3.74.49 \(\int \frac {360-1800 x-744 x^2-408 x^3-528 x^4-104 x^5-32 x^6-32 x^7+e^{2 x} (-72 x^3-72 x^4-18 x^5)+e^x (828 x^2+540 x^3+120 x^4+120 x^5+48 x^6)}{900+360 x+156 x^2+264 x^3+52 x^4+16 x^5+16 x^6+e^{2 x} (36 x^2+36 x^3+9 x^4)+e^x (-360 x-252 x^2-60 x^3-60 x^4-24 x^5)} \, dx\)

Optimal. Leaf size=34 \[ -x^2+\frac {4}{-3+x^2-\left (e^x-\frac {5}{x}-\frac {x}{3}\right ) (2+x)} \]

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Rubi [F]  time = 2.78, 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 {360-1800 x-744 x^2-408 x^3-528 x^4-104 x^5-32 x^6-32 x^7+e^{2 x} \left (-72 x^3-72 x^4-18 x^5\right )+e^x \left (828 x^2+540 x^3+120 x^4+120 x^5+48 x^6\right )}{900+360 x+156 x^2+264 x^3+52 x^4+16 x^5+16 x^6+e^{2 x} \left (36 x^2+36 x^3+9 x^4\right )+e^x \left (-360 x-252 x^2-60 x^3-60 x^4-24 x^5\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(360 - 1800*x - 744*x^2 - 408*x^3 - 528*x^4 - 104*x^5 - 32*x^6 - 32*x^7 + E^(2*x)*(-72*x^3 - 72*x^4 - 18*x
^5) + E^x*(828*x^2 + 540*x^3 + 120*x^4 + 120*x^5 + 48*x^6))/(900 + 360*x + 156*x^2 + 264*x^3 + 52*x^4 + 16*x^5
+ 16*x^6 + E^(2*x)*(36*x^2 + 36*x^3 + 9*x^4) + E^x*(-360*x - 252*x^2 - 60*x^3 - 60*x^4 - 24*x^5)),x]

[Out]

