3.13.31 \(\int \frac {-e^{7+8 x}+e^{7+4 x} (-3000+2300 x-660 x^2+84 x^3-4 x^4)}{390625-625000 x+437500 x^2-175000 x^3+43750 x^4-7000 x^5+700 x^6-40 x^7+x^8+e^{8 x} (4+4 x+x^2)+e^{4 x} (-2500+750 x+400 x^2-220 x^3+36 x^4-2 x^5)} \, dx\)

Optimal. Leaf size=21 \[ \frac {e^7}{2-e^{-4 x} (-5+x)^4+x} \]

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Rubi [F]  time = 6.65, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {-e^{7+8 x}+e^{7+4 x} \left (-3000+2300 x-660 x^2+84 x^3-4 x^4\right )}{390625-625000 x+437500 x^2-175000 x^3+43750 x^4-7000 x^5+700 x^6-40 x^7+x^8+e^{8 x} \left (4+4 x+x^2\right )+e^{4 x} \left (-2500+750 x+400 x^2-220 x^3+36 x^4-2 x^5\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-E^(7 + 8*x) + E^(7 + 4*x)*(-3000 + 2300*x - 660*x^2 + 84*x^3 - 4*x^4))/(390625 - 625000*x + 437500*x^2 -
175000*x^3 + 43750*x^4 - 7000*x^5 + 700*x^6 - 40*x^7 + x^8 + E^(8*x)*(4 + 4*x + x^2) + E^(4*x)*(-2500 + 750*x
+ 400*x^2 - 220*x^3 + 36*x^4 - 2*x^5)),x]

[Out]

