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

Optimal. Leaf size=84 \[ \frac{625 x^7}{14}+\frac{3625 x^6}{24}+\frac{1855 x^5}{8}+\frac{6245 x^4}{64}-\frac{21229 x^3}{96}-\frac{28747 x^2}{128}+\frac{307461}{512} \log \left (2 x^2-x+3\right )+\frac{122691 x}{128}+\frac{1156639 \tan ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{256 \sqrt{23}} \]

[Out]

(122691*x)/128 - (28747*x^2)/128 - (21229*x^3)/96 + (6245*x^4)/64 + (1855*x^5)/8 + (3625*x^6)/24 + (625*x^7)/1
4 + (1156639*ArcTan[(1 - 4*x)/Sqrt[23]])/(256*Sqrt[23]) + (307461*Log[3 - x + 2*x^2])/512

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Rubi [A]  time = 0.056764, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {1657, 634, 618, 204, 628} \[ \frac{625 x^7}{14}+\frac{3625 x^6}{24}+\frac{1855 x^5}{8}+\frac{6245 x^4}{64}-\frac{21229 x^3}{96}-\frac{28747 x^2}{128}+\frac{307461}{512} \log \left (2 x^2-x+3\right )+\frac{122691 x}{128}+\frac{1156639 \tan ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{256 \sqrt{23}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(122691*x)/128 - (28747*x^2)/128 - (21229*x^3)/96 + (6245*x^4)/64 + (1855*x^5)/8 + (3625*x^6)/24 + (625*x^7)/1
4 + (1156639*ArcTan[(1 - 4*x)/Sqrt[23]])/(256*Sqrt[23]) + (307461*Log[3 - x + 2*x^2])/512

Rule 1657

Int[(Pq_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x + c*x^2)^p, x
], x] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rubi steps

\begin{align*} \int \frac{\left (2+3 x+5 x^2\right )^4}{3-x+2 x^2} \, dx &=\int \left (\frac{122691}{128}-\frac{28747 x}{64}-\frac{21229 x^2}{32}+\frac{6245 x^3}{16}+\frac{9275 x^4}{8}+\frac{3625 x^5}{4}+\frac{625 x^6}{2}-\frac{14641 (25-21 x)}{128 \left (3-x+2 x^2\right )}\right ) \, dx\\ &=\frac{122691 x}{128}-\frac{28747 x^2}{128}-\frac{21229 x^3}{96}+\frac{6245 x^4}{64}+\frac{1855 x^5}{8}+\frac{3625 x^6}{24}+\frac{625 x^7}{14}-\frac{14641}{128} \int \frac{25-21 x}{3-x+2 x^2} \, dx\\ &=\frac{122691 x}{128}-\frac{28747 x^2}{128}-\frac{21229 x^3}{96}+\frac{6245 x^4}{64}+\frac{1855 x^5}{8}+\frac{3625 x^6}{24}+\frac{625 x^7}{14}+\frac{307461}{512} \int \frac{-1+4 x}{3-x+2 x^2} \, dx-\frac{1156639}{512} \int \frac{1}{3-x+2 x^2} \, dx\\ &=\frac{122691 x}{128}-\frac{28747 x^2}{128}-\frac{21229 x^3}{96}+\frac{6245 x^4}{64}+\frac{1855 x^5}{8}+\frac{3625 x^6}{24}+\frac{625 x^7}{14}+\frac{307461}{512} \log \left (3-x+2 x^2\right )+\frac{1156639}{256} \operatorname{Subst}\left (\int \frac{1}{-23-x^2} \, dx,x,-1+4 x\right )\\ &=\frac{122691 x}{128}-\frac{28747 x^2}{128}-\frac{21229 x^3}{96}+\frac{6245 x^4}{64}+\frac{1855 x^5}{8}+\frac{3625 x^6}{24}+\frac{625 x^7}{14}+\frac{1156639 \tan ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{256 \sqrt{23}}+\frac{307461}{512} \log \left (3-x+2 x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.0268885, size = 72, normalized size = 0.86 \[ \frac{x \left (120000 x^6+406000 x^5+623280 x^4+262290 x^3-594412 x^2-603687 x+2576511\right )}{2688}+\frac{307461}{512} \log \left (2 x^2-x+3\right )-\frac{1156639 \tan ^{-1}\left (\frac{4 x-1}{\sqrt{23}}\right )}{256 \sqrt{23}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(2576511 - 603687*x - 594412*x^2 + 262290*x^3 + 623280*x^4 + 406000*x^5 + 120000*x^6))/2688 - (1156639*ArcT
an[(-1 + 4*x)/Sqrt[23]])/(256*Sqrt[23]) + (307461*Log[3 - x + 2*x^2])/512

