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

Optimal. Leaf size=124 \[ \frac{25}{12} \left (2 x^2-x+3\right )^{3/2} x^3+\frac{63}{16} \left (2 x^2-x+3\right )^{3/2} x^2+\frac{769}{256} \left (2 x^2-x+3\right )^{3/2} x-\frac{2107 \left (2 x^2-x+3\right )^{3/2}}{3072}+\frac{12371 (1-4 x) \sqrt{2 x^2-x+3}}{16384}+\frac{284533 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{32768 \sqrt{2}} \]

[Out]

(12371*(1 - 4*x)*Sqrt[3 - x + 2*x^2])/16384 - (2107*(3 - x + 2*x^2)^(3/2))/3072 + (769*x*(3 - x + 2*x^2)^(3/2)
)/256 + (63*x^2*(3 - x + 2*x^2)^(3/2))/16 + (25*x^3*(3 - x + 2*x^2)^(3/2))/12 + (284533*ArcSinh[(1 - 4*x)/Sqrt
[23]])/(32768*Sqrt[2])

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Rubi [A]  time = 0.0990913, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {1661, 640, 612, 619, 215} \[ \frac{25}{12} \left (2 x^2-x+3\right )^{3/2} x^3+\frac{63}{16} \left (2 x^2-x+3\right )^{3/2} x^2+\frac{769}{256} \left (2 x^2-x+3\right )^{3/2} x-\frac{2107 \left (2 x^2-x+3\right )^{3/2}}{3072}+\frac{12371 (1-4 x) \sqrt{2 x^2-x+3}}{16384}+\frac{284533 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{32768 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(12371*(1 - 4*x)*Sqrt[3 - x + 2*x^2])/16384 - (2107*(3 - x + 2*x^2)^(3/2))/3072 + (769*x*(3 - x + 2*x^2)^(3/2)
)/256 + (63*x^2*(3 - x + 2*x^2)^(3/2))/16 + (25*x^3*(3 - x + 2*x^2)^(3/2))/12 + (284533*ArcSinh[(1 - 4*x)/Sqrt
[23]])/(32768*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 612

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

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 \sqrt{3-x+2 x^2} \left (2+3 x+5 x^2\right )^2 \, dx &=\frac{25}{12} x^3 \left (3-x+2 x^2\right )^{3/2}+\frac{1}{12} \int \sqrt{3-x+2 x^2} \left (48+144 x+123 x^2+\frac{945 x^3}{2}\right ) \, dx\\ &=\frac{63}{16} x^2 \left (3-x+2 x^2\right )^{3/2}+\frac{25}{12} x^3 \left (3-x+2 x^2\right )^{3/2}+\frac{1}{120} \int \sqrt{3-x+2 x^2} \left (480-1395 x+\frac{11535 x^2}{4}\right ) \, dx\\ &=\frac{769}{256} x \left (3-x+2 x^2\right )^{3/2}+\frac{63}{16} x^2 \left (3-x+2 x^2\right )^{3/2}+\frac{25}{12} x^3 \left (3-x+2 x^2\right )^{3/2}+\frac{1}{960} \int \left (-\frac{19245}{4}-\frac{31605 x}{8}\right ) \sqrt{3-x+2 x^2} \, dx\\ &=-\frac{2107 \left (3-x+2 x^2\right )^{3/2}}{3072}+\frac{769}{256} x \left (3-x+2 x^2\right )^{3/2}+\frac{63}{16} x^2 \left (3-x+2 x^2\right )^{3/2}+\frac{25}{12} x^3 \left (3-x+2 x^2\right )^{3/2}-\frac{12371 \int \sqrt{3-x+2 x^2} \, dx}{2048}\\ &=\frac{12371 (1-4 x) \sqrt{3-x+2 x^2}}{16384}-\frac{2107 \left (3-x+2 x^2\right )^{3/2}}{3072}+\frac{769}{256} x \left (3-x+2 x^2\right )^{3/2}+\frac{63}{16} x^2 \left (3-x+2 x^2\right )^{3/2}+\frac{25}{12} x^3 \left (3-x+2 x^2\right )^{3/2}-\frac{284533 \int \frac{1}{\sqrt{3-x+2 x^2}} \, dx}{32768}\\ &=\frac{12371 (1-4 x) \sqrt{3-x+2 x^2}}{16384}-\frac{2107 \left (3-x+2 x^2\right )^{3/2}}{3072}+\frac{769}{256} x \left (3-x+2 x^2\right )^{3/2}+\frac{63}{16} x^2 \left (3-x+2 x^2\right )^{3/2}+\frac{25}{12} x^3 \left (3-x+2 x^2\right )^{3/2}-\frac{\left (12371 \sqrt{\frac{23}{2}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{23}}} \, dx,x,-1+4 x\right )}{32768}\\ &=\frac{12371 (1-4 x) \sqrt{3-x+2 x^2}}{16384}-\frac{2107 \left (3-x+2 x^2\right )^{3/2}}{3072}+\frac{769}{256} x \left (3-x+2 x^2\right )^{3/2}+\frac{63}{16} x^2 \left (3-x+2 x^2\right )^{3/2}+\frac{25}{12} x^3 \left (3-x+2 x^2\right )^{3/2}+\frac{284533 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{32768 \sqrt{2}}\\ \end{align*}

