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

Optimal. Leaf size=91 \[ \frac{125 x^5}{4}+\frac{2125 x^4}{16}+\frac{9775 x^3}{48}-\frac{1185 x^2}{8}-\frac{14641 (79 x+101)}{2944 \left (2 x^2-x+3\right )}-\frac{30613}{128} \log \left (2 x^2-x+3\right )-\frac{89359 x}{64}-\frac{13292697 \tan ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{1472 \sqrt{23}} \]

[Out]

(-89359*x)/64 - (1185*x^2)/8 + (9775*x^3)/48 + (2125*x^4)/16 + (125*x^5)/4 - (14641*(101 + 79*x))/(2944*(3 - x
+ 2*x^2)) - (13292697*ArcTan[(1 - 4*x)/Sqrt[23]])/(1472*Sqrt[23]) - (30613*Log[3 - x + 2*x^2])/128

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Rubi [A]  time = 0.0863451, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ \frac{125 x^5}{4}+\frac{2125 x^4}{16}+\frac{9775 x^3}{48}-\frac{1185 x^2}{8}-\frac{14641 (79 x+101)}{2944 \left (2 x^2-x+3\right )}-\frac{30613}{128} \log \left (2 x^2-x+3\right )-\frac{89359 x}{64}-\frac{13292697 \tan ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{1472 \sqrt{23}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-89359*x)/64 - (1185*x^2)/8 + (9775*x^3)/48 + (2125*x^4)/16 + (125*x^5)/4 - (14641*(101 + 79*x))/(2944*(3 - x
+ 2*x^2)) - (13292697*ArcTan[(1 - 4*x)/Sqrt[23]])/(1472*Sqrt[23]) - (30613*Log[3 - x + 2*x^2])/128

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 )^2} \, dx &=-\frac{14641 (101+79 x)}{2944 \left (3-x+2 x^2\right )}+\frac{1}{23} \int \frac{\frac{832627}{64}-\frac{661181 x}{64}-\frac{488267 x^2}{32}+\frac{143635 x^3}{16}+\frac{213325 x^4}{8}+\frac{83375 x^5}{4}+\frac{14375 x^6}{2}}{3-x+2 x^2} \, dx\\ &=-\frac{14641 (101+79 x)}{2944 \left (3-x+2 x^2\right )}+\frac{1}{23} \int \left (-\frac{2055257}{64}-\frac{27255 x}{4}+\frac{224825 x^2}{16}+\frac{48875 x^3}{4}+\frac{14375 x^4}{4}+\frac{1331 (2629-529 x)}{32 \left (3-x+2 x^2\right )}\right ) \, dx\\ &=-\frac{89359 x}{64}-\frac{1185 x^2}{8}+\frac{9775 x^3}{48}+\frac{2125 x^4}{16}+\frac{125 x^5}{4}-\frac{14641 (101+79 x)}{2944 \left (3-x+2 x^2\right )}+\frac{1331}{736} \int \frac{2629-529 x}{3-x+2 x^2} \, dx\\ &=-\frac{89359 x}{64}-\frac{1185 x^2}{8}+\frac{9775 x^3}{48}+\frac{2125 x^4}{16}+\frac{125 x^5}{4}-\frac{14641 (101+79 x)}{2944 \left (3-x+2 x^2\right )}-\frac{30613}{128} \int \frac{-1+4 x}{3-x+2 x^2} \, dx+\frac{13292697 \int \frac{1}{3-x+2 x^2} \, dx}{2944}\\ &=-\frac{89359 x}{64}-\frac{1185 x^2}{8}+\frac{9775 x^3}{48}+\frac{2125 x^4}{16}+\frac{125 x^5}{4}-\frac{14641 (101+79 x)}{2944 \left (3-x+2 x^2\right )}-\frac{30613}{128} \log \left (3-x+2 x^2\right )-\frac{13292697 \operatorname{Subst}\left (\int \frac{1}{-23-x^2} \, dx,x,-1+4 x\right )}{1472}\\ &=-\frac{89359 x}{64}-\frac{1185 x^2}{8}+\frac{9775 x^3}{48}+\frac{2125 x^4}{16}+\frac{125 x^5}{4}-\frac{14641 (101+79 x)}{2944 \left (3-x+2 x^2\right )}-\frac{13292697 \tan ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{1472 \sqrt{23}}-\frac{30613}{128} \log \left (3-x+2 x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.0499269, size = 91, normalized size = 1. \[ \frac{125 x^5}{4}+\frac{2125 x^4}{16}+\frac{9775 x^3}{48}-\frac{1185 x^2}{8}-\frac{14641 (79 x+101)}{2944 \left (2 x^2-x+3\right )}-\frac{30613}{128} \log \left (2 x^2-x+3\right )-\frac{89359 x}{64}+\frac{13292697 \tan ^{-1}\left (\frac{4 x-1}{\sqrt{23}}\right )}{1472 \sqrt{23}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-89359*x)/64 - (1185*x^2)/8 + (9775*x^3)/48 + (2125*x^4)/16 + (125*x^5)/4 - (14641*(101 + 79*x))/(2944*(3 - x
+ 2*x^2)) + (13292697*ArcTan[(-1 + 4*x)/Sqrt[23]])/(1472*Sqrt[23]) - (30613*Log[3 - x + 2*x^2])/128

