\(\int \frac {e^{40} (841 x+348 x^2-432 x^3-80 x^4+48 x^5)}{32-128 e^{20}+128 e^{40}} \, dx\) [1430]

Optimal result

Integrand size = 42, antiderivative size = 25 \[ \int \frac {e^{40} \left (841 x+348 x^2-432 x^3-80 x^4+48 x^5\right )}{32-128 e^{20}+128 e^{40}} \, dx=\frac {x^2 \left (\frac {29}{4}+x-x^2\right )^2}{\left (-4+\frac {2}{e^{20}}\right )^2} \]

[Out]

(29/4-x^2+x)^2*x^2/(2/exp(20)-4)^2

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(96\) vs. \(2(25)=50\).

Time = 0.01 (sec) , antiderivative size = 96, normalized size of antiderivative = 3.84, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.024, Rules used = {12} \[ \int \frac {e^{40} \left (841 x+348 x^2-432 x^3-80 x^4+48 x^5\right )}{32-128 e^{20}+128 e^{40}} \, dx=\frac {e^{40} x^6}{4 \left (1-2 e^{20}\right )^2}-\frac {e^{40} x^5}{2 \left (1-2 e^{20}\right )^2}-\frac {27 e^{40} x^4}{8 \left (1-2 e^{20}\right )^2}+\frac {29 e^{40} x^3}{8 \left (1-2 e^{20}\right )^2}+\frac {841 e^{40} x^2}{64 \left (1-2 e^{20}\right )^2} \]

[In]

Int[(E^40*(841*x + 348*x^2 - 432*x^3 - 80*x^4 + 48*x^5))/(32 - 128*E^20 + 128*E^40),x]

[Out]

(841*E^40*x^2)/(64*(1 - 2*E^20)^2) + (29*E^40*x^3)/(8*(1 - 2*E^20)^2) - (27*E^40*x^4)/(8*(1 - 2*E^20)^2) - (E^
40*x^5)/(2*(1 - 2*E^20)^2) + (E^40*x^6)/(4*(1 - 2*E^20)^2)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rubi steps \begin{align*} \text {integral}& = \frac {e^{40} \int \left (841 x+348 x^2-432 x^3-80 x^4+48 x^5\right ) \, dx}{32 \left (1-2 e^{20}\right )^2} \\ & = \frac {841 e^{40} x^2}{64 \left (1-2 e^{20}\right )^2}+\frac {29 e^{40} x^3}{8 \left (1-2 e^{20}\right )^2}-\frac {27 e^{40} x^4}{8 \left (1-2 e^{20}\right )^2}-\frac {e^{40} x^5}{2 \left (1-2 e^{20}\right )^2}+\frac {e^{40} x^6}{4 \left (1-2 e^{20}\right )^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.76 \[ \int \frac {e^{40} \left (841 x+348 x^2-432 x^3-80 x^4+48 x^5\right )}{32-128 e^{20}+128 e^{40}} \, dx=\frac {e^{40} \left (\frac {841 x^2}{2}+116 x^3-108 x^4-16 x^5+8 x^6\right )}{32 \left (1-2 e^{20}\right )^2} \]

[In]

Integrate[(E^40*(841*x + 348*x^2 - 432*x^3 - 80*x^4 + 48*x^5))/(32 - 128*E^20 + 128*E^40),x]

[Out]

(E^40*((841*x^2)/2 + 116*x^3 - 108*x^4 - 16*x^5 + 8*x^6))/(32*(1 - 2*E^20)^2)

Maple [A] (verified)

