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

Optimal. Leaf size=68 \[ \frac{121 (19-7 x)}{276 \left (2 x^2-x+3\right )^{3/2}}-\frac{11 (2336 x+7351)}{6348 \sqrt{2 x^2-x+3}}-\frac{25 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{4 \sqrt{2}} \]

[Out]

(121*(19 - 7*x))/(276*(3 - x + 2*x^2)^(3/2)) - (11*(7351 + 2336*x))/(6348*Sqrt[3 - x + 2*x^2]) - (25*ArcSinh[(
1 - 4*x)/Sqrt[23]])/(4*Sqrt[2])

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Rubi [A]  time = 0.0612398, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148, Rules used = {1660, 12, 619, 215} \[ \frac{121 (19-7 x)}{276 \left (2 x^2-x+3\right )^{3/2}}-\frac{11 (2336 x+7351)}{6348 \sqrt{2 x^2-x+3}}-\frac{25 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{4 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(121*(19 - 7*x))/(276*(3 - x + 2*x^2)^(3/2)) - (11*(7351 + 2336*x))/(6348*Sqrt[3 - x + 2*x^2]) - (25*ArcSinh[(
1 - 4*x)/Sqrt[23]])/(4*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 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 )^{5/2}} \, dx &=\frac{121 (19-7 x)}{276 \left (3-x+2 x^2\right )^{3/2}}+\frac{2}{69} \int \frac{\frac{131}{16}+\frac{5865 x}{8}+\frac{1725 x^2}{4}}{\left (3-x+2 x^2\right )^{3/2}} \, dx\\ &=\frac{121 (19-7 x)}{276 \left (3-x+2 x^2\right )^{3/2}}-\frac{11 (7351+2336 x)}{6348 \sqrt{3-x+2 x^2}}+\frac{4 \int \frac{39675}{16 \sqrt{3-x+2 x^2}} \, dx}{1587}\\ &=\frac{121 (19-7 x)}{276 \left (3-x+2 x^2\right )^{3/2}}-\frac{11 (7351+2336 x)}{6348 \sqrt{3-x+2 x^2}}+\frac{25}{4} \int \frac{1}{\sqrt{3-x+2 x^2}} \, dx\\ &=\frac{121 (19-7 x)}{276 \left (3-x+2 x^2\right )^{3/2}}-\frac{11 (7351+2336 x)}{6348 \sqrt{3-x+2 x^2}}+\frac{25 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{23}}} \, dx,x,-1+4 x\right )}{4 \sqrt{46}}\\ &=\frac{121 (19-7 x)}{276 \left (3-x+2 x^2\right )^{3/2}}-\frac{11 (7351+2336 x)}{6348 \sqrt{3-x+2 x^2}}-\frac{25 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{4 \sqrt{2}}\\ \end{align*}

Mathematica [A]  time = 0.240769, size = 55, normalized size = 0.81 \[ \frac{25 \sinh ^{-1}\left (\frac{4 x-1}{\sqrt{23}}\right )}{4 \sqrt{2}}-\frac{11 \left (2336 x^3+6183 x^2+714 x+8623\right )}{3174 \left (2 x^2-x+3\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-11*(8623 + 714*x + 6183*x^2 + 2336*x^3))/(3174*(3 - x + 2*x^2)^(3/2)) + (25*ArcSinh[(-1 + 4*x)/Sqrt[23]])/(4
*Sqrt[2])

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Maple [B]  time = 0.052, size = 146, normalized size = 2.2 \begin{align*} -{\frac{25\,{x}^{3}}{6} \left ( 2\,{x}^{2}-x+3 \right ) ^{-{\frac{3}{2}}}}-{\frac{145\,{x}^{2}}{8} \left ( 2\,{x}^{2}-x+3 \right ) ^{-{\frac{3}{2}}}}-{\frac{319\,x}{64} \left ( 2\,{x}^{2}-x+3 \right ) ^{-{\frac{3}{2}}}}-{\frac{15775}{768} \left ( 2\,{x}^{2}-x+3 \right ) ^{-{\frac{3}{2}}}}+{\frac{-8493+33972\,x}{5888} \left ( 2\,{x}^{2}-x+3 \right ) ^{-{\frac{3}{2}}}}+{\frac{-2267+9068\,x}{2116}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}-{\frac{25\,x}{4}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}-{\frac{25}{16}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}+{\frac{25\,\sqrt{2}}{8}{\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)^(5/2),x)