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Maple [A]  time = 0.048, size = 71, normalized size = 0.8 \begin{align*}{\frac{125\,{x}^{5}}{4}}+{\frac{2125\,{x}^{4}}{16}}+{\frac{9775\,{x}^{3}}{48}}-{\frac{1185\,{x}^{2}}{8}}-{\frac{89359\,x}{64}}-{\frac{1331}{64} \left ({\frac{869\,x}{92}}+{\frac{1111}{92}} \right ) \left ({x}^{2}-{\frac{x}{2}}+{\frac{3}{2}} \right ) ^{-1}}-{\frac{30613\,\ln \left ( 2\,{x}^{2}-x+3 \right ) }{128}}+{\frac{13292697\,\sqrt{23}}{33856}\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)^2,x)

[Out]

125/4*x^5+2125/16*x^4+9775/48*x^3-1185/8*x^2-89359/64*x-1331/64*(869/92*x+1111/92)/(x^2-1/2*x+3/2)-30613/128*l
n(2*x^2-x+3)[phone]/33856*23^(1/2)*arctan(1/23*(-1+4*x)*23^(1/2))

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Maxima [A]  time = 1.42549, size = 97, normalized size = 1.07 \begin{align*} \frac{125}{4} \, x^{5} + \frac{2125}{16} \, x^{4} + \frac{9775}{48} \, x^{3} - \frac{1185}{8} \, x^{2} + \frac{13292697}{33856} \, \sqrt{23} \arctan \left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) - \frac{89359}{64} \, x - \frac{14641 \,{\left (79 \, x + 101\right )}}{2944 \,{\left (2 \, x^{2} - x + 3\right )}} - \frac{30613}{128} \, \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)^2,x, algorithm="maxima")

[Out]

125/4*x^5 + 2125/16*x^4 + 9775/48*x^3 - 1185/8*x^2 + 13292697/33856*sqrt(23)*arctan(1/23*sqrt(23)*(4*x - 1)) -
89359/64*x - 14641/2944*(79*x + 101)/(2*x^2 - x + 3) - 30613/128*log(2*x^2 - x + 3)

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Fricas [A]  time = 0.9736, size = 348, normalized size = 3.82 \begin{align*} \frac{12696000 \, x^{7} + 47610000 \, x^{6} + 74800600 \, x^{5} - 20609840 \, x^{4} - 413058012 \, x^{3} + 79756182 \, \sqrt{23}{\left (2 \, x^{2} - x + 3\right )} \arctan \left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) + 193356906 \, x^{2} - 48582831 \,{\left (2 \, x^{2} - x + 3\right )} \log \left (2 \, x^{2} - x + 3\right ) - 930684489 \, x - 102033129}{203136 \,{\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)^2,x, algorithm="fricas")

[Out]

1/203136*(12696000*x^7 + 47610000*x^6 + 74800600*x^5 - 20609840*x^4 - 413058012*x^3 + 79756182*sqrt(23)*(2*x^2
- x + 3)*arctan(1/23*sqrt(23)*(4*x - 1)) + 193356906*x^2 - 48582831*(2*x^2 - x + 3)*log(2*x^2 - x + 3) - 9306
84489*x - 102033129)/(2*x^2 - x + 3)

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Sympy [A]  time = 0.245003, size = 88, normalized size = 0.97 \begin{align*} \frac{125 x^{5}}{4} + \frac{2125 x^{4}}{16} + \frac{9775 x^{3}}{48} - \frac{1185 x^{2}}{8} - \frac{89359 x}{64} - \frac{1156639 x + 1478741}{5888 x^{2} - 2944 x + 8832} - \frac{30613 \log{\left (x^{2} - \frac{x}{2} + \frac{3}{2} \right )}}{128} + \frac{13292697 \sqrt{23} \operatorname{atan}{\left (\frac{4 \sqrt{23} x}{23} - \frac{\sqrt{23}}{23} \right )}}{33856} \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)**2,x)

[Out]

125*x**5/4 + 2125*x**4/16 + 9775*x**3/48 - 1185*x**2/8 - 89359*x/64 - (1156639*x + 1478741)/(5888*x**2 - 2944*
x + 8832) - 30613*log(x**2 - x/2 + 3/2)/128 + 13292697*sqrt(23)*atan(4*sqrt(23)*x/23 - sqrt(23)/23)/33856

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Giac [A]  time = 1.1539, size = 97, normalized size = 1.07 \begin{align*} \frac{125}{4} \, x^{5} + \frac{2125}{16} \, x^{4} + \frac{9775}{48} \, x^{3} - \frac{1185}{8} \, x^{2} + \frac{13292697}{33856} \, \sqrt{23} \arctan \left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) - \frac{89359}{64} \, x - \frac{14641 \,{\left (79 \, x + 101\right )}}{2944 \,{\left (2 \, x^{2} - x + 3\right )}} - \frac{30613}{128} \, \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)^2,x, algorithm="giac")

[Out]

125/4*x^5 + 2125/16*x^4 + 9775/48*x^3 - 1185/8*x^2 + 13292697/33856*sqrt(23)*arctan(1/23*sqrt(23)*(4*x - 1)) -
89359/64*x - 14641/2944*(79*x + 101)/(2*x^2 - x + 3) - 30613/128*log(2*x^2 - x + 3)