### 3.352 $$\int \frac{(5+2 x) (2+x+3 x^2-x^3+5 x^4)}{(3-x+2 x^2)^{3/2}} \, dx$$

Optimal. Leaf size=103 $\frac{5}{6} \sqrt{2 x^2-x+3} x^2+\frac{193}{48} \sqrt{2 x^2-x+3} x+\frac{33}{64} \sqrt{2 x^2-x+3}-\frac{53-373 x}{23 \sqrt{2 x^2-x+3}}+\frac{3111 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{128 \sqrt{2}}$

[Out]

-(53 - 373*x)/(23*Sqrt[3 - x + 2*x^2]) + (33*Sqrt[3 - x + 2*x^2])/64 + (193*x*Sqrt[3 - x + 2*x^2])/48 + (5*x^2
*Sqrt[3 - x + 2*x^2])/6 + (3111*ArcSinh[(1 - 4*x)/Sqrt[23]])/(128*Sqrt[2])

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Rubi [A]  time = 0.101666, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 38, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.132, Rules used = {1660, 1661, 640, 619, 215} $\frac{5}{6} \sqrt{2 x^2-x+3} x^2+\frac{193}{48} \sqrt{2 x^2-x+3} x+\frac{33}{64} \sqrt{2 x^2-x+3}-\frac{53-373 x}{23 \sqrt{2 x^2-x+3}}+\frac{3111 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{128 \sqrt{2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-(53 - 373*x)/(23*Sqrt[3 - x + 2*x^2]) + (33*Sqrt[3 - x + 2*x^2])/64 + (193*x*Sqrt[3 - x + 2*x^2])/48 + (5*x^2
*Sqrt[3 - x + 2*x^2])/6 + (3111*ArcSinh[(1 - 4*x)/Sqrt[23]])/(128*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{(5+2 x) \left (2+x+3 x^2-x^3+5 x^4\right )}{\left (3-x+2 x^2\right )^{3/2}} \, dx &=-\frac{53-373 x}{23 \sqrt{3-x+2 x^2}}+\frac{2}{23} \int \frac{-\frac{575}{4}+161 x^2+\frac{115 x^3}{2}}{\sqrt{3-x+2 x^2}} \, dx\\ &=-\frac{53-373 x}{23 \sqrt{3-x+2 x^2}}+\frac{5}{6} x^2 \sqrt{3-x+2 x^2}+\frac{1}{69} \int \frac{-\frac{1725}{2}-345 x+\frac{4439 x^2}{4}}{\sqrt{3-x+2 x^2}} \, dx\\ &=-\frac{53-373 x}{23 \sqrt{3-x+2 x^2}}+\frac{193}{48} x \sqrt{3-x+2 x^2}+\frac{5}{6} x^2 \sqrt{3-x+2 x^2}+\frac{1}{276} \int \frac{-\frac{27117}{4}+\frac{2277 x}{8}}{\sqrt{3-x+2 x^2}} \, dx\\ &=-\frac{53-373 x}{23 \sqrt{3-x+2 x^2}}+\frac{33}{64} \sqrt{3-x+2 x^2}+\frac{193}{48} x \sqrt{3-x+2 x^2}+\frac{5}{6} x^2 \sqrt{3-x+2 x^2}-\frac{3111}{128} \int \frac{1}{\sqrt{3-x+2 x^2}} \, dx\\ &=-\frac{53-373 x}{23 \sqrt{3-x+2 x^2}}+\frac{33}{64} \sqrt{3-x+2 x^2}+\frac{193}{48} x \sqrt{3-x+2 x^2}+\frac{5}{6} x^2 \sqrt{3-x+2 x^2}-\frac{3111 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{23}}} \, dx,x,-1+4 x\right )}{128 \sqrt{46}}\\ &=-\frac{53-373 x}{23 \sqrt{3-x+2 x^2}}+\frac{33}{64} \sqrt{3-x+2 x^2}+\frac{193}{48} x \sqrt{3-x+2 x^2}+\frac{5}{6} x^2 \sqrt{3-x+2 x^2}+\frac{3111 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{128 \sqrt{2}}\\ \end{align*}

