3.218 \(\int \frac {8-13 x^2-7 x^3+12 x^5}{4-20 x+41 x^2-80 x^3+116 x^4-80 x^5+100 x^6} \, dx\)

Optimal. Leaf size=86 \[ -\frac {502 x+313}{1452 \left (2 x^2+1\right )}+\frac {2843 \log \left (2 x^2+1\right )}{7986}+\frac {5828}{9075 (2-5 x)}-\frac {59096 \log (2-5 x)}{99825}+\frac {272 \sqrt {2} \tan ^{-1}\left (\sqrt {2} x\right )}{1331}-\frac {251 \tan ^{-1}\left (\sqrt {2} x\right )}{726 \sqrt {2}} \]

[Out]

5828/9075/(2-5*x)+1/1452*(-313-502*x)/(2*x^2+1)-59096/99825*ln(2-5*x)+2843/7986*ln(2*x^2+1)+503/15972*arctan(x
*2^(1/2))*2^(1/2)

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Rubi [A]  time = 0.10, antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 50, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2074, 639, 203, 635, 260} \[ -\frac {502 x+313}{1452 \left (2 x^2+1\right )}+\frac {2843 \log \left (2 x^2+1\right )}{7986}+\frac {5828}{9075 (2-5 x)}-\frac {59096 \log (2-5 x)}{99825}+\frac {272 \sqrt {2} \tan ^{-1}\left (\sqrt {2} x\right )}{1331}-\frac {251 \tan ^{-1}\left (\sqrt {2} x\right )}{726 \sqrt {2}} \]

Antiderivative was successfully verified.

[In]

Int[(8 - 13*x^2 - 7*x^3 + 12*x^5)/(4 - 20*x + 41*x^2 - 80*x^3 + 116*x^4 - 80*x^5 + 100*x^6),x]

[Out]

5828/(9075*(2 - 5*x)) - (313 + 502*x)/(1452*(1 + 2*x^2)) - (251*ArcTan[Sqrt[2]*x])/(726*Sqrt[2]) + (272*Sqrt[2
]*ArcTan[Sqrt[2]*x])/1331 - (59096*Log[2 - 5*x])/99825 + (2843*Log[1 + 2*x^2])/7986

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 635

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[d, Int[1/(a + c*x^2), x], x] + Dist[e, Int[x/
(a + c*x^2), x], x] /; FreeQ[{a, c, d, e}, x] &&  !NiceSqrtQ[-(a*c)]

Rule 639

Int[((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((a*e - c*d*x)*(a + c*x^2)^(p + 1))/(2*a
*c*(p + 1)), x] + Dist[(d*(2*p + 3))/(2*a*(p + 1)), Int[(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e}, x]
&& LtQ[p, -1] && NeQ[p, -3/2]

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 {8-13 x^2-7 x^3+12 x^5}{4-20 x+41 x^2-80 x^3+116 x^4-80 x^5+100 x^6} \, dx &=\int \left (\frac {5828}{1815 (-2+5 x)^2}-\frac {59096}{19965 (-2+5 x)}+\frac {-251+313 x}{363 \left (1+2 x^2\right )^2}+\frac {2 (816+2843 x)}{3993 \left (1+2 x^2\right )}\right ) \, dx\\ &=\frac {5828}{9075 (2-5 x)}-\frac {59096 \log (2-5 x)}{99825}+\frac {2 \int \frac {816+2843 x}{1+2 x^2} \, dx}{3993}+\frac {1}{363} \int \frac {-251+313 x}{\left (1+2 x^2\right )^2} \, dx\\ &=\frac {5828}{9075 (2-5 x)}-\frac {313+502 x}{1452 \left (1+2 x^2\right )}-\frac {59096 \log (2-5 x)}{99825}-\frac {251}{726} \int \frac {1}{1+2 x^2} \, dx+\frac {544 \int \frac {1}{1+2 x^2} \, dx}{1331}+\frac {5686 \int \frac {x}{1+2 x^2} \, dx}{3993}\\ &=\frac {5828}{9075 (2-5 x)}-\frac {313+502 x}{1452 \left (1+2 x^2\right )}-\frac {251 \tan ^{-1}\left (\sqrt {2} x\right )}{726 \sqrt {2}}+\frac {272 \sqrt {2} \tan ^{-1}\left (\sqrt {2} x\right )}{1331}-\frac {59096 \log (2-5 x)}{99825}+\frac {2843 \log \left (1+2 x^2\right )}{7986}\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 67, normalized size = 0.78 \[ \frac {142150 \log \left (2 x^2+1\right )-\frac {33 \left (36458 x^2+4675 x+2554\right )}{10 x^3-4 x^2+5 x-2}-236384 \log (2-5 x)+12575 \sqrt {2} \tan ^{-1}\left (\sqrt {2} x\right )}{399300} \]

