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

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

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Rubi [A]  time = 0.06, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1657, 634, 618, 204, 628} \begin {gather*} \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}} \end {gather*}

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 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 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 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]

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 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]

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*}

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Mathematica [A]  time = 0.03, size = 72, normalized size = 0.86 \begin {gather*} \frac {307461}{512} \log \left (2 x^2-x+3\right )+\frac {x \left (120000 x^6+406000 x^5+623280 x^4+262290 x^3-594412 x^2-603687 x+2576511\right )}{2688}-\frac {1156639 \tan ^{-1}\left (\frac {4 x-1}{\sqrt {23}}\right )}{256 \sqrt {23}} \end {gather*}

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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (2+3 x+5 x^2\right )^4}{3-x+2 x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.40, size = 63, normalized size = 0.75 \begin {gather*} \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 {gather*}

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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giac [A]  time = 0.21, size = 63, normalized size = 0.75 \begin {gather*} \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 {gather*}

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)

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maple [A]  time = 0.01, size = 64, normalized size = 0.76 \begin {gather*} \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 {1156639 \sqrt {23}\, \arctan \left (\frac {\left (4 x -1\right ) \sqrt {23}}{23}\right )}{5888}+\frac {307461 \ln \left (2 x^{2}-x +3\right )}{512} \end {gather*}

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*(4*x-1)*23^(1/2))

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maxima [A]  time = 0.98, size = 63, normalized size = 0.75 \begin {gather*} \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 {gather*}

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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mupad [B]  time = 3.44, size = 65, normalized size = 0.77 \begin {gather*} \frac {122691\,x}{128}+\frac {307461\,\ln \left (2\,x^2-x+3\right )}{512}-\frac {1156639\,\sqrt {23}\,\mathrm {atan}\left (\frac {4\,\sqrt {23}\,x}{23}-\frac {\sqrt {23}}{23}\right )}{5888}-\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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(122691*x)/128 + (307461*log(2*x^2 - x + 3))/512 - (1156639*23^(1/2)*atan((4*23^(1/2)*x)/23 - 23^(1/2)/23))/58
88 - (28747*x^2)/128 - (21229*x^3)/96 + (6245*x^4)/64 + (1855*x^5)/8 + (3625*x^6)/24 + (625*x^7)/14

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sympy [A]  time = 0.17, size = 87, normalized size = 1.04 \begin {gather*} \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 {gather*}

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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