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

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

[Out]

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

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Rubi [A]  time = 0.10, antiderivative size = 124, normalized size of antiderivative = 1.00, 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 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]

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 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 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]

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*}

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Mathematica [A]  time = 0.10, 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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fricas [A]  time = 0.85, size = 78, normalized size = 0.63 \[ \frac {1}{49152} \, {\left (204800 \, x^{5} + 284672 \, x^{4} + 408960 \, x^{3} + 365536 \, x^{2} + 328204 \, x - 64023\right )} \sqrt {2 \, x^{2} - x + 3} + \frac {284533}{131072} \, \sqrt {2} \log \left (4 \, \sqrt {2} \sqrt {2 \, x^{2} - x + 3} {\left (4 \, x - 1\right )} - 32 \, x^{2} + 16 \, x - 25\right ) \]

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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giac [A]  time = 0.22, size = 73, normalized size = 0.59 \[ \frac {1}{49152} \, {\left (4 \, {\left (8 \, {\left (4 \, {\left (16 \, {\left (100 \, x + 139\right )} x + 3195\right )} x + 11423\right )} x + 82051\right )} x - 64023\right )} \sqrt {2 \, x^{2} - x + 3} + \frac {284533}{65536} \, \sqrt {2} \log \left (-2 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )} + 1\right ) \]

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)

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maple [A]  time = 0.01, size = 98, normalized size = 0.79 \[ \frac {25 \left (2 x^{2}-x +3\right )^{\frac {3}{2}} x^{3}}{12}+\frac {63 \left (2 x^{2}-x +3\right )^{\frac {3}{2}} x^{2}}{16}+\frac {769 \left (2 x^{2}-x +3\right )^{\frac {3}{2}} x}{256}-\frac {284533 \sqrt {2}\, \arcsinh \left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{65536}-\frac {2107 \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}{3072}-\frac {12371 \left (4 x -1\right ) \sqrt {2 x^{2}-x +3}}{16384} \]

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*(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-2107/3072*(2*x^2-x+3)^(3/2
)-12371/16384*(4*x-1)*(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 = 0.98, size = 109, normalized size = 0.88 \[ \frac {25}{12} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{3} + \frac {63}{16} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} + \frac {769}{256} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x - \frac {2107}{3072} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} - \frac {12371}{4096} \, \sqrt {2 \, x^{2} - x + 3} x - \frac {284533}{65536} \, \sqrt {2} \operatorname {arsinh}\left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) + \frac {12371}{16384} \, \sqrt {2 \, x^{2} - x + 3} \]

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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mupad [B]  time = 4.19, size = 153, normalized size = 1.23 \[ \frac {63\,x^2\,{\left (2\,x^2-x+3\right )}^{3/2}}{16}+\frac {25\,x^3\,{\left (2\,x^2-x+3\right )}^{3/2}}{12}-\frac {29509\,\sqrt {2}\,\ln \left (\sqrt {2\,x^2-x+3}+\frac {\sqrt {2}\,\left (2\,x-\frac {1}{2}\right )}{2}\right )}{8192}-\frac {1283\,\left (\frac {x}{2}-\frac {1}{8}\right )\,\sqrt {2\,x^2-x+3}}{256}-\frac {2107\,\sqrt {2\,x^2-x+3}\,\left (32\,x^2-4\,x+45\right )}{49152}+\frac {769\,x\,{\left (2\,x^2-x+3\right )}^{3/2}}{256}-\frac {48461\,\sqrt {2}\,\ln \left (2\,\sqrt {2\,x^2-x+3}+\frac {\sqrt {2}\,\left (4\,x-1\right )}{2}\right )}{65536} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x^2 - x + 3)^(1/2)*(3*x + 5*x^2 + 2)^2,x)

[Out]

(63*x^2*(2*x^2 - x + 3)^(3/2))/16 + (25*x^3*(2*x^2 - x + 3)^(3/2))/12 - (29509*2^(1/2)*log((2*x^2 - x + 3)^(1/
2) + (2^(1/2)*(2*x - 1/2))/2))/8192 - (1283*(x/2 - 1/8)*(2*x^2 - x + 3)^(1/2))/256 - (2107*(2*x^2 - x + 3)^(1/
2)*(32*x^2 - 4*x + 45))/49152 + (769*x*(2*x^2 - x + 3)^(3/2))/256 - (48461*2^(1/2)*log(2*(2*x^2 - x + 3)^(1/2)
+ (2^(1/2)*(4*x - 1))/2))/65536

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {2 x^{2} - x + 3} \left (5 x^{2} + 3 x + 2\right )^{2}\, dx \]

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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