac{1}{192} \int \frac{-\frac{7257}{4}-\frac{34119 x}{8}}{\sqrt{3-x+2 x^2}} \, dx\\ &=-\frac{11373 \sqrt{3-x+2 x^2}}{1024}+\frac{3443}{768} x \sqrt{3-x+2 x^2}+\frac{655}{96} x^2 \sqrt{3-x+2 x^2}+\frac{25}{8} x^3 \sqrt{3-x+2 x^2}-\frac{30725 \int \frac{1}{\sqrt{3-x+2 x^2}} \, dx}{2048}\\ &=-\frac{11373 \sqrt{3-x+2 x^2}}{1024}+\frac{3443}{768} x \sqrt{3-x+2 x^2}+\frac{655}{96} x^2 \sqrt{3-x+2 x^2}+\frac{25}{8} x^3 \sqrt{3-x+2 x^2}-\frac{30725 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{23}}} \, dx,x,-1+4 x\right )}{2048 \sqrt{46}}\\ &=-\frac{11373 \sqrt{3-x+2 x^2}}{1024}+\frac{3443}{768} x \sqrt{3-x+2 x^2}+\frac{655}{96} x^2 \sqrt{3-x+2 x^2}+\frac{25}{8} x^3 \sqrt{3-x+2 x^2}+\frac{30725 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{2048 \sqrt{2}}\\ \end{align*}

Mathematica [A]  time = 0.0755216, size = 55, normalized size = 0.54 \[ \frac{4 \sqrt{2 x^2-x+3} \left (9600 x^3+20960 x^2+13772 x-34119\right )+92175 \sqrt{2} \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{12288} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*Sqrt[3 - x + 2*x^2]*(-34119 + 13772*x + 20960*x^2 + 9600*x^3) + 92175*Sqrt[2]*ArcSinh[(1 - 4*x)/Sqrt[23]])/
12288

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Maple [A]  time = 0.051, size = 79, normalized size = 0.8 \begin{align*}{\frac{25\,{x}^{3}}{8}\sqrt{2\,{x}^{2}-x+3}}+{\frac{655\,{x}^{2}}{96}\sqrt{2\,{x}^{2}-x+3}}+{\frac{3443\,x}{768}\sqrt{2\,{x}^{2}-x+3}}-{\frac{11373}{1024}\sqrt{2\,{x}^{2}-x+3}}-{\frac{30725\,\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^2+3*x+2)^2/(2*x^2-x+3)^(1/2),x)

[Out]

25/8*x^3*(2*x^2-x+3)^(1/2)+655/96*x^2*(2*x^2-x+3)^(1/2)+3443/768*x*(2*x^2-x+3)^(1/2)-11373/1024*(2*x^2-x+3)^(1
/2)-30725/4096*2^(1/2)*arcsinh(4/23*23^(1/2)*(x-1/4))

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Maxima [A]  time = 1.48942, size = 108, normalized size = 1.07 \begin{align*} \frac{25}{8} \, \sqrt{2 \, x^{2} - x + 3} x^{3} + \frac{655}{96} \, \sqrt{2 \, x^{2} - x + 3} x^{2} + \frac{3443}{768} \, \sqrt{2 \, x^{2} - x + 3} x - \frac{30725}{4096} \, \sqrt{2} \operatorname{arsinh}\left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) - \frac{11373}{1024} \, \sqrt{2 \, x^{2} - x + 3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

25/8*sqrt(2*x^2 - x + 3)*x^3 + 655/96*sqrt(2*x^2 - x + 3)*x^2 + 3443/768*sqrt(2*x^2 - x + 3)*x - 30725/4096*sq
rt(2)*arcsinh(1/23*sqrt(23)*(4*x - 1)) - 11373/1024*sqrt(2*x^2 - x + 3)

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Fricas [A]  time = 1.37724, size = 211, normalized size = 2.09 \begin{align*} \frac{1}{3072} \,{\left (9600 \, x^{3} + 20960 \, x^{2} + 13772 \, x - 34119\right )} \sqrt{2 \, x^{2} - x + 3} + \frac{30725}{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^2+3*x+2)^2/(2*x^2-x+3)^(1/2),x, algorithm="fricas")

[Out]

1/3072*(9600*x^3 + 20960*x^2 + 13772*x - 34119)*sqrt(2*x^2 - x + 3) + 30725/8192*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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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (5 x^{2} + 3 x + 2\right )^{2}}{\sqrt{2 x^{2} - x + 3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.16484, size = 85, normalized size = 0.84 \begin{align*} \frac{1}{3072} \,{\left (4 \,{\left (40 \,{\left (60 \, x + 131\right )} x + 3443\right )} x - 34119\right )} \sqrt{2 \, x^{2} - x + 3} + \frac{30725}{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^2+3*x+2)^2/(2*x^2-x+3)^(1/2),x, algorithm="giac")

[Out]

1/3072*(4*(40*(60*x + 131)*x + 3443)*x - 34119)*sqrt(2*x^2 - x + 3) + 30725/4096*sqrt(2)*log(-2*sqrt(2)*(sqrt(
2)*x - sqrt(2*x^2 - x + 3)) + 1)