### 3.325 $$\int \sqrt{3-x+2 x^2} (2+x+3 x^2-x^3+5 x^4) \, dx$$

Optimal. Leaf size=124 $\frac{5}{12} \left (2 x^2-x+3\right )^{3/2} x^3+\frac{7}{80} \left (2 x^2-x+3\right )^{3/2} x^2-\frac{71 \left (2 x^2-x+3\right )^{3/2} x}{1280}+\frac{287 \left (2 x^2-x+3\right )^{3/2}}{5120}-\frac{4609 (1-4 x) \sqrt{2 x^2-x+3}}{16384}-\frac{106007 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{32768 \sqrt{2}}$

[Out]

(-4609*(1 - 4*x)*Sqrt[3 - x + 2*x^2])/16384 + (287*(3 - x + 2*x^2)^(3/2))/5120 - (71*x*(3 - x + 2*x^2)^(3/2))/
1280 + (7*x^2*(3 - x + 2*x^2)^(3/2))/80 + (5*x^3*(3 - x + 2*x^2)^(3/2))/12 - (106007*ArcSinh[(1 - 4*x)/Sqrt[23
]])/(32768*Sqrt[2])

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Rubi [A]  time = 0.0921227, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 33, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.152, Rules used = {1661, 640, 612, 619, 215} $\frac{5}{12} \left (2 x^2-x+3\right )^{3/2} x^3+\frac{7}{80} \left (2 x^2-x+3\right )^{3/2} x^2-\frac{71 \left (2 x^2-x+3\right )^{3/2} x}{1280}+\frac{287 \left (2 x^2-x+3\right )^{3/2}}{5120}-\frac{4609 (1-4 x) \sqrt{2 x^2-x+3}}{16384}-\frac{106007 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{32768 \sqrt{2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-4609*(1 - 4*x)*Sqrt[3 - x + 2*x^2])/16384 + (287*(3 - x + 2*x^2)^(3/2))/5120 - (71*x*(3 - x + 2*x^2)^(3/2))/
1280 + (7*x^2*(3 - x + 2*x^2)^(3/2))/80 + (5*x^3*(3 - x + 2*x^2)^(3/2))/12 - (106007*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+x+3 x^2-x^3+5 x^4\right ) \, dx &=\frac{5}{12} x^3 \left (3-x+2 x^2\right )^{3/2}+\frac{1}{12} \int \sqrt{3-x+2 x^2} \left (24+12 x-9 x^2+\frac{21 x^3}{2}\right ) \, dx\\ &=\frac{7}{80} x^2 \left (3-x+2 x^2\right )^{3/2}+\frac{5}{12} x^3 \left (3-x+2 x^2\right )^{3/2}+\frac{1}{120} \int \left (240+57 x-\frac{213 x^2}{4}\right ) \sqrt{3-x+2 x^2} \, dx\\ &=-\frac{71 x \left (3-x+2 x^2\right )^{3/2}}{1280}+\frac{7}{80} x^2 \left (3-x+2 x^2\right )^{3/2}+\frac{5}{12} x^3 \left (3-x+2 x^2\right )^{3/2}+\frac{1}{960} \int \left (\frac{8319}{4}+\frac{2583 x}{8}\right ) \sqrt{3-x+2 x^2} \, dx\\ &=\frac{287 \left (3-x+2 x^2\right )^{3/2}}{5120}-\frac{71 x \left (3-x+2 x^2\right )^{3/2}}{1280}+\frac{7}{80} x^2 \left (3-x+2 x^2\right )^{3/2}+\frac{5}{12} x^3 \left (3-x+2 x^2\right )^{3/2}+\frac{4609 \int \sqrt{3-x+2 x^2} \, dx}{2048}\\ &=-\frac{4609 (1-4 x) \sqrt{3-x+2 x^2}}{16384}+\frac{287 \left (3-x+2 x^2\right )^{3/2}}{5120}-\frac{71 x \left (3-x+2 x^2\right )^{3/2}}{1280}+\frac{7}{80} x^2 \left (3-x+2 x^2\right )^{3/2}+\frac{5}{12} x^3 \left (3-x+2 x^2\right )^{3/2}+\frac{106007 \int \frac{1}{\sqrt{3-x+2 x^2}} \, dx}{32768}\\ &=-\frac{4609 (1-4 x) \sqrt{3-x+2 x^2}}{16384}+\frac{287 \left (3-x+2 x^2\right )^{3/2}}{5120}-\frac{71 x \left (3-x+2 x^2\right )^{3/2}}{1280}+\frac{7}{80} x^2 \left (3-x+2 x^2\right )^{3/2}+\frac{5}{12} x^3 \left (3-x+2 x^2\right )^{3/2}+\frac{\left (4609 \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{4609 (1-4 x) \sqrt{3-x+2 x^2}}{16384}+\frac{287 \left (3-x+2 x^2\right )^{3/2}}{5120}-\frac{71 x \left (3-x+2 x^2\right )^{3/2}}{1280}+\frac{7}{80} x^2 \left (3-x+2 x^2\right )^{3/2}+\frac{5}{12} x^3 \left (3-x+2 x^2\right )^{3/2}-\frac{106007 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{32768 \sqrt{2}}\\ \end{align*}

