3.29.29 \(\int \frac {720-192 x-86 x^2+28 x^3+2 x^4+e^x (-180+228 x-56 x^2-12 x^3+4 x^4)}{576-336 x+x^2+14 x^3+x^4+e^{2 x} (36-24 x+4 x^2)+e^x (-288+180 x-16 x^2-4 x^3)} \, dx\)

Optimal. Leaf size=28 \[ \frac {x (5+x)}{5-e^x-\frac {3}{3-x}+\frac {x}{2}} \]

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Rubi [F]  time = 1.39, 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 {720-192 x-86 x^2+28 x^3+2 x^4+e^x \left (-180+228 x-56 x^2-12 x^3+4 x^4\right )}{576-336 x+x^2+14 x^3+x^4+e^{2 x} \left (36-24 x+4 x^2\right )+e^x \left (-288+180 x-16 x^2-4 x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(720 - 192*x - 86*x^2 + 28*x^3 + 2*x^4 + E^x*(-180 + 228*x - 56*x^2 - 12*x^3 + 4*x^4))/(576 - 336*x + x^2
+ 14*x^3 + x^4 + E^(2*x)*(36 - 24*x + 4*x^2) + E^x*(-288 + 180*x - 16*x^2 - 4*x^3)),x]

[Out]

30*Defer[Int][(24 - 6*E^x - 7*x + 2*E^x*x - x^2)^(-1), x] + 690*Defer[Int][x/(-24 + 6*E^x + 7*x - 2*E^x*x + x^
2)^2, x] - 252*Defer[Int][x^2/(-24 + 6*E^x + 7*x - 2*E^x*x + x^2)^2, x] - 48*Defer[Int][x^3/(-24 + 6*E^x + 7*x
- 2*E^x*x + x^2)^2, x] + 16*Defer[Int][x^4/(-24 + 6*E^x + 7*x - 2*E^x*x + x^2)^2, x] + 2*Defer[Int][x^5/(-24
+ 6*E^x + 7*x - 2*E^x*x + x^2)^2, x] + 28*Defer[Int][x/(-24 + 6*E^x + 7*x - 2*E^x*x + x^2), x] - 2*Defer[Int][
x^3/(-24 + 6*E^x + 7*x - 2*E^x*x + x^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 \left (360-96 x-43 x^2+14 x^3+x^4+2 e^x (-3+x)^2 \left (-5+3 x+x^2\right )\right )}{\left (24+2 e^x (-3+x)-7 x-x^2\right )^2} \, dx\\ &=2 \int \frac {360-96 x-43 x^2+14 x^3+x^4+2 e^x (-3+x)^2 \left (-5+3 x+x^2\right )}{\left (24+2 e^x (-3+x)-7 x-x^2\right )^2} \, dx\\ &=2 \int \left (\frac {15-14 x+x^3}{24-6 e^x-7 x+2 e^x x-x^2}+\frac {x \left (345-126 x-24 x^2+8 x^3+x^4\right )}{\left (-24+6 e^x+7 x-2 e^x x+x^2\right )^2}\right ) \, dx\\ &=2 \int \frac {15-14 x+x^3}{24-6 e^x-7 x+2 e^x x-x^2} \, dx+2 \int \frac {x \left (345-126 x-24 x^2+8 x^3+x^4\right )}{\left (-24+6 e^x+7 x-2 e^x x+x^2\right )^2} \, dx\\ &=2 \int \left (\frac {345 x}{\left (-24+6 e^x+7 x-2 e^x x+x^2\right )^2}-\frac {126 x^2}{\left (-24+6 e^x+7 x-2 e^x x+x^2\right )^2}-\frac {24 x^3}{\left (-24+6 e^x+7 x-2 e^x x+x^2\right )^2}+\frac {8 x^4}{\left (-24+6 e^x+7 x-2 e^x x+x^2\right )^2}+\frac {x^5}{\left (-24+6 e^x+7 x-2 e^x x+x^2\right )^2}\right ) \, dx+2 \int \left (\frac {15}{24-6 e^x-7 x+2 e^x x-x^2}+\frac {14 x}{-24+6 e^x+7 x-2 e^x x+x^2}-\frac {x^3}{-24+6 e^x+7 x-2 e^x x+x^2}\right ) \, dx\\ &=2 \int \frac {x^5}{\left (-24+6 e^x+7 x-2 e^x x+x^2\right )^2} \, dx-2 \int \frac {x^3}{-24+6 e^x+7 x-2 e^x x+x^2} \, dx+16 \int \frac {x^4}{\left (-24+6 e^x+7 x-2 e^x x+x^2\right )^2} \, dx+28 \int \frac {x}{-24+6 e^x+7 x-2 e^x x+x^2} \, dx+30 \int \frac {1}{24-6 e^x-7 x+2 e^x x-x^2} \, dx-48 \int \frac {x^3}{\left (-24+6 e^x+7 x-2 e^x x+x^2\right )^2} \, dx-252 \int \frac {x^2}{\left (-24+6 e^x+7 x-2 e^x x+x^2\right )^2} \, dx+690 \int \frac {x}{\left (-24+6 e^x+7 x-2 e^x x+x^2\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.43, size = 27, normalized size = 0.96 \begin {gather*} \frac {2 (-3+x) x (5+x)}{-24-2 e^x (-3+x)+7 x+x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(720 - 192*x - 86*x^2 + 28*x^3 + 2*x^4 + E^x*(-180 + 228*x - 56*x^2 - 12*x^3 + 4*x^4))/(576 - 336*x
+ x^2 + 14*x^3 + x^4 + E^(2*x)*(36 - 24*x + 4*x^2) + E^x*(-288 + 180*x - 16*x^2 - 4*x^3)),x]