Antiderivative was successfully verified.

[In]

Integrate[(8 - 13*x^2 - 7*x^3 + 12*x^5)/(4 - 20*x + 41*x^2 - 80*x^3 + 116*x^4 - 80*x^5 + 100*x^6),x]

[Out]

((-33*(2554 + 4675*x + 36458*x^2))/(-2 + 5*x - 4*x^2 + 10*x^3) + 12575*Sqrt[2]*ArcTan[Sqrt[2]*x] - 236384*Log[
2 - 5*x] + 142150*Log[1 + 2*x^2])/399300

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fricas [A]  time = 0.41, size = 103, normalized size = 1.20 \[ \frac {12575 \, \sqrt {2} {\left (10 \, x^{3} - 4 \, x^{2} + 5 \, x - 2\right )} \arctan \left (\sqrt {2} x\right ) - 1203114 \, x^{2} + 142150 \, {\left (10 \, x^{3} - 4 \, x^{2} + 5 \, x - 2\right )} \log \left (2 \, x^{2} + 1\right ) - 236384 \, {\left (10 \, x^{3} - 4 \, x^{2} + 5 \, x - 2\right )} \log \left (5 \, x - 2\right ) - 154275 \, x - 84282}{399300 \, {\left (10 \, x^{3} - 4 \, x^{2} + 5 \, x - 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((12*x^5-7*x^3-13*x^2+8)/(100*x^6-80*x^5+116*x^4-80*x^3+41*x^2-20*x+4),x, algorithm="fricas")

[Out]

1/399300*(12575*sqrt(2)*(10*x^3 - 4*x^2 + 5*x - 2)*arctan(sqrt(2)*x) - 1203114*x^2 + 142150*(10*x^3 - 4*x^2 +
5*x - 2)*log(2*x^2 + 1) - 236384*(10*x^3 - 4*x^2 + 5*x - 2)*log(5*x - 2) - 154275*x - 84282)/(10*x^3 - 4*x^2 +
5*x - 2)

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giac [A]  time = 0.90, size = 59, normalized size = 0.69 \[ \frac {503}{15972} \, \sqrt {2} \arctan \left (\sqrt {2} x\right ) - \frac {36458 \, x^{2} + 4675 \, x + 2554}{12100 \, {\left (2 \, x^{2} + 1\right )} {\left (5 \, x - 2\right )}} + \frac {2843}{7986} \, \log \left (2 \, x^{2} + 1\right ) - \frac {59096}{99825} \, \log \left ({\left | 5 \, x - 2 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((12*x^5-7*x^3-13*x^2+8)/(100*x^6-80*x^5+116*x^4-80*x^3+41*x^2-20*x+4),x, algorithm="giac")

[Out]

503/15972*sqrt(2)*arctan(sqrt(2)*x) - 1/12100*(36458*x^2 + 4675*x + 2554)/((2*x^2 + 1)*(5*x - 2)) + 2843/7986*
log(2*x^2 + 1) - 59096/99825*log(abs(5*x - 2))

