3.17.22 \(\int \frac {-12 x^2-1203 x^4+e^5 (12-6 x-3609 x^2)}{e^{10}+2 e^5 x^2+x^4} \, dx\)

Optimal. Leaf size=22 \[ \frac {3 x \left (4-x-401 x^2\right )}{e^5+x^2} \]

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Rubi [A]  time = 0.03, antiderivative size = 28, normalized size of antiderivative = 1.27, number of steps used = 4, number of rules used = 4, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.093, Rules used = {28, 1814, 21, 8} \begin {gather*} \frac {3 \left (\left (4+401 e^5\right ) x+e^5\right )}{x^2+e^5}-1203 x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-12*x^2 - 1203*x^4 + E^5*(12 - 6*x - 3609*x^2))/(E^10 + 2*E^5*x^2 + x^4),x]

[Out]

-1203*x + (3*(E^5 + (4 + 401*E^5)*x))/(E^5 + x^2)

Rule 8

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

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 28

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/c^p, Int[u*(b/2 + c*x^n)^(2*
p), x], x] /; FreeQ[{a, b, c, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

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} &=\int \frac {-12 x^2-1203 x^4+e^5 \left (12-6 x-3609 x^2\right )}{\left (e^5+x^2\right )^2} \, dx\\ &=\frac {3 \left (e^5+\left (4+401 e^5\right ) x\right )}{e^5+x^2}-\frac {\int \frac {2406 e^{10}+2406 e^5 x^2}{e^5+x^2} \, dx}{2 e^5}\\ &=\frac {3 \left (e^5+\left (4+401 e^5\right ) x\right )}{e^5+x^2}-1203 \int 1 \, dx\\ &=-1203 x+\frac {3 \left (e^5+\left (4+401 e^5\right ) x\right )}{e^5+x^2}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 23, normalized size = 1.05 \begin {gather*} \frac {3 \left (e^5+4 x-401 x^3\right )}{e^5+x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-12*x^2 - 1203*x^4 + E^5*(12 - 6*x - 3609*x^2))/(E^10 + 2*E^5*x^2 + x^4),x]

[Out]

(3*(E^5 + 4*x - 401*x^3))/(E^5 + x^2)

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fricas [A]  time = 0.85, size = 23, normalized size = 1.05 \begin {gather*} -\frac {3 \, {\left (401 \, x^{3} - 4 \, x - e^{5}\right )}}{x^{2} + e^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3609*x^2-6*x+12)*exp(5)-1203*x^4-12*x^2)/(exp(5)^2+2*x^2*exp(5)+x^4),x, algorithm="fricas")

[Out]

-3*(401*x^3 - 4*x - e^5)/(x^2 + e^5)

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3609*x^2-6*x+12)*exp(5)-1203*x^4-12*x^2)/(exp(5)^2+2*x^2*exp(5)+x^4),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: -3*(sqrt(exp(5)^2-exp(10))*(4*exp(5)^3-4
*exp(5)*exp(10)+exp(5)-4*exp(10))/(8*exp(10)^2-16*exp(10)*exp(5)^2+8*exp(10)*exp(5)-2*exp(10)+8*exp(5)^4-8*exp
(5)^3+2*exp(5)^2)*ln(

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maple [A]  time = 0.07, size = 22, normalized size = 1.00

method	result	size
gosper	\(\frac {12 x -1203 x^{3}+3 \,{\mathrm e}^{5}}{x^{2}+{\mathrm e}^{5}}\)	\(22\)
norman	\(\frac {12 x -1203 x^{3}+3 \,{\mathrm e}^{5}}{x^{2}+{\mathrm e}^{5}}\)	\(23\)
risch	\(-1203 x +\frac {\left (1203 \,{\mathrm e}^{5}+12\right ) x +3 \,{\mathrm e}^{5}}{x^{2}+{\mathrm e}^{5}}\)	\(27\)
default	\(-1203 x -\frac {3 \left (-\frac {{\mathrm e}^{-5} \left (802 \,{\mathrm e}^{10}+8 \,{\mathrm e}^{5}\right ) x}{2}-{\mathrm e}^{5}\right )}{x^{2}+{\mathrm e}^{5}}\)	\(68\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-3609*x^2-6*x+12)*exp(5)-1203*x^4-12*x^2)/(exp(5)^2+2*x^2*exp(5)+x^4),x,method=_RETURNVERBOSE)

[Out]

3*(-401*x^3+exp(5)+4*x)/(x^2+exp(5))

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maxima [A]  time = 0.41, size = 25, normalized size = 1.14 \begin {gather*} -1203 \, x + \frac {3 \, {\left (x {\left (401 \, e^{5} + 4\right )} + e^{5}\right )}}{x^{2} + e^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3609*x^2-6*x+12)*exp(5)-1203*x^4-12*x^2)/(exp(5)^2+2*x^2*exp(5)+x^4),x, algorithm="maxima")

[Out]

-1203*x + 3*(x*(401*e^5 + 4) + e^5)/(x^2 + e^5)

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mupad [B]  time = 0.09, size = 21, normalized size = 0.95 \begin {gather*} \frac {3\,\left (-401\,x^3+4\,x+{\mathrm {e}}^5\right )}{x^2+{\mathrm {e}}^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(5)*(6*x + 3609*x^2 - 12) + 12*x^2 + 1203*x^4)/(exp(10) + 2*x^2*exp(5) + x^4),x)

[Out]

(3*(4*x + exp(5) - 401*x^3))/(exp(5) + x^2)

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sympy [A]  time = 0.41, size = 26, normalized size = 1.18 \begin {gather*} - 1203 x - \frac {x \left (- 1203 e^{5} - 12\right ) - 3 e^{5}}{x^{2} + e^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3609*x**2-6*x+12)*exp(5)-1203*x**4-12*x**2)/(exp(5)**2+2*x**2*exp(5)+x**4),x)

[Out]

-1203*x - (x*(-1203*exp(5) - 12) - 3*exp(5))/(x**2 + exp(5))

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