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Maple [A]  time = 0.046, size = 64, normalized size = 0.8 \begin{align*}{\frac{625\,{x}^{7}}{14}}+{\frac{3625\,{x}^{6}}{24}}+{\frac{1855\,{x}^{5}}{8}}+{\frac{6245\,{x}^{4}}{64}}-{\frac{21229\,{x}^{3}}{96}}-{\frac{28747\,{x}^{2}}{128}}+{\frac{122691\,x}{128}}+{\frac{307461\,\ln \left ( 2\,{x}^{2}-x+3 \right ) }{512}}-{\frac{1156639\,\sqrt{23}}{5888}\arctan \left ({\frac{ \left ( -1+4\,x \right ) \sqrt{23}}{23}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

625/14*x^7+3625/24*x^6+1855/8*x^5+6245/64*x^4-21229/96*x^3-28747/128*x^2+122691/128*x+307461/512*ln(2*x^2-x+3)
-1156639/5888*23^(1/2)*arctan(1/23*(-1+4*x)*23^(1/2))

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Maxima [A]  time = 1.43462, size = 85, normalized size = 1.01 \begin{align*} \frac{625}{14} \, x^{7} + \frac{3625}{24} \, x^{6} + \frac{1855}{8} \, x^{5} + \frac{6245}{64} \, x^{4} - \frac{21229}{96} \, x^{3} - \frac{28747}{128} \, x^{2} - \frac{1156639}{5888} \, \sqrt{23} \arctan \left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) + \frac{122691}{128} \, x + \frac{307461}{512} \, \log \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)^4/(2*x^2-x+3),x, algorithm="maxima")

[Out]

625/14*x^7 + 3625/24*x^6 + 1855/8*x^5 + 6245/64*x^4 - 21229/96*x^3 - 28747/128*x^2 - 1156639/5888*sqrt(23)*arc
tan(1/23*sqrt(23)*(4*x - 1)) + 122691/128*x + 307461/512*log(2*x^2 - x + 3)

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Fricas [A]  time = 0.975225, size = 252, normalized size = 3. \begin{align*} \frac{625}{14} \, x^{7} + \frac{3625}{24} \, x^{6} + \frac{1855}{8} \, x^{5} + \frac{6245}{64} \, x^{4} - \frac{21229}{96} \, x^{3} - \frac{28747}{128} \, x^{2} - \frac{1156639}{5888} \, \sqrt{23} \arctan \left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) + \frac{122691}{128} \, x + \frac{307461}{512} \, \log \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)^4/(2*x^2-x+3),x, algorithm="fricas")

[Out]

625/14*x^7 + 3625/24*x^6 + 1855/8*x^5 + 6245/64*x^4 - 21229/96*x^3 - 28747/128*x^2 - 1156639/5888*sqrt(23)*arc
tan(1/23*sqrt(23)*(4*x - 1)) + 122691/128*x + 307461/512*log(2*x^2 - x + 3)

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Sympy [A]  time = 0.156462, size = 87, normalized size = 1.04 \begin{align*} \frac{625 x^{7}}{14} + \frac{3625 x^{6}}{24} + \frac{1855 x^{5}}{8} + \frac{6245 x^{4}}{64} - \frac{21229 x^{3}}{96} - \frac{28747 x^{2}}{128} + \frac{122691 x}{128} + \frac{307461 \log{\left (x^{2} - \frac{x}{2} + \frac{3}{2} \right )}}{512} - \frac{1156639 \sqrt{23} \operatorname{atan}{\left (\frac{4 \sqrt{23} x}{23} - \frac{\sqrt{23}}{23} \right )}}{5888} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

625*x**7/14 + 3625*x**6/24 + 1855*x**5/8 + 6245*x**4/64 - 21229*x**3/96 - 28747*x**2/128 + 122691*x/128 + 3074
61*log(x**2 - x/2 + 3/2)/512 - 1156639*sqrt(23)*atan(4*sqrt(23)*x/23 - sqrt(23)/23)/5888

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Giac [A]  time = 1.25004, size = 85, normalized size = 1.01 \begin{align*} \frac{625}{14} \, x^{7} + \frac{3625}{24} \, x^{6} + \frac{1855}{8} \, x^{5} + \frac{6245}{64} \, x^{4} - \frac{21229}{96} \, x^{3} - \frac{28747}{128} \, x^{2} - \frac{1156639}{5888} \, \sqrt{23} \arctan \left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) + \frac{122691}{128} \, x + \frac{307461}{512} \, \log \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)^4/(2*x^2-x+3),x, algorithm="giac")

[Out]

625/14*x^7 + 3625/24*x^6 + 1855/8*x^5 + 6245/64*x^4 - 21229/96*x^3 - 28747/128*x^2 - 1156639/5888*sqrt(23)*arc
tan(1/23*sqrt(23)*(4*x - 1)) + 122691/128*x + 307461/512*log(2*x^2 - x + 3)