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

Optimal. Leaf size=147 \[ \frac{5}{16} \left (2 x^2-x+3\right )^{5/2} x^3+\frac{23}{448} \left (2 x^2-x+3\right )^{5/2} x^2+\frac{125 \left (2 x^2-x+3\right )^{5/2} x}{3584}+\frac{1167 \left (2 x^2-x+3\right )^{5/2}}{14336}-\frac{8597 (1-4 x) \left (2 x^2-x+3\right )^{3/2}}{65536}-\frac{593193 (1-4 x) \sqrt{2 x^2-x+3}}{1048576}-\frac{13643439 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{2097152 \sqrt{2}} \]

[Out]

(-593193*(1 - 4*x)*Sqrt[3 - x + 2*x^2])/1048576 - (8597*(1 - 4*x)*(3 - x + 2*x^2)^(3/2))/65536 + (1167*(3 - x
+ 2*x^2)^(5/2))/14336 + (125*x*(3 - x + 2*x^2)^(5/2))/3584 + (23*x^2*(3 - x + 2*x^2)^(5/2))/448 + (5*x^3*(3 -
x + 2*x^2)^(5/2))/16 - (13643439*ArcSinh[(1 - 4*x)/Sqrt[23]])/(2097152*Sqrt[2])

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Rubi [A]  time = 0.121456, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 8, 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} \[ \frac{5}{16} \left (2 x^2-x+3\right )^{5/2} x^3+\frac{23}{448} \left (2 x^2-x+3\right )^{5/2} x^2+\frac{125 \left (2 x^2-x+3\right )^{5/2} x}{3584}+\frac{1167 \left (2 x^2-x+3\right )^{5/2}}{14336}-\frac{8597 (1-4 x) \left (2 x^2-x+3\right )^{3/2}}{65536}-\frac{593193 (1-4 x) \sqrt{2 x^2-x+3}}{1048576}-\frac{13643439 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{2097152 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-593193*(1 - 4*x)*Sqrt[3 - x + 2*x^2])/1048576 - (8597*(1 - 4*x)*(3 - x + 2*x^2)^(3/2))/65536 + (1167*(3 - x
+ 2*x^2)^(5/2))/14336 + (125*x*(3 - x + 2*x^2)^(5/2))/3584 + (23*x^2*(3 - x + 2*x^2)^(5/2))/448 + (5*x^3*(3 -
x + 2*x^2)^(5/2))/16 - (13643439*ArcSinh[(1 - 4*x)/Sqrt[23]])/(2097152*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 \left (3-x+2 x^2\right )^{3/2} \left (2+x+3 x^2-x^3+5 x^4\right ) \, dx &=\frac{5}{16} x^3 \left (3-x+2 x^2\right )^{5/2}+\frac{1}{16} \int \left (3-x+2 x^2\right )^{3/2} \left (32+16 x+3 x^2+\frac{23 x^3}{2}\right ) \, dx\\ &=\frac{23}{448} x^2 \left (3-x+2 x^2\right )^{5/2}+\frac{5}{16} x^3 \left (3-x+2 x^2\right )^{5/2}+\frac{1}{224} \int \left (3-x+2 x^2\right )^{3/2} \left (448+155 x+\frac{375 x^2}{4}\right ) \, dx\\ &=\frac{125 x \left (3-x+2 x^2\right )^{5/2}}{3584}+\frac{23}{448} x^2 \left (3-x+2 x^2\right )^{5/2}+\frac{5}{16} x^3 \left (3-x+2 x^2\right )^{5/2}+\frac{\int \left (\frac{20379}{4}+\frac{17505 x}{8}\right ) \left (3-x+2 x^2\right )^{3/2} \, dx}{2688}\\ &=\frac{1167 \left (3-x+2 x^2\right )^{5/2}}{14336}+\frac{125 x \left (3-x+2 x^2\right )^{5/2}}{3584}+\frac{23}{448} x^2 \left (3-x+2 x^2\right )^{5/2}+\frac{5}{16} x^3 \left (3-x+2 x^2\right )^{5/2}+\frac{8597 \int \left (3-x+2 x^2\right )^{3/2} \, dx}{4096}\\ &=-\frac{8597 (1-4 x) \left (3-x+2 x^2\right )^{3/2}}{65536}+\frac{1167 \left (3-x+2 x^2\right )^{5/2}}{14336}+\frac{125 x \left (3-x+2 x^2\right )^{5/2}}{3584}+\frac{23}{448} x^2 \left (3-x+2 x^2\right )^{5/2}+\frac{5}{16} x^3 \left (3-x+2 x^2\right )^{5/2}+\frac{593193 \int \sqrt{3-x+2 x^2} \, dx}{131072}\\ &=-\frac{593193 (1-4 x) \sqrt{3-x+2 x^2}}{1048576}-\frac{8597 (1-4 x) \left (3-x+2 x^2\right )^{3/2}}{65536}+\frac{1167 \left (3-x+2 x^2\right )^{5/2}}{14336}+\frac{125 x \left (3-x+2 x^2\right )^{5/2}}{3584}+\frac{23}{448} x^2 \left (3-x+2 x^2\right )^{5/2}+\frac{5}{16} x^3 \left (3-x+2 x^2\right )^{5/2}+\frac{13643439 \int \frac{1}{\sqrt{3-x+2 x^2}} \, dx}{2097152}\\ &=-\frac{593193 (1-4 x) \sqrt{3-x+2 x^2}}{1048576}-\frac{8597 (1-4 x) \left (3-x+2 x^2\right )^{3/2}}{65536}+\frac{1167 \left (3-x+2 x^2\right )^{5/2}}{14336}+\frac{125 x \left (3-x+2 x^2\right )^{5/2}}{3584}+\frac{23}{448} x^2 \left (3-x+2 x^2\right )^{5/2}+\frac{5}{16} x^3 \left (3-x+2 x^2\right )^{5/2}+\frac{\left (593193 \sqrt{\frac{23}{2}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{23}}} \, dx,x,-1+4 x\right )}{2097152}\\ &=-\frac{593193 (1-4 x) \sqrt{3-x+2 x^2}}{1048576}-\frac{8597 (1-4 x) \left (3-x+2 x^2\right )^{3/2}}{65536}+\frac{1167 \left (3-x+2 x^2\right )^{5/2}}{14336}+\frac{125 x \left (3-x+2 x^2\right )^{5/2}}{3584}+\frac{23}{448} x^2 \left (3-x+2 x^2\right )^{5/2}+\frac{5}{16} x^3 \left (3-x+2 x^2\right )^{5/2}-\frac{13643439 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{2097152 \sqrt{2}}\\ \end{align*}