-2112*Defer[Int][E^(7 + 4*x)/((-5 + x)^4 - E^(4*x)*(2 + x))^2, x] + 2106*Defer[Int][(E^(7 + 4*x)*x)/((-5 + x)^
4 - E^(4*x)*(2 + x))^2, x] - 638*Defer[Int][(E^(7 + 4*x)*x^2)/((-5 + x)^4 - E^(4*x)*(2 + x))^2, x] + 83*Defer[
Int][(E^(7 + 4*x)*x^3)/((-5 + x)^4 - E^(4*x)*(2 + x))^2, x] - 4*Defer[Int][(E^(7 + 4*x)*x^4)/((-5 + x)^4 - E^(
4*x)*(2 + x))^2, x] - 2401*Defer[Int][E^(7 + 4*x)/((2 + x)*((-5 + x)^4 - E^(4*x)*(2 + x))^2), x] + Defer[Int][
E^(7 + 4*x)/((2 + x)*((-5 + x)^4 - E^(4*x)*(2 + x))), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{7+4 x} \left (-e^{4 x}-4 (-6+x) (-5+x)^3\right )}{\left ((-5+x)^4-e^{4 x} (2+x)\right )^2} \, dx\\ &=\int \left (-\frac {e^{7+4 x} (-5+x)^3 \left (-53-15 x+4 x^2\right )}{(2+x) \left (625-2 e^{4 x}-500 x-e^{4 x} x+150 x^2-20 x^3+x^4\right )^2}+\frac {e^{7+4 x}}{(2+x) \left (625-2 e^{4 x}-500 x-e^{4 x} x+150 x^2-20 x^3+x^4\right )}\right ) \, dx\\ &=-\int \frac {e^{7+4 x} (-5+x)^3 \left (-53-15 x+4 x^2\right )}{(2+x) \left (625-2 e^{4 x}-500 x-e^{4 x} x+150 x^2-20 x^3+x^4\right )^2} \, dx+\int \frac {e^{7+4 x}}{(2+x) \left (625-2 e^{4 x}-500 x-e^{4 x} x+150 x^2-20 x^3+x^4\right )} \, dx\\ &=-\int \frac {e^{7+4 x} (5-x)^3 \left (53+15 x-4 x^2\right )}{(2+x) \left ((-5+x)^4-e^{4 x} (2+x)\right )^2} \, dx+\int \frac {e^{7+4 x}}{(2+x) \left ((-5+x)^4-e^{4 x} (2+x)\right )} \, dx\\ &=\int \frac {e^{7+4 x}}{(2+x) \left ((-5+x)^4-e^{4 x} (2+x)\right )} \, dx-\int \left (\frac {2112 e^{7+4 x}}{\left (-625+2 e^{4 x}+500 x+e^{4 x} x-150 x^2+20 x^3-x^4\right )^2}-\frac {2106 e^{7+4 x} x}{\left (625-2 e^{4 x}-500 x-e^{4 x} x+150 x^2-20 x^3+x^4\right )^2}+\frac {638 e^{7+4 x} x^2}{\left (625-2 e^{4 x}-500 x-e^{4 x} x+150 x^2-20 x^3+x^4\right )^2}-\frac {83 e^{7+4 x} x^3}{\left (625-2 e^{4 x}-500 x-e^{4 x} x+150 x^2-20 x^3+x^4\right )^2}+\frac {4 e^{7+4 x} x^4}{\left (625-2 e^{4 x}-500 x-e^{4 x} x+150 x^2-20 x^3+x^4\right )^2}+\frac {2401 e^{7+4 x}}{(2+x) \left (625-2 e^{4 x}-500 x-e^{4 x} x+150 x^2-20 x^3+x^4\right )^2}\right ) \, dx\\ &=-\left (4 \int \frac {e^{7+4 x} x^4}{\left (625-2 e^{4 x}-500 x-e^{4 x} x+150 x^2-20 x^3+x^4\right )^2} \, dx\right )+83 \int \frac {e^{7+4 x} x^3}{\left (625-2 e^{4 x}-500 x-e^{4 x} x+150 x^2-20 x^3+x^4\right )^2} \, dx-638 \int \frac {e^{7+4 x} x^2}{\left (625-2 e^{4 x}-500 x-e^{4 x} x+150 x^2-20 x^3+x^4\right )^2} \, dx+2106 \int \frac {e^{7+4 x} x}{\left (625-2 e^{4 x}-500 x-e^{4 x} x+150 x^2-20 x^3+x^4\right )^2} \, dx-2112 \int \frac {e^{7+4 x}}{\left (-625+2 e^{4 x}+500 x+e^{4 x} x-150 x^2+20 x^3-x^4\right )^2} \, dx-2401 \int \frac {e^{7+4 x}}{(2+x) \left (625-2 e^{4 x}-500 x-e^{4 x} x+150 x^2-20 x^3+x^4\right )^2} \, dx+\int \frac {e^{7+4 x}}{(2+x) \left ((-5+x)^4-e^{4 x} (2+x)\right )} \, dx\\ &=-\left (4 \int \frac {e^{7+4 x} x^4}{\left ((-5+x)^4-e^{4 x} (2+x)\right )^2} \, dx\right )+83 \int \frac {e^{7+4 x} x^3}{\left ((-5+x)^4-e^{4 x} (2+x)\right )^2} \, dx-638 \int \frac {e^{7+4 x} x^2}{\left ((-5+x)^4-e^{4 x} (2+x)\right )^2} \, dx+2106 \int \frac {e^{7+4 x} x}{\left ((-5+x)^4-e^{4 x} (2+x)\right )^2} \, dx-2112 \int \frac {e^{7+4 x}}{\left ((-5+x)^4-e^{4 x} (2+x)\right )^2} \, dx-2401 \int \frac {e^{7+4 x}}{(2+x) \left ((-5+x)^4-e^{4 x} (2+x)\right )^2} \, dx+\int \frac {e^{7+4 x}}{(2+x) \left ((-5+x)^4-e^{4 x} (2+x)\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 1.34, size = 27, normalized size = 1.29 \begin {gather*} \frac {e^{7+4 x}}{-(-5+x)^4+e^{4 x} (2+x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-E^(7 + 8*x) + E^(7 + 4*x)*(-3000 + 2300*x - 660*x^2 + 84*x^3 - 4*x^4))/(390625 - 625000*x + 437500
*x^2 - 175000*x^3 + 43750*x^4 - 7000*x^5 + 700*x^6 - 40*x^7 + x^8 + E^(8*x)*(4 + 4*x + x^2) + E^(4*x)*(-2500 +
750*x + 400*x^2 - 220*x^3 + 36*x^4 - 2*x^5)),x]

