3.26.16 \(\int \frac {9+24 x^4+6 x^5+16 x^8+8 x^9+x^{10}+e^2 x (-16 x^2-5 x^3)}{9+24 x^4+6 x^5+16 x^8+8 x^9+x^{10}} \, dx\)

Optimal. Leaf size=18 \[ e+x+\frac {e^2}{3+x^4 (4+x)} \]

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Rubi [A]  time = 0.12, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 69, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.029, Rules used = {2074, 1588} \begin {gather*} \frac {e^2}{x^5+4 x^4+3}+x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(9 + 24*x^4 + 6*x^5 + 16*x^8 + 8*x^9 + x^10 + E^2*x*(-16*x^2 - 5*x^3))/(9 + 24*x^4 + 6*x^5 + 16*x^8 + 8*x^
9 + x^10),x]

[Out]

x + E^2/(3 + 4*x^4 + x^5)

Rule 1588

Int[(Pp_)*(Qq_)^(m_.), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x]}, Simp[(Coeff[Pp, x, p]*x^(p - q
+ 1)*Qq^(m + 1))/((p + m*q + 1)*Coeff[Qq, x, q]), x] /; NeQ[p + m*q + 1, 0] && EqQ[(p + m*q + 1)*Coeff[Qq, x,
q]*Pp, Coeff[Pp, x, p]*x^(p - q)*((p - q + 1)*Qq + (m + 1)*x*D[Qq, x])]] /; FreeQ[m, x] && PolyQ[Pp, x] && Pol
yQ[Qq, x] && NeQ[m, -1]

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 (1-\frac {e^2 x^3 (16+5 x)}{\left (3+4 x^4+x^5\right )^2}\right ) \, dx\\ &=x-e^2 \int \frac {x^3 (16+5 x)}{\left (3+4 x^4+x^5\right )^2} \, dx\\ &=x+\frac {e^2}{3+4 x^4+x^5}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 18, normalized size = 1.00 \begin {gather*} x+\frac {e^2}{3+4 x^4+x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(9 + 24*x^4 + 6*x^5 + 16*x^8 + 8*x^9 + x^10 + E^2*x*(-16*x^2 - 5*x^3))/(9 + 24*x^4 + 6*x^5 + 16*x^8
+ 8*x^9 + x^10),x]

[Out]

x + E^2/(3 + 4*x^4 + x^5)

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fricas [A]  time = 1.00, size = 27, normalized size = 1.50 \begin {gather*} \frac {x^{6} + 4 \, x^{5} + 3 \, x + e^{2}}{x^{5} + 4 \, x^{4} + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-5*x^3-16*x^2)*exp(log(x)+2)+x^10+8*x^9+16*x^8+6*x^5+24*x^4+9)/(x^10+8*x^9+16*x^8+6*x^5+24*x^4+9),
x, algorithm="fricas")

[Out]

(x^6 + 4*x^5 + 3*x + e^2)/(x^5 + 4*x^4 + 3)

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giac [A]  time = 0.14, size = 17, normalized size = 0.94 \begin {gather*} x + \frac {e^{2}}{x^{5} + 4 \, x^{4} + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-5*x^3-16*x^2)*exp(log(x)+2)+x^10+8*x^9+16*x^8+6*x^5+24*x^4+9)/(x^10+8*x^9+16*x^8+6*x^5+24*x^4+9),
x, algorithm="giac")

[Out]

x + e^2/(x^5 + 4*x^4 + 3)

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maple [A]  time = 0.05, size = 18, normalized size = 1.00

method	result	size
default	\(x +\frac {{\mathrm e}^{2}}{x^{5}+4 x^{4}+3}\)	\(18\)
risch	\(x +\frac {{\mathrm e}^{2}}{x^{5}+4 x^{4}+3}\)	\(18\)
norman	\(\frac {x^{6}-16 x^{4}+3 x -12+{\mathrm e}^{2}}{x^{5}+4 x^{4}+3}\)	\(29\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-5*x^3-16*x^2)*exp(ln(x)+2)+x^10+8*x^9+16*x^8+6*x^5+24*x^4+9)/(x^10+8*x^9+16*x^8+6*x^5+24*x^4+9),x,metho
d=_RETURNVERBOSE)

[Out]

x+exp(2)/(x^5+4*x^4+3)

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maxima [A]  time = 0.43, size = 17, normalized size = 0.94 \begin {gather*} x + \frac {e^{2}}{x^{5} + 4 \, x^{4} + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-5*x^3-16*x^2)*exp(log(x)+2)+x^10+8*x^9+16*x^8+6*x^5+24*x^4+9)/(x^10+8*x^9+16*x^8+6*x^5+24*x^4+9),
x, algorithm="maxima")

[Out]

x + e^2/(x^5 + 4*x^4 + 3)

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mupad [B]  time = 1.45, size = 17, normalized size = 0.94 \begin {gather*} x+\frac {{\mathrm {e}}^2}{x^5+4\,x^4+3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((24*x^4 - exp(log(x) + 2)*(16*x^2 + 5*x^3) + 6*x^5 + 16*x^8 + 8*x^9 + x^10 + 9)/(24*x^4 + 6*x^5 + 16*x^8 +
8*x^9 + x^10 + 9),x)

[Out]

x + exp(2)/(4*x^4 + x^5 + 3)

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sympy [A]  time = 0.27, size = 14, normalized size = 0.78 \begin {gather*} x + \frac {e^{2}}{x^{5} + 4 x^{4} + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-5*x**3-16*x**2)*exp(ln(x)+2)+x**10+8*x**9+16*x**8+6*x**5+24*x**4+9)/(x**10+8*x**9+16*x**8+6*x**5+
24*x**4+9),x)

[Out]

x + exp(2)/(x**5 + 4*x**4 + 3)

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