3.21.86 \(\int \frac {-270-232 x-24 x^2+4 x^3+1800 x^4+2520 x^5+1242 x^6+252 x^7+18 x^8}{90 x+53 x^2+2 x^3-x^4+1800 x^5+2520 x^6+1242 x^7+252 x^8+18 x^9} \, dx\)

Optimal. Leaf size=28 \[ \log \left (2 x+\frac {x-\frac {x^2}{9}}{x^4 \left (1+x+(3+x)^2\right )}\right ) \]

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Rubi [A]  time = 0.30, antiderivative size = 38, normalized size of antiderivative = 1.36, number of steps used = 3, number of rules used = 2, integrand size = 87, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.023, Rules used = {2074, 1587} \begin {gather*} \log \left (18 x^6+126 x^5+180 x^4-x+9\right )-3 \log (x)-\log (x+2)-\log (x+5) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-270 - 232*x - 24*x^2 + 4*x^3 + 1800*x^4 + 2520*x^5 + 1242*x^6 + 252*x^7 + 18*x^8)/(90*x + 53*x^2 + 2*x^3
- x^4 + 1800*x^5 + 2520*x^6 + 1242*x^7 + 252*x^8 + 18*x^9),x]

[Out]

-3*Log[x] - Log[2 + x] - Log[5 + x] + Log[9 - x + 180*x^4 + 126*x^5 + 18*x^6]

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 {1}{-5-x}+\frac {1}{-2-x}-\frac {3}{x}+\frac {-1+720 x^3+630 x^4+108 x^5}{9-x+180 x^4+126 x^5+18 x^6}\right ) \, dx\\ &=-3 \log (x)-\log (2+x)-\log (5+x)+\int \frac {-1+720 x^3+630 x^4+108 x^5}{9-x+180 x^4+126 x^5+18 x^6} \, dx\\ &=-3 \log (x)-\log (2+x)-\log (5+x)+\log \left (9-x+180 x^4+126 x^5+18 x^6\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.03, size = 47, normalized size = 1.68 \begin {gather*} 2 \left (-\frac {3 \log (x)}{2}-\frac {1}{2} \log \left (10+7 x+x^2\right )+\frac {1}{2} \log \left (9-x+180 x^4+126 x^5+18 x^6\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-270 - 232*x - 24*x^2 + 4*x^3 + 1800*x^4 + 2520*x^5 + 1242*x^6 + 252*x^7 + 18*x^8)/(90*x + 53*x^2 +
2*x^3 - x^4 + 1800*x^5 + 2520*x^6 + 1242*x^7 + 252*x^8 + 18*x^9),x]

[Out]

2*((-3*Log[x])/2 - Log[10 + 7*x + x^2]/2 + Log[9 - x + 180*x^4 + 126*x^5 + 18*x^6]/2)

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fricas [A]  time = 0.68, size = 37, normalized size = 1.32 \begin {gather*} \log \left (18 \, x^{6} + 126 \, x^{5} + 180 \, x^{4} - x + 9\right ) - \log \left (x^{2} + 7 \, x + 10\right ) - 3 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((18*x^8+252*x^7+1242*x^6+2520*x^5+1800*x^4+4*x^3-24*x^2-232*x-270)/(18*x^9+252*x^8+1242*x^7+2520*x^6
+1800*x^5-x^4+2*x^3+53*x^2+90*x),x, algorithm="fricas")

[Out]

log(18*x^6 + 126*x^5 + 180*x^4 - x + 9) - log(x^2 + 7*x + 10) - 3*log(x)

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giac [A]  time = 0.24, size = 42, normalized size = 1.50 \begin {gather*} \log \left ({\left | 18 \, x^{6} + 126 \, x^{5} + 180 \, x^{4} - x + 9 \right |}\right ) - \log \left ({\left | x + 5 \right |}\right ) - \log \left ({\left | x + 2 \right |}\right ) - 3 \, \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((18*x^8+252*x^7+1242*x^6+2520*x^5+1800*x^4+4*x^3-24*x^2-232*x-270)/(18*x^9+252*x^8+1242*x^7+2520*x^6
+1800*x^5-x^4+2*x^3+53*x^2+90*x),x, algorithm="giac")

