3.96.68 \(\int \frac {-132 e^5+9000 x+180 x^2}{900 x^4+e^{10} (9+6 x+x^2)+e^5 (-180 x^2-60 x^3)} \, dx\)

Optimal. Leaf size=22 \[ \frac {25+x}{-5 x^2+\frac {1}{6} e^5 (3+x)} \]

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Rubi [A]  time = 0.08, antiderivative size = 23, normalized size of antiderivative = 1.05, number of steps used = 4, number of rules used = 4, integrand size = 50, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {1680, 12, 1814, 8} \begin {gather*} \frac {6 (x+25)}{-30 x^2+e^5 x+3 e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-132*E^5 + 9000*x + 180*x^2)/(900*x^4 + E^10*(9 + 6*x + x^2) + E^5*(-180*x^2 - 60*x^3)),x]

[Out]

(6*(25 + x))/(3*E^5 + E^5*x - 30*x^2)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 1680

Int[(Pq_)*(Q4_)^(p_), x_Symbol] :> With[{a = Coeff[Q4, x, 0], b = Coeff[Q4, x, 1], c = Coeff[Q4, x, 2], d = Co
eff[Q4, x, 3], e = Coeff[Q4, x, 4]}, Subst[Int[SimplifyIntegrand[(Pq /. x -> -(d/(4*e)) + x)*(a + d^4/(256*e^3
) - (b*d)/(8*e) + (c - (3*d^2)/(8*e))*x^2 + e*x^4)^p, x], x], x, d/(4*e) + x] /; EqQ[d^3 - 4*c*d*e + 8*b*e^2,
0] && NeQ[d, 0]] /; FreeQ[p, x] && PolyQ[Pq, x] && PolyQ[Q4, x, 4] &&  !IGtQ[p, 0]

Rule 1814

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, a + b*x^2, x], f = Coeff[P
olynomialRemainder[Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 1]}, Simp[((a
*g - b*f*x)*(a + b*x^2)^(p + 1))/(2*a*b*(p + 1)), x] + Dist[1/(2*a*(p + 1)), Int[(a + b*x^2)^(p + 1)*ExpandToS
um[2*a*(p + 1)*Q + f*(2*p + 3), x], x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && LtQ[p, -1]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\operatorname {Subst}\left (\int \frac {720 \left (e^5 \left (360+e^5\right )+120 \left (1500+e^5\right ) x+3600 x^2\right )}{\left (360 e^5+e^{10}-3600 x^2\right )^2} \, dx,x,-\frac {e^5}{60}+x\right )\\ &=720 \operatorname {Subst}\left (\int \frac {e^5 \left (360+e^5\right )+120 \left (1500+e^5\right ) x+3600 x^2}{\left (360 e^5+e^{10}-3600 x^2\right )^2} \, dx,x,-\frac {e^5}{60}+x\right )\\ &=\frac {6 (25+x)}{3 e^5+e^5 x-30 x^2}-\frac {360 \operatorname {Subst}\left (\int 0 \, dx,x,-\frac {e^5}{60}+x\right )}{e^5 \left (360+e^5\right )}\\ &=\frac {6 (25+x)}{3 e^5+e^5 x-30 x^2}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 20, normalized size = 0.91 \begin {gather*} \frac {6 (25+x)}{-30 x^2+e^5 (3+x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-132*E^5 + 9000*x + 180*x^2)/(900*x^4 + E^10*(9 + 6*x + x^2) + E^5*(-180*x^2 - 60*x^3)),x]

[Out]

(6*(25 + x))/(-30*x^2 + E^5*(3 + x))

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fricas [A]  time = 0.61, size = 20, normalized size = 0.91 \begin {gather*} -\frac {6 \, {\left (x + 25\right )}}{30 \, x^{2} - {\left (x + 3\right )} e^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-132*exp(5)+180*x^2+9000*x)/((x^2+6*x+9)*exp(5)^2+(-60*x^3-180*x^2)*exp(5)+900*x^4),x, algorithm="f
ricas")

