### 3.360 $$\int \frac{2+x+3 x^2-x^3+5 x^4}{(3-x+2 x^2)^{5/2}} \, dx$$

Optimal. Leaf size=68 $\frac{219 x+89}{276 \left (2 x^2-x+3\right )^{3/2}}-\frac{2604 x+1465}{2116 \sqrt{2 x^2-x+3}}-\frac{5 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{4 \sqrt{2}}$

[Out]

(89 + 219*x)/(276*(3 - x + 2*x^2)^(3/2)) - (1465 + 2604*x)/(2116*Sqrt[3 - x + 2*x^2]) - (5*ArcSinh[(1 - 4*x)/S
qrt[23]])/(4*Sqrt[2])

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Rubi [A]  time = 0.0518717, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 33, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.121, Rules used = {1660, 12, 619, 215} $\frac{219 x+89}{276 \left (2 x^2-x+3\right )^{3/2}}-\frac{2604 x+1465}{2116 \sqrt{2 x^2-x+3}}-\frac{5 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{4 \sqrt{2}}$

Antiderivative was successfully veriﬁed.

[In]

Int[(2 + x + 3*x^2 - x^3 + 5*x^4)/(3 - x + 2*x^2)^(5/2),x]

[Out]

(89 + 219*x)/(276*(3 - x + 2*x^2)^(3/2)) - (1465 + 2604*x)/(2116*Sqrt[3 - x + 2*x^2]) - (5*ArcSinh[(1 - 4*x)/S
qrt[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{2+x+3 x^2-x^3+5 x^4}{\left (3-x+2 x^2\right )^{5/2}} \, dx &=\frac{89+219 x}{276 \left (3-x+2 x^2\right )^{3/2}}+\frac{2}{69} \int \frac{-\frac{159}{16}+\frac{207 x}{8}+\frac{345 x^2}{4}}{\left (3-x+2 x^2\right )^{3/2}} \, dx\\ &=\frac{89+219 x}{276 \left (3-x+2 x^2\right )^{3/2}}-\frac{1465+2604 x}{2116 \sqrt{3-x+2 x^2}}+\frac{4 \int \frac{7935}{16 \sqrt{3-x+2 x^2}} \, dx}{1587}\\ &=\frac{89+219 x}{276 \left (3-x+2 x^2\right )^{3/2}}-\frac{1465+2604 x}{2116 \sqrt{3-x+2 x^2}}+\frac{5}{4} \int \frac{1}{\sqrt{3-x+2 x^2}} \, dx\\ &=\frac{89+219 x}{276 \left (3-x+2 x^2\right )^{3/2}}-\frac{1465+2604 x}{2116 \sqrt{3-x+2 x^2}}+\frac{5 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{23}}} \, dx,x,-1+4 x\right )}{4 \sqrt{46}}\\ &=\frac{89+219 x}{276 \left (3-x+2 x^2\right )^{3/2}}-\frac{1465+2604 x}{2116 \sqrt{3-x+2 x^2}}-\frac{5 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{4 \sqrt{2}}\\ \end{align*}

Mathematica [A]  time = 0.226112, size = 55, normalized size = 0.81 $\frac{5 \sinh ^{-1}\left (\frac{4 x-1}{\sqrt{23}}\right )}{4 \sqrt{2}}-\frac{7812 x^3+489 x^2+7002 x+5569}{3174 \left (2 x^2-x+3\right )^{3/2}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(2 + x + 3*x^2 - x^3 + 5*x^4)/(3 - x + 2*x^2)^(5/2),x]

[Out]

-(5569 + 7002*x + 489*x^2 + 7812*x^3)/(3174*(3 - x + 2*x^2)^(3/2)) + (5*ArcSinh[(-1 + 4*x)/Sqrt[23]])/(4*Sqrt[
2])

