3.66.68 \(\int \frac {1996 x^3+e^2 (-15968+9624 x)}{996004 x^3-800396 x^4+160801 x^5} \, dx\)

Optimal. Leaf size=30 \[ \frac {\frac {4 e^2}{x^2}+x}{100 (5-2 x)+\frac {1}{2} (-2-x)} \]

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Rubi [A]  time = 0.07, antiderivative size = 39, normalized size of antiderivative = 1.30, number of steps used = 4, number of rules used = 3, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.088, Rules used = {1594, 27, 1620} \begin {gather*} \frac {4 e^2}{499 x^2}+\frac {802 e^2}{249001 x}+\frac {2 \left (248502998+64481201 e^2\right )}{99849401 (998-401 x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1996*x^3 + E^2*(-15968 + 9624*x))/(996004*x^3 - 800396*x^4 + 160801*x^5),x]

[Out]

(2*(248502998 + 64481201*E^2))/(99849401*(998 - 401*x)) + (4*E^2)/(499*x^2) + (802*E^2)/(249001*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 {1996 x^3+e^2 (-15968+9624 x)}{x^3 \left (996004-800396 x+160801 x^2\right )} \, dx\\ &=\int \frac {1996 x^3+e^2 (-15968+9624 x)}{x^3 (-998+401 x)^2} \, dx\\ &=\int \left (-\frac {8 e^2}{499 x^3}-\frac {802 e^2}{249001 x^2}+\frac {2 \left (248502998+64481201 e^2\right )}{249001 (-998+401 x)^2}\right ) \, dx\\ &=\frac {2 \left (248502998+64481201 e^2\right )}{99849401 (998-401 x)}+\frac {4 e^2}{499 x^2}+\frac {802 e^2}{249001 x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 25, normalized size = 0.83 \begin {gather*} -\frac {4 \left (802 e^2+499 x^2\right )}{401 x^2 (-998+401 x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1996*x^3 + E^2*(-15968 + 9624*x))/(996004*x^3 - 800396*x^4 + 160801*x^5),x]

[Out]

(-4*(802*E^2 + 499*x^2))/(401*x^2*(-998 + 401*x))

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fricas [A]  time = 0.84, size = 25, normalized size = 0.83 \begin {gather*} -\frac {4 \, {\left (499 \, x^{2} + 802 \, e^{2}\right )}}{401 \, {\left (401 \, x^{3} - 998 \, x^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((9624*x-15968)*exp(2)+1996*x^3)/(160801*x^5-800396*x^4+996004*x^3),x, algorithm="fricas")

[Out]

-4/401*(499*x^2 + 802*e^2)/(401*x^3 - 998*x^2)

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giac [A]  time = 0.15, size = 31, normalized size = 1.03 \begin {gather*} -\frac {2 \, {\left (64481201 \, e^{2} + 248502998\right )}}{99849401 \, {\left (401 \, x - 998\right )}} + \frac {2 \, {\left (401 \, x e^{2} + 998 \, e^{2}\right )}}{249001 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((9624*x-15968)*exp(2)+1996*x^3)/(160801*x^5-800396*x^4+996004*x^3),x, algorithm="giac")

[Out]

-2/99849401*(64481201*e^2 + 248502998)/(401*x - 998) + 2/249001*(401*x*e^2 + 998*e^2)/x^2

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maple [A]  time = 0.06, size = 22, normalized size = 0.73

method	result	size
norman	\(\frac {-\frac {1996 x^{2}}{401}-8 \,{\mathrm e}^{2}}{x^{2} \left (401 x -998\right )}\)	\(22\)
gosper	\(-\frac {4 \left (499 x^{2}+802 \,{\mathrm e}^{2}\right )}{401 x^{2} \left (401 x -998\right )}\)	\(23\)
risch	\(\frac {-\frac {1996 x^{2}}{401}-8 \,{\mathrm e}^{2}}{x^{2} \left (401 x -998\right )}\)	\(23\)
default	\(-\frac {4 \left (\frac {499}{401}+\frac {160801 \,{\mathrm e}^{2}}{498002}\right )}{401 x -998}+\frac {4 \,{\mathrm e}^{2}}{499 x^{2}}+\frac {802 \,{\mathrm e}^{2}}{249001 x}\)	\(31\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((9624*x-15968)*exp(2)+1996*x^3)/(160801*x^5-800396*x^4+996004*x^3),x,method=_RETURNVERBOSE)

[Out]

(-1996/401*x^2-8*exp(2))/x^2/(401*x-998)

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maxima [A]  time = 0.37, size = 25, normalized size = 0.83 \begin {gather*} -\frac {4 \, {\left (499 \, x^{2} + 802 \, e^{2}\right )}}{401 \, {\left (401 \, x^{3} - 998 \, x^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((9624*x-15968)*exp(2)+1996*x^3)/(160801*x^5-800396*x^4+996004*x^3),x, algorithm="maxima")

[Out]

-4/401*(499*x^2 + 802*e^2)/(401*x^3 - 998*x^2)

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mupad [B]  time = 0.09, size = 24, normalized size = 0.80 \begin {gather*} \frac {\frac {1996\,x^2}{401}+8\,{\mathrm {e}}^2}{998\,x^2-401\,x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1996*x^3 + exp(2)*(9624*x - 15968))/(996004*x^3 - 800396*x^4 + 160801*x^5),x)

[Out]

(8*exp(2) + (1996*x^2)/401)/(998*x^2 - 401*x^3)

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sympy [A]  time = 0.33, size = 20, normalized size = 0.67 \begin {gather*} \frac {- 1996 x^{2} - 3208 e^{2}}{160801 x^{3} - 400198 x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((9624*x-15968)*exp(2)+1996*x**3)/(160801*x**5-800396*x**4+996004*x**3),x)

[Out]

(-1996*x**2 - 3208*exp(2))/(160801*x**3 - 400198*x**2)

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