Time = 0.09 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.76

method	result	size
gosper	\(\frac {{\mathrm e}^{40} x^{2} \left (16 x^{4}-32 x^{3}-216 x^{2}+232 x +841\right )}{256 \,{\mathrm e}^{40}-256 \,{\mathrm e}^{20}+64}\)	\(44\)
parallelrisch	\(\frac {{\mathrm e}^{40} \left (8 x^{6}-16 x^{5}-108 x^{4}+116 x^{3}+\frac {841}{2} x^{2}\right )}{128 \,{\mathrm e}^{40}-128 \,{\mathrm e}^{20}+32}\)	\(46\)
default	\(\frac {{\mathrm e}^{40} \left (8 x^{6}-16 x^{5}-108 x^{4}+116 x^{3}+\frac {841}{2} x^{2}\right )}{128 \,{\mathrm e}^{40}-128 \,{\mathrm e}^{20}+32}\)	\(47\)
norman	\(\frac {\frac {841 \,{\mathrm e}^{40} x^{2}}{64 \left (2 \,{\mathrm e}^{20}-1\right )}+\frac {29 \,{\mathrm e}^{40} x^{3}}{8 \left (2 \,{\mathrm e}^{20}-1\right )}-\frac {27 \,{\mathrm e}^{40} x^{4}}{8 \left (2 \,{\mathrm e}^{20}-1\right )}-\frac {{\mathrm e}^{40} x^{5}}{2 \left (2 \,{\mathrm e}^{20}-1\right )}+\frac {{\mathrm e}^{40} x^{6}}{8 \,{\mathrm e}^{20}-4}}{2 \,{\mathrm e}^{20}-1}\)	\(96\)
risch	\(\frac {8 \,{\mathrm e}^{40} x^{6}}{128 \,{\mathrm e}^{40}-128 \,{\mathrm e}^{20}+32}-\frac {16 \,{\mathrm e}^{40} x^{5}}{128 \,{\mathrm e}^{40}-128 \,{\mathrm e}^{20}+32}-\frac {108 \,{\mathrm e}^{40} x^{4}}{128 \,{\mathrm e}^{40}-128 \,{\mathrm e}^{20}+32}+\frac {116 \,{\mathrm e}^{40} x^{3}}{128 \,{\mathrm e}^{40}-128 \,{\mathrm e}^{20}+32}+\frac {841 \,{\mathrm e}^{40} x^{2}}{2 \left (128 \,{\mathrm e}^{40}-128 \,{\mathrm e}^{20}+32\right )}\)	\(97\)

[In]

int((48*x^5-80*x^4-432*x^3+348*x^2+841*x)*exp(20)^2/(128*exp(20)^2-128*exp(20)+32),x,method=_RETURNVERBOSE)

[Out]

1/64*x^2*(16*x^4-32*x^3-216*x^2+232*x+841)*exp(20)^2/(4*exp(20)^2-4*exp(20)+1)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.68 \[ \int \frac {e^{40} \left (841 x+348 x^2-432 x^3-80 x^4+48 x^5\right )}{32-128 e^{20}+128 e^{40}} \, dx=\frac {{\left (16 \, x^{6} - 32 \, x^{5} - 216 \, x^{4} + 232 \, x^{3} + 841 \, x^{2}\right )} e^{40}}{64 \, {\left (4 \, e^{40} - 4 \, e^{20} + 1\right )}} \]

[In]

integrate((48*x^5-80*x^4-432*x^3+348*x^2+841*x)*exp(20)^2/(128*exp(20)^2-128*exp(20)+32),x, algorithm="fricas"
)

[Out]

1/64*(16*x^6 - 32*x^5 - 216*x^4 + 232*x^3 + 841*x^2)*e^40/(4*e^40 - 4*e^20 + 1)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 97 vs. \(2 (20) = 40\).

Time = 0.03 (sec) , antiderivative size = 97, normalized size of antiderivative = 3.88 \[ \int \frac {e^{40} \left (841 x+348 x^2-432 x^3-80 x^4+48 x^5\right )}{32-128 e^{20}+128 e^{40}} \, dx=\frac {x^{6} e^{40}}{- 16 e^{20} + 4 + 16 e^{40}} - \frac {x^{5} e^{40}}{- 8 e^{20} + 2 + 8 e^{40}} - \frac {27 x^{4} e^{40}}{- 32 e^{20} + 8 + 32 e^{40}} + \frac {29 x^{3} e^{40}}{- 32 e^{20} + 8 + 32 e^{40}} + \frac {841 x^{2} e^{40}}{- 256 e^{20} + 64 + 256 e^{40}} \]

