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

Optimal. Leaf size=77 \[ \frac {8 x^3}{75}-\frac {54 x^2}{125}+\frac {1331 (247 x+443)}{96875 \left (5 x^2+3 x+2\right )}-\frac {10769 \log \left (5 x^2+3 x+2\right )}{6250}+\frac {1466 x}{625}+\frac {3819607 \tan ^{-1}\left (\frac {10 x+3}{\sqrt {31}}\right )}{96875 \sqrt {31}} \]

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Rubi [A]  time = 0.07, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 7, 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 {8 x^3}{75}-\frac {54 x^2}{125}+\frac {1331 (247 x+443)}{96875 \left (5 x^2+3 x+2\right )}-\frac {10769 \log \left (5 x^2+3 x+2\right )}{6250}+\frac {1466 x}{625}+\frac {3819607 \tan ^{-1}\left (\frac {10 x+3}{\sqrt {31}}\right )}{96875 \sqrt {31}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(1466*x)/625 - (54*x^2)/125 + (8*x^3)/75 + (1331*(443 + 247*x))/(96875*(2 + 3*x + 5*x^2)) + (3819607*ArcTan[(3
+ 10*x)/Sqrt[31]])/(96875*Sqrt[31]) - (10769*Log[2 + 3*x + 5*x^2])/6250

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 )^2} \, dx &=\frac {1331 (443+247 x)}{96875 \left (2+3 x+5 x^2\right )}+\frac {1}{31} \int \frac {\frac {372701}{625}-\frac {230981 x}{625}+\frac {37882 x^2}{125}-\frac {2604 x^3}{25}+\frac {248 x^4}{5}}{2+3 x+5 x^2} \, dx\\ &=\frac {1331 (443+247 x)}{96875 \left (2+3 x+5 x^2\right )}+\frac {1}{31} \int \left (\frac {45446}{625}-\frac {3348 x}{125}+\frac {248 x^2}{25}+\frac {121 (2329-2759 x)}{625 \left (2+3 x+5 x^2\right )}\right ) \, dx\\ &=\frac {1466 x}{625}-\frac {54 x^2}{125}+\frac {8 x^3}{75}+\frac {1331 (443+247 x)}{96875 \left (2+3 x+5 x^2\right )}+\frac {121 \int \frac {2329-2759 x}{2+3 x+5 x^2} \, dx}{19375}\\ &=\frac {1466 x}{625}-\frac {54 x^2}{125}+\frac {8 x^3}{75}+\frac {1331 (443+247 x)}{96875 \left (2+3 x+5 x^2\right )}-\frac {10769 \int \frac {3+10 x}{2+3 x+5 x^2} \, dx}{6250}+\frac {3819607 \int \frac {1}{2+3 x+5 x^2} \, dx}{193750}\\ &=\frac {1466 x}{625}-\frac {54 x^2}{125}+\frac {8 x^3}{75}+\frac {1331 (443+247 x)}{96875 \left (2+3 x+5 x^2\right )}-\frac {10769 \log \left (2+3 x+5 x^2\right )}{6250}-\frac {3819607 \operatorname {Subst}\left (\int \frac {1}{-31-x^2} \, dx,x,3+10 x\right )}{96875}\\ &=\frac {1466 x}{625}-\frac {54 x^2}{125}+\frac {8 x^3}{75}+\frac {1331 (443+247 x)}{96875 \left (2+3 x+5 x^2\right )}+\frac {3819607 \tan ^{-1}\left (\frac {3+10 x}{\sqrt {31}}\right )}{96875 \sqrt {31}}-\frac {10769 \log \left (2+3 x+5 x^2\right )}{6250}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 77, normalized size = 1.00 \begin {gather*} \frac {8 x^3}{75}-\frac {54 x^2}{125}+\frac {1331 (247 x+443)}{96875 \left (5 x^2+3 x+2\right )}-\frac {10769 \log \left (5 x^2+3 x+2\right )}{6250}+\frac {1466 x}{625}+\frac {3819607 \tan ^{-1}\left (\frac {10 x+3}{\sqrt {31}}\right )}{96875 \sqrt {31}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(1466*x)/625 - (54*x^2)/125 + (8*x^3)/75 + (1331*(443 + 247*x))/(96875*(2 + 3*x + 5*x^2)) + (3819607*ArcTan[(3
+ 10*x)/Sqrt[31]])/(96875*Sqrt[31]) - (10769*Log[2 + 3*x + 5*x^2])/6250

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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 )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.41, size = 88, normalized size = 1.14 \begin {gather*} \frac {9610000 \, x^{5} - 33154500 \, x^{4} + 191815600 \, x^{3} + 22917642 \, \sqrt {31} {\left (5 \, x^{2} + 3 \, x + 2\right )} \arctan \left (\frac {1}{31} \, \sqrt {31} {\left (10 \, x + 3\right )}\right ) + 111226140 \, x^{2} - 31047027 \, {\left (5 \, x^{2} + 3 \, x + 2\right )} \log \left (5 \, x^{2} + 3 \, x + 2\right ) + 145678362 \, x + 109671738}{18018750 \, {\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)^2,x, algorithm="fricas")

[Out]

