3.104.15 \(\int \frac {4 x+2 e^{40} x^2+2 x^4+e^{20} (-2-4 x^3)}{e^{40} x^2-2 e^{20} x^3+x^4} \, dx\)

Optimal. Leaf size=27 \[ -\frac {e^{4/3}}{3}+2 \left (\frac {1}{\left (e^{20}-x\right ) x}+x\right ) \]

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Rubi [A]  time = 0.07, antiderivative size = 26, normalized size of antiderivative = 0.96, number of steps used = 4, number of rules used = 3, integrand size = 50, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.060, Rules used = {1594, 27, 1620} \begin {gather*} 2 x+\frac {2}{e^{20} \left (e^{20}-x\right )}+\frac {2}{e^{20} x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(4*x + 2*E^40*x^2 + 2*x^4 + E^20*(-2 - 4*x^3))/(E^40*x^2 - 2*E^20*x^3 + x^4),x]

[Out]

2/(E^20*(E^20 - x)) + 2/(E^20*x) + 2*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 {4 x+2 e^{40} x^2+2 x^4+e^{20} \left (-2-4 x^3\right )}{x^2 \left (e^{40}-2 e^{20} x+x^2\right )} \, dx\\ &=\int \frac {4 x+2 e^{40} x^2+2 x^4+e^{20} \left (-2-4 x^3\right )}{x^2 \left (-e^{20}+x\right )^2} \, dx\\ &=\int \left (2+\frac {2}{e^{20} \left (e^{20}-x\right )^2}-\frac {2}{e^{20} x^2}\right ) \, dx\\ &=\frac {2}{e^{20} \left (e^{20}-x\right )}+\frac {2}{e^{20} x}+2 x\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 25, normalized size = 0.93 \begin {gather*} 2 \left (\frac {1}{e^{20} x}+x-\frac {1}{e^{20} \left (-e^{20}+x\right )}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(4*x + 2*E^40*x^2 + 2*x^4 + E^20*(-2 - 4*x^3))/(E^40*x^2 - 2*E^20*x^3 + x^4),x]

[Out]

2*(1/(E^20*x) + x - 1/(E^20*(-E^20 + x)))

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fricas [A]  time = 0.71, size = 25, normalized size = 0.93 \begin {gather*} \frac {2 \, {\left (x^{3} - x^{2} e^{20} - 1\right )}}{x^{2} - x e^{20}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^2*exp(5)^8+(-4*x^3-2)*exp(5)^4+2*x^4+4*x)/(x^2*exp(5)^8-2*x^3*exp(5)^4+x^4),x, algorithm="frica
s")

[Out]

2*(x^3 - x^2*e^20 - 1)/(x^2 - x*e^20)

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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((2*x^2*exp(5)^8+(-4*x^3-2)*exp(5)^4+2*x^4+4*x)/(x^2*exp(5)^8-2*x^3*exp(5)^4+x^4),x, algorithm="giac"
)

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: 2*(sageVARx+exp(20)*exp(40)/exp(40)^2/sa
geVARx+(exp(20)^2-exp(40))/exp(40)^2*ln(sageVARx^2-2*sageVARx*exp(20)+exp(40))+(-2*exp(20)^3+3*exp(20)*exp(40)
)*1/2/exp(40)^2/sqrt(

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maple [A]  time = 0.09, size = 18, normalized size = 0.67

method	result	size
risch	\(2 x +\frac {2}{x \left ({\mathrm e}^{20}-x \right )}\)	\(18\)
gosper	\(\frac {-2 x^{3}+2+2 \,{\mathrm e}^{40} x}{x \left ({\mathrm e}^{20}-x \right )}\)	\(29\)
norman	\(\frac {-2 x^{3}+2+2 \,{\mathrm e}^{40} x}{x \left ({\mathrm e}^{20}-x \right )}\)	\(29\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x^2*exp(5)^8+(-4*x^3-2)*exp(5)^4+2*x^4+4*x)/(x^2*exp(5)^8-2*x^3*exp(5)^4+x^4),x,method=_RETURNVERBOSE)

[Out]

2*x+2/x/(exp(20)-x)

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maxima [A]  time = 0.35, size = 17, normalized size = 0.63 \begin {gather*} 2 \, x - \frac {2}{x^{2} - x e^{20}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^2*exp(5)^8+(-4*x^3-2)*exp(5)^4+2*x^4+4*x)/(x^2*exp(5)^8-2*x^3*exp(5)^4+x^4),x, algorithm="maxim
a")

[Out]

2*x - 2/(x^2 - x*e^20)

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mupad [B]  time = 7.12, size = 17, normalized size = 0.63 \begin {gather*} 2\,x-\frac {2}{x\,\left (x-{\mathrm {e}}^{20}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x - exp(20)*(4*x^3 + 2) + 2*x^2*exp(40) + 2*x^4)/(x^2*exp(40) - 2*x^3*exp(20) + x^4),x)

[Out]

2*x - 2/(x*(x - exp(20)))

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sympy [A]  time = 0.17, size = 12, normalized size = 0.44 \begin {gather*} 2 x - \frac {2}{x^{2} - x e^{20}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x**2*exp(5)**8+(-4*x**3-2)*exp(5)**4+2*x**4+4*x)/(x**2*exp(5)**8-2*x**3*exp(5)**4+x**4),x)

[Out]

2*x - 2/(x**2 - x*exp(20))

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