Mathematica [A]  time = 0.10554, size = 65, normalized size = 0.52 \[ \frac{4 \sqrt{2 x^2-x+3} \left (204800 x^5+284672 x^4+408960 x^3+365536 x^2+328204 x-64023\right )+853599 \sqrt{2} \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{196608} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*Sqrt[3 - x + 2*x^2]*(-64023 + 328204*x + 365536*x^2 + 408960*x^3 + 284672*x^4 + 204800*x^5) + 853599*Sqrt[2
]*ArcSinh[(1 - 4*x)/Sqrt[23]])/196608

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Maple [A]  time = 0.053, size = 98, normalized size = 0.8 \begin{align*}{\frac{25\,{x}^{3}}{12} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{3}{2}}}}+{\frac{63\,{x}^{2}}{16} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{3}{2}}}}+{\frac{769\,x}{256} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{3}{2}}}}-{\frac{2107}{3072} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{3}{2}}}}-{\frac{-12371+49484\,x}{16384}\sqrt{2\,{x}^{2}-x+3}}-{\frac{284533\,\sqrt{2}}{65536}{\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]

25/12*x^3*(2*x^2-x+3)^(3/2)+63/16*x^2*(2*x^2-x+3)^(3/2)+769/256*x*(2*x^2-x+3)^(3/2)-2107/3072*(2*x^2-x+3)^(3/2
)-12371/16384*(-1+4*x)*(2*x^2-x+3)^(1/2)-284533/65536*2^(1/2)*arcsinh(4/23*23^(1/2)*(x-1/4))

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Maxima [A]  time = 1.53431, size = 147, normalized size = 1.19 \begin{align*} \frac{25}{12} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{3} + \frac{63}{16} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{2} + \frac{769}{256} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x - \frac{2107}{3072} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} - \frac{12371}{4096} \, \sqrt{2 \, x^{2} - x + 3} x - \frac{284533}{65536} \, \sqrt{2} \operatorname{arsinh}\left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) + \frac{12371}{16384} \, \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]

25/12*(2*x^2 - x + 3)^(3/2)*x^3 + 63/16*(2*x^2 - x + 3)^(3/2)*x^2 + 769/256*(2*x^2 - x + 3)^(3/2)*x - 2107/307
2*(2*x^2 - x + 3)^(3/2) - 12371/4096*sqrt(2*x^2 - x + 3)*x - 284533/65536*sqrt(2)*arcsinh(1/23*sqrt(23)*(4*x -
1)) + 12371/16384*sqrt(2*x^2 - x + 3)

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Fricas [A]  time = 1.52922, size = 257, normalized size = 2.07 \begin{align*} \frac{1}{49152} \,{\left (204800 \, x^{5} + 284672 \, x^{4} + 408960 \, x^{3} + 365536 \, x^{2} + 328204 \, x - 64023\right )} \sqrt{2 \, x^{2} - x + 3} + \frac{284533}{131072} \, \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]

1/49152*(204800*x^5 + 284672*x^4 + 408960*x^3 + 365536*x^2 + 328204*x - 64023)*sqrt(2*x^2 - x + 3) + 284533/13
1072*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 \sqrt{2 x^{2} - x + 3} \left (5 x^{2} + 3 x + 2\right )^{2}\, 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(sqrt(2*x**2 - x + 3)*(5*x**2 + 3*x + 2)**2, x)

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Giac [A]  time = 1.19608, size = 99, normalized size = 0.8 \begin{align*} \frac{1}{49152} \,{\left (4 \,{\left (8 \,{\left (4 \,{\left (16 \,{\left (100 \, x + 139\right )} x + 3195\right )} x + 11423\right )} x + 82051\right )} x - 64023\right )} \sqrt{2 \, x^{2} - x + 3} + \frac{284533}{65536} \, \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]

1/49152*(4*(8*(4*(16*(100*x + 139)*x + 3195)*x + 11423)*x + 82051)*x - 64023)*sqrt(2*x^2 - x + 3) + 284533/655
36*sqrt(2)*log(-2*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3)) + 1)