3.62.11 \(\int \frac {-2 x^2-12 x^5+e^{10} (-2-12 x^3)+e^5 (-4 x-18 x^2-24 x^4)}{e^{10} x^2+2 e^5 x^3+x^4} \, dx\)

Optimal. Leaf size=24 \[ \frac {2 \left (1+x-3 x^2 \left (x+\frac {3}{e^5+x}\right )\right )}{x} \]

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Rubi [A]  time = 0.07, antiderivative size = 23, normalized size of antiderivative = 0.96, number of steps used = 4, number of rules used = 3, integrand size = 62, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {1594, 27, 1620} \begin {gather*} -6 x^2+\frac {18 e^5}{x+e^5}+\frac {2}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-2*x^2 - 12*x^5 + E^10*(-2 - 12*x^3) + E^5*(-4*x - 18*x^2 - 24*x^4))/(E^10*x^2 + 2*E^5*x^3 + x^4),x]

[Out]

2/x - 6*x^2 + (18*E^5)/(E^5 + x)

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 1594

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^
(q - p) + c*x^(r - p))^n, x] /; FreeQ[{a, b, c, p, q, r}, x] && IntegerQ[n] && PosQ[q - p] && PosQ[r - p]

Rule 1620

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)
^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2]) &&
GtQ[Expon[Px, x], 2]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-2 x^2-12 x^5+e^{10} \left (-2-12 x^3\right )+e^5 \left (-4 x-18 x^2-24 x^4\right )}{x^2 \left (e^{10}+2 e^5 x+x^2\right )} \, dx\\ &=\int \frac {-2 x^2-12 x^5+e^{10} \left (-2-12 x^3\right )+e^5 \left (-4 x-18 x^2-24 x^4\right )}{x^2 \left (e^5+x\right )^2} \, dx\\ &=\int \left (-\frac {2}{x^2}-12 x-\frac {18 e^5}{\left (e^5+x\right )^2}\right ) \, dx\\ &=\frac {2}{x}-6 x^2+\frac {18 e^5}{e^5+x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 25, normalized size = 1.04 \begin {gather*} -2 \left (-\frac {1}{x}+3 x^2-\frac {9 e^5}{e^5+x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-2*x^2 - 12*x^5 + E^10*(-2 - 12*x^3) + E^5*(-4*x - 18*x^2 - 24*x^4))/(E^10*x^2 + 2*E^5*x^3 + x^4),x
]

[Out]

-2*(-x^(-1) + 3*x^2 - (9*E^5)/(E^5 + x))

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fricas [A]  time = 0.65, size = 34, normalized size = 1.42 \begin {gather*} -\frac {2 \, {\left (3 \, x^{4} + {\left (3 \, x^{3} - 9 \, x - 1\right )} e^{5} - x\right )}}{x^{2} + x e^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-12*x^3-2)*exp(5)^2+(-24*x^4-18*x^2-4*x)*exp(5)-12*x^5-2*x^2)/(x^2*exp(5)^2+2*x^3*exp(5)+x^4),x, a
lgorithm="fricas")

[Out]

-2*(3*x^4 + (3*x^3 - 9*x - 1)*e^5 - x)/(x^2 + x*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(((-12*x^3-2)*exp(5)^2+(-24*x^4-18*x^2-4*x)*exp(5)-12*x^5-2*x^2)/(x^2*exp(5)^2+2*x^3*exp(5)+x^4),x, a
lgorithm="giac")

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: -2*(3*sageVARx^2-1/sageVARx+9*exp(5)*1/2
/sqrt(-exp(10)+exp(5)^2)*ln(abs(2*sageVARx+2*exp(5)-2*sqrt(-exp(10)+exp(5)^2))/abs(2*sageVARx+2*exp(5)+2*sqrt(
-exp(10)+exp(5)^2))))

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maple [A]  time = 0.09, size = 30, normalized size = 1.25

method	result	size
risch	\(-6 x^{2}+\frac {\left (18 \,{\mathrm e}^{5}+2\right ) x +2 \,{\mathrm e}^{5}}{\left ({\mathrm e}^{5}+x \right ) x}\)	\(30\)
norman	\(\frac {\left (18 \,{\mathrm e}^{5}+2\right ) x -6 x^{4}-6 x^{3} {\mathrm e}^{5}+2 \,{\mathrm e}^{5}}{\left ({\mathrm e}^{5}+x \right ) x}\)	\(36\)
gosper	\(-\frac {2 \left (3 x^{3} {\mathrm e}^{5}+3 x^{4}-9 x \,{\mathrm e}^{5}-{\mathrm e}^{5}-x \right )}{x \left ({\mathrm e}^{5}+x \right )}\)	\(37\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-12*x^3-2)*exp(5)^2+(-24*x^4-18*x^2-4*x)*exp(5)-12*x^5-2*x^2)/(x^2*exp(5)^2+2*x^3*exp(5)+x^4),x,method=_
RETURNVERBOSE)

[Out]

-6*x^2+((18*exp(5)+2)*x+2*exp(5))/(exp(5)+x)/x

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maxima [A]  time = 0.48, size = 29, normalized size = 1.21 \begin {gather*} -6 \, x^{2} + \frac {2 \, {\left (x {\left (9 \, e^{5} + 1\right )} + e^{5}\right )}}{x^{2} + x e^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-12*x^3-2)*exp(5)^2+(-24*x^4-18*x^2-4*x)*exp(5)-12*x^5-2*x^2)/(x^2*exp(5)^2+2*x^3*exp(5)+x^4),x, a
lgorithm="maxima")

[Out]

-6*x^2 + 2*(x*(9*e^5 + 1) + e^5)/(x^2 + x*e^5)

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mupad [B]  time = 0.20, size = 29, normalized size = 1.21 \begin {gather*} \frac {2\,{\mathrm {e}}^5+x\,\left (18\,{\mathrm {e}}^5+2\right )}{x\,\left (x+{\mathrm {e}}^5\right )}-6\,x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(10)*(12*x^3 + 2) + exp(5)*(4*x + 18*x^2 + 24*x^4) + 2*x^2 + 12*x^5)/(2*x^3*exp(5) + x^2*exp(10) + x^
4),x)

[Out]

(2*exp(5) + x*(18*exp(5) + 2))/(x*(x + exp(5))) - 6*x^2

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sympy [A]  time = 0.29, size = 29, normalized size = 1.21 \begin {gather*} - 6 x^{2} - \frac {x \left (- 18 e^{5} - 2\right ) - 2 e^{5}}{x^{2} + x e^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-12*x**3-2)*exp(5)**2+(-24*x**4-18*x**2-4*x)*exp(5)-12*x**5-2*x**2)/(x**2*exp(5)**2+2*x**3*exp(5)+
x**4),x)

[Out]

-6*x**2 - (x*(-18*exp(5) - 2) - 2*exp(5))/(x**2 + x*exp(5))

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