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

Optimal. Leaf size=105 \[ \frac{5}{12} x \left (2 x^2-x+3\right )^{5/2}+\frac{107}{240} \left (2 x^2-x+3\right )^{5/2}-\frac{179 (1-4 x) \left (2 x^2-x+3\right )^{3/2}}{1536}-\frac{4117 (1-4 x) \sqrt{2 x^2-x+3}}{8192}-\frac{94691 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{16384 \sqrt{2}} \]

[Out]

(-4117*(1 - 4*x)*Sqrt[3 - x + 2*x^2])/8192 - (179*(1 - 4*x)*(3 - x + 2*x^2)^(3/2))/1536 + (107*(3 - x + 2*x^2)
^(5/2))/240 + (5*x*(3 - x + 2*x^2)^(5/2))/12 - (94691*ArcSinh[(1 - 4*x)/Sqrt[23]])/(16384*Sqrt[2])

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Rubi [A]  time = 0.0504904, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {1661, 640, 612, 619, 215} \[ \frac{5}{12} x \left (2 x^2-x+3\right )^{5/2}+\frac{107}{240} \left (2 x^2-x+3\right )^{5/2}-\frac{179 (1-4 x) \left (2 x^2-x+3\right )^{3/2}}{1536}-\frac{4117 (1-4 x) \sqrt{2 x^2-x+3}}{8192}-\frac{94691 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{16384 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-4117*(1 - 4*x)*Sqrt[3 - x + 2*x^2])/8192 - (179*(1 - 4*x)*(3 - x + 2*x^2)^(3/2))/1536 + (107*(3 - x + 2*x^2)
^(5/2))/240 + (5*x*(3 - x + 2*x^2)^(5/2))/12 - (94691*ArcSinh[(1 - 4*x)/Sqrt[23]])/(16384*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+3 x+5 x^2\right ) \, dx &=\frac{5}{12} x \left (3-x+2 x^2\right )^{5/2}+\frac{1}{12} \int \left (9+\frac{107 x}{2}\right ) \left (3-x+2 x^2\right )^{3/2} \, dx\\ &=\frac{107}{240} \left (3-x+2 x^2\right )^{5/2}+\frac{5}{12} x \left (3-x+2 x^2\right )^{5/2}+\frac{179}{96} \int \left (3-x+2 x^2\right )^{3/2} \, dx\\ &=-\frac{179 (1-4 x) \left (3-x+2 x^2\right )^{3/2}}{1536}+\frac{107}{240} \left (3-x+2 x^2\right )^{5/2}+\frac{5}{12} x \left (3-x+2 x^2\right )^{5/2}+\frac{4117 \int \sqrt{3-x+2 x^2} \, dx}{1024}\\ &=-\frac{4117 (1-4 x) \sqrt{3-x+2 x^2}}{8192}-\frac{179 (1-4 x) \left (3-x+2 x^2\right )^{3/2}}{1536}+\frac{107}{240} \left (3-x+2 x^2\right )^{5/2}+\frac{5}{12} x \left (3-x+2 x^2\right )^{5/2}+\frac{94691 \int \frac{1}{\sqrt{3-x+2 x^2}} \, dx}{16384}\\ &=-\frac{4117 (1-4 x) \sqrt{3-x+2 x^2}}{8192}-\frac{179 (1-4 x) \left (3-x+2 x^2\right )^{3/2}}{1536}+\frac{107}{240} \left (3-x+2 x^2\right )^{5/2}+\frac{5}{12} x \left (3-x+2 x^2\right )^{5/2}+\frac{\left (4117 \sqrt{\frac{23}{2}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{23}}} \, dx,x,-1+4 x\right )}{16384}\\ &=-\frac{4117 (1-4 x) \sqrt{3-x+2 x^2}}{8192}-\frac{179 (1-4 x) \left (3-x+2 x^2\right )^{3/2}}{1536}+\frac{107}{240} \left (3-x+2 x^2\right )^{5/2}+\frac{5}{12} x \left (3-x+2 x^2\right )^{5/2}-\frac{94691 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{16384 \sqrt{2}}\\ \end{align*}

