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

Optimal. Leaf size=84 \[ \frac{121 (342840 x+188381)}{6006250 \left (5 x^2+3 x+2\right )}+\frac{1331 (247 x+443)}{193750 \left (5 x^2+3 x+2\right )^2}-\frac{66}{625} \log \left (5 x^2+3 x+2\right )+\frac{8 x}{125}+\frac{11341176 \tan ^{-1}\left (\frac{10 x+3}{\sqrt{31}}\right )}{600625 \sqrt{31}} \]

[Out]

(8*x)/125 + (1331*(443 + 247*x))/(193750*(2 + 3*x + 5*x^2)^2) + (121*(188381 + 342840*x))/(6006250*(2 + 3*x +
5*x^2)) + (11341176*ArcTan[(3 + 10*x)/Sqrt[31]])/(600625*Sqrt[31]) - (66*Log[2 + 3*x + 5*x^2])/625

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Rubi [A]  time = 0.0865293, antiderivative size = 84, 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} \[ \frac{121 (342840 x+188381)}{6006250 \left (5 x^2+3 x+2\right )}+\frac{1331 (247 x+443)}{193750 \left (5 x^2+3 x+2\right )^2}-\frac{66}{625} \log \left (5 x^2+3 x+2\right )+\frac{8 x}{125}+\frac{11341176 \tan ^{-1}\left (\frac{10 x+3}{\sqrt{31}}\right )}{600625 \sqrt{31}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(8*x)/125 + (1331*(443 + 247*x))/(193750*(2 + 3*x + 5*x^2)^2) + (121*(188381 + 342840*x))/(6006250*(2 + 3*x +
5*x^2)) + (11341176*ArcTan[(3 + 10*x)/Sqrt[31]])/(600625*Sqrt[31]) - (66*Log[2 + 3*x + 5*x^2])/625

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 (3-x+2 x^2\right )^3}{\left (2+3 x+5 x^2\right )^3} \, dx &=\frac{1331 (443+247 x)}{193750 \left (2+3 x+5 x^2\right )^2}+\frac{1}{62} \int \frac{\frac{4055767}{3125}-\frac{461962 x}{625}+\frac{75764 x^2}{125}-\frac{5208 x^3}{25}+\frac{496 x^4}{5}}{\left (2+3 x+5 x^2\right )^2} \, dx\\ &=\frac{1331 (443+247 x)}{193750 \left (2+3 x+5 x^2\right )^2}+\frac{121 (188381+342840 x)}{6006250 \left (2+3 x+5 x^2\right )}+\frac{\int \frac{\frac{2222876}{125}-\frac{207576 x}{125}+\frac{15376 x^2}{25}}{2+3 x+5 x^2} \, dx}{1922}\\ &=\frac{1331 (443+247 x)}{193750 \left (2+3 x+5 x^2\right )^2}+\frac{121 (188381+342840 x)}{6006250 \left (2+3 x+5 x^2\right )}+\frac{\int \left (\frac{15376}{125}+\frac{132 (16607-1922 x)}{125 \left (2+3 x+5 x^2\right )}\right ) \, dx}{1922}\\ &=\frac{8 x}{125}+\frac{1331 (443+247 x)}{193750 \left (2+3 x+5 x^2\right )^2}+\frac{121 (188381+342840 x)}{6006250 \left (2+3 x+5 x^2\right )}+\frac{66 \int \frac{16607-1922 x}{2+3 x+5 x^2} \, dx}{120125}\\ &=\frac{8 x}{125}+\frac{1331 (443+247 x)}{193750 \left (2+3 x+5 x^2\right )^2}+\frac{121 (188381+342840 x)}{6006250 \left (2+3 x+5 x^2\right )}-\frac{66}{625} \int \frac{3+10 x}{2+3 x+5 x^2} \, dx+\frac{5670588 \int \frac{1}{2+3 x+5 x^2} \, dx}{600625}\\ &=\frac{8 x}{125}+\frac{1331 (443+247 x)}{193750 \left (2+3 x+5 x^2\right )^2}+\frac{121 (188381+342840 x)}{6006250 \left (2+3 x+5 x^2\right )}-\frac{66}{625} \log \left (2+3 x+5 x^2\right )-\frac{11341176 \operatorname{Subst}\left (\int \frac{1}{-31-x^2} \, dx,x,3+10 x\right )}{600625}\\ &=\frac{8 x}{125}+\frac{1331 (443+247 x)}{193750 \left (2+3 x+5 x^2\right )^2}+\frac{121 (188381+342840 x)}{6006250 \left (2+3 x+5 x^2\right )}+\frac{11341176 \tan ^{-1}\left (\frac{3+10 x}{\sqrt{31}}\right )}{600625 \sqrt{31}}-\frac{66}{625} \log \left (2+3 x+5 x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.037298, size = 78, normalized size = 0.93 \[ \frac{\frac{3751 (342840 x+188381)}{5 x^2+3 x+2}+\frac{1279091 (247 x+443)}{\left (5 x^2+3 x+2\right )^2}-19662060 \log \left (5 x^2+3 x+2\right )[phone] x+113411760 \sqrt{31} \tan ^{-1}\left (\frac{10 x+3}{\sqrt{31}}\right )}{186193750} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(11916400*x + (1279091*(443 + 247*x))/(2 + 3*x + 5*x^2)^2 + (3751*(188381 + 342840*x))/(2 + 3*x + 5*x^2) + 113
411760*Sqrt[31]*ArcTan[(3 + 10*x)/Sqrt[31]] - 19662060*Log[2 + 3*x + 5*x^2])/186193750