Mathematica [A]  time = 0.129866, size = 75, normalized size = 0.51 \[ \frac{4 \sqrt{2 x^2-x+3} \left (9175040 x^7-7667712 x^6+29335552 x^5-7497728 x^4+27023744 x^3+3845856 x^2+27845612 x-1663407\right )-95504073 \sqrt{2} \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{29360128} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*Sqrt[3 - x + 2*x^2]*(-1663407 + 27845612*x + 3845856*x^2 + 27023744*x^3 - 7497728*x^4 + 29335552*x^5 - 7667
712*x^6 + 9175040*x^7) - 95504073*Sqrt[2]*ArcSinh[(1 - 4*x)/Sqrt[23]])/29360128

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Maple [A]  time = 0.055, size = 117, normalized size = 0.8 \begin{align*}{\frac{5\,{x}^{3}}{16} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{5}{2}}}}+{\frac{23\,{x}^{2}}{448} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{5}{2}}}}+{\frac{125\,x}{3584} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{5}{2}}}}+{\frac{-593193+2372772\,x}{1048576}\sqrt{2\,{x}^{2}-x+3}}+{\frac{13643439\,\sqrt{2}}{4194304}{\it Arcsinh} \left ({\frac{4\,\sqrt{23}}{23} \left ( x-{\frac{1}{4}} \right ) } \right ) }+{\frac{-8597+34388\,x}{65536} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{3}{2}}}}+{\frac{1167}{14336} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x^2-x+3)^(3/2)*(5*x^4-x^3+3*x^2+x+2),x)

