3.30.3 \(\int \frac {-200-120 x^2+50 x^3-56 x^4+12 x^5-10 x^6+2 x^7+x^9+2 x^{10}}{100 x+80 x^3-40 x^4+56 x^5-16 x^6+20 x^7-8 x^8+4 x^9+4 x^{10}+4 x^{11}} \, dx\)

Optimal. Leaf size=34 \[ 1+\frac {1}{4} \log \left (x+x^2+\frac {\left (\frac {2+\frac {5}{x^2}-x}{x}+x\right )^2}{x^2}\right ) \]

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Rubi [A]  time = 0.43, antiderivative size = 51, normalized size of antiderivative = 1.50, number of steps used = 3, number of rules used = 2, integrand size = 92, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.022, Rules used = {2074, 1587} \begin {gather*} \frac {1}{4} \log \left (x^{10}+x^9+x^8-2 x^7+5 x^6-4 x^5+14 x^4-10 x^3+20 x^2+25\right )-2 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-200 - 120*x^2 + 50*x^3 - 56*x^4 + 12*x^5 - 10*x^6 + 2*x^7 + x^9 + 2*x^10)/(100*x + 80*x^3 - 40*x^4 + 56*
x^5 - 16*x^6 + 20*x^7 - 8*x^8 + 4*x^9 + 4*x^10 + 4*x^11),x]

[Out]

-2*Log[x] + Log[25 + 20*x^2 - 10*x^3 + 14*x^4 - 4*x^5 + 5*x^6 - 2*x^7 + x^8 + x^9 + x^10]/4

Rule 1587

Int[(Pp_)/(Qq_), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x]}, Simp[(Coeff[Pp, x, p]*Log[RemoveConte
nt[Qq, x]])/(q*Coeff[Qq, x, q]), x] /; EqQ[p, q - 1] && EqQ[Pp, Simplify[(Coeff[Pp, x, p]*D[Qq, x])/(q*Coeff[Q
q, x, q])]]] /; PolyQ[Pp, x] && PolyQ[Qq, x]

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 {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {2}{x}+\frac {x \left (40-30 x+56 x^2-20 x^3+30 x^4-14 x^5+8 x^6+9 x^7+10 x^8\right )}{4 \left (25+20 x^2-10 x^3+14 x^4-4 x^5+5 x^6-2 x^7+x^8+x^9+x^{10}\right )}\right ) \, dx\\ &=-2 \log (x)+\frac {1}{4} \int \frac {x \left (40-30 x+56 x^2-20 x^3+30 x^4-14 x^5+8 x^6+9 x^7+10 x^8\right )}{25+20 x^2-10 x^3+14 x^4-4 x^5+5 x^6-2 x^7+x^8+x^9+x^{10}} \, dx\\ &=-2 \log (x)+\frac {1}{4} \log \left (25+20 x^2-10 x^3+14 x^4-4 x^5+5 x^6-2 x^7+x^8+x^9+x^{10}\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 51, normalized size = 1.50 \begin {gather*} \frac {1}{4} \left (-8 \log (x)+\log \left (25+20 x^2-10 x^3+14 x^4-4 x^5+5 x^6-2 x^7+x^8+x^9+x^{10}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-200 - 120*x^2 + 50*x^3 - 56*x^4 + 12*x^5 - 10*x^6 + 2*x^7 + x^9 + 2*x^10)/(100*x + 80*x^3 - 40*x^4
+ 56*x^5 - 16*x^6 + 20*x^7 - 8*x^8 + 4*x^9 + 4*x^10 + 4*x^11),x]

[Out]

(-8*Log[x] + Log[25 + 20*x^2 - 10*x^3 + 14*x^4 - 4*x^5 + 5*x^6 - 2*x^7 + x^8 + x^9 + x^10])/4

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fricas [A]  time = 0.96, size = 49, normalized size = 1.44 \begin {gather*} \frac {1}{4} \, \log \left (x^{10} + x^{9} + x^{8} - 2 \, x^{7} + 5 \, x^{6} - 4 \, x^{5} + 14 \, x^{4} - 10 \, x^{3} + 20 \, x^{2} + 25\right ) - 2 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^10+x^9+2*x^7-10*x^6+12*x^5-56*x^4+50*x^3-120*x^2-200)/(4*x^11+4*x^10+4*x^9-8*x^8+20*x^7-16*x^6+
56*x^5-40*x^4+80*x^3+100*x),x, algorithm="fricas")

[Out]

1/4*log(x^10 + x^9 + x^8 - 2*x^7 + 5*x^6 - 4*x^5 + 14*x^4 - 10*x^3 + 20*x^2 + 25) - 2*log(x)

