3.73.97 \(\int \frac {64+e^2 (40-40 x)-64 x+56 x^2}{64+25 e^4+e^2 (80-140 x)-224 x+196 x^2} \, dx\)

Optimal. Leaf size=24 \[ \frac {x}{1-\frac {5 \left (e^2-2 x\right )}{2 (-4+2 x)}} \]

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Rubi [A]  time = 0.06, antiderivative size = 34, normalized size of antiderivative = 1.42, number of steps used = 4, number of rules used = 3, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {1984, 27, 683} \begin {gather*} \frac {2 x}{7}+\frac {5 \left (32+12 e^2-5 e^4\right )}{49 \left (-14 x+5 e^2+8\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(64 + E^2*(40 - 40*x) - 64*x + 56*x^2)/(64 + 25*E^4 + E^2*(80 - 140*x) - 224*x + 196*x^2),x]

[Out]

(5*(32 + 12*E^2 - 5*E^4))/(49*(8 + 5*E^2 - 14*x)) + (2*x)/7

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 683

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[2*c*d - b*e,
0] && IGtQ[p, 0] &&  !(EqQ[m, 3] && NeQ[p, 1])

Rule 1984

Int[(u_)^(p_.)*(v_)^(q_.), x_Symbol] :> Int[ExpandToSum[u, x]^p*ExpandToSum[v, x]^q, x] /; FreeQ[{p, q}, x] &&
QuadraticQ[{u, v}, x] &&  !QuadraticMatchQ[{u, v}, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {8 \left (8+5 e^2\right )-8 \left (8+5 e^2\right ) x+56 x^2}{\left (8+5 e^2\right )^2-28 \left (8+5 e^2\right ) x+196 x^2} \, dx\\ &=\int \frac {8 \left (8+5 e^2\right )-8 \left (8+5 e^2\right ) x+56 x^2}{\left (8+5 e^2-14 x\right )^2} \, dx\\ &=\int \left (\frac {2}{7}-\frac {10 \left (-32-12 e^2+5 e^4\right )}{7 \left (8+5 e^2-14 x\right )^2}\right ) \, dx\\ &=\frac {5 \left (32+12 e^2-5 e^4\right )}{49 \left (8+5 e^2-14 x\right )}+\frac {2 x}{7}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 40, normalized size = 1.67 \begin {gather*} -\frac {2 \left (-48+25 e^4+e^2 (10-70 x)-112 x+98 x^2\right )}{49 \left (8+5 e^2-14 x\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(64 + E^2*(40 - 40*x) - 64*x + 56*x^2)/(64 + 25*E^4 + E^2*(80 - 140*x) - 224*x + 196*x^2),x]

[Out]

(-2*(-48 + 25*E^4 + E^2*(10 - 70*x) - 112*x + 98*x^2))/(49*(8 + 5*E^2 - 14*x))

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fricas [A]  time = 0.84, size = 36, normalized size = 1.50 \begin {gather*} \frac {196 \, x^{2} - 10 \, {\left (7 \, x + 6\right )} e^{2} - 112 \, x + 25 \, e^{4} - 160}{49 \, {\left (14 \, x - 5 \, e^{2} - 8\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-40*x+40)*exp(2)+56*x^2-64*x+64)/(25*exp(2)^2+(-140*x+80)*exp(2)+196*x^2-224*x+64),x, algorithm="f
ricas")

[Out]

1/49*(196*x^2 - 10*(7*x + 6)*e^2 - 112*x + 25*e^4 - 160)/(14*x - 5*e^2 - 8)

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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(((-40*x+40)*exp(2)+56*x^2-64*x+64)/(25*exp(2)^2+(-140*x+80)*exp(2)+196*x^2-224*x+64),x, algorithm="g
iac")

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: 8*(7*sageVARx/196+(60*exp(2)-25*exp(4)+1
60)*1/3920/sqrt(exp(2)^2-exp(4))*ln(sqrt((392*sageVARx-140*exp(2)-224)^2+(-140*sqrt(-exp(2)^2+exp(4)))^2)/sqrt
((392*sageVARx-140*ex