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Maple [B]  time = 0.053, size = 146, normalized size = 2.2 \begin{align*} -{\frac{5\,{x}^{3}}{6} \left ( 2\,{x}^{2}-x+3 \right ) ^{-{\frac{3}{2}}}}-{\frac{{x}^{2}}{8} \left ( 2\,{x}^{2}-x+3 \right ) ^{-{\frac{3}{2}}}}-{\frac{47\,x}{64} \left ( 2\,{x}^{2}-x+3 \right ) ^{-{\frac{3}{2}}}}-{\frac{271}{768} \left ( 2\,{x}^{2}-x+3 \right ) ^{-{\frac{3}{2}}}}+{\frac{-2423+9692\,x}{17664} \left ( 2\,{x}^{2}-x+3 \right ) ^{-{\frac{3}{2}}}}+{\frac{-173+692\,x}{1587}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}-{\frac{5\,x}{4}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}-{\frac{5}{16}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}+{\frac{5\,\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^4-x^3+3*x^2+x+2)/(2*x^2-x+3)^(5/2),x)

[Out]

-5/6*x^3/(2*x^2-x+3)^(3/2)-1/8*x^2/(2*x^2-x+3)^(3/2)-47/64*x/(2*x^2-x+3)^(3/2)-271/768/(2*x^2-x+3)^(3/2)+2423/
17664*(-1+4*x)/(2*x^2-x+3)^(3/2)+173/1587*(-1+4*x)/(2*x^2-x+3)^(1/2)-5/4*x/(2*x^2-x+3)^(1/2)-5/16/(2*x^2-x+3)^
(1/2)+5/8*2^(1/2)*arcsinh(4/23*23^(1/2)*(x-1/4))

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Maxima [B]  time = 1.46495, size = 250, normalized size = 3.68 \begin{align*} \frac{5}{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{5}{8} \, \sqrt{2} \operatorname{arsinh}\left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) - \frac{355}{3174} \, \sqrt{2 \, x^{2} - x + 3} - \frac{58 \, x}{1587 \, \sqrt{2 \, x^{2} - x + 3}} + \frac{x^{2}}{2 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} - \frac{1897}{6348 \, \sqrt{2 \, x^{2} - x + 3}} - \frac{95 \, x}{276 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} + \frac{41}{276 \,{\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^4-x^3+3*x^2+x+2)/(2*x^2-x+3)^(5/2),x, algorithm="maxima")

[Out]

5/6348*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)) + 5/8*sqrt(2)*arcsinh(1/23*sqrt(23)*(4*x - 1)) - 355/3174*sqrt(2*x
^2 - x + 3) - 58/1587*x/sqrt(2*x^2 - x + 3) + 1/2*x^2/(2*x^2 - x + 3)^(3/2) - 1897/6348/sqrt(2*x^2 - x + 3) -
95/276*x/(2*x^2 - x + 3)^(3/2) + 41/276/(2*x^2 - x + 3)^(3/2)

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Fricas [B]  time = 1.34762, size = 300, normalized size = 4.41 \begin{align*} \frac{7935 \, \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 ) - 8 \,{\left (7812 \, x^{3} + 489 \, x^{2} + 7002 \, x + 5569\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^4-x^3+3*x^2+x+2)/(2*x^2-x+3)^(5/2),x, algorithm="fricas")

[Out]

1/25392*(7935*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) - 8*(7812*x^3 + 489*x^2 + 7002*x + 5569)*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{5 x^{4} - x^{3} + 3 x^{2} + x + 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**4-x**3+3*x**2+x+2)/(2*x**2-x+3)**(5/2),x)

[Out]

Integral((5*x**4 - x**3 + 3*x**2 + x + 2)/(2*x**2 - x + 3)**(5/2), x)

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Giac [A]  time = 1.14788, size = 84, normalized size = 1.24 \begin{align*} -\frac{5}{8} \, \sqrt{2} \log \left (-2 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )} + 1\right ) - \frac{3 \,{\left ({\left (2604 \, x + 163\right )} x + 2334\right )} x + 5569}{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^4-x^3+3*x^2+x+2)/(2*x^2-x+3)^(5/2),x, algorithm="giac")

[Out]

-5/8*sqrt(2)*log(-2*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3)) + 1) - 1/3174*(3*((2604*x + 163)*x + 2334)*x + 5
569)/(2*x^2 - x + 3)^(3/2)