3.253 \(\int \frac{(1+2 x)^2 (1+3 x+4 x^2)}{(2-x+3 x^2)^{5/2}} \, dx\)

Optimal. Leaf size=68 \[ \frac{2 (1249-2273 x)}{1863 \left (3 x^2-x+2\right )^{3/2}}-\frac{8 (23257-1473 x)}{42849 \sqrt{3 x^2-x+2}}-\frac{16 \sinh ^{-1}\left (\frac{1-6 x}{\sqrt{23}}\right )}{9 \sqrt{3}} \]

[Out]

(2*(1249 - 2273*x))/(1863*(2 - x + 3*x^2)^(3/2)) - (8*(23257 - 1473*x))/(42849*Sqrt[2 - x + 3*x^2]) - (16*ArcS
inh[(1 - 6*x)/Sqrt[23]])/(9*Sqrt[3])

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Rubi [A]  time = 0.0946948, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {1660, 12, 619, 215} \[ \frac{2 (1249-2273 x)}{1863 \left (3 x^2-x+2\right )^{3/2}}-\frac{8 (23257-1473 x)}{42849 \sqrt{3 x^2-x+2}}-\frac{16 \sinh ^{-1}\left (\frac{1-6 x}{\sqrt{23}}\right )}{9 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Int[((1 + 2*x)^2*(1 + 3*x + 4*x^2))/(2 - x + 3*x^2)^(5/2),x]

[Out]

(2*(1249 - 2273*x))/(1863*(2 - x + 3*x^2)^(3/2)) - (8*(23257 - 1473*x))/(42849*Sqrt[2 - x + 3*x^2]) - (16*ArcS
inh[(1 - 6*x)/Sqrt[23]])/(9*Sqrt[3])

Rule 1660

Int[(Pq_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, a + b*x + c*
x^2, x], f = Coeff[PolynomialRemainder[Pq, a + b*x + c*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b
*x + c*x^2, x], x, 1]}, Simp[((b*f - 2*a*g + (2*c*f - b*g)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c
)), x] + Dist[1/((p + 1)*(b^2 - 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1)*ExpandToSum[(p + 1)*(b^2 - 4*a*c)*Q - (
2*p + 3)*(2*c*f - b*g), x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1
]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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)^2 \left (1+3 x+4 x^2\right )}{\left (2-x+3 x^2\right )^{5/2}} \, dx &=\frac{2 (1249-2273 x)}{1863 \left (2-x+3 x^2\right )^{3/2}}+\frac{2}{69} \int \frac{\frac{1802}{27}+\frac{1150 x}{3}+184 x^2}{\left (2-x+3 x^2\right )^{3/2}} \, dx\\ &=\frac{2 (1249-2273 x)}{1863 \left (2-x+3 x^2\right )^{3/2}}-\frac{8 (23257-1473 x)}{42849 \sqrt{2-x+3 x^2}}+\frac{4 \int \frac{2116}{3 \sqrt{2-x+3 x^2}} \, dx}{1587}\\ &=\frac{2 (1249-2273 x)}{1863 \left (2-x+3 x^2\right )^{3/2}}-\frac{8 (23257-1473 x)}{42849 \sqrt{2-x+3 x^2}}+\frac{16}{9} \int \frac{1}{\sqrt{2-x+3 x^2}} \, dx\\ &=\frac{2 (1249-2273 x)}{1863 \left (2-x+3 x^2\right )^{3/2}}-\frac{8 (23257-1473 x)}{42849 \sqrt{2-x+3 x^2}}+\frac{16 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{23}}} \, dx,x,-1+6 x\right )}{9 \sqrt{69}}\\ &=\frac{2 (1249-2273 x)}{1863 \left (2-x+3 x^2\right )^{3/2}}-\frac{8 (23257-1473 x)}{42849 \sqrt{2-x+3 x^2}}-\frac{16 \sinh ^{-1}\left (\frac{1-6 x}{\sqrt{23}}\right )}{9 \sqrt{3}}\\ \end{align*}

Mathematica [A]  time = 0.148035, size = 66, normalized size = 0.97 \[ \frac{2 \left (5892 x^3-94992 x^2+4232 \sqrt{3} \left (3 x^2-x+2\right )^{3/2} \sinh ^{-1}\left (\frac{6 x-1}{\sqrt{23}}\right )+17511 x-52443\right )}{14283 \left (3 x^2-x+2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[((1 + 2*x)^2*(1 + 3*x + 4*x^2))/(2 - x + 3*x^2)^(5/2),x]

[Out]

(2*(-52443 + 17511*x - 94992*x^2 + 5892*x^3 + 4232*Sqrt[3]*(2 - x + 3*x^2)^(3/2)*ArcSinh[(-1 + 6*x)/Sqrt[23]])
)/(14283*(2 - x + 3*x^2)^(3/2))

