3.217 \(\int \frac {-32+5 x-27 x^2+4 x^3}{-70-299 x-286 x^2+50 x^3-13 x^4+30 x^5} \, dx\)

Optimal. Leaf size=63 \[ \frac {11049 \log \left (x^2+x+5\right )}{260015}-\frac {3146 \log (7-3 x)}{80155}-\frac {334}{323} \log (2 x+1)+\frac {4822 \log (5 x+2)}{4879}+\frac {3988 \tan ^{-1}\left (\frac {2 x+1}{\sqrt {19}}\right )}{13685 \sqrt {19}} \]

[Out]

-3146/80155*ln(7-3*x)-334/323*ln(1+2*x)+4822/4879*ln(2+5*x)+11049/260015*ln(x^2+x+5)+3988/260015*arctan(1/19*(
1+2*x)*19^(1/2))*19^(1/2)

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Rubi [A]  time = 0.09, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.116, Rules used = {2074, 634, 618, 204, 628} \[ \frac {11049 \log \left (x^2+x+5\right )}{260015}-\frac {3146 \log (7-3 x)}{80155}-\frac {334}{323} \log (2 x+1)+\frac {4822 \log (5 x+2)}{4879}+\frac {3988 \tan ^{-1}\left (\frac {2 x+1}{\sqrt {19}}\right )}{13685 \sqrt {19}} \]

Antiderivative was successfully verified.

[In]

Int[(-32 + 5*x - 27*x^2 + 4*x^3)/(-70 - 299*x - 286*x^2 + 50*x^3 - 13*x^4 + 30*x^5),x]

[Out]

(3988*ArcTan[(1 + 2*x)/Sqrt[19]])/(13685*Sqrt[19]) - (3146*Log[7 - 3*x])/80155 - (334*Log[1 + 2*x])/323 + (482
2*Log[2 + 5*x])/4879 + (11049*Log[5 + x + x^2])/260015

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 2074

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P]}, Int[ExpandIntegrand[PP^p*Q^q, x], x] /;  !SumQ[N
onfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x] && PolyQ[Q, x] && IntegerQ[p] && NeQ[P, x]

Rubi steps

\begin {align*} \int \frac {-32+5 x-27 x^2+4 x^3}{-70-299 x-286 x^2+50 x^3-13 x^4+30 x^5} \, dx &=\int \left (-\frac {668}{323 (1+2 x)}-\frac {9438}{80155 (-7+3 x)}+\frac {24110}{4879 (2+5 x)}+\frac {48935+22098 x}{260015 \left (5+x+x^2\right )}\right ) \, dx\\ &=-\frac {3146 \log (7-3 x)}{80155}-\frac {334}{323} \log (1+2 x)+\frac {4822 \log (2+5 x)}{4879}+\frac {\int \frac {48935+22098 x}{5+x+x^2} \, dx}{260015}\\ &=-\frac {3146 \log (7-3 x)}{80155}-\frac {334}{323} \log (1+2 x)+\frac {4822 \log (2+5 x)}{4879}+\frac {11049 \int \frac {1+2 x}{5+x+x^2} \, dx}{260015}+\frac {1994 \int \frac {1}{5+x+x^2} \, dx}{13685}\\ &=-\frac {3146 \log (7-3 x)}{80155}-\frac {334}{323} \log (1+2 x)+\frac {4822 \log (2+5 x)}{4879}+\frac {11049 \log \left (5+x+x^2\right )}{260015}-\frac {3988 \operatorname {Subst}\left (\int \frac {1}{-19-x^2} \, dx,x,1+2 x\right )}{13685}\\ &=\frac {3988 \tan ^{-1}\left (\frac {1+2 x}{\sqrt {19}}\right )}{13685 \sqrt {19}}-\frac {3146 \log (7-3 x)}{80155}-\frac {334}{323} \log (1+2 x)+\frac {4822 \log (2+5 x)}{4879}+\frac {11049 \log \left (5+x+x^2\right )}{260015}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 57, normalized size = 0.90 \[ \frac {453009 \log \left (x^2+x+5\right )-418418 \log (7-3 x)-11023670 \log (2 x+1)[phone] \log (5 x+2)+163508 \sqrt {19} \tan ^{-1}\left (\frac {2 x+1}{\sqrt {19}}\right )}{10660615} \]

Antiderivative was successfully verified.

[In]

Integrate[(-32 + 5*x - 27*x^2 + 4*x^3)/(-70 - 299*x - 286*x^2 + 50*x^3 - 13*x^4 + 30*x^5),x]

[Out]

(163508*Sqrt[19]*ArcTan[(1 + 2*x)/Sqrt[19]] - 418418*Log[7 - 3*x] - 11023670*Log[1 + 2*x] + 10536070*Log[2 + 5
*x] + 453009*Log[5 + x + x^2])/10660615

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fricas [A]  time = 0.42, size = 50, normalized size = 0.79 \[ \frac {3988}{260015} \, \sqrt {19} \arctan \left (\frac {1}{19} \, \sqrt {19} {\left (2 \, x + 1\right )}\right ) + \frac {11049}{260015} \, \log \left (x^{2} + x + 5\right ) + \frac {4822}{4879} \, \log \left (5 \, x + 2\right ) - \frac {3146}{80155} \, \log \left (3 \, x - 7\right ) - \frac {334}{323} \, \log \left (2 \, x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^3-27*x^2+5*x-32)/(30*x^5-13*x^4+50*x^3-286*x^2-299*x-70),x, algorithm="fricas")

[Out]