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giac [A]  time = 0.26, size = 50, normalized size = 1.47 \begin {gather*} \frac {1}{4} \, \log \left (x^{10} + x^{9} + x^{8} - 2 \, x^{7} + 5 \, x^{6} - 4 \, x^{5} + 14 \, x^{4} - 10 \, x^{3} + 20 \, x^{2} + 25\right ) - 2 \, \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^10+x^9+2*x^7-10*x^6+12*x^5-56*x^4+50*x^3-120*x^2-200)/(4*x^11+4*x^10+4*x^9-8*x^8+20*x^7-16*x^6+
56*x^5-40*x^4+80*x^3+100*x),x, algorithm="giac")

[Out]

1/4*log(x^10 + x^9 + x^8 - 2*x^7 + 5*x^6 - 4*x^5 + 14*x^4 - 10*x^3 + 20*x^2 + 25) - 2*log(abs(x))

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maple [A]  time = 0.04, size = 50, normalized size = 1.47

method	result	size
default	\(-2 \ln \relax (x )+\frac {\ln \left (x^{10}+x^{9}+x^{8}-2 x^{7}+5 x^{6}-4 x^{5}+14 x^{4}-10 x^{3}+20 x^{2}+25\right )}{4}\)	\(50\)
norman	\(-2 \ln \relax (x )+\frac {\ln \left (x^{10}+x^{9}+x^{8}-2 x^{7}+5 x^{6}-4 x^{5}+14 x^{4}-10 x^{3}+20 x^{2}+25\right )}{4}\)	\(50\)
risch	\(-2 \ln \relax (x )+\frac {\ln \left (x^{10}+x^{9}+x^{8}-2 x^{7}+5 x^{6}-4 x^{5}+14 x^{4}-10 x^{3}+20 x^{2}+25\right )}{4}\)	\(50\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x^10+x^9+2*x^7-10*x^6+12*x^5-56*x^4+50*x^3-120*x^2-200)/(4*x^11+4*x^10+4*x^9-8*x^8+20*x^7-16*x^6+56*x^5
-40*x^4+80*x^3+100*x),x,method=_RETURNVERBOSE)

[Out]

-2*ln(x)+1/4*ln(x^10+x^9+x^8-2*x^7+5*x^6-4*x^5+14*x^4-10*x^3+20*x^2+25)

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maxima [A]  time = 0.44, size = 49, normalized size = 1.44 \begin {gather*} \frac {1}{4} \, \log \left (x^{10} + x^{9} + x^{8} - 2 \, x^{7} + 5 \, x^{6} - 4 \, x^{5} + 14 \, x^{4} - 10 \, x^{3} + 20 \, x^{2} + 25\right ) - 2 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^10+x^9+2*x^7-10*x^6+12*x^5-56*x^4+50*x^3-120*x^2-200)/(4*x^11+4*x^10+4*x^9-8*x^8+20*x^7-16*x^6+
56*x^5-40*x^4+80*x^3+100*x),x, algorithm="maxima")

[Out]

1/4*log(x^10 + x^9 + x^8 - 2*x^7 + 5*x^6 - 4*x^5 + 14*x^4 - 10*x^3 + 20*x^2 + 25) - 2*log(x)

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mupad [B]  time = 0.14, size = 49, normalized size = 1.44 \begin {gather*} \frac {\ln \left (x^{10}+x^9+x^8-2\,x^7+5\,x^6-4\,x^5+14\,x^4-10\,x^3+20\,x^2+25\right )}{4}-2\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((50*x^3 - 120*x^2 - 56*x^4 + 12*x^5 - 10*x^6 + 2*x^7 + x^9 + 2*x^10 - 200)/(100*x + 80*x^3 - 40*x^4 + 56*x
^5 - 16*x^6 + 20*x^7 - 8*x^8 + 4*x^9 + 4*x^10 + 4*x^11),x)

[Out]

log(20*x^2 - 10*x^3 + 14*x^4 - 4*x^5 + 5*x^6 - 2*x^7 + x^8 + x^9 + x^10 + 25)/4 - 2*log(x)

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sympy [A]  time = 0.18, size = 49, normalized size = 1.44 \begin {gather*} - 2 \log {\relax (x )} + \frac {\log {\left (x^{10} + x^{9} + x^{8} - 2 x^{7} + 5 x^{6} - 4 x^{5} + 14 x^{4} - 10 x^{3} + 20 x^{2} + 25 \right )}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x**10+x**9+2*x**7-10*x**6+12*x**5-56*x**4+50*x**3-120*x**2-200)/(4*x**11+4*x**10+4*x**9-8*x**8+20
*x**7-16*x**6+56*x**5-40*x**4+80*x**3+100*x),x)

[Out]

-2*log(x) + log(x**10 + x**9 + x**8 - 2*x**7 + 5*x**6 - 4*x**5 + 14*x**4 - 10*x**3 + 20*x**2 + 25)/4

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