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

Optimal. Leaf size=98 \[ \frac{625 x^3}{24}+\frac{4875 x^2}{32}+\frac{1331 (76420 x+5229)}{135424 \left (2 x^2-x+3\right )}-\frac{14641 (79 x+101)}{5888 \left (2 x^2-x+3\right )^2}-\frac{13915}{64} \log \left (2 x^2-x+3\right )+\frac{2725 x}{8}+\frac{63799791 \tan ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{16928 \sqrt{23}} \]

[Out]

(2725*x)/8 + (4875*x^2)/32 + (625*x^3)/24 - (14641*(101 + 79*x))/(5888*(3 - x + 2*x^2)^2) + (1331*(5229 + 7642
0*x))/(135424*(3 - x + 2*x^2)) + (63799791*ArcTan[(1 - 4*x)/Sqrt[23]])/(16928*Sqrt[23]) - (13915*Log[3 - x + 2
*x^2])/64

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Rubi [A]  time = 0.11253, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24, Rules used = {1660, 1657, 634, 618, 204, 628} \[ \frac{625 x^3}{24}+\frac{4875 x^2}{32}+\frac{1331 (76420 x+5229)}{135424 \left (2 x^2-x+3\right )}-\frac{14641 (79 x+101)}{5888 \left (2 x^2-x+3\right )^2}-\frac{13915}{64} \log \left (2 x^2-x+3\right )+\frac{2725 x}{8}+\frac{63799791 \tan ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{16928 \sqrt{23}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2725*x)/8 + (4875*x^2)/32 + (625*x^3)/24 - (14641*(101 + 79*x))/(5888*(3 - x + 2*x^2)^2) + (1331*(5229 + 7642
0*x))/(135424*(3 - x + 2*x^2)) + (63799791*ArcTan[(1 - 4*x)/Sqrt[23]])/(16928*Sqrt[23]) - (13915*Log[3 - x + 2
*x^2])/64

