3.242 \(\int \frac{(1+2 x) (1+3 x+4 x^2)}{\sqrt{2-x+3 x^2}} \, dx\)

Optimal. Leaf size=70 \[ \frac{2}{9} \sqrt{3 x^2-x+2} (2 x+1)^2+\frac{1}{54} (62 x+69) \sqrt{3 x^2-x+2}+\frac{251 \sinh ^{-1}\left (\frac{1-6 x}{\sqrt{23}}\right )}{108 \sqrt{3}} \]

[Out]

(2*(1 + 2*x)^2*Sqrt[2 - x + 3*x^2])/9 + ((69 + 62*x)*Sqrt[2 - x + 3*x^2])/54 + (251*ArcSinh[(1 - 6*x)/Sqrt[23]
])/(108*Sqrt[3])

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Rubi [A]  time = 0.0580893, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {1653, 779, 619, 215} \[ \frac{2}{9} \sqrt{3 x^2-x+2} (2 x+1)^2+\frac{1}{54} (62 x+69) \sqrt{3 x^2-x+2}+\frac{251 \sinh ^{-1}\left (\frac{1-6 x}{\sqrt{23}}\right )}{108 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Int[((1 + 2*x)*(1 + 3*x + 4*x^2))/Sqrt[2 - x + 3*x^2],x]

[Out]

(2*(1 + 2*x)^2*Sqrt[2 - x + 3*x^2])/9 + ((69 + 62*x)*Sqrt[2 - x + 3*x^2])/54 + (251*ArcSinh[(1 - 6*x)/Sqrt[23]
])/(108*Sqrt[3])

Rule 1653

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq
, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[(f*(d + e*x)^(m + q - 1)*(a + b*x + c*x^2)^(p + 1))/(c*e^(q - 1)*(
m + q + 2*p + 1)), x] + Dist[1/(c*e^q*(m + q + 2*p + 1)), Int[(d + e*x)^m*(a + b*x + c*x^2)^p*ExpandToSum[c*e^
q*(m + q + 2*p + 1)*Pq - c*f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x)^(q - 2)*(b*d*e*(p + 1) + a*e^2*(m + q
- 1) - c*d^2*(m + q + 2*p + 1) - e*(2*c*d - b*e)*(m + q + p)*x), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p +
1, 0]] /; FreeQ[{a, b, c, d, e, m, p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2
, 0] &&  !(IGtQ[m, 0] && RationalQ[a, b, c, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))

Rule 779

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((b
*e*g*(p + 2) - c*(e*f + d*g)*(2*p + 3) - 2*c*e*g*(p + 1)*x)*(a + b*x + c*x^2)^(p + 1))/(2*c^2*(p + 1)*(2*p + 3
)), x] + Dist[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p + 3))/(2*c^2*(2*p + 3)), Int[(a
+ b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[b^2 - 4*a*c, 0] &&  !LeQ[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{(1+2 x) \left (1+3 x+4 x^2\right )}{\sqrt{2-x+3 x^2}} \, dx &=\frac{2}{9} (1+2 x)^2 \sqrt{2-x+3 x^2}+\frac{1}{36} \int \frac{(1+2 x) (-24+124 x)}{\sqrt{2-x+3 x^2}} \, dx\\ &=\frac{2}{9} (1+2 x)^2 \sqrt{2-x+3 x^2}+\frac{1}{54} (69+62 x) \sqrt{2-x+3 x^2}-\frac{251}{108} \int \frac{1}{\sqrt{2-x+3 x^2}} \, dx\\ &=\frac{2}{9} (1+2 x)^2 \sqrt{2-x+3 x^2}+\frac{1}{54} (69+62 x) \sqrt{2-x+3 x^2}-\frac{251 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{23}}} \, dx,x,-1+6 x\right )}{108 \sqrt{69}}\\ &=\frac{2}{9} (1+2 x)^2 \sqrt{2-x+3 x^2}+\frac{1}{54} (69+62 x) \sqrt{2-x+3 x^2}+\frac{251 \sinh ^{-1}\left (\frac{1-6 x}{\sqrt{23}}\right )}{108 \sqrt{3}}\\ \end{align*}