-x^2 + 288*Defer[Int][(30 + 6*x - 6*E^x*x + 2*x^2 - 3*E^x*x^2 + 4*x^3)^(-2), x] + 576*Defer[Int][x/(30 + 6*x -
6*E^x*x + 2*x^2 - 3*E^x*x^2 + 4*x^3)^2, x] - 24*Defer[Int][x^2/(30 + 6*x - 6*E^x*x + 2*x^2 - 3*E^x*x^2 + 4*x^
3)^2, x] - 24*Defer[Int][x^3/(30 + 6*x - 6*E^x*x + 2*x^2 - 3*E^x*x^2 + 4*x^3)^2, x] + 48*Defer[Int][x^4/(30 +
6*x - 6*E^x*x + 2*x^2 - 3*E^x*x^2 + 4*x^3)^2, x] + 144*Defer[Int][1/((2 + x)*(30 + 6*x - 6*E^x*x + 2*x^2 - 3*E
^x*x^2 + 4*x^3)^2), x] - 12*Defer[Int][(30 + 6*x - 6*E^x*x + 2*x^2 - 3*E^x*x^2 + 4*x^3)^(-1), x] - 12*Defer[In
t][x/(30 + 6*x - 6*E^x*x + 2*x^2 - 3*E^x*x^2 + 4*x^3), x] + 24*Defer[Int][1/((2 + x)*(30 + 6*x - 6*E^x*x + 2*x
^2 - 3*E^x*x^2 + 4*x^3)), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 \left (180-900 x+6 \left (-62+69 e^x\right ) x^2-6 \left (34-45 e^x+6 e^{2 x}\right ) x^3-12 \left (22-5 e^x+3 e^{2 x}\right ) x^4-\left (52-60 e^x+9 e^{2 x}\right ) x^5+8 \left (-2+3 e^x\right ) x^6-16 x^7\right )}{\left (30-6 \left (-1+e^x\right ) x+\left (2-3 e^x\right ) x^2+4 x^3\right )^2} \, dx\\ &=2 \int \frac {180-900 x+6 \left (-62+69 e^x\right ) x^2-6 \left (34-45 e^x+6 e^{2 x}\right ) x^3-12 \left (22-5 e^x+3 e^{2 x}\right ) x^4-\left (52-60 e^x+9 e^{2 x}\right ) x^5+8 \left (-2+3 e^x\right ) x^6-16 x^7}{\left (30-6 \left (-1+e^x\right ) x+\left (2-3 e^x\right ) x^2+4 x^3\right )^2} \, dx\\ &=2 \int \left (-x-\frac {6 x (3+x)}{(2+x) \left (30+6 x-6 e^x x+2 x^2-3 e^x x^2+4 x^3\right )}+\frac {12 \left (30+60 x+22 x^2-3 x^3+3 x^4+2 x^5\right )}{(2+x) \left (30+6 x-6 e^x x+2 x^2-3 e^x x^2+4 x^3\right )^2}\right ) \, dx\\ &=-x^2-12 \int \frac {x (3+x)}{(2+x) \left (30+6 x-6 e^x x+2 x^2-3 e^x x^2+4 x^3\right )} \, dx+24 \int \frac {30+60 x+22 x^2-3 x^3+3 x^4+2 x^5}{(2+x) \left (30+6 x-6 e^x x+2 x^2-3 e^x x^2+4 x^3\right )^2} \, dx\\ &=-x^2-12 \int \left (\frac {1}{30+6 x-6 e^x x+2 x^2-3 e^x x^2+4 x^3}+\frac {x}{30+6 x-6 e^x x+2 x^2-3 e^x x^2+4 x^3}-\frac {2}{(2+x) \left (30+6 x-6 e^x x+2 x^2-3 e^x x^2+4 x^3\right )}\right ) \, dx+24 \int \left (\frac {12}{\left (30+6 x-6 e^x x+2 x^2-3 e^x x^2+4 x^3\right )^2}+\frac {24 x}{\left (30+6 x-6 e^x x+2 x^2-3 e^x x^2+4 x^3\right )^2}-\frac {x^2}{\left (30+6 x-6 e^x x+2 x^2-3 e^x x^2+4 x^3\right )^2}-\frac {x^3}{\left (30+6 x-6 e^x x+2 x^2-3 e^x x^2+4 x^3\right )^2}+\frac {2 x^4}{\left (30+6 x-6 e^x x+2 x^2-3 e^x x^2+4 x^3\right )^2}+\frac {6}{(2+x) \left (30+6 x-6 e^x x+2 x^2-3 e^x x^2+4 x^3\right )^2}\right ) \, dx\\ &=-x^2-12 \int \frac {1}{30+6 x-6 e^x x+2 x^2-3 e^x x^2+4 x^3} \, dx-12 \int \frac {x}{30+6 x-6 e^x x+2 x^2-3 e^x x^2+4 x^3} \, dx-24 \int \frac {x^2}{\left (30+6 x-6 e^x x+2 x^2-3 e^x x^2+4 x^3\right )^2} \, dx-24 \int \frac {x^3}{\left (30+6 x-6 e^x x+2 x^2-3 e^x x^2+4 x^3\right )^2} \, dx+24 \int \frac {1}{(2+x) \left (30+6 x-6 e^x x+2 x^2-3 e^x x^2+4 x^3\right )} \, dx+48 \int \frac {x^4}{\left (30+6 x-6 e^x x+2 x^2-3 e^x x^2+4 x^3\right )^2} \, dx+144 \int \frac {1}{(2+x) \left (30+6 x-6 e^x x+2 x^2-3 e^x x^2+4 x^3\right )^2} \, dx+288 \int \frac {1}{\left (30+6 x-6 e^x x+2 x^2-3 e^x x^2+4 x^3\right )^2} \, dx+576 \int \frac {x}{\left (30+6 x-6 e^x x+2 x^2-3 e^x x^2+4 x^3\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.08, size = 35, normalized size = 1.03 \begin {gather*} -x \left (x-\frac {12}{30-6 \left (-1+e^x\right ) x+\left (2-3 e^x\right ) x^2+4 x^3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(360 - 1800*x - 744*x^2 - 408*x^3 - 528*x^4 - 104*x^5 - 32*x^6 - 32*x^7 + E^(2*x)*(-72*x^3 - 72*x^4
- 18*x^5) + E^x*(828*x^2 + 540*x^3 + 120*x^4 + 120*x^5 + 48*x^6))/(900 + 360*x + 156*x^2 + 264*x^3 + 52*x^4 +
16*x^5 + 16*x^6 + E^(2*x)*(36*x^2 + 36*x^3 + 9*x^4) + E^x*(-360*x - 252*x^2 - 60*x^3 - 60*x^4 - 24*x^5)),x]