[Out]

-6*(x + 25)/(30*x^2 - (x + 3)*e^5)

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giac [A]  time = 0.14, size = 13, normalized size = 0.59 \begin {gather*} 2.02587824616667 \times 10^{13} \, \log \left (x + 2.10463242876030\right ) + 11344289279736 \, \log \left (x - 7.05173773218000\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-132*exp(5)+180*x^2+9000*x)/((x^2+6*x+9)*exp(5)^2+(-60*x^3-180*x^2)*exp(5)+900*x^4),x, algorithm="g
iac")

[Out]

2.02587824616667e13*log(x + 2.10463242876030) + 11344289279736*log(x - 7.05173773218000)

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maple [A]  time = 0.12, size = 22, normalized size = 1.00

method	result	size
gosper	\(\frac {6 x +150}{x \,{\mathrm e}^{5}-30 x^{2}+3 \,{\mathrm e}^{5}}\)	\(22\)
norman	\(\frac {6 x +150}{x \,{\mathrm e}^{5}-30 x^{2}+3 \,{\mathrm e}^{5}}\)	\(23\)
risch	\(\frac {6 x +150}{x \,{\mathrm e}^{5}-30 x^{2}+3 \,{\mathrm e}^{5}}\)	\(23\)
default	\(-6 \left (\munderset {\textit {\_R} =\RootOf \left (900 \textit {\_Z}^{4}-60 \textit {\_Z}^{3} {\mathrm e}^{5}+\left (-180 \,{\mathrm e}^{5}+{\mathrm e}^{10}\right ) \textit {\_Z}^{2}+6 \textit {\_Z} \,{\mathrm e}^{10}+9 \,{\mathrm e}^{10}\right )}{\sum }\frac {\left (-11 \,{\mathrm e}^{5}+15 \textit {\_R}^{2}+750 \textit {\_R} \right ) \ln \left (x -\textit {\_R} \right )}{90 \textit {\_R}^{2} {\mathrm e}^{5}-1800 \textit {\_R}^{3}+180 \textit {\_R} \,{\mathrm e}^{5}-\textit {\_R} \,{\mathrm e}^{10}-3 \,{\mathrm e}^{10}}\right )\)	\(89\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-132*exp(5)+180*x^2+9000*x)/((x^2+6*x+9)*exp(5)^2+(-60*x^3-180*x^2)*exp(5)+900*x^4),x,method=_RETURNVERBO
SE)

[Out]

6*(x+25)/(x*exp(5)-30*x^2+3*exp(5))

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maxima [A]  time = 0.37, size = 22, normalized size = 1.00 \begin {gather*} -\frac {6 \, {\left (x + 25\right )}}{30 \, x^{2} - x e^{5} - 3 \, e^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-132*exp(5)+180*x^2+9000*x)/((x^2+6*x+9)*exp(5)^2+(-60*x^3-180*x^2)*exp(5)+900*x^4),x, algorithm="m
axima")

[Out]

-6*(x + 25)/(30*x^2 - x*e^5 - 3*e^5)

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mupad [B]  time = 9.27, size = 22, normalized size = 1.00 \begin {gather*} \frac {6\,x+150}{-30\,x^2+{\mathrm {e}}^5\,x+3\,{\mathrm {e}}^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((9000*x - 132*exp(5) + 180*x^2)/(exp(10)*(6*x + x^2 + 9) - exp(5)*(180*x^2 + 60*x^3) + 900*x^4),x)

[Out]

(6*x + 150)/(3*exp(5) + x*exp(5) - 30*x^2)

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sympy [A]  time = 0.39, size = 20, normalized size = 0.91 \begin {gather*} \frac {- 6 x - 150}{30 x^{2} - x e^{5} - 3 e^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-132*exp(5)+180*x**2+9000*x)/((x**2+6*x+9)*exp(5)**2+(-60*x**3-180*x**2)*exp(5)+900*x**4),x)

[Out]

(-6*x - 150)/(30*x**2 - x*exp(5) - 3*exp(5))

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