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 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}{\left (3-x+2 x^2\right )^3} \, dx &=-\frac{14641 (101+79 x)}{5888 \left (3-x+2 x^2\right )^2}+\frac{1}{46} \int \frac{\frac{2173869}{128}-\frac{661181 x}{32}-\frac{488267 x^2}{16}+\frac{143635 x^3}{8}+\frac{213325 x^4}{4}+\frac{83375 x^5}{2}+14375 x^6}{\left (3-x+2 x^2\right )^2} \, dx\\ &=-\frac{14641 (101+79 x)}{5888 \left (3-x+2 x^2\right )^2}+\frac{1331 (5229+76420 x)}{135424 \left (3-x+2 x^2\right )}+\frac{\int \frac{-\frac{5460539}{8}-\frac{626865 x}{2}+\frac{5170975 x^2}{8}+\frac{1124125 x^3}{2}+\frac{330625 x^4}{2}}{3-x+2 x^2} \, dx}{1058}\\ &=-\frac{14641 (101+79 x)}{5888 \left (3-x+2 x^2\right )^2}+\frac{1331 (5229+76420 x)}{135424 \left (3-x+2 x^2\right )}+\frac{\int \left (\frac{1441525}{4}+\frac{2578875 x}{8}+\frac{330625 x^2}{4}-\frac{121 (116609+60835 x)}{8 \left (3-x+2 x^2\right )}\right ) \, dx}{1058}\\ &=\frac{2725 x}{8}+\frac{4875 x^2}{32}+\frac{625 x^3}{24}-\frac{14641 (101+79 x)}{5888 \left (3-x+2 x^2\right )^2}+\frac{1331 (5229+76420 x)}{135424 \left (3-x+2 x^2\right )}-\frac{121 \int \frac{116609+60835 x}{3-x+2 x^2} \, dx}{8464}\\ &=\frac{2725 x}{8}+\frac{4875 x^2}{32}+\frac{625 x^3}{24}-\frac{14641 (101+79 x)}{5888 \left (3-x+2 x^2\right )^2}+\frac{1331 (5229+76420 x)}{135424 \left (3-x+2 x^2\right )}-\frac{13915}{64} \int \frac{-1+4 x}{3-x+2 x^2} \, dx-\frac{63799791 \int \frac{1}{3-x+2 x^2} \, dx}{33856}\\ &=\frac{2725 x}{8}+\frac{4875 x^2}{32}+\frac{625 x^3}{24}-\frac{14641 (101+79 x)}{5888 \left (3-x+2 x^2\right )^2}+\frac{1331 (5229+76420 x)}{135424 \left (3-x+2 x^2\right )}-\frac{13915}{64} \log \left (3-x+2 x^2\right )+\frac{63799791 \operatorname{Subst}\left (\int \frac{1}{-23-x^2} \, dx,x,-1+4 x\right )}{16928}\\ &=\frac{2725 x}{8}+\frac{4875 x^2}{32}+\frac{625 x^3}{24}-\frac{14641 (101+79 x)}{5888 \left (3-x+2 x^2\right )^2}+\frac{1331 (5229+76420 x)}{135424 \left (3-x+2 x^2\right )}+\frac{63799791 \tan ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{16928 \sqrt{23}}-\frac{13915}{64} \log \left (3-x+2 x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.0370471, size = 98, normalized size = 1. \[ \frac{625 x^3}{24}+\frac{4875 x^2}{32}+\frac{1331 (76420 x+5229)}{135424 \left (2 x^2-x+3\right )}-\frac{14641 (79 x+101)}{5888 \left (2 x^2-x+3\right )^2}-\frac{13915}{64} \log \left (2 x^2-x+3\right )+\frac{2725 x}{8}-\frac{63799791 \tan ^{-1}\left (\frac{4 x-1}{\sqrt{23}}\right )}{16928 \sqrt{23}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2725*x)/8 + (4875*x^2)/32 + (625*x^3)/24 - (14641*(101 + 79*x))/(5888*(3 - x + 2*x^2)^2) + (1331*(5229 + 7642
0*x))/(135424*(3 - x + 2*x^2)) - (63799791*ArcTan[(-1 + 4*x)/Sqrt[23]])/(16928*Sqrt[23]) - (13915*Log[3 - x +
2*x^2])/64

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Maple [A]  time = 0.054, size = 73, normalized size = 0.7 \begin{align*}{\frac{625\,{x}^{3}}{24}}+{\frac{4875\,{x}^{2}}{32}}+{\frac{2725\,x}{8}}-{\frac{121}{4\, \left ( 2\,{x}^{2}-x+3 \right ) ^{2}} \left ( -{\frac{210155\,{x}^{3}}{4232}}+{\frac{362791\,{x}^{2}}{16928}}-{\frac{561121\,x}{8464}}+{\frac{54263}{16928}} \right ) }-{\frac{13915\,\ln \left ( 2\,{x}^{2}-x+3 \right ) }{64}}-{\frac{63799791\,\sqrt{23}}{389344}\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)^3,x)

[Out]

625/24*x^3+4875/32*x^2+2725/8*x-121/4*(-210155/4232*x^3+362791/16928*x^2-561121/8464*x+54263/16928)/(2*x^2-x+3
)^2-13915/64*ln(2*x^2-x+3)-63799791/389344*23^(1/2)*arctan(1/23*(-1+4*x)*23^(1/2))

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Maxima [A]  time = 1.45374, size = 111, normalized size = 1.13 \begin{align*} \frac{625}{24} \, x^{3} + \frac{4875}{32} \, x^{2} - \frac{63799791}{389344} \, \sqrt{23} \arctan \left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) + \frac{2725}{8} \, x + \frac{1331 \,{\left (76420 \, x^{3} - 32981 \, x^{2} + 102022 \, x - 4933\right )}}{67712 \,{\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\right )}} - \frac{13915}{64} \, \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)^3,x, algorithm="maxima")