[In]

integrate((48*x**5-80*x**4-432*x**3+348*x**2+841*x)*exp(20)**2/(128*exp(20)**2-128*exp(20)+32),x)

[Out]

x**6*exp(40)/(-16*exp(20) + 4 + 16*exp(40)) - x**5*exp(40)/(-8*exp(20) + 2 + 8*exp(40)) - 27*x**4*exp(40)/(-32
*exp(20) + 8 + 32*exp(40)) + 29*x**3*exp(40)/(-32*exp(20) + 8 + 32*exp(40)) + 841*x**2*exp(40)/(-256*exp(20) +
64 + 256*exp(40))

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.68 \[ \int \frac {e^{40} \left (841 x+348 x^2-432 x^3-80 x^4+48 x^5\right )}{32-128 e^{20}+128 e^{40}} \, dx=\frac {{\left (16 \, x^{6} - 32 \, x^{5} - 216 \, x^{4} + 232 \, x^{3} + 841 \, x^{2}\right )} e^{40}}{64 \, {\left (4 \, e^{40} - 4 \, e^{20} + 1\right )}} \]

[In]

integrate((48*x^5-80*x^4-432*x^3+348*x^2+841*x)*exp(20)^2/(128*exp(20)^2-128*exp(20)+32),x, algorithm="maxima"
)

[Out]

1/64*(16*x^6 - 32*x^5 - 216*x^4 + 232*x^3 + 841*x^2)*e^40/(4*e^40 - 4*e^20 + 1)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.68 \[ \int \frac {e^{40} \left (841 x+348 x^2-432 x^3-80 x^4+48 x^5\right )}{32-128 e^{20}+128 e^{40}} \, dx=\frac {{\left (16 \, x^{6} - 32 \, x^{5} - 216 \, x^{4} + 232 \, x^{3} + 841 \, x^{2}\right )} e^{40}}{64 \, {\left (4 \, e^{40} - 4 \, e^{20} + 1\right )}} \]

[In]

integrate((48*x^5-80*x^4-432*x^3+348*x^2+841*x)*exp(20)^2/(128*exp(20)^2-128*exp(20)+32),x, algorithm="giac")

[Out]

1/64*(16*x^6 - 32*x^5 - 216*x^4 + 232*x^3 + 841*x^2)*e^40/(4*e^40 - 4*e^20 + 1)

Mupad [B] (verification not implemented)

Time = 8.23 (sec) , antiderivative size = 76, normalized size of antiderivative = 3.04 \[ \int \frac {e^{40} \left (841 x+348 x^2-432 x^3-80 x^4+48 x^5\right )}{32-128 e^{20}+128 e^{40}} \, dx=\frac {{\mathrm {e}}^{40}\,x^6}{4\,{\left (2\,{\mathrm {e}}^{20}-1\right )}^2}-\frac {{\mathrm {e}}^{40}\,x^5}{2\,{\left (2\,{\mathrm {e}}^{20}-1\right )}^2}-\frac {27\,{\mathrm {e}}^{40}\,x^4}{8\,{\left (2\,{\mathrm {e}}^{20}-1\right )}^2}+\frac {29\,{\mathrm {e}}^{40}\,x^3}{8\,{\left (2\,{\mathrm {e}}^{20}-1\right )}^2}+\frac {841\,{\mathrm {e}}^{40}\,x^2}{64\,{\left (2\,{\mathrm {e}}^{20}-1\right )}^2} \]

[In]

int((exp(40)*(841*x + 348*x^2 - 432*x^3 - 80*x^4 + 48*x^5))/(128*exp(40) - 128*exp(20) + 32),x)

[Out]

(841*x^2*exp(40))/(64*(2*exp(20) - 1)^2) + (29*x^3*exp(40))/(8*(2*exp(20) - 1)^2) - (27*x^4*exp(40))/(8*(2*exp
(20) - 1)^2) - (x^5*exp(40))/(2*(2*exp(20) - 1)^2) + (x^6*exp(40))/(4*(2*exp(20) - 1)^2)