[Out]

(2*(-3 + x)*x*(5 + x))/(-24 - 2*E^x*(-3 + x) + 7*x + x^2)

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fricas [A]  time = 0.54, size = 31, normalized size = 1.11 \begin {gather*} \frac {2 \, {\left (x^{3} + 2 \, x^{2} - 15 \, x\right )}}{x^{2} - 2 \, {\left (x - 3\right )} e^{x} + 7 \, x - 24} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^4-12*x^3-56*x^2+228*x-180)*exp(x)+2*x^4+28*x^3-86*x^2-192*x+720)/((4*x^2-24*x+36)*exp(x)^2+(-4
*x^3-16*x^2+180*x-288)*exp(x)+x^4+14*x^3+x^2-336*x+576),x, algorithm="fricas")

[Out]

2*(x^3 + 2*x^2 - 15*x)/(x^2 - 2*(x - 3)*e^x + 7*x - 24)

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giac [A]  time = 0.36, size = 33, normalized size = 1.18 \begin {gather*} \frac {2 \, {\left (x^{3} + 2 \, x^{2} - 15 \, x\right )}}{x^{2} - 2 \, x e^{x} + 7 \, x + 6 \, e^{x} - 24} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^4-12*x^3-56*x^2+228*x-180)*exp(x)+2*x^4+28*x^3-86*x^2-192*x+720)/((4*x^2-24*x+36)*exp(x)^2+(-4
*x^3-16*x^2+180*x-288)*exp(x)+x^4+14*x^3+x^2-336*x+576),x, algorithm="giac")

[Out]

2*(x^3 + 2*x^2 - 15*x)/(x^2 - 2*x*e^x + 7*x + 6*e^x - 24)

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maple [A]  time = 0.07, size = 29, normalized size = 1.04

method	result	size
risch	\(\frac {2 \left (x -3\right ) \left (5+x \right ) x}{x^{2}-2 \,{\mathrm e}^{x} x +7 x +6 \,{\mathrm e}^{x}-24}\)	\(29\)
norman	\(\frac {2 x^{3}+4 x^{2}-30 x}{x^{2}-2 \,{\mathrm e}^{x} x +7 x +6 \,{\mathrm e}^{x}-24}\)	\(35\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((4*x^4-12*x^3-56*x^2+228*x-180)*exp(x)+2*x^4+28*x^3-86*x^2-192*x+720)/((4*x^2-24*x+36)*exp(x)^2+(-4*x^3-1
6*x^2+180*x-288)*exp(x)+x^4+14*x^3+x^2-336*x+576),x,method=_RETURNVERBOSE)

[Out]

2*(x-3)*(5+x)*x/(x^2-2*exp(x)*x+7*x+6*exp(x)-24)

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maxima [A]  time = 0.49, size = 31, normalized size = 1.11 \begin {gather*} \frac {2 \, {\left (x^{3} + 2 \, x^{2} - 15 \, x\right )}}{x^{2} - 2 \, {\left (x - 3\right )} e^{x} + 7 \, x - 24} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^4-12*x^3-56*x^2+228*x-180)*exp(x)+2*x^4+28*x^3-86*x^2-192*x+720)/((4*x^2-24*x+36)*exp(x)^2+(-4
*x^3-16*x^2+180*x-288)*exp(x)+x^4+14*x^3+x^2-336*x+576),x, algorithm="maxima")

[Out]

2*(x^3 + 2*x^2 - 15*x)/(x^2 - 2*(x - 3)*e^x + 7*x - 24)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} -\int \frac {192\,x+{\mathrm {e}}^x\,\left (-4\,x^4+12\,x^3+56\,x^2-228\,x+180\right )+86\,x^2-28\,x^3-2\,x^4-720}{{\mathrm {e}}^{2\,x}\,\left (4\,x^2-24\,x+36\right )-336\,x+x^2+14\,x^3+x^4-{\mathrm {e}}^x\,\left (4\,x^3+16\,x^2-180\,x+288\right )+576} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(192*x + exp(x)*(56*x^2 - 228*x + 12*x^3 - 4*x^4 + 180) + 86*x^2 - 28*x^3 - 2*x^4 - 720)/(exp(2*x)*(4*x^2
- 24*x + 36) - 336*x + x^2 + 14*x^3 + x^4 - exp(x)*(16*x^2 - 180*x + 4*x^3 + 288) + 576),x)

[Out]

-int((192*x + exp(x)*(56*x^2 - 228*x + 12*x^3 - 4*x^4 + 180) + 86*x^2 - 28*x^3 - 2*x^4 - 720)/(exp(2*x)*(4*x^2
- 24*x + 36) - 336*x + x^2 + 14*x^3 + x^4 - exp(x)*(16*x^2 - 180*x + 4*x^3 + 288) + 576), x)

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sympy [A]  time = 0.24, size = 29, normalized size = 1.04 \begin {gather*} \frac {- 2 x^{3} - 4 x^{2} + 30 x}{- x^{2} - 7 x + \left (2 x - 6\right ) e^{x} + 24} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x**4-12*x**3-56*x**2+228*x-180)*exp(x)+2*x**4+28*x**3-86*x**2-192*x+720)/((4*x**2-24*x+36)*exp(x
)**2+(-4*x**3-16*x**2+180*x-288)*exp(x)+x**4+14*x**3+x**2-336*x+576),x)

[Out]

(-2*x**3 - 4*x**2 + 30*x)/(-x**2 - 7*x + (2*x - 6)*exp(x) + 24)

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