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maple [A]  time = 0.02, size = 54, normalized size = 0.63 \[ \frac {503 \sqrt {2}\, \arctan \left (\sqrt {2}\, x \right )}{15972}-\frac {59096 \ln \left (5 x -2\right )}{99825}+\frac {2843 \ln \left (2 x^{2}+1\right )}{7986}+\frac {-\frac {2761 x}{4}-\frac {3443}{8}}{3993 x^{2}+\frac {3993}{2}}-\frac {5828}{9075 \left (5 x -2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((12*x^5-7*x^3-13*x^2+8)/(100*x^6-80*x^5+116*x^4-80*x^3+41*x^2-20*x+4),x)

[Out]

1/3993*(-2761/4*x-3443/8)/(x^2+1/2)+2843/7986*ln(2*x^2+1)+503/15972*arctan(2^(1/2)*x)*2^(1/2)-5828/9075/(5*x-2
)-59096/99825*ln(5*x-2)

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maxima [A]  time = 1.20, size = 59, normalized size = 0.69 \[ \frac {503}{15972} \, \sqrt {2} \arctan \left (\sqrt {2} x\right ) - \frac {36458 \, x^{2} + 4675 \, x + 2554}{12100 \, {\left (10 \, x^{3} - 4 \, x^{2} + 5 \, x - 2\right )}} + \frac {2843}{7986} \, \log \left (2 \, x^{2} + 1\right ) - \frac {59096}{99825} \, \log \left (5 \, x - 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((12*x^5-7*x^3-13*x^2+8)/(100*x^6-80*x^5+116*x^4-80*x^3+41*x^2-20*x+4),x, algorithm="maxima")

[Out]

503/15972*sqrt(2)*arctan(sqrt(2)*x) - 1/12100*(36458*x^2 + 4675*x + 2554)/(10*x^3 - 4*x^2 + 5*x - 2) + 2843/79
86*log(2*x^2 + 1) - 59096/99825*log(5*x - 2)

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mupad [B]  time = 0.13, size = 71, normalized size = 0.83 \[ -\frac {59096\,\ln \left (x-\frac {2}{5}\right )}{99825}-\frac {\frac {18229\,x^2}{60500}+\frac {17\,x}{440}+\frac {1277}{60500}}{x^3-\frac {2\,x^2}{5}+\frac {x}{2}-\frac {1}{5}}-\ln \left (x-\frac {\sqrt {2}\,1{}\mathrm {i}}{2}\right )\,\left (-\frac {2843}{7986}+\frac {\sqrt {2}\,503{}\mathrm {i}}{31944}\right )+\ln \left (x+\frac {\sqrt {2}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {2843}{7986}+\frac {\sqrt {2}\,503{}\mathrm {i}}{31944}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(13*x^2 + 7*x^3 - 12*x^5 - 8)/(41*x^2 - 20*x - 80*x^3 + 116*x^4 - 80*x^5 + 100*x^6 + 4),x)

[Out]

log(x + (2^(1/2)*1i)/2)*((2^(1/2)*503i)/31944 + 2843/7986) - ((17*x)/440 + (18229*x^2)/60500 + 1277/60500)/(x/
2 - (2*x^2)/5 + x^3 - 1/5) - log(x - (2^(1/2)*1i)/2)*((2^(1/2)*503i)/31944 - 2843/7986) - (59096*log(x - 2/5))
/99825

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sympy [A]  time = 0.23, size = 65, normalized size = 0.76 \[ \frac {- 36458 x^{2} - 4675 x - 2554}{121000 x^{3} - 48400 x^{2} + 60500 x - 24200} - \frac {59096 \log {\left (x - \frac {2}{5} \right )}}{99825} + \frac {2843 \log {\left (x^{2} + \frac {1}{2} \right )}}{7986} + \frac {503 \sqrt {2} \operatorname {atan}{\left (\sqrt {2} x \right )}}{15972} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((12*x**5-7*x**3-13*x**2+8)/(100*x**6-80*x**5+116*x**4-80*x**3+41*x**2-20*x+4),x)

[Out]

(-36458*x**2 - 4675*x - 2554)/(121000*x**3 - 48400*x**2 + 60500*x - 24200) - 59096*log(x - 2/5)/99825 + 2843*l
og(x**2 + 1/2)/7986 + 503*sqrt(2)*atan(sqrt(2)*x)/15972

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