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

Optimal. Leaf size=101 \[ \frac {655}{96} \sqrt {2 x^2-x+3} x^2+\frac {3443}{768} \sqrt {2 x^2-x+3} x-\frac {11373 \sqrt {2 x^2-x+3}}{1024}+\frac {25}{8} \sqrt {2 x^2-x+3} x^3+\frac {30725 \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{2048 \sqrt {2}} \]

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Rubi [A]  time = 0.09, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {1661, 640, 619, 215} \begin {gather*} \frac {25}{8} \sqrt {2 x^2-x+3} x^3+\frac {655}{96} \sqrt {2 x^2-x+3} x^2+\frac {3443}{768} \sqrt {2 x^2-x+3} x-\frac {11373 \sqrt {2 x^2-x+3}}{1024}+\frac {30725 \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{2048 \sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-11373*Sqrt[3 - x + 2*x^2])/1024 + (3443*x*Sqrt[3 - x + 2*x^2])/768 + (655*x^2*Sqrt[3 - x + 2*x^2])/96 + (25*
x^3*Sqrt[3 - x + 2*x^2])/8 + (30725*ArcSinh[(1 - 4*x)/Sqrt[23]])/(2048*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 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 \frac {\left (2+3 x+5 x^2\right )^2}{\sqrt {3-x+2 x^2}} \, dx &=\frac {25}{8} x^3 \sqrt {3-x+2 x^2}+\frac {1}{8} \int \frac {32+96 x+7 x^2+\frac {655 x^3}{2}}{\sqrt {3-x+2 x^2}} \, dx\\ &=\frac {655}{96} x^2 \sqrt {3-x+2 x^2}+\frac {25}{8} x^3 \sqrt {3-x+2 x^2}+\frac {1}{48} \int \frac {192-1389 x+\frac {3443 x^2}{4}}{\sqrt {3-x+2 x^2}} \, dx\\ &=\frac {3443}{768} x \sqrt {3-x+2 x^2}+\frac {655}{96} x^2 \sqrt {3-x+2 x^2}+\frac {25}{8} x^3 \sqrt {3-x+2 x^2}+\frac {1}{192} \int \frac {-\frac {7257}{4}-\frac {34119 x}{8}}{\sqrt {3-x+2 x^2}} \, dx\\ &=-\frac {11373 \sqrt {3-x+2 x^2}}{1024}+\frac {3443}{768} x \sqrt {3-x+2 x^2}+\frac {655}{96} x^2 \sqrt {3-x+2 x^2}+\frac {25}{8} x^3 \sqrt {3-x+2 x^2}-\frac {30725 \int \frac {1}{\sqrt {3-x+2 x^2}} \, dx}{2048}\\ &=-\frac {11373 \sqrt {3-x+2 x^2}}{1024}+\frac {3443}{768} x \sqrt {3-x+2 x^2}+\frac {655}{96} x^2 \sqrt {3-x+2 x^2}+\frac {25}{8} x^3 \sqrt {3-x+2 x^2}-\frac {30725 \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{23}}} \, dx,x,-1+4 x\right )}{2048 \sqrt {46}}\\ &=-\frac {11373 \sqrt {3-x+2 x^2}}{1024}+\frac {3443}{768} x \sqrt {3-x+2 x^2}+\frac {655}{96} x^2 \sqrt {3-x+2 x^2}+\frac {25}{8} x^3 \sqrt {3-x+2 x^2}+\frac {30725 \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{2048 \sqrt {2}}\\ \end {align*}

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Mathematica [A]  time = 0.08, size = 55, normalized size = 0.54 \begin {gather*} \frac {4 \sqrt {2 x^2-x+3} \left (9600 x^3+20960 x^2+13772 x-34119\right )+92175 \sqrt {2} \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{12288} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4*Sqrt[3 - x + 2*x^2]*(-34119 + 13772*x + 20960*x^2 + 9600*x^3) + 92175*Sqrt[2]*ArcSinh[(1 - 4*x)/Sqrt[23]])/
12288

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IntegrateAlgebraic [A]  time = 0.49, size = 70, normalized size = 0.69 \begin {gather*} \frac {30725 \log \left (2 \sqrt {2} \sqrt {2 x^2-x+3}-4 x+1\right )}{2048 \sqrt {2}}+\frac {\sqrt {2 x^2-x+3} \left (9600 x^3+20960 x^2+13772 x-34119\right )}{3072} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[3 - x + 2*x^2]*(-34119 + 13772*x + 20960*x^2 + 9600*x^3))/3072 + (30725*Log[1 - 4*x + 2*Sqrt[2]*Sqrt[3 -
x + 2*x^2]])/(2048*Sqrt[2])

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fricas [A]  time = 0.41, size = 68, normalized size = 0.67 \begin {gather*} \frac {1}{3072} \, {\left (9600 \, x^{3} + 20960 \, x^{2} + 13772 \, x - 34119\right )} \sqrt {2 \, x^{2} - x + 3} + \frac {30725}{8192} \, \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 {gather*}

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]

1/3072*(9600*x^3 + 20960*x^2 + 13772*x - 34119)*sqrt(2*x^2 - x + 3) + 30725/8192*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 = 0.49, size = 63, normalized size = 0.62 \begin {gather*} \frac {1}{3072} \, {\left (4 \, {\left (40 \, {\left (60 \, x + 131\right )} x + 3443\right )} x - 34119\right )} \sqrt {2 \, x^{2} - x + 3} + \frac {30725}{4096} \, \sqrt {2} \log \left (-2 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )} + 1\right ) \end {gather*}

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]

1/3072*(4*(40*(60*x + 131)*x + 3443)*x - 34119)*sqrt(2*x^2 - x + 3) + 30725/4096*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 = 79, normalized size = 0.78 \begin {gather*} \frac {25 \sqrt {2 x^{2}-x +3}\, x^{3}}{8}+\frac {655 \sqrt {2 x^{2}-x +3}\, x^{2}}{96}+\frac {3443 \sqrt {2 x^{2}-x +3}\, x}{768}-\frac {30725 \sqrt {2}\, \arcsinh \left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{4096}-\frac {11373 \sqrt {2 x^{2}-x +3}}{1024} \end {gather*}

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]

25/8*(2*x^2-x+3)^(1/2)*x^3+655/96*(2*x^2-x+3)^(1/2)*x^2+3443/768*(2*x^2-x+3)^(1/2)*x-11373/1024*(2*x^2-x+3)^(1
/2)-30725/4096*2^(1/2)*arcsinh(4/23*23^(1/2)*(x-1/4))

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maxima [A]  time = 0.97, size = 80, normalized size = 0.79 \begin {gather*} \frac {25}{8} \, \sqrt {2 \, x^{2} - x + 3} x^{3} + \frac {655}{96} \, \sqrt {2 \, x^{2} - x + 3} x^{2} + \frac {3443}{768} \, \sqrt {2 \, x^{2} - x + 3} x - \frac {30725}{4096} \, \sqrt {2} \operatorname {arsinh}\left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) - \frac {11373}{1024} \, \sqrt {2 \, x^{2} - x + 3} \end {gather*}

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]

25/8*sqrt(2*x^2 - x + 3)*x^3 + 655/96*sqrt(2*x^2 - x + 3)*x^2 + 3443/768*sqrt(2*x^2 - x + 3)*x - 30725/4096*sq
rt(2)*arcsinh(1/23*sqrt(23)*(4*x - 1)) - 11373/1024*sqrt(2*x^2 - x + 3)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (5\,x^2+3\,x+2\right )}^2}{\sqrt {2\,x^2-x+3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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

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