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Maple [A]  time = 0.048, size = 63, normalized size = 0.8 \begin{align*}{\frac{8\,x}{125}}-{\frac{11}{5\, \left ( 5\,{x}^{2}+3\,x+2 \right ) ^{2}} \left ( -{\frac{377124\,{x}^{3}}{24025}}-{\frac{866987\,{x}^{2}}{48050}}-{\frac{293711\,x}{24025}}-{\frac{232243}{48050}} \right ) }-{\frac{66\,\ln \left ( 5\,{x}^{2}+3\,x+2 \right ) }{625}}+{\frac{11341176\,\sqrt{31}}{18619375}\arctan \left ({\frac{ \left ( 3+10\,x \right ) \sqrt{31}}{31}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

8/125*x-11/5*(-377124/24025*x^3-866987/48050*x^2-293711/24025*x-232243/48050)/(5*x^2+3*x+2)^2-66/625*ln(5*x^2+
3*x+2)[phone]/18619375*arctan(1/31*(3+10*x)*31^(1/2))*31^(1/2)

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Maxima [A]  time = 1.44198, size = 97, normalized size = 1.15 \begin{align*} \frac{11341176}{18619375} \, \sqrt{31} \arctan \left (\frac{1}{31} \, \sqrt{31}{\left (10 \, x + 3\right )}\right ) + \frac{8}{125} \, x + \frac{121 \,{\left (68568 \, x^{3} + 78817 \, x^{2} + 53402 \, x + 21113\right )}}{240250 \,{\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )}} - \frac{66}{625} \, \log \left (5 \, x^{2} + 3 \, x + 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

11341176/18619375*sqrt(31)*arctan(1/31*sqrt(31)*(10*x + 3)) + 8/125*x + 121/240250*(68568*x^3 + 78817*x^2 + 53
402*x + 21113)/(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4) - 66/625*log(5*x^2 + 3*x + 2)

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Fricas [A]  time = 1.01191, size = 406, normalized size = 4.83 \begin{align*} \frac{59582000 \, x^{5} + 71498400 \, x^{4} + 1355107960 \, x^{3} + 22682352 \, \sqrt{31}{\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} \arctan \left (\frac{1}{31} \, \sqrt{31}{\left (10 \, x + 3\right )}\right ) + 1506812195 \, x^{2} - 3932412 \,{\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} \log \left (5 \, x^{2} + 3 \, x + 2\right ) + 1011087630 \, x + 395974315}{37238750 \,{\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/37238750*(59582000*x^5 + 71498400*x^4 + 1355107960*x^3 + 22682352*sqrt(31)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x
+ 4)*arctan(1/31*sqrt(31)*(10*x + 3)) + 1506812195*x^2 - 3932412*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)*log(5*x
^2 + 3*x + 2) + 1011087630*x + 395974315)/(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)

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Sympy [A]  time = 0.216081, size = 85, normalized size = 1.01 \begin{align*} \frac{8 x}{125} + \frac{8296728 x^{3} + 9536857 x^{2} + 6461642 x + 2554673}{6006250 x^{4} + 7207500 x^{3} + 6967250 x^{2} + 2883000 x + 961000} - \frac{66 \log{\left (x^{2} + \frac{3 x}{5} + \frac{2}{5} \right )}}{625} + \frac{11341176 \sqrt{31} \operatorname{atan}{\left (\frac{10 \sqrt{31} x}{31} + \frac{3 \sqrt{31}}{31} \right )}}{18619375} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

8*x/125 + (8296728*x**3 + 9536857*x**2 + 6461642*x + 2554673)/(6006250*x**4 + 7207500*x**3 + 6967250*x**2 + 28
83000*x + 961000) - 66*log(x**2 + 3*x/5 + 2/5)/625 + 11341176*sqrt(31)*atan(10*sqrt(31)*x/31 + 3*sqrt(31)/31)/
18619375

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Giac [A]  time = 1.21183, size = 84, normalized size = 1. \begin{align*} \frac{11341176}{18619375} \, \sqrt{31} \arctan \left (\frac{1}{31} \, \sqrt{31}{\left (10 \, x + 3\right )}\right ) + \frac{8}{125} \, x + \frac{121 \,{\left (68568 \, x^{3} + 78817 \, x^{2} + 53402 \, x + 21113\right )}}{240250 \,{\left (5 \, x^{2} + 3 \, x + 2\right )}^{2}} - \frac{66}{625} \, \log \left (5 \, x^{2} + 3 \, x + 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

11341176/18619375*sqrt(31)*arctan(1/31*sqrt(31)*(10*x + 3)) + 8/125*x + 121/240250*(68568*x^3 + 78817*x^2 + 53
402*x + 21113)/(5*x^2 + 3*x + 2)^2 - 66/625*log(5*x^2 + 3*x + 2)