Mathematica [A]  time = 0.0214987, size = 50, normalized size = 0.71 \[ \frac{1}{324} \left (6 \sqrt{3 x^2-x+2} \left (48 x^2+110 x+81\right )-251 \sqrt{3} \sinh ^{-1}\left (\frac{6 x-1}{\sqrt{23}}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[((1 + 2*x)*(1 + 3*x + 4*x^2))/Sqrt[2 - x + 3*x^2],x]

[Out]

(6*Sqrt[2 - x + 3*x^2]*(81 + 110*x + 48*x^2) - 251*Sqrt[3]*ArcSinh[(-1 + 6*x)/Sqrt[23]])/324

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Maple [A]  time = 0.057, size = 62, normalized size = 0.9 \begin{align*}{\frac{8\,{x}^{2}}{9}\sqrt{3\,{x}^{2}-x+2}}+{\frac{55\,x}{27}\sqrt{3\,{x}^{2}-x+2}}+{\frac{3}{2}\sqrt{3\,{x}^{2}-x+2}}-{\frac{251\,\sqrt{3}}{324}{\it Arcsinh} \left ({\frac{6\,\sqrt{23}}{23} \left ( x-{\frac{1}{6}} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

8/9*x^2*(3*x^2-x+2)^(1/2)+55/27*x*(3*x^2-x+2)^(1/2)+3/2*(3*x^2-x+2)^(1/2)-251/324*3^(1/2)*arcsinh(6/23*23^(1/2
)*(x-1/6))

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Maxima [A]  time = 1.53746, size = 85, normalized size = 1.21 \begin{align*} \frac{8}{9} \, \sqrt{3 \, x^{2} - x + 2} x^{2} + \frac{55}{27} \, \sqrt{3 \, x^{2} - x + 2} x - \frac{251}{324} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{23} \, \sqrt{23}{\left (6 \, x - 1\right )}\right ) + \frac{3}{2} \, \sqrt{3 \, x^{2} - x + 2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

8/9*sqrt(3*x^2 - x + 2)*x^2 + 55/27*sqrt(3*x^2 - x + 2)*x - 251/324*sqrt(3)*arcsinh(1/23*sqrt(23)*(6*x - 1)) +
3/2*sqrt(3*x^2 - x + 2)

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Fricas [A]  time = 1.61625, size = 178, normalized size = 2.54 \begin{align*} \frac{1}{54} \,{\left (48 \, x^{2} + 110 \, x + 81\right )} \sqrt{3 \, x^{2} - x + 2} + \frac{251}{648} \, \sqrt{3} \log \left (4 \, \sqrt{3} \sqrt{3 \, x^{2} - x + 2}{\left (6 \, x - 1\right )} - 72 \, x^{2} + 24 \, x - 25\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/54*(48*x^2 + 110*x + 81)*sqrt(3*x^2 - x + 2) + 251/648*sqrt(3)*log(4*sqrt(3)*sqrt(3*x^2 - x + 2)*(6*x - 1) -
72*x^2 + 24*x - 25)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (2 x + 1\right ) \left (4 x^{2} + 3 x + 1\right )}{\sqrt{3 x^{2} - x + 2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((2*x + 1)*(4*x**2 + 3*x + 1)/sqrt(3*x**2 - x + 2), x)

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Giac [A]  time = 1.16874, size = 78, normalized size = 1.11 \begin{align*} \frac{1}{54} \,{\left (2 \,{\left (24 \, x + 55\right )} x + 81\right )} \sqrt{3 \, x^{2} - x + 2} + \frac{251}{324} \, \sqrt{3} \log \left (-2 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} - x + 2}\right )} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/54*(2*(24*x + 55)*x + 81)*sqrt(3*x^2 - x + 2) + 251/324*sqrt(3)*log(-2*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 - x +
2)) + 1)