3.100.67 \(\int \frac {-120+e^{2 x} (-100-80 x-16 x^2)+e^x (220+116 x+8 x^2)}{36-60 x+25 x^2+e^x (-60+86 x-6 x^2-20 x^3)+e^{2 x} (25-30 x-11 x^2+12 x^3+4 x^4)} \, dx\)

Optimal. Leaf size=31 \[ 16+\frac {4}{1+\frac {-x+\frac {x}{-5+e^x (5+2 x)}}{x^2}} \]

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Rubi [F]  time = 3.87, 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 {-120+e^{2 x} \left (-100-80 x-16 x^2\right )+e^x \left (220+116 x+8 x^2\right )}{36-60 x+25 x^2+e^x \left (-60+86 x-6 x^2-20 x^3\right )+e^{2 x} \left (25-30 x-11 x^2+12 x^3+4 x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-120 + E^(2*x)*(-100 - 80*x - 16*x^2) + E^x*(220 + 116*x + 8*x^2))/(36 - 60*x + 25*x^2 + E^x*(-60 + 86*x
- 6*x^2 - 20*x^3) + E^(2*x)*(25 - 30*x - 11*x^2 + 12*x^3 + 4*x^4)),x]

[Out]

-4/(1 - x) + 16*Defer[Int][(6 - 5*E^x - 5*x + 3*E^x*x + 2*E^x*x^2)^(-2), x] - 4*Defer[Int][1/((-1 + x)^2*(6 -
5*E^x - 5*x + 3*E^x*x + 2*E^x*x^2)^2), x] - (64*Defer[Int][1/((-1 + x)*(6 - 5*E^x - 5*x + 3*E^x*x + 2*E^x*x^2)
^2), x])/7 + 20*Defer[Int][x/(6 - 5*E^x - 5*x + 3*E^x*x + 2*E^x*x^2)^2, x] - (740*Defer[Int][1/((5 + 2*x)*(6 -
5*E^x - 5*x + 3*E^x*x + 2*E^x*x^2)^2), x])/7 + 4*Defer[Int][(6 - 5*E^x - 5*x + 3*E^x*x + 2*E^x*x^2)^(-1), x]
+ 8*Defer[Int][1/((-1 + x)^2*(6 - 5*E^x - 5*x + 3*E^x*x + 2*E^x*x^2)), x] + (64*Defer[Int][1/((-1 + x)*(6 - 5*
E^x - 5*x + 3*E^x*x + 2*E^x*x^2)), x])/7 + (40*Defer[Int][1/((5 + 2*x)*(6 - 5*E^x - 5*x + 3*E^x*x + 2*E^x*x^2)
), x])/7