[Out]

log(abs(18*x^6 + 126*x^5 + 180*x^4 - x + 9)) - log(abs(x + 5)) - log(abs(x + 2)) - 3*log(abs(x))

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maple [A]  time = 0.05, size = 38, normalized size = 1.36

method	result	size
risch	\(-3 \ln \relax (x )-\ln \left (x^{2}+7 x +10\right )+\ln \left (18 x^{6}+126 x^{5}+180 x^{4}-x +9\right )\)	\(38\)
default	\(-3 \ln \relax (x )-\ln \left (5+x \right )-\ln \left (2+x \right )+\ln \left (18 x^{6}+126 x^{5}+180 x^{4}-x +9\right )\)	\(39\)
norman	\(-3 \ln \relax (x )-\ln \left (5+x \right )-\ln \left (2+x \right )+\ln \left (18 x^{6}+126 x^{5}+180 x^{4}-x +9\right )\)	\(39\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((18*x^8+252*x^7+1242*x^6+2520*x^5+1800*x^4+4*x^3-24*x^2-232*x-270)/(18*x^9+252*x^8+1242*x^7+2520*x^6+1800*
x^5-x^4+2*x^3+53*x^2+90*x),x,method=_RETURNVERBOSE)

[Out]

-3*ln(x)-ln(x^2+7*x+10)+ln(18*x^6+126*x^5+180*x^4-x+9)

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maxima [A]  time = 0.55, size = 38, normalized size = 1.36 \begin {gather*} \log \left (18 \, x^{6} + 126 \, x^{5} + 180 \, x^{4} - x + 9\right ) - \log \left (x + 5\right ) - \log \left (x + 2\right ) - 3 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((18*x^8+252*x^7+1242*x^6+2520*x^5+1800*x^4+4*x^3-24*x^2-232*x-270)/(18*x^9+252*x^8+1242*x^7+2520*x^6
+1800*x^5-x^4+2*x^3+53*x^2+90*x),x, algorithm="maxima")

[Out]

log(18*x^6 + 126*x^5 + 180*x^4 - x + 9) - log(x + 5) - log(x + 2) - 3*log(x)

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mupad [B]  time = 0.20, size = 37, normalized size = 1.32 \begin {gather*} \ln \left (18\,x^6+126\,x^5+180\,x^4-x+9\right )-\ln \left (x^2+7\,x+10\right )-3\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x^3 - 24*x^2 - 232*x + 1800*x^4 + 2520*x^5 + 1242*x^6 + 252*x^7 + 18*x^8 - 270)/(90*x + 53*x^2 + 2*x^3
- x^4 + 1800*x^5 + 2520*x^6 + 1242*x^7 + 252*x^8 + 18*x^9),x)

[Out]

log(180*x^4 - x + 126*x^5 + 18*x^6 + 9) - log(7*x + x^2 + 10) - 3*log(x)

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sympy [A]  time = 0.18, size = 34, normalized size = 1.21 \begin {gather*} - 3 \log {\relax (x )} - \log {\left (x^{2} + 7 x + 10 \right )} + \log {\left (18 x^{6} + 126 x^{5} + 180 x^{4} - x + 9 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((18*x**8+252*x**7+1242*x**6+2520*x**5+1800*x**4+4*x**3-24*x**2-232*x-270)/(18*x**9+252*x**8+1242*x**
7+2520*x**6+1800*x**5-x**4+2*x**3+53*x**2+90*x),x)

[Out]

-3*log(x) - log(x**2 + 7*x + 10) + log(18*x**6 + 126*x**5 + 180*x**4 - x + 9)

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