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

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

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Rubi [A]  time = 0.09, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {1660, 1657, 634, 618, 204, 628} \begin {gather*} \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}} \end {gather*}

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 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]

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
]

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*}

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Mathematica [A]  time = 0.04, size = 78, normalized size = 0.93 \begin {gather*} \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} \end {gather*}

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.39, size = 118, normalized size = 1.40 \begin {gather*} \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 {gather*}

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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giac [A]  time = 0.19, size = 62, normalized size = 0.74 \begin {gather*} \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 {gather*}

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)

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maple [A]  time = 0.01, size = 63, normalized size = 0.75 \begin {gather*} \frac {8 x}{125}+\frac {11341176 \sqrt {31}\, \arctan \left (\frac {\left (10 x +3\right ) \sqrt {31}}{31}\right )}{18619375}-\frac {66 \ln \left (5 x^{2}+3 x +2\right )}{625}-\frac {11 \left (-\frac {377124}{24025} x^{3}-\frac {866987}{48050} x^{2}-\frac {293711}{24025} x -\frac {232243}{48050}\right )}{5 \left (5 x^{2}+3 x +2\right )^{2}} \end {gather*}

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*31^(1/2)*arctan(1/31*(10*x+3)*31^(1/2))

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maxima [A]  time = 0.97, size = 72, normalized size = 0.86 \begin {gather*} \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 {gather*}

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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mupad [B]  time = 3.43, size = 71, normalized size = 0.85 \begin {gather*} \frac {8\,x}{125}-\frac {66\,\ln \left (5\,x^2+3\,x+2\right )}{625}+\frac {11341176\,\sqrt {31}\,\mathrm {atan}\left (\frac {10\,\sqrt {31}\,x}{31}+\frac {3\,\sqrt {31}}{31}\right )}{18619375}+\frac {\frac {4148364\,x^3}{3003125}+\frac {9536857\,x^2}{6006250}+\frac {3230821\,x}{3003125}+\frac {2554673}{6006250}}{x^4+\frac {6\,x^3}{5}+\frac {29\,x^2}{25}+\frac {12\,x}{25}+\frac {4}{25}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(8*x)/125 - (66*log(3*x + 5*x^2 + 2))/625 + (11341176*31^(1/2)*atan((10*31^(1/2)*x)/31 + (3*31^(1/2))/31))/186
19375 + ((3230821*x)/3003125 + (9536857*x^2)/6006250 + (4148364*x^3)/3003125 + 2554673/6006250)/((12*x)/25 + (
29*x^2)/25 + (6*x^3)/5 + x^4 + 4/25)

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sympy [A]  time = 0.23, size = 85, normalized size = 1.01 \begin {gather*} \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 {gather*}

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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