1/18018750*(9610000*x^5 - 33154500*x^4 + 191815600*x^3 + 22917642*sqrt(31)*(5*x^2 + 3*x + 2)*arctan(1/31*sqrt(
31)*(10*x + 3)) + 111226140*x^2 - 31047027*(5*x^2 + 3*x + 2)*log(5*x^2 + 3*x + 2) + 145678362*x + 109671738)/(
5*x^2 + 3*x + 2)

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giac [A]  time = 0.21, size = 62, normalized size = 0.81 \begin {gather*} \frac {8}{75} \, x^{3} - \frac {54}{125} \, x^{2} + \frac {3819607}{3003125} \, \sqrt {31} \arctan \left (\frac {1}{31} \, \sqrt {31} {\left (10 \, x + 3\right )}\right ) + \frac {1466}{625} \, x + \frac {1331 \, {\left (247 \, x + 443\right )}}{96875 \, {\left (5 \, x^{2} + 3 \, x + 2\right )}} - \frac {10769}{6250} \, \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)^2,x, algorithm="giac")

[Out]

8/75*x^3 - 54/125*x^2 + 3819607/3003125*sqrt(31)*arctan(1/31*sqrt(31)*(10*x + 3)) + 1466/625*x + 1331/96875*(2
47*x + 443)/(5*x^2 + 3*x + 2) - 10769/6250*log(5*x^2 + 3*x + 2)

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maple [A]  time = 0.01, size = 61, normalized size = 0.79 \begin {gather*} \frac {8 x^{3}}{75}-\frac {54 x^{2}}{125}+\frac {1466 x}{625}+\frac {3819607 \sqrt {31}\, \arctan \left (\frac {\left (10 x +3\right ) \sqrt {31}}{31}\right )}{3003125}-\frac {10769 \ln \left (5 x^{2}+3 x +2\right )}{6250}-\frac {121 \left (-\frac {2717 x}{775}-\frac {4873}{775}\right )}{625 \left (x^{2}+\frac {3}{5} x +\frac {2}{5}\right )} \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)^2,x)

[Out]

8/75*x^3-54/125*x^2+1466/625*x-121/625*(-2717/775*x-4873/775)/(x^2+3/5*x+2/5)-10769/6250*ln(5*x^2+3*x+2)+38196
07/3003125*31^(1/2)*arctan(1/31*(10*x+3)*31^(1/2))

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maxima [A]  time = 0.96, size = 62, normalized size = 0.81 \begin {gather*} \frac {8}{75} \, x^{3} - \frac {54}{125} \, x^{2} + \frac {3819607}{3003125} \, \sqrt {31} \arctan \left (\frac {1}{31} \, \sqrt {31} {\left (10 \, x + 3\right )}\right ) + \frac {1466}{625} \, x + \frac {1331 \, {\left (247 \, x + 443\right )}}{96875 \, {\left (5 \, x^{2} + 3 \, x + 2\right )}} - \frac {10769}{6250} \, \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)^2,x, algorithm="maxima")

[Out]

8/75*x^3 - 54/125*x^2 + 3819607/3003125*sqrt(31)*arctan(1/31*sqrt(31)*(10*x + 3)) + 1466/625*x + 1331/96875*(2
47*x + 443)/(5*x^2 + 3*x + 2) - 10769/6250*log(5*x^2 + 3*x + 2)

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mupad [B]  time = 3.43, size = 61, normalized size = 0.79 \begin {gather*} \frac {1466\,x}{625}-\frac {10769\,\ln \left (5\,x^2+3\,x+2\right )}{6250}+\frac {\frac {328757\,x}{484375}+\frac {589633}{484375}}{x^2+\frac {3\,x}{5}+\frac {2}{5}}+\frac {3819607\,\sqrt {31}\,\mathrm {atan}\left (\frac {10\,\sqrt {31}\,x}{31}+\frac {3\,\sqrt {31}}{31}\right )}{3003125}-\frac {54\,x^2}{125}+\frac {8\,x^3}{75} \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)^2,x)

[Out]

(1466*x)/625 - (10769*log(3*x + 5*x^2 + 2))/6250 + ((328757*x)/484375 + 589633/484375)/((3*x)/5 + x^2 + 2/5) +
(3819607*31^(1/2)*atan((10*31^(1/2)*x)/31 + (3*31^(1/2))/31))/3003125 - (54*x^2)/125 + (8*x^3)/75

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sympy [A]  time = 0.19, size = 78, normalized size = 1.01 \begin {gather*} \frac {8 x^{3}}{75} - \frac {54 x^{2}}{125} + \frac {1466 x}{625} + \frac {328757 x + 589633}{484375 x^{2} + 290625 x + 193750} - \frac {10769 \log {\left (x^{2} + \frac {3 x}{5} + \frac {2}{5} \right )}}{6250} + \frac {3819607 \sqrt {31} \operatorname {atan}{\left (\frac {10 \sqrt {31} x}{31} + \frac {3 \sqrt {31}}{31} \right )}}{3003125} \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)**2,x)

[Out]

8*x**3/75 - 54*x**2/125 + 1466*x/625 + (328757*x + 589633)/(484375*x**2 + 290625*x + 193750) - 10769*log(x**2
+ 3*x/5 + 2/5)/6250 + 3819607*sqrt(31)*atan(10*sqrt(31)*x/31 + 3*sqrt(31)/31)/3003125

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