Mathematica [A]  time = 0.18094, size = 60, normalized size = 0.58 $\frac{7360 x^4+31832 x^3-2162 x^2+122607 x-3345}{4416 \sqrt{2 x^2-x+3}}-\frac{3111 \sinh ^{-1}\left (\frac{4 x-1}{\sqrt{23}}\right )}{128 \sqrt{2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-3345 + 122607*x - 2162*x^2 + 31832*x^3 + 7360*x^4)/(4416*Sqrt[3 - x + 2*x^2]) - (3111*ArcSinh[(-1 + 4*x)/Sqr
t[23]])/(128*Sqrt[2])

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Maple [A]  time = 0.064, size = 115, normalized size = 1.1 \begin{align*}{\frac{5\,{x}^{4}}{3}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}-{\frac{3111\,\sqrt{2}}{256}{\it Arcsinh} \left ({\frac{4\,\sqrt{23}}{23} \left ( x-{\frac{1}{4}} \right ) } \right ) }+{\frac{-10185+40740\,x}{11776}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}+{\frac{3111\,x}{128}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}+{\frac{173\,{x}^{3}}{24}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}-{\frac{47\,{x}^{2}}{96}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}+{\frac{55}{512}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5/3*x^4/(2*x^2-x+3)^(1/2)-3111/256*2^(1/2)*arcsinh(4/23*23^(1/2)*(x-1/4))+10185/11776*(-1+4*x)/(2*x^2-x+3)^(1/
2)+3111/128*x/(2*x^2-x+3)^(1/2)+173/24*x^3/(2*x^2-x+3)^(1/2)-47/96*x^2/(2*x^2-x+3)^(1/2)+55/512/(2*x^2-x+3)^(1
/2)

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Maxima [A]  time = 1.4559, size = 131, normalized size = 1.27 \begin{align*} \frac{5 \, x^{4}}{3 \, \sqrt{2 \, x^{2} - x + 3}} + \frac{173 \, x^{3}}{24 \, \sqrt{2 \, x^{2} - x + 3}} - \frac{47 \, x^{2}}{96 \, \sqrt{2 \, x^{2} - x + 3}} - \frac{3111}{256} \, \sqrt{2} \operatorname{arsinh}\left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) + \frac{40869 \, x}{1472 \, \sqrt{2 \, x^{2} - x + 3}} - \frac{1115}{1472 \, \sqrt{2 \, x^{2} - x + 3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((5+2*x)*(5*x^4-x^3+3*x^2+x+2)/(2*x^2-x+3)^(3/2),x, algorithm="maxima")

[Out]

5/3*x^4/sqrt(2*x^2 - x + 3) + 173/24*x^3/sqrt(2*x^2 - x + 3) - 47/96*x^2/sqrt(2*x^2 - x + 3) - 3111/256*sqrt(2
)*arcsinh(1/23*sqrt(23)*(4*x - 1)) + 40869/1472*x/sqrt(2*x^2 - x + 3) - 1115/1472/sqrt(2*x^2 - x + 3)

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Fricas [A]  time = 1.32192, size = 270, normalized size = 2.62 \begin{align*} \frac{214659 \, \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 (7360 \, x^{4} + 31832 \, x^{3} - 2162 \, x^{2} + 122607 \, x - 3345\right )} \sqrt{2 \, x^{2} - x + 3}}{35328 \,{\left (2 \, x^{2} - x + 3\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((5+2*x)*(5*x^4-x^3+3*x^2+x+2)/(2*x^2-x+3)^(3/2),x, algorithm="fricas")

[Out]

1/35328*(214659*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*
(7360*x^4 + 31832*x^3 - 2162*x^2 + 122607*x - 3345)*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 (2 x + 5\right ) \left (5 x^{4} - x^{3} + 3 x^{2} + x + 2\right )}{\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+2*x)*(5*x**4-x**3+3*x**2+x+2)/(2*x**2-x+3)**(3/2),x)

[Out]

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

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Giac [A]  time = 1.14613, size = 90, normalized size = 0.87 \begin{align*} \frac{3111}{256} \, \sqrt{2} \log \left (-2 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )} + 1\right ) + \frac{{\left (46 \,{\left (4 \,{\left (40 \, x + 173\right )} x - 47\right )} x + 122607\right )} x - 3345}{4416 \, \sqrt{2 \, x^{2} - x + 3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((5+2*x)*(5*x^4-x^3+3*x^2+x+2)/(2*x^2-x+3)^(3/2),x, algorithm="giac")

[Out]

3111/256*sqrt(2)*log(-2*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3)) + 1) + 1/4416*((46*(4*(40*x + 173)*x - 47)*x
+ 122607)*x - 3345)/sqrt(2*x^2 - x + 3)