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

Optimal. Leaf size=82 \[ \frac{121 (19-7 x)}{92 \sqrt{2 x^2-x+3}}+\frac{25}{8} x \sqrt{2 x^2-x+3}+\frac{415}{32} \sqrt{2 x^2-x+3}-\frac{223 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{64 \sqrt{2}} \]

[Out]

(121*(19 - 7*x))/(92*Sqrt[3 - x + 2*x^2]) + (415*Sqrt[3 - x + 2*x^2])/32 + (25*x*Sqrt[3 - x + 2*x^2])/8 - (223
*ArcSinh[(1 - 4*x)/Sqrt[23]])/(64*Sqrt[2])

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Rubi [A]  time = 0.0708539, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {1660, 1661, 640, 619, 215} \[ \frac{121 (19-7 x)}{92 \sqrt{2 x^2-x+3}}+\frac{25}{8} x \sqrt{2 x^2-x+3}+\frac{415}{32} \sqrt{2 x^2-x+3}-\frac{223 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{64 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(121*(19 - 7*x))/(92*Sqrt[3 - x + 2*x^2]) + (415*Sqrt[3 - x + 2*x^2])/32 + (25*x*Sqrt[3 - x + 2*x^2])/8 - (223
*ArcSinh[(1 - 4*x)/Sqrt[23]])/(64*Sqrt[2])

Rule 1660

Int[(Pq_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, a + b*x + c*
x^2, x], f = Coeff[PolynomialRemainder[Pq, a + b*x + c*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b
*x + c*x^2, x], x, 1]}, Simp[((b*f - 2*a*g + (2*c*f - b*g)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c
)), x] + Dist[1/((p + 1)*(b^2 - 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1)*ExpandToSum[(p + 1)*(b^2 - 4*a*c)*Q - (
2*p + 3)*(2*c*f - b*g), x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[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]

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 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 \frac{\left (2+3 x+5 x^2\right )^2}{\left (3-x+2 x^2\right )^{3/2}} \, dx &=\frac{121 (19-7 x)}{92 \sqrt{3-x+2 x^2}}+\frac{2}{23} \int \frac{\frac{1173}{16}+\frac{1955 x}{8}+\frac{575 x^2}{4}}{\sqrt{3-x+2 x^2}} \, dx\\ &=\frac{121 (19-7 x)}{92 \sqrt{3-x+2 x^2}}+\frac{25}{8} x \sqrt{3-x+2 x^2}+\frac{1}{46} \int \frac{-138+\frac{9545 x}{8}}{\sqrt{3-x+2 x^2}} \, dx\\ &=\frac{121 (19-7 x)}{92 \sqrt{3-x+2 x^2}}+\frac{415}{32} \sqrt{3-x+2 x^2}+\frac{25}{8} x \sqrt{3-x+2 x^2}+\frac{223}{64} \int \frac{1}{\sqrt{3-x+2 x^2}} \, dx\\ &=\frac{121 (19-7 x)}{92 \sqrt{3-x+2 x^2}}+\frac{415}{32} \sqrt{3-x+2 x^2}+\frac{25}{8} x \sqrt{3-x+2 x^2}+\frac{223 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{23}}} \, dx,x,-1+4 x\right )}{64 \sqrt{46}}\\ &=\frac{121 (19-7 x)}{92 \sqrt{3-x+2 x^2}}+\frac{415}{32} \sqrt{3-x+2 x^2}+\frac{25}{8} x \sqrt{3-x+2 x^2}-\frac{223 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{64 \sqrt{2}}\\ \end{align*}

Mathematica [A]  time = 0.140896, size = 55, normalized size = 0.67 \[ \frac{4600 x^3+16790 x^2-9421 x+47027}{736 \sqrt{2 x^2-x+3}}+\frac{223 \sinh ^{-1}\left (\frac{4 x-1}{\sqrt{23}}\right )}{64 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(47027 - 9421*x + 16790*x^2 + 4600*x^3)/(736*Sqrt[3 - x + 2*x^2]) + (223*ArcSinh[(-1 + 4*x)/Sqrt[23]])/(64*Sqr
t[2])

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Maple [A]  time = 0.052, size = 98, normalized size = 1.2 \begin{align*}{\frac{25\,{x}^{3}}{4}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}+{\frac{365\,{x}^{2}}{16}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}-{\frac{223\,x}{64}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}+{\frac{15761}{256}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}-{\frac{-13713+54852\,x}{5888}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}+{\frac{223\,\sqrt{2}}{128}{\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^2+3*x+2)^2/(2*x^2-x+3)^(3/2),x)

[Out]

25/4*x^3/(2*x^2-x+3)^(1/2)+365/16*x^2/(2*x^2-x+3)^(1/2)-223/64*x/(2*x^2-x+3)^(1/2)+15761/256/(2*x^2-x+3)^(1/2)
-13713/5888*(-1+4*x)/(2*x^2-x+3)^(1/2)+223/128*2^(1/2)*arcsinh(4/23*23^(1/2)*(x-1/4))

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Maxima [A]  time = 1.48402, size = 108, normalized size = 1.32 \begin{align*} \frac{25 \, x^{3}}{4 \, \sqrt{2 \, x^{2} - x + 3}} + \frac{365 \, x^{2}}{16 \, \sqrt{2 \, x^{2} - x + 3}} + \frac{223}{128} \, \sqrt{2} \operatorname{arsinh}\left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) - \frac{9421 \, x}{736 \, \sqrt{2 \, x^{2} - x + 3}} + \frac{47027}{736 \, \sqrt{2 \, x^{2} - x + 3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

25/4*x^3/sqrt(2*x^2 - x + 3) + 365/16*x^2/sqrt(2*x^2 - x + 3) + 223/128*sqrt(2)*arcsinh(1/23*sqrt(23)*(4*x - 1
)) - 9421/736*x/sqrt(2*x^2 - x + 3) + 47027/736/sqrt(2*x^2 - x + 3)

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Fricas [A]  time = 1.3002, size = 251, normalized size = 3.06 \begin{align*} \frac{5129 \, \sqrt{2}{\left (2 \, x^{2} - x + 3\right )} \log \left (-4 \, \sqrt{2} \sqrt{2 \, x^{2} - x + 3}{\left (4 \, x - 1\right )} - 32 \, x^{2} + 16 \, x - 25\right ) + 8 \,{\left (4600 \, x^{3} + 16790 \, x^{2} - 9421 \, x + 47027\right )} \sqrt{2 \, x^{2} - x + 3}}{5888 \,{\left (2 \, x^{2} - x + 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5888*(5129*sqrt(2)*(2*x^2 - x + 3)*log(-4*sqrt(2)*sqrt(2*x^2 - x + 3)*(4*x - 1) - 32*x^2 + 16*x - 25) + 8*(4
600*x^3 + 16790*x^2 - 9421*x + 47027)*sqrt(2*x^2 - x + 3))/(2*x^2 - x + 3)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.18148, size = 84, normalized size = 1.02 \begin{align*} -\frac{223}{128} \, \sqrt{2} \log \left (-2 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )} + 1\right ) + \frac{{\left (230 \,{\left (20 \, x + 73\right )} x - 9421\right )} x + 47027}{736 \, \sqrt{2 \, x^{2} - x + 3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-223/128*sqrt(2)*log(-2*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3)) + 1) + 1/736*((230*(20*x + 73)*x - 9421)*x +
47027)/sqrt(2*x^2 - x + 3)