[Out]

-(x*(x - 12/(30 - 6*(-1 + E^x)*x + (2 - 3*E^x)*x^2 + 4*x^3)))

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fricas [B]  time = 1.04, size = 67, normalized size = 1.97 \begin {gather*} -\frac {4 \, x^{5} + 2 \, x^{4} + 6 \, x^{3} + 30 \, x^{2} - 3 \, {\left (x^{4} + 2 \, x^{3}\right )} e^{x} - 12 \, x}{4 \, x^{3} + 2 \, x^{2} - 3 \, {\left (x^{2} + 2 \, x\right )} e^{x} + 6 \, x + 30} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-18*x^5-72*x^4-72*x^3)*exp(x)^2+(48*x^6+120*x^5+120*x^4+540*x^3+828*x^2)*exp(x)-32*x^7-32*x^6-104*
x^5-528*x^4-408*x^3-744*x^2-1800*x+360)/((9*x^4+36*x^3+36*x^2)*exp(x)^2+(-24*x^5-60*x^4-60*x^3-252*x^2-360*x)*
exp(x)+16*x^6+16*x^5+52*x^4+264*x^3+156*x^2+360*x+900),x, algorithm="fricas")

[Out]

-(4*x^5 + 2*x^4 + 6*x^3 + 30*x^2 - 3*(x^4 + 2*x^3)*e^x - 12*x)/(4*x^3 + 2*x^2 - 3*(x^2 + 2*x)*e^x + 6*x + 30)

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giac [B]  time = 0.17, size = 69, normalized size = 2.03 \begin {gather*} -\frac {4 \, x^{5} - 3 \, x^{4} e^{x} + 2 \, x^{4} - 6 \, x^{3} e^{x} + 6 \, x^{3} + 30 \, x^{2} - 12 \, x}{4 \, x^{3} - 3 \, x^{2} e^{x} + 2 \, x^{2} - 6 \, x e^{x} + 6 \, x + 30} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-18*x^5-72*x^4-72*x^3)*exp(x)^2+(48*x^6+120*x^5+120*x^4+540*x^3+828*x^2)*exp(x)-32*x^7-32*x^6-104*
x^5-528*x^4-408*x^3-744*x^2-1800*x+360)/((9*x^4+36*x^3+36*x^2)*exp(x)^2+(-24*x^5-60*x^4-60*x^3-252*x^2-360*x)*
exp(x)+16*x^6+16*x^5+52*x^4+264*x^3+156*x^2+360*x+900),x, algorithm="giac")

[Out]

-(4*x^5 - 3*x^4*e^x + 2*x^4 - 6*x^3*e^x + 6*x^3 + 30*x^2 - 12*x)/(4*x^3 - 3*x^2*e^x + 2*x^2 - 6*x*e^x + 6*x +
30)

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maple [A]  time = 0.09, size = 39, normalized size = 1.15

method	result	size
risch	\(-x^{2}+\frac {12 x}{4 x^{3}-3 \,{\mathrm e}^{x} x^{2}+2 x^{2}-6 \,{\mathrm e}^{x} x +6 x +30}\)	\(39\)
norman	\(\frac {-30 x^{2}-6 x^{3}+12 x -2 x^{4}-4 x^{5}+6 \,{\mathrm e}^{x} x^{3}+3 \,{\mathrm e}^{x} x^{4}}{4 x^{3}-3 \,{\mathrm e}^{x} x^{2}+2 x^{2}-6 \,{\mathrm e}^{x} x +6 x +30}\)	\(69\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-18*x^5-72*x^4-72*x^3)*exp(x)^2+(48*x^6+120*x^5+120*x^4+540*x^3+828*x^2)*exp(x)-32*x^7-32*x^6-104*x^5-52
8*x^4-408*x^3-744*x^2-1800*x+360)/((9*x^4+36*x^3+36*x^2)*exp(x)^2+(-24*x^5-60*x^4-60*x^3-252*x^2-360*x)*exp(x)
+16*x^6+16*x^5+52*x^4+264*x^3+156*x^2+360*x+900),x,method=_RETURNVERBOSE)

[Out]

-x^2+12*x/(4*x^3-3*exp(x)*x^2+2*x^2-6*exp(x)*x+6*x+30)