Mathematica [A]  time = 0.100818, size = 65, normalized size = 0.52 $\frac{4 \sqrt{2 x^2-x+3} \left (204800 x^5-59392 x^4+258432 x^3+105696 x^2+221868 x-27807\right )-1590105 \sqrt{2} \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{983040}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(4*Sqrt[3 - x + 2*x^2]*(-27807 + 221868*x + 105696*x^2 + 258432*x^3 - 59392*x^4 + 204800*x^5) - 1590105*Sqrt[2
]*ArcSinh[(1 - 4*x)/Sqrt[23]])/983040

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Maple [A]  time = 0.055, size = 98, normalized size = 0.8 \begin{align*}{\frac{5\,{x}^{3}}{12} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{3}{2}}}}+{\frac{7\,{x}^{2}}{80} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{3}{2}}}}-{\frac{71\,x}{1280} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{3}{2}}}}+{\frac{287}{5120} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{3}{2}}}}+{\frac{-4609+18436\,x}{16384}\sqrt{2\,{x}^{2}-x+3}}+{\frac{106007\,\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^4-x^3+3*x^2+x+2)*(2*x^2-x+3)^(1/2),x)

[Out]

5/12*x^3*(2*x^2-x+3)^(3/2)+7/80*x^2*(2*x^2-x+3)^(3/2)-71/1280*x*(2*x^2-x+3)^(3/2)+287/5120*(2*x^2-x+3)^(3/2)+4
609/16384*(-1+4*x)*(2*x^2-x+3)^(1/2)+106007/65536*2^(1/2)*arcsinh(4/23*23^(1/2)*(x-1/4))

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Maxima [A]  time = 1.57703, size = 147, normalized size = 1.19 \begin{align*} \frac{5}{12} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{3} + \frac{7}{80} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{2} - \frac{71}{1280} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + \frac{287}{5120} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{4609}{4096} \, \sqrt{2 \, x^{2} - x + 3} x + \frac{106007}{65536} \, \sqrt{2} \operatorname{arsinh}\left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) - \frac{4609}{16384} \, \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]

5/12*(2*x^2 - x + 3)^(3/2)*x^3 + 7/80*(2*x^2 - x + 3)^(3/2)*x^2 - 71/1280*(2*x^2 - x + 3)^(3/2)*x + 287/5120*(
2*x^2 - x + 3)^(3/2) + 4609/4096*sqrt(2*x^2 - x + 3)*x + 106007/65536*sqrt(2)*arcsinh(1/23*sqrt(23)*(4*x - 1))
- 4609/16384*sqrt(2*x^2 - x + 3)

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Fricas [A]  time = 1.35483, size = 258, normalized size = 2.08 \begin{align*} \frac{1}{245760} \,{\left (204800 \, x^{5} - 59392 \, x^{4} + 258432 \, x^{3} + 105696 \, x^{2} + 221868 \, x - 27807\right )} \sqrt{2 \, x^{2} - x + 3} + \frac{106007}{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^4-x^3+3*x^2+x+2)*(2*x^2-x+3)^(1/2),x, algorithm="fricas")

[Out]

1/245760*(204800*x^5 - 59392*x^4 + 258432*x^3 + 105696*x^2 + 221868*x - 27807)*sqrt(2*x^2 - x + 3) + 106007/13
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^{4} - x^{3} + 3 x^{2} + x + 2\right )\, 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(sqrt(2*x**2 - x + 3)*(5*x**4 - x**3 + 3*x**2 + x + 2), x)

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Giac [A]  time = 1.17368, size = 99, normalized size = 0.8 \begin{align*} \frac{1}{245760} \,{\left (4 \,{\left (8 \,{\left (4 \,{\left (16 \,{\left (100 \, x - 29\right )} x + 2019\right )} x + 3303\right )} x + 55467\right )} x - 27807\right )} \sqrt{2 \, x^{2} - x + 3} - \frac{106007}{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^4-x^3+3*x^2+x+2)*(2*x^2-x+3)^(1/2),x, algorithm="giac")

[Out]

1/245760*(4*(8*(4*(16*(100*x - 29)*x + 2019)*x + 3303)*x + 55467)*x - 27807)*sqrt(2*x^2 - x + 3) - 106007/6553
6*sqrt(2)*log(-2*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3)) + 1)