3988/260015*sqrt(19)*arctan(1/19*sqrt(19)*(2*x + 1)) + 11049/260015*log(x^2 + x + 5) + 4822/4879*log(5*x + 2)
- 3146/80155*log(3*x - 7) - 334/323*log(2*x + 1)

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giac [A]  time = 1.03, size = 53, normalized size = 0.84 \[ \frac {3988}{260015} \, \sqrt {19} \arctan \left (\frac {1}{19} \, \sqrt {19} {\left (2 \, x + 1\right )}\right ) + \frac {11049}{260015} \, \log \left (x^{2} + x + 5\right ) + \frac {4822}{4879} \, \log \left ({\left | 5 \, x + 2 \right |}\right ) - \frac {3146}{80155} \, \log \left ({\left | 3 \, x - 7 \right |}\right ) - \frac {334}{323} \, \log \left ({\left | 2 \, x + 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^3-27*x^2+5*x-32)/(30*x^5-13*x^4+50*x^3-286*x^2-299*x-70),x, algorithm="giac")

[Out]

3988/260015*sqrt(19)*arctan(1/19*sqrt(19)*(2*x + 1)) + 11049/260015*log(x^2 + x + 5) + 4822/4879*log(abs(5*x +
2)) - 3146/80155*log(abs(3*x - 7)) - 334/323*log(abs(2*x + 1))

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maple [A]  time = 0.01, size = 51, normalized size = 0.81 \[ \frac {3988 \sqrt {19}\, \arctan \left (\frac {\left (2 x +1\right ) \sqrt {19}}{19}\right )}{260015}-\frac {334 \ln \left (2 x +1\right )}{323}+\frac {4822 \ln \left (5 x +2\right )}{4879}-\frac {3146 \ln \left (3 x -7\right )}{80155}+\frac {11049 \ln \left (x^{2}+x +5\right )}{260015} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x^3-27*x^2+5*x-32)/(30*x^5-13*x^4+50*x^3-286*x^2-299*x-70),x)

[Out]

4822/4879*ln(2+5*x)+11049/260015*ln(x^2+x+5)+3988/260015*arctan(1/19*(2*x+1)*19^(1/2))*19^(1/2)-334/323*ln(2*x
+1)-3146/80155*ln(3*x-7)

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maxima [A]  time = 1.27, size = 50, normalized size = 0.79 \[ \frac {3988}{260015} \, \sqrt {19} \arctan \left (\frac {1}{19} \, \sqrt {19} {\left (2 \, x + 1\right )}\right ) + \frac {11049}{260015} \, \log \left (x^{2} + x + 5\right ) + \frac {4822}{4879} \, \log \left (5 \, x + 2\right ) - \frac {3146}{80155} \, \log \left (3 \, x - 7\right ) - \frac {334}{323} \, \log \left (2 \, x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^3-27*x^2+5*x-32)/(30*x^5-13*x^4+50*x^3-286*x^2-299*x-70),x, algorithm="maxima")

[Out]

3988/260015*sqrt(19)*arctan(1/19*sqrt(19)*(2*x + 1)) + 11049/260015*log(x^2 + x + 5) + 4822/4879*log(5*x + 2)
- 3146/80155*log(3*x - 7) - 334/323*log(2*x + 1)

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mupad [B]  time = 0.28, size = 58, normalized size = 0.92 \[ \frac {4822\,\ln \left (x+\frac {2}{5}\right )}{4879}-\frac {334\,\ln \left (x+\frac {1}{2}\right )}{323}-\frac {3146\,\ln \left (x-\frac {7}{3}\right )}{80155}-\ln \left (x+\frac {1}{2}-\frac {\sqrt {19}\,1{}\mathrm {i}}{2}\right )\,\left (-\frac {11049}{260015}+\frac {\sqrt {19}\,1994{}\mathrm {i}}{260015}\right )+\ln \left (x+\frac {1}{2}+\frac {\sqrt {19}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {11049}{260015}+\frac {\sqrt {19}\,1994{}\mathrm {i}}{260015}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(5*x - 27*x^2 + 4*x^3 - 32)/(299*x + 286*x^2 - 50*x^3 + 13*x^4 - 30*x^5 + 70),x)

[Out]

(4822*log(x + 2/5))/4879 - (334*log(x + 1/2))/323 - (3146*log(x - 7/3))/80155 - log(x - (19^(1/2)*1i)/2 + 1/2)
*((19^(1/2)*1994i)/260015 - 11049/260015) + log(x + (19^(1/2)*1i)/2 + 1/2)*((19^(1/2)*1994i)/260015 + 11049/26
0015)

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sympy [A]  time = 0.37, size = 68, normalized size = 1.08 \[ - \frac {3146 \log {\left (x - \frac {7}{3} \right )}}{80155} + \frac {4822 \log {\left (x + \frac {2}{5} \right )}}{4879} - \frac {334 \log {\left (x + \frac {1}{2} \right )}}{323} + \frac {11049 \log {\left (x^{2} + x + 5 \right )}}{260015} + \frac {3988 \sqrt {19} \operatorname {atan}{\left (\frac {2 \sqrt {19} x}{19} + \frac {\sqrt {19}}{19} \right )}}{260015} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x**3-27*x**2+5*x-32)/(30*x**5-13*x**4+50*x**3-286*x**2-299*x-70),x)

[Out]

-3146*log(x - 7/3)/80155 + 4822*log(x + 2/5)/4879 - 334*log(x + 1/2)/323 + 11049*log(x**2 + x + 5)/260015 + 39
88*sqrt(19)*atan(2*sqrt(19)*x/19 + sqrt(19)/19)/260015

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