[Out]

-25/6*x^3/(2*x^2-x+3)^(3/2)-145/8*x^2/(2*x^2-x+3)^(3/2)-319/64*x/(2*x^2-x+3)^(3/2)-15775/768/(2*x^2-x+3)^(3/2)
+8493/5888*(-1+4*x)/(2*x^2-x+3)^(3/2)+2267/2116*(-1+4*x)/(2*x^2-x+3)^(1/2)-25/4*x/(2*x^2-x+3)^(1/2)-25/16/(2*x
^2-x+3)^(1/2)+25/8*2^(1/2)*arcsinh(4/23*23^(1/2)*(x-1/4))

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Maxima [B]  time = 1.77602, size = 250, normalized size = 3.68 \begin{align*} \frac{25}{6348} \, x{\left (\frac{284 \, x}{\sqrt{2 \, x^{2} - x + 3}} - \frac{3174 \, x^{2}}{{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} - \frac{71}{\sqrt{2 \, x^{2} - x + 3}} + \frac{805 \, x}{{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} - \frac{3243}{{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}\right )} + \frac{25}{8} \, \sqrt{2} \operatorname{arsinh}\left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) - \frac{1775}{3174} \, \sqrt{2 \, x^{2} - x + 3} + \frac{1017 \, x}{529 \, \sqrt{2 \, x^{2} - x + 3}} - \frac{15 \, x^{2}}{{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} - \frac{6413}{3174 \, \sqrt{2 \, x^{2} - x + 3}} - \frac{x}{138 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} - \frac{2593}{138 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} \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)^(5/2),x, algorithm="maxima")

[Out]

25/6348*x*(284*x/sqrt(2*x^2 - x + 3) - 3174*x^2/(2*x^2 - x + 3)^(3/2) - 71/sqrt(2*x^2 - x + 3) + 805*x/(2*x^2
- x + 3)^(3/2) - 3243/(2*x^2 - x + 3)^(3/2)) + 25/8*sqrt(2)*arcsinh(1/23*sqrt(23)*(4*x - 1)) - 1775/3174*sqrt(
2*x^2 - x + 3) + 1017/529*x/sqrt(2*x^2 - x + 3) - 15*x^2/(2*x^2 - x + 3)^(3/2) - 6413/3174/sqrt(2*x^2 - x + 3)
- 1/138*x/(2*x^2 - x + 3)^(3/2) - 2593/138/(2*x^2 - x + 3)^(3/2)

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Fricas [B]  time = 1.37113, size = 302, normalized size = 4.44 \begin{align*} \frac{39675 \, \sqrt{2}{\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\right )} \log \left (-4 \, \sqrt{2} \sqrt{2 \, x^{2} - x + 3}{\left (4 \, x - 1\right )} - 32 \, x^{2} + 16 \, x - 25\right ) - 88 \,{\left (2336 \, x^{3} + 6183 \, x^{2} + 714 \, x + 8623\right )} \sqrt{2 \, x^{2} - x + 3}}{25392 \,{\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\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)^(5/2),x, algorithm="fricas")

[Out]

1/25392*(39675*sqrt(2)*(4*x^4 - 4*x^3 + 13*x^2 - 6*x + 9)*log(-4*sqrt(2)*sqrt(2*x^2 - x + 3)*(4*x - 1) - 32*x^
2 + 16*x - 25) - 88*(2336*x^3 + 6183*x^2 + 714*x + 8623)*sqrt(2*x^2 - x + 3))/(4*x^4 - 4*x^3 + 13*x^2 - 6*x +
9)

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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{5}{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)**(5/2),x)

[Out]

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

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Giac [A]  time = 1.26898, size = 82, normalized size = 1.21 \begin{align*} -\frac{25}{8} \, \sqrt{2} \log \left (-2 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )} + 1\right ) - \frac{11 \,{\left ({\left ({\left (2336 \, x + 6183\right )} x + 714\right )} x + 8623\right )}}{3174 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} \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)^(5/2),x, algorithm="giac")

[Out]

-25/8*sqrt(2)*log(-2*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3)) + 1) - 11/3174*(((2336*x + 6183)*x + 714)*x + 8
623)/(2*x^2 - x + 3)^(3/2)