[Out]

625/24*x^3 + 4875/32*x^2 - 63799791/389344*sqrt(23)*arctan(1/23*sqrt(23)*(4*x - 1)) + 2725/8*x + 1331/67712*(7
6420*x^3 - 32981*x^2 + 102022*x - 4933)/(4*x^4 - 4*x^3 + 13*x^2 - 6*x + 9) - 13915/64*log(2*x^2 - x + 3)

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Fricas [A]  time = 0.905483, size = 447, normalized size = 4.56 \begin{align*} \frac{486680000 \, x^{7} + 2360398000 \, x^{6} + 5100406400 \, x^{5} + 2157209100 \, x^{4} + 24531516180 \, x^{3} - 765597492 \, \sqrt{23}{\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\right )} \arctan \left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) - 6171678159 \, x^{2} - 1015822830 \,{\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\right )} \log \left (2 \, x^{2} - x + 3\right ) + 23692590858 \, x - 453041787}{4672128 \,{\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^2+3*x+2)^4/(2*x^2-x+3)^3,x, algorithm="fricas")

[Out]

1/4672128*(486680000*x^7 + 2360398000*x^6 + 5100406400*x^5 + 2157209100*x^4 + 24531516180*x^3 - 765597492*sqrt
(23)*(4*x^4 - 4*x^3 + 13*x^2 - 6*x + 9)*arctan(1/23*sqrt(23)*(4*x - 1)) - 6171678159*x^2 - 1015822830*(4*x^4 -
4*x^3 + 13*x^2 - 6*x + 9)*log(2*x^2 - x + 3) + 23692590858*x - 453041787)/(4*x^4 - 4*x^3 + 13*x^2 - 6*x + 9)

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Sympy [A]  time = 0.277565, size = 95, normalized size = 0.97 \begin{align*} \frac{625 x^{3}}{24} + \frac{4875 x^{2}}{32} + \frac{2725 x}{8} + \frac{101715020 x^{3} - 43897711 x^{2} + 135791282 x - 6565823}{270848 x^{4} - 270848 x^{3} + 880256 x^{2} - 406272 x + 609408} - \frac{13915 \log{\left (x^{2} - \frac{x}{2} + \frac{3}{2} \right )}}{64} - \frac{63799791 \sqrt{23} \operatorname{atan}{\left (\frac{4 \sqrt{23} x}{23} - \frac{\sqrt{23}}{23} \right )}}{389344} \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)**3,x)

[Out]

625*x**3/24 + 4875*x**2/32 + 2725*x/8 + (101715020*x**3 - 43897711*x**2 + 135791282*x - 6565823)/(270848*x**4
- 270848*x**3 + 880256*x**2 - 406272*x + 609408) - 13915*log(x**2 - x/2 + 3/2)/64 - 63799791*sqrt(23)*atan(4*s
qrt(23)*x/23 - sqrt(23)/23)/389344

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Giac [A]  time = 1.14796, size = 97, normalized size = 0.99 \begin{align*} \frac{625}{24} \, x^{3} + \frac{4875}{32} \, x^{2} - \frac{63799791}{389344} \, \sqrt{23} \arctan \left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) + \frac{2725}{8} \, x + \frac{1331 \,{\left (76420 \, x^{3} - 32981 \, x^{2} + 102022 \, x - 4933\right )}}{67712 \,{\left (2 \, x^{2} - x + 3\right )}^{2}} - \frac{13915}{64} \, \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)^3,x, algorithm="giac")

[Out]

625/24*x^3 + 4875/32*x^2 - 63799791/389344*sqrt(23)*arctan(1/23*sqrt(23)*(4*x - 1)) + 2725/8*x + 1331/67712*(7
6420*x^3 - 32981*x^2 + 102022*x - 4933)/(2*x^2 - x + 3)^2 - 13915/64*log(2*x^2 - x + 3)