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maple [A]  time = 11.85, size = 24, normalized size = 1.00

method	result	size
norman	\(\frac {-4 x^{2}+\frac {20 \,{\mathrm e}^{2}}{7}+\frac {32}{7}}{5 \,{\mathrm e}^{2}-14 x +8}\)	\(24\)
gosper	\(\frac {-4 x^{2}+\frac {20 \,{\mathrm e}^{2}}{7}+\frac {32}{7}}{5 \,{\mathrm e}^{2}-14 x +8}\)	\(25\)
risch	\(\frac {2 x}{7}-\frac {5 \,{\mathrm e}^{4}}{49 \left ({\mathrm e}^{2}-\frac {14 x}{5}+\frac {8}{5}\right )}+\frac {12 \,{\mathrm e}^{2}}{49 \left ({\mathrm e}^{2}-\frac {14 x}{5}+\frac {8}{5}\right )}+\frac {32}{49 \left ({\mathrm e}^{2}-\frac {14 x}{5}+\frac {8}{5}\right )}\)	\(42\)
meijerg	\(-\frac {32 x}{7 \left (-\frac {5 \,{\mathrm e}^{2}}{14}-\frac {4}{7}\right ) \left (1-\frac {14 x}{5 \,{\mathrm e}^{2}+8}\right ) \left (5 \,{\mathrm e}^{2}+8\right )}+\frac {\left (-40 \,{\mathrm e}^{2}-64\right ) \left (\frac {14 x}{\left (1-\frac {14 x}{5 \,{\mathrm e}^{2}+8}\right ) \left (5 \,{\mathrm e}^{2}+8\right )}+\ln \left (1-\frac {14 x}{5 \,{\mathrm e}^{2}+8}\right )\right )}{196}+\frac {\left (5 \,{\mathrm e}^{2}+8\right )^{2} \left (-\frac {14 x \left (-\frac {42 x}{5 \,{\mathrm e}^{2}+8}+6\right )}{3 \left (5 \,{\mathrm e}^{2}+8\right ) \left (1-\frac {14 x}{5 \,{\mathrm e}^{2}+8}\right )}-2 \ln \left (1-\frac {14 x}{5 \,{\mathrm e}^{2}+8}\right )\right )}{-245 \,{\mathrm e}^{2}-392}-\frac {20 \,{\mathrm e}^{2} x}{7 \left (-\frac {5 \,{\mathrm e}^{2}}{14}-\frac {4}{7}\right ) \left (1-\frac {14 x}{5 \,{\mathrm e}^{2}+8}\right ) \left (5 \,{\mathrm e}^{2}+8\right )}\)	\(195\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-40*x+40)*exp(2)+56*x^2-64*x+64)/(25*exp(2)^2+(-140*x+80)*exp(2)+196*x^2-224*x+64),x,method=_RETURNVERBO
SE)

[Out]

(-4*x^2+20/7*exp(2)+32/7)/(5*exp(2)-14*x+8)

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maxima [A]  time = 0.37, size = 27, normalized size = 1.12 \begin {gather*} \frac {2}{7} \, x + \frac {5 \, {\left (5 \, e^{4} - 12 \, e^{2} - 32\right )}}{49 \, {\left (14 \, x - 5 \, e^{2} - 8\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-40*x+40)*exp(2)+56*x^2-64*x+64)/(25*exp(2)^2+(-140*x+80)*exp(2)+196*x^2-224*x+64),x, algorithm="m
axima")

[Out]

2/7*x + 5/49*(5*e^4 - 12*e^2 - 32)/(14*x - 5*e^2 - 8)

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mupad [B]  time = 4.49, size = 26, normalized size = 1.08 \begin {gather*} \frac {2\,x}{7}+\frac {\frac {60\,{\mathrm {e}}^2}{49}-\frac {25\,{\mathrm {e}}^4}{49}+\frac {160}{49}}{5\,{\mathrm {e}}^2-14\,x+8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(64*x - 56*x^2 + exp(2)*(40*x - 40) - 64)/(25*exp(4) - 224*x + 196*x^2 - exp(2)*(140*x - 80) + 64),x)

[Out]

(2*x)/7 + ((60*exp(2))/49 - (25*exp(4))/49 + 160/49)/(5*exp(2) - 14*x + 8)

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sympy [A]  time = 0.25, size = 26, normalized size = 1.08 \begin {gather*} \frac {2 x}{7} + \frac {- 60 e^{2} - 160 + 25 e^{4}}{686 x - 245 e^{2} - 392} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-40*x+40)*exp(2)+56*x**2-64*x+64)/(25*exp(2)**2+(-140*x+80)*exp(2)+196*x**2-224*x+64),x)

[Out]

2*x/7 + (-60*exp(2) - 160 + 25*exp(4))/(686*x - 245*exp(2) - 392)

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