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

Optimal. Leaf size=124 \[ \frac{5}{4} \sqrt{2 x^2-x+3} x^3+\frac{153}{16} \sqrt{2 x^2-x+3} x^2+\frac{2645}{128} \sqrt{2 x^2-x+3} x-\frac{13153}{512} \sqrt{2 x^2-x+3}-\frac{4 (346-533 x)}{23 \sqrt{2 x^2-x+3}}+\frac{144217 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{1024 \sqrt{2}} \]

[Out]

(-4*(346 - 533*x))/(23*Sqrt[3 - x + 2*x^2]) - (13153*Sqrt[3 - x + 2*x^2])/512 + (2645*x*Sqrt[3 - x + 2*x^2])/1
28 + (153*x^2*Sqrt[3 - x + 2*x^2])/16 + (5*x^3*Sqrt[3 - x + 2*x^2])/4 + (144217*ArcSinh[(1 - 4*x)/Sqrt[23]])/(
1024*Sqrt[2])

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Rubi [A]  time = 0.152252, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {1660, 1661, 640, 619, 215} \[ \frac{5}{4} \sqrt{2 x^2-x+3} x^3+\frac{153}{16} \sqrt{2 x^2-x+3} x^2+\frac{2645}{128} \sqrt{2 x^2-x+3} x-\frac{13153}{512} \sqrt{2 x^2-x+3}-\frac{4 (346-533 x)}{23 \sqrt{2 x^2-x+3}}+\frac{144217 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{1024 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*(346 - 533*x))/(23*Sqrt[3 - x + 2*x^2]) - (13153*Sqrt[3 - x + 2*x^2])/512 + (2645*x*Sqrt[3 - x + 2*x^2])/1
28 + (153*x^2*Sqrt[3 - x + 2*x^2])/16 + (5*x^3*Sqrt[3 - x + 2*x^2])/4 + (144217*ArcSinh[(1 - 4*x)/Sqrt[23]])/(
1024*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)^2 \left (2+x+3 x^2-x^3+5 x^4\right )}{\left (3-x+2 x^2\right )^{3/2}} \, dx &=-\frac{4 (346-533 x)}{23 \sqrt{3-x+2 x^2}}+\frac{2}{23} \int \frac{-759-\frac{575 x}{2}+805 x^2+\frac{1219 x^3}{2}+115 x^4}{\sqrt{3-x+2 x^2}} \, dx\\ &=-\frac{4 (346-533 x)}{23 \sqrt{3-x+2 x^2}}+\frac{5}{4} x^3 \sqrt{3-x+2 x^2}+\frac{1}{92} \int \frac{-6072-2300 x+5405 x^2+\frac{10557 x^3}{2}}{\sqrt{3-x+2 x^2}} \, dx\\ &=-\frac{4 (346-533 x)}{23 \sqrt{3-x+2 x^2}}+\frac{153}{16} x^2 \sqrt{3-x+2 x^2}+\frac{5}{4} x^3 \sqrt{3-x+2 x^2}+\frac{1}{552} \int \frac{-36432-45471 x+\frac{182505 x^2}{4}}{\sqrt{3-x+2 x^2}} \, dx\\ &=-\frac{4 (346-533 x)}{23 \sqrt{3-x+2 x^2}}+\frac{2645}{128} x \sqrt{3-x+2 x^2}+\frac{153}{16} x^2 \sqrt{3-x+2 x^2}+\frac{5}{4} x^3 \sqrt{3-x+2 x^2}+\frac{\int \frac{-\frac{1130427}{4}-\frac{907557 x}{8}}{\sqrt{3-x+2 x^2}} \, dx}{2208}\\ &=-\frac{4 (346-533 x)}{23 \sqrt{3-x+2 x^2}}-\frac{13153}{512} \sqrt{3-x+2 x^2}+\frac{2645}{128} x \sqrt{3-x+2 x^2}+\frac{153}{16} x^2 \sqrt{3-x+2 x^2}+\frac{5}{4} x^3 \sqrt{3-x+2 x^2}-\frac{144217 \int \frac{1}{\sqrt{3-x+2 x^2}} \, dx}{1024}\\ &=-\frac{4 (346-533 x)}{23 \sqrt{3-x+2 x^2}}-\frac{13153}{512} \sqrt{3-x+2 x^2}+\frac{2645}{128} x \sqrt{3-x+2 x^2}+\frac{153}{16} x^2 \sqrt{3-x+2 x^2}+\frac{5}{4} x^3 \sqrt{3-x+2 x^2}-\frac{144217 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{23}}} \, dx,x,-1+4 x\right )}{1024 \sqrt{46}}\\ &=-\frac{4 (346-533 x)}{23 \sqrt{3-x+2 x^2}}-\frac{13153}{512} \sqrt{3-x+2 x^2}+\frac{2645}{128} x \sqrt{3-x+2 x^2}+\frac{153}{16} x^2 \sqrt{3-x+2 x^2}+\frac{5}{4} x^3 \sqrt{3-x+2 x^2}+\frac{144217 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{1024 \sqrt{2}}\\ \end{align*}