[Out]

E^(7 + 4*x)/(-(-5 + x)^4 + E^(4*x)*(2 + x))

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fricas [B]  time = 0.69, size = 43, normalized size = 2.05 \begin {gather*} -\frac {e^{\left (4 \, x + 14\right )}}{{\left (x^{4} - 20 \, x^{3} + 150 \, x^{2} - 500 \, x + 625\right )} e^{7} - {\left (x + 2\right )} e^{\left (4 \, x + 7\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-exp(7)*exp(x)^8+(-4*x^4+84*x^3-660*x^2+2300*x-3000)*exp(7)*exp(x)^4)/((x^2+4*x+4)*exp(x)^8+(-2*x^5
+36*x^4-220*x^3+400*x^2+750*x-2500)*exp(x)^4+x^8-40*x^7+700*x^6-7000*x^5+43750*x^4-175000*x^3+437500*x^2-62500
0*x+390625),x, algorithm="fricas")

[Out]

-e^(4*x + 14)/((x^4 - 20*x^3 + 150*x^2 - 500*x + 625)*e^7 - (x + 2)*e^(4*x + 7))

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giac [A]  time = 0.39, size = 41, normalized size = 1.95 \begin {gather*} -\frac {e^{\left (4 \, x + 7\right )}}{x^{4} - 20 \, x^{3} + 150 \, x^{2} - x e^{\left (4 \, x\right )} - 500 \, x - 2 \, e^{\left (4 \, x\right )} + 625} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-exp(7)*exp(x)^8+(-4*x^4+84*x^3-660*x^2+2300*x-3000)*exp(7)*exp(x)^4)/((x^2+4*x+4)*exp(x)^8+(-2*x^5
+36*x^4-220*x^3+400*x^2+750*x-2500)*exp(x)^4+x^8-40*x^7+700*x^6-7000*x^5+43750*x^4-175000*x^3+437500*x^2-62500
0*x+390625),x, algorithm="giac")

[Out]

-e^(4*x + 7)/(x^4 - 20*x^3 + 150*x^2 - x*e^(4*x) - 500*x - 2*e^(4*x) + 625)

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maple [B]  time = 0.05, size = 70, normalized size = 3.33

method	result	size
risch	\(\frac {{\mathrm e}^{7}}{2+x}-\frac {{\mathrm e}^{7} \left (x^{4}-20 x^{3}+150 x^{2}-500 x +625\right )}{\left (2+x \right ) \left (-x \,{\mathrm e}^{4 x}+x^{4}-2 \,{\mathrm e}^{4 x}-20 x^{3}+150 x^{2}-500 x +625\right )}\)	\(70\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-exp(7)*exp(x)^8+(-4*x^4+84*x^3-660*x^2+2300*x-3000)*exp(7)*exp(x)^4)/((x^2+4*x+4)*exp(x)^8+(-2*x^5+36*x^
4-220*x^3+400*x^2+750*x-2500)*exp(x)^4+x^8-40*x^7+700*x^6-7000*x^5+43750*x^4-175000*x^3+437500*x^2-625000*x+39
0625),x,method=_RETURNVERBOSE)

[Out]

exp(7)/(2+x)-exp(7)*(x^4-20*x^3+150*x^2-500*x+625)/(2+x)/(-x*exp(4*x)+x^4-2*exp(4*x)-20*x^3+150*x^2-500*x+625)