[Out]

5/16*x^3*(2*x^2-x+3)^(5/2)+23/448*x^2*(2*x^2-x+3)^(5/2)+125/3584*x*(2*x^2-x+3)^(5/2)+593193/1048576*(-1+4*x)*(
2*x^2-x+3)^(1/2)[phone]/4194304*2^(1/2)*arcsinh(4/23*23^(1/2)*(x-1/4))+8597/65536*(-1+4*x)*(2*x^2-x+3)^(3/2)
+1167/14336*(2*x^2-x+3)^(5/2)

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Maxima [A]  time = 1.47795, size = 186, normalized size = 1.27 \begin{align*} \frac{5}{16} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}} x^{3} + \frac{23}{448} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}} x^{2} + \frac{125}{3584} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}} x + \frac{1167}{14336} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}} + \frac{8597}{16384} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x - \frac{8597}{65536} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{593193}{262144} \, \sqrt{2 \, x^{2} - x + 3} x + \frac{13643439}{4194304} \, \sqrt{2} \operatorname{arsinh}\left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) - \frac{593193}{1048576} \, \sqrt{2 \, x^{2} - x + 3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5/16*(2*x^2 - x + 3)^(5/2)*x^3 + 23/448*(2*x^2 - x + 3)^(5/2)*x^2 + 125/3584*(2*x^2 - x + 3)^(5/2)*x + 1167/14
336*(2*x^2 - x + 3)^(5/2) + 8597/16384*(2*x^2 - x + 3)^(3/2)*x - 8597/65536*(2*x^2 - x + 3)^(3/2) + 593193/262
144*sqrt(2*x^2 - x + 3)*x + 13643439/4194304*sqrt(2)*arcsinh(1/23*sqrt(23)*(4*x - 1)) - 593193/1048576*sqrt(2*
x^2 - x + 3)

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Fricas [A]  time = 1.2855, size = 316, normalized size = 2.15 \begin{align*} \frac{1}{7340032} \,{\left (9175040 \, x^{7} - 7667712 \, x^{6} + 29335552 \, x^{5} - 7497728 \, x^{4} + 27023744 \, x^{3} + 3845856 \, x^{2} + 27845612 \, x - 1663407\right )} \sqrt{2 \, x^{2} - x + 3} + \frac{13643439}{8388608} \, \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((2*x^2-x+3)^(3/2)*(5*x^4-x^3+3*x^2+x+2),x, algorithm="fricas")

[Out]

1/7340032*(9175040*x^7 - 7667712*x^6 + 29335552*x^5 - 7497728*x^4 + 27023744*x^3 + 3845856*x^2 + 27845612*x -
1663407)*sqrt(2*x^2 - x + 3) + 13643439/8388608*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 \left (2 x^{2} - x + 3\right )^{\frac{3}{2}} \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((2*x**2-x+3)**(3/2)*(5*x**4-x**3+3*x**2+x+2),x)

[Out]

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

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Giac [A]  time = 1.38808, size = 112, normalized size = 0.76 \begin{align*} \frac{1}{7340032} \,{\left (4 \,{\left (8 \,{\left (4 \,{\left (16 \,{\left (4 \,{\left (8 \,{\left (140 \, x - 117\right )} x + 3581\right )} x - 3661\right )} x + 211123\right )} x + 120183\right )} x + 6961403\right )} x - 1663407\right )} \sqrt{2 \, x^{2} - x + 3} - \frac{13643439}{4194304} \, \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((2*x^2-x+3)^(3/2)*(5*x^4-x^3+3*x^2+x+2),x, algorithm="giac")

[Out]

1/7340032*(4*(8*(4*(16*(4*(8*(140*x - 117)*x + 3581)*x - 3661)*x + 211123)*x + 120183)*x + 6961403)*x - 166340
7)*sqrt(2*x^2 - x + 3) - 13643439/4194304*sqrt(2)*log(-2*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3)) + 1)