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Maple [B]  time = 0.054, size = 146, normalized size = 2.2 \begin{align*} -{\frac{16\,{x}^{3}}{9} \left ( 3\,{x}^{2}-x+2 \right ) ^{-{\frac{3}{2}}}}-{\frac{92\,{x}^{2}}{9} \left ( 3\,{x}^{2}-x+2 \right ) ^{-{\frac{3}{2}}}}-{\frac{67\,x}{27} \left ( 3\,{x}^{2}-x+2 \right ) ^{-{\frac{3}{2}}}}-{\frac{2653}{486} \left ( 3\,{x}^{2}-x+2 \right ) ^{-{\frac{3}{2}}}}+{\frac{-4585+27510\,x}{11178} \left ( 3\,{x}^{2}-x+2 \right ) ^{-{\frac{3}{2}}}}+{\frac{-18892+113352\,x}{42849}{\frac{1}{\sqrt{3\,{x}^{2}-x+2}}}}-{\frac{16\,x}{9}{\frac{1}{\sqrt{3\,{x}^{2}-x+2}}}}-{\frac{8}{27}{\frac{1}{\sqrt{3\,{x}^{2}-x+2}}}}+{\frac{16\,\sqrt{3}}{27}{\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)^2*(4*x^2+3*x+1)/(3*x^2-x+2)^(5/2),x)

[Out]

-16/9*x^3/(3*x^2-x+2)^(3/2)-92/9*x^2/(3*x^2-x+2)^(3/2)-67/27*x/(3*x^2-x+2)^(3/2)-2653/486/(3*x^2-x+2)^(3/2)+45
85/11178*(-1+6*x)/(3*x^2-x+2)^(3/2)+18892/42849*(-1+6*x)/(3*x^2-x+2)^(1/2)-16/9*x/(3*x^2-x+2)^(1/2)-8/27/(3*x^
2-x+2)^(1/2)+16/27*3^(1/2)*arcsinh(6/23*23^(1/2)*(x-1/6))

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Maxima [B]  time = 1.4618, size = 250, normalized size = 3.68 \begin{align*} \frac{16}{14283} \, x{\left (\frac{426 \, x}{\sqrt{3 \, x^{2} - x + 2}} - \frac{4761 \, x^{2}}{{\left (3 \, x^{2} - x + 2\right )}^{\frac{3}{2}}} - \frac{71}{\sqrt{3 \, x^{2} - x + 2}} + \frac{805 \, x}{{\left (3 \, x^{2} - x + 2\right )}^{\frac{3}{2}}} - \frac{2162}{{\left (3 \, x^{2} - x + 2\right )}^{\frac{3}{2}}}\right )} + \frac{16}{27} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{23} \, \sqrt{23}{\left (6 \, x - 1\right )}\right ) - \frac{2272}{14283} \, \sqrt{3 \, x^{2} - x + 2} + \frac{28184 \, x}{14283 \, \sqrt{3 \, x^{2} - x + 2}} - \frac{28 \, x^{2}}{3 \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{3}{2}}} - \frac{2956}{4761 \, \sqrt{3 \, x^{2} - x + 2}} - \frac{106 \, x}{621 \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{3}{2}}} - \frac{3394}{621 \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

16/14283*x*(426*x/sqrt(3*x^2 - x + 2) - 4761*x^2/(3*x^2 - x + 2)^(3/2) - 71/sqrt(3*x^2 - x + 2) + 805*x/(3*x^2
- x + 2)^(3/2) - 2162/(3*x^2 - x + 2)^(3/2)) + 16/27*sqrt(3)*arcsinh(1/23*sqrt(23)*(6*x - 1)) - 2272/14283*sq
rt(3*x^2 - x + 2) + 28184/14283*x/sqrt(3*x^2 - x + 2) - 28/3*x^2/(3*x^2 - x + 2)^(3/2) - 2956/4761/sqrt(3*x^2
- x + 2) - 106/621*x/(3*x^2 - x + 2)^(3/2) - 3394/621/(3*x^2 - x + 2)^(3/2)

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Fricas [B]  time = 1.0629, size = 304, normalized size = 4.47 \begin{align*} \frac{2 \,{\left (2116 \, \sqrt{3}{\left (9 \, x^{4} - 6 \, x^{3} + 13 \, x^{2} - 4 \, x + 4\right )} \log \left (-4 \, \sqrt{3} \sqrt{3 \, x^{2} - x + 2}{\left (6 \, x - 1\right )} - 72 \, x^{2} + 24 \, x - 25\right ) + 3 \,{\left (1964 \, x^{3} - 31664 \, x^{2} + 5837 \, x - 17481\right )} \sqrt{3 \, x^{2} - x + 2}\right )}}{14283 \,{\left (9 \, x^{4} - 6 \, x^{3} + 13 \, x^{2} - 4 \, x + 4\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/14283*(2116*sqrt(3)*(9*x^4 - 6*x^3 + 13*x^2 - 4*x + 4)*log(-4*sqrt(3)*sqrt(3*x^2 - x + 2)*(6*x - 1) - 72*x^2
+ 24*x - 25) + 3*(1964*x^3 - 31664*x^2 + 5837*x - 17481)*sqrt(3*x^2 - x + 2))/(9*x^4 - 6*x^3 + 13*x^2 - 4*x +
4)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (2 x + 1\right )^{2} \left (4 x^{2} + 3 x + 1\right )}{\left (3 x^{2} - x + 2\right )^{\frac{5}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.15909, size = 84, normalized size = 1.24 \begin{align*} -\frac{16}{27} \, \sqrt{3} \log \left (-2 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} - x + 2}\right )} + 1\right ) + \frac{2 \,{\left ({\left (4 \,{\left (491 \, x - 7916\right )} x + 5837\right )} x - 17481\right )}}{4761 \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-16/27*sqrt(3)*log(-2*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 - x + 2)) + 1) + 2/4761*((4*(491*x - 7916)*x + 5837)*x -
17481)/(3*x^2 - x + 2)^(3/2)