Mathematica [A]  time = 0.0837233, size = 65, normalized size = 0.62 \[ \frac{4 \sqrt{2 x^2-x+3} \left (204800 x^5+14336 x^4+561024 x^3+319072 x^2+565276 x+388341\right )-1420365 \sqrt{2} \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{491520} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*Sqrt[3 - x + 2*x^2]*(388341 + 565276*x + 319072*x^2 + 561024*x^3 + 14336*x^4 + 204800*x^5) - 1420365*Sqrt[2
]*ArcSinh[(1 - 4*x)/Sqrt[23]])/491520

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Maple [A]  time = 0.058, size = 83, normalized size = 0.8 \begin{align*}{\frac{5\,x}{12} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{5}{2}}}}+{\frac{107}{240} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{5}{2}}}}+{\frac{-179+716\,x}{1536} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{3}{2}}}}+{\frac{-4117+16468\,x}{8192}\sqrt{2\,{x}^{2}-x+3}}+{\frac{94691\,\sqrt{2}}{32768}{\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((2*x^2-x+3)^(3/2)*(5*x^2+3*x+2),x)

[Out]

5/12*x*(2*x^2-x+3)^(5/2)+107/240*(2*x^2-x+3)^(5/2)+179/1536*(-1+4*x)*(2*x^2-x+3)^(3/2)+4117/8192*(-1+4*x)*(2*x
^2-x+3)^(1/2)+94691/32768*2^(1/2)*arcsinh(4/23*23^(1/2)*(x-1/4))

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Maxima [A]  time = 1.49874, size = 140, normalized size = 1.33 \begin{align*} \frac{5}{12} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}} x + \frac{107}{240} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}} + \frac{179}{384} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x - \frac{179}{1536} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{4117}{2048} \, \sqrt{2 \, x^{2} - x + 3} x + \frac{94691}{32768} \, \sqrt{2} \operatorname{arsinh}\left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) - \frac{4117}{8192} \, \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^2+3*x+2),x, algorithm="maxima")

[Out]

5/12*(2*x^2 - x + 3)^(5/2)*x + 107/240*(2*x^2 - x + 3)^(5/2) + 179/384*(2*x^2 - x + 3)^(3/2)*x - 179/1536*(2*x
^2 - x + 3)^(3/2) + 4117/2048*sqrt(2*x^2 - x + 3)*x + 94691/32768*sqrt(2)*arcsinh(1/23*sqrt(23)*(4*x - 1)) - 4
117/8192*sqrt(2*x^2 - x + 3)

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Fricas [A]  time = 1.60156, size = 257, normalized size = 2.45 \begin{align*} \frac{1}{122880} \,{\left (204800 \, x^{5} + 14336 \, x^{4} + 561024 \, x^{3} + 319072 \, x^{2} + 565276 \, x + 388341\right )} \sqrt{2 \, x^{2} - x + 3} + \frac{94691}{65536} \, \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^2+3*x+2),x, algorithm="fricas")

[Out]

1/122880*(204800*x^5 + 14336*x^4 + 561024*x^3 + 319072*x^2 + 565276*x + 388341)*sqrt(2*x^2 - x + 3) + 94691/65
536*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^{2} + 3 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**2+3*x+2),x)

[Out]

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

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Giac [A]  time = 1.24798, size = 99, normalized size = 0.94 \begin{align*} \frac{1}{122880} \,{\left (4 \,{\left (8 \,{\left (4 \,{\left (16 \,{\left (100 \, x + 7\right )} x + 4383\right )} x + 9971\right )} x + 141319\right )} x + 388341\right )} \sqrt{2 \, x^{2} - x + 3} - \frac{94691}{32768} \, \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^2+3*x+2),x, algorithm="giac")

[Out]

1/122880*(4*(8*(4*(16*(100*x + 7)*x + 4383)*x + 9971)*x + 141319)*x + 388341)*sqrt(2*x^2 - x + 3) - 94691/3276
8*sqrt(2)*log(-2*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3)) + 1)