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maxima [A]  time = 0.57, size = 37, normalized size = 1.76 \begin {gather*} -\frac {e^{\left (4 \, x + 7\right )}}{x^{4} - 20 \, x^{3} + 150 \, x^{2} - {\left (x + 2\right )} e^{\left (4 \, x\right )} - 500 \, x + 625} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-exp(7)*exp(x)^8+(-4*x^4+84*x^3-660*x^2+2300*x-3000)*exp(7)*exp(x)^4)/((x^2+4*x+4)*exp(x)^8+(-2*x^5
+36*x^4-220*x^3+400*x^2+750*x-2500)*exp(x)^4+x^8-40*x^7+700*x^6-7000*x^5+43750*x^4-175000*x^3+437500*x^2-62500
0*x+390625),x, algorithm="maxima")

[Out]

-e^(4*x + 7)/(x^4 - 20*x^3 + 150*x^2 - (x + 2)*e^(4*x) - 500*x + 625)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} \int -\frac {{\mathrm {e}}^{8\,x}\,{\mathrm {e}}^7+{\mathrm {e}}^{4\,x}\,{\mathrm {e}}^7\,\left (4\,x^4-84\,x^3+660\,x^2-2300\,x+3000\right )}{{\mathrm {e}}^{8\,x}\,\left (x^2+4\,x+4\right )-625000\,x+{\mathrm {e}}^{4\,x}\,\left (-2\,x^5+36\,x^4-220\,x^3+400\,x^2+750\,x-2500\right )+437500\,x^2-175000\,x^3+43750\,x^4-7000\,x^5+700\,x^6-40\,x^7+x^8+390625} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(8*x)*exp(7) + exp(4*x)*exp(7)*(660*x^2 - 2300*x - 84*x^3 + 4*x^4 + 3000))/(exp(8*x)*(4*x + x^2 + 4)
- 625000*x + exp(4*x)*(750*x + 400*x^2 - 220*x^3 + 36*x^4 - 2*x^5 - 2500) + 437500*x^2 - 175000*x^3 + 43750*x^
4 - 7000*x^5 + 700*x^6 - 40*x^7 + x^8 + 390625),x)

[Out]

int(-(exp(8*x)*exp(7) + exp(4*x)*exp(7)*(660*x^2 - 2300*x - 84*x^3 + 4*x^4 + 3000))/(exp(8*x)*(4*x + x^2 + 4)
- 625000*x + exp(4*x)*(750*x + 400*x^2 - 220*x^3 + 36*x^4 - 2*x^5 - 2500) + 437500*x^2 - 175000*x^3 + 43750*x^
4 - 7000*x^5 + 700*x^6 - 40*x^7 + x^8 + 390625), x)

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sympy [B]  time = 0.39, size = 78, normalized size = 3.71 \begin {gather*} \frac {x^{4} e^{7} - 20 x^{3} e^{7} + 150 x^{2} e^{7} - 500 x e^{7} + 625 e^{7}}{- x^{5} + 18 x^{4} - 110 x^{3} + 200 x^{2} + 375 x + \left (x^{2} + 4 x + 4\right ) e^{4 x} - 1250} + \frac {e^{7}}{x + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-exp(7)*exp(x)**8+(-4*x**4+84*x**3-660*x**2+2300*x-3000)*exp(7)*exp(x)**4)/((x**2+4*x+4)*exp(x)**8+
(-2*x**5+36*x**4-220*x**3+400*x**2+750*x-2500)*exp(x)**4+x**8-40*x**7+700*x**6-7000*x**5+43750*x**4-175000*x**
3+437500*x**2-625000*x+390625),x)

[Out]

(x**4*exp(7) - 20*x**3*exp(7) + 150*x**2*exp(7) - 500*x*exp(7) + 625*exp(7))/(-x**5 + 18*x**4 - 110*x**3 + 200
*x**2 + 375*x + (x**2 + 4*x + 4)*exp(4*x) - 1250) + exp(7)/(x + 2)

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