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maxima [B]  time = 0.42, size = 67, normalized size = 1.97 \begin {gather*} -\frac {4 \, x^{5} + 2 \, x^{4} + 6 \, x^{3} + 30 \, x^{2} - 3 \, {\left (x^{4} + 2 \, x^{3}\right )} e^{x} - 12 \, x}{4 \, x^{3} + 2 \, x^{2} - 3 \, {\left (x^{2} + 2 \, x\right )} e^{x} + 6 \, x + 30} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-18*x^5-72*x^4-72*x^3)*exp(x)^2+(48*x^6+120*x^5+120*x^4+540*x^3+828*x^2)*exp(x)-32*x^7-32*x^6-104*
x^5-528*x^4-408*x^3-744*x^2-1800*x+360)/((9*x^4+36*x^3+36*x^2)*exp(x)^2+(-24*x^5-60*x^4-60*x^3-252*x^2-360*x)*
exp(x)+16*x^6+16*x^5+52*x^4+264*x^3+156*x^2+360*x+900),x, algorithm="maxima")

[Out]

-(4*x^5 + 2*x^4 + 6*x^3 + 30*x^2 - 3*(x^4 + 2*x^3)*e^x - 12*x)/(4*x^3 + 2*x^2 - 3*(x^2 + 2*x)*e^x + 6*x + 30)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int -\frac {1800\,x+{\mathrm {e}}^{2\,x}\,\left (18\,x^5+72\,x^4+72\,x^3\right )-{\mathrm {e}}^x\,\left (48\,x^6+120\,x^5+120\,x^4+540\,x^3+828\,x^2\right )+744\,x^2+408\,x^3+528\,x^4+104\,x^5+32\,x^6+32\,x^7-360}{360\,x-{\mathrm {e}}^x\,\left (24\,x^5+60\,x^4+60\,x^3+252\,x^2+360\,x\right )+{\mathrm {e}}^{2\,x}\,\left (9\,x^4+36\,x^3+36\,x^2\right )+156\,x^2+264\,x^3+52\,x^4+16\,x^5+16\,x^6+900} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(1800*x + exp(2*x)*(72*x^3 + 72*x^4 + 18*x^5) - exp(x)*(828*x^2 + 540*x^3 + 120*x^4 + 120*x^5 + 48*x^6) +
744*x^2 + 408*x^3 + 528*x^4 + 104*x^5 + 32*x^6 + 32*x^7 - 360)/(360*x - exp(x)*(360*x + 252*x^2 + 60*x^3 + 60
*x^4 + 24*x^5) + exp(2*x)*(36*x^2 + 36*x^3 + 9*x^4) + 156*x^2 + 264*x^3 + 52*x^4 + 16*x^5 + 16*x^6 + 900),x)

[Out]

int(-(1800*x + exp(2*x)*(72*x^3 + 72*x^4 + 18*x^5) - exp(x)*(828*x^2 + 540*x^3 + 120*x^4 + 120*x^5 + 48*x^6) +
744*x^2 + 408*x^3 + 528*x^4 + 104*x^5 + 32*x^6 + 32*x^7 - 360)/(360*x - exp(x)*(360*x + 252*x^2 + 60*x^3 + 60
*x^4 + 24*x^5) + exp(2*x)*(36*x^2 + 36*x^3 + 9*x^4) + 156*x^2 + 264*x^3 + 52*x^4 + 16*x^5 + 16*x^6 + 900), x)

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sympy [A]  time = 0.34, size = 34, normalized size = 1.00 \begin {gather*} - x^{2} - \frac {12 x}{- 4 x^{3} - 2 x^{2} - 6 x + \left (3 x^{2} + 6 x\right ) e^{x} - 30} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-18*x**5-72*x**4-72*x**3)*exp(x)**2+(48*x**6+120*x**5+120*x**4+540*x**3+828*x**2)*exp(x)-32*x**7-3
2*x**6-104*x**5-528*x**4-408*x**3-744*x**2-1800*x+360)/((9*x**4+36*x**3+36*x**2)*exp(x)**2+(-24*x**5-60*x**4-6
0*x**3-252*x**2-360*x)*exp(x)+16*x**6+16*x**5+52*x**4+264*x**3+156*x**2+360*x+900),x)

[Out]

-x**2 - 12*x/(-4*x**3 - 2*x**2 - 6*x + (3*x**2 + 6*x)*exp(x) - 30)

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