Mathematica [A]  time = 0.491766, size = 74, normalized size = 0.6 \[ \frac{4 \left (29440 x^5+210496 x^4+418232 x^3-510554 x^2+2124123 x-1616165\right )+3316991 \sqrt{4 x^2-2 x+6} \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{47104 \sqrt{2 x^2-x+3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*(-1616165 + 2124123*x - 510554*x^2 + 418232*x^3 + 210496*x^4 + 29440*x^5) + 3316991*Sqrt[6 - 2*x + 4*x^2]*A
rcSinh[(1 - 4*x)/Sqrt[23]])/(47104*Sqrt[3 - x + 2*x^2])

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Maple [A]  time = 0.056, size = 132, normalized size = 1.1 \begin{align*}{\frac{5\,{x}^{5}}{2}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}+{\frac{143\,{x}^{4}}{8}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}-{\frac{144217\,\sqrt{2}}{2048}{\it Arcsinh} \left ({\frac{4\,\sqrt{23}}{23} \left ( x-{\frac{1}{4}} \right ) } \right ) }+{\frac{-931255+3725020\,x}{94208}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}+{\frac{144217\,x}{1024}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}+{\frac{2273\,{x}^{3}}{64}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}-{\frac{11099\,{x}^{2}}{256}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}-{\frac{521655}{4096}{\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)^2*(5*x^4-x^3+3*x^2+x+2)/(2*x^2-x+3)^(3/2),x)

[Out]

5/2*x^5/(2*x^2-x+3)^(1/2)+143/8*x^4/(2*x^2-x+3)^(1/2)-144217/2048*2^(1/2)*arcsinh(4/23*23^(1/2)*(x-1/4))+93125
5/94208*(-1+4*x)/(2*x^2-x+3)^(1/2)+144217/1024*x/(2*x^2-x+3)^(1/2)+2273/64*x^3/(2*x^2-x+3)^(1/2)-11099/256*x^2
/(2*x^2-x+3)^(1/2)-521655/4096/(2*x^2-x+3)^(1/2)

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Maxima [A]  time = 1.64788, size = 154, normalized size = 1.24 \begin{align*} \frac{5 \, x^{5}}{2 \, \sqrt{2 \, x^{2} - x + 3}} + \frac{143 \, x^{4}}{8 \, \sqrt{2 \, x^{2} - x + 3}} + \frac{2273 \, x^{3}}{64 \, \sqrt{2 \, x^{2} - x + 3}} - \frac{11099 \, x^{2}}{256 \, \sqrt{2 \, x^{2} - x + 3}} - \frac{144217}{2048} \, \sqrt{2} \operatorname{arsinh}\left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) + \frac{2124123 \, x}{11776 \, \sqrt{2 \, x^{2} - x + 3}} - \frac{1616165}{11776 \, \sqrt{2 \, x^{2} - x + 3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5/2*x^5/sqrt(2*x^2 - x + 3) + 143/8*x^4/sqrt(2*x^2 - x + 3) + 2273/64*x^3/sqrt(2*x^2 - x + 3) - 11099/256*x^2/
sqrt(2*x^2 - x + 3) - 144217/2048*sqrt(2)*arcsinh(1/23*sqrt(23)*(4*x - 1)) + 2124123/11776*x/sqrt(2*x^2 - x +
3) - 1616165/11776/sqrt(2*x^2 - x + 3)

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Fricas [A]  time = 1.36305, size = 300, normalized size = 2.42 \begin{align*} \frac{3316991 \, \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 (29440 \, x^{5} + 210496 \, x^{4} + 418232 \, x^{3} - 510554 \, x^{2} + 2124123 \, x - 1616165\right )} \sqrt{2 \, x^{2} - x + 3}}{94208 \,{\left (2 \, x^{2} - x + 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/94208*(3316991*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
*(29440*x^5 + 210496*x^4 + 418232*x^3 - 510554*x^2 + 2124123*x - 1616165)*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 )^{2} \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)**2*(5*x**4-x**3+3*x**2+x+2)/(2*x**2-x+3)**(3/2),x)

[Out]

Integral((2*x + 5)**2*(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 = 1.15011, size = 97, normalized size = 0.78 \begin{align*} \frac{144217}{2048} \, \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 (8 \,{\left (20 \, x + 143\right )} x + 2273\right )} x - 11099\right )} x + 2124123\right )} x - 1616165}{11776 \, \sqrt{2 \, x^{2} - x + 3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

144217/2048*sqrt(2)*log(-2*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3)) + 1) + 1/11776*((46*(4*(8*(20*x + 143)*x
+ 2273)*x - 11099)*x + 2124123)*x - 1616165)/sqrt(2*x^2 - x + 3)