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {4 \left (-30-e^{2 x} (5+2 x)^2+e^x \left (55+29 x+2 x^2\right )\right )}{\left (6-5 x+e^x \left (-5+3 x+2 x^2\right )\right )^2} \, dx\\ &=4 \int \frac {-30-e^{2 x} (5+2 x)^2+e^x \left (55+29 x+2 x^2\right )}{\left (6-5 x+e^x \left (-5+3 x+2 x^2\right )\right )^2} \, dx\\ &=4 \int \left (-\frac {1}{(-1+x)^2}+\frac {5+7 x^2+2 x^3}{(-1+x)^2 (5+2 x) \left (6-5 e^x-5 x+3 e^x x+2 e^x x^2\right )}+\frac {x \left (37-67 x+13 x^2+10 x^3\right )}{(-1+x)^2 (5+2 x) \left (6-5 e^x-5 x+3 e^x x+2 e^x x^2\right )^2}\right ) \, dx\\ &=-\frac {4}{1-x}+4 \int \frac {5+7 x^2+2 x^3}{(-1+x)^2 (5+2 x) \left (6-5 e^x-5 x+3 e^x x+2 e^x x^2\right )} \, dx+4 \int \frac {x \left (37-67 x+13 x^2+10 x^3\right )}{(-1+x)^2 (5+2 x) \left (6-5 e^x-5 x+3 e^x x+2 e^x x^2\right )^2} \, dx\\ &=-\frac {4}{1-x}+4 \int \left (\frac {4}{\left (6-5 e^x-5 x+3 e^x x+2 e^x x^2\right )^2}-\frac {1}{(-1+x)^2 \left (6-5 e^x-5 x+3 e^x x+2 e^x x^2\right )^2}-\frac {16}{7 (-1+x) \left (6-5 e^x-5 x+3 e^x x+2 e^x x^2\right )^2}+\frac {5 x}{\left (6-5 e^x-5 x+3 e^x x+2 e^x x^2\right )^2}-\frac {185}{7 (5+2 x) \left (6-5 e^x-5 x+3 e^x x+2 e^x x^2\right )^2}\right ) \, dx+4 \int \left (\frac {1}{6-5 e^x-5 x+3 e^x x+2 e^x x^2}+\frac {2}{(-1+x)^2 \left (6-5 e^x-5 x+3 e^x x+2 e^x x^2\right )}+\frac {16}{7 (-1+x) \left (6-5 e^x-5 x+3 e^x x+2 e^x x^2\right )}+\frac {10}{7 (5+2 x) \left (6-5 e^x-5 x+3 e^x x+2 e^x x^2\right )}\right ) \, dx\\ &=-\frac {4}{1-x}-4 \int \frac {1}{(-1+x)^2 \left (6-5 e^x-5 x+3 e^x x+2 e^x x^2\right )^2} \, dx+4 \int \frac {1}{6-5 e^x-5 x+3 e^x x+2 e^x x^2} \, dx+\frac {40}{7} \int \frac {1}{(5+2 x) \left (6-5 e^x-5 x+3 e^x x+2 e^x x^2\right )} \, dx+8 \int \frac {1}{(-1+x)^2 \left (6-5 e^x-5 x+3 e^x x+2 e^x x^2\right )} \, dx-\frac {64}{7} \int \frac {1}{(-1+x) \left (6-5 e^x-5 x+3 e^x x+2 e^x x^2\right )^2} \, dx+\frac {64}{7} \int \frac {1}{(-1+x) \left (6-5 e^x-5 x+3 e^x x+2 e^x x^2\right )} \, dx+16 \int \frac {1}{\left (6-5 e^x-5 x+3 e^x x+2 e^x x^2\right )^2} \, dx+20 \int \frac {x}{\left (6-5 e^x-5 x+3 e^x x+2 e^x x^2\right )^2} \, dx-\frac {740}{7} \int \frac {1}{(5+2 x) \left (6-5 e^x-5 x+3 e^x x+2 e^x x^2\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.57, size = 35, normalized size = 1.13 \begin {gather*} -\frac {4 \left (6-e^x (5+2 x)\right )}{6-5 x+e^x \left (-5+3 x+2 x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-120 + E^(2*x)*(-100 - 80*x - 16*x^2) + E^x*(220 + 116*x + 8*x^2))/(36 - 60*x + 25*x^2 + E^x*(-60 +
86*x - 6*x^2 - 20*x^3) + E^(2*x)*(25 - 30*x - 11*x^2 + 12*x^3 + 4*x^4)),x]

[Out]

(-4*(6 - E^x*(5 + 2*x)))/(6 - 5*x + E^x*(-5 + 3*x + 2*x^2))

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fricas [A]  time = 0.62, size = 32, normalized size = 1.03 \begin {gather*} \frac {4 \, {\left ({\left (2 \, x + 5\right )} e^{x} - 6\right )}}{{\left (2 \, x^{2} + 3 \, x - 5\right )} e^{x} - 5 \, x + 6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-16*x^2-80*x-100)*exp(x)^2+(8*x^2+116*x+220)*exp(x)-120)/((4*x^4+12*x^3-11*x^2-30*x+25)*exp(x)^2+(
-20*x^3-6*x^2+86*x-60)*exp(x)+25*x^2-60*x+36),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.19, size = 36, normalized size = 1.16 \begin {gather*} \frac {4 \, {\left (2 \, x e^{x} + 5 \, e^{x} - 6\right )}}{2 \, x^{2} e^{x} + 3 \, x e^{x} - 5 \, x - 5 \, e^{x} + 6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-16*x^2-80*x-100)*exp(x)^2+(8*x^2+116*x+220)*exp(x)-120)/((4*x^4+12*x^3-11*x^2-30*x+25)*exp(x)^2+(
-20*x^3-6*x^2+86*x-60)*exp(x)+25*x^2-60*x+36),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.08, size = 36, normalized size = 1.16

method	result	size
norman	\(\frac {20 \,{\mathrm e}^{x}+8 \,{\mathrm e}^{x} x -24}{2 \,{\mathrm e}^{x} x^{2}+3 \,{\mathrm e}^{x} x -5 \,{\mathrm e}^{x}-5 x +6}\)	\(36\)
risch	\(\frac {4}{x -1}-\frac {4 x}{\left (x -1\right ) \left (2 \,{\mathrm e}^{x} x^{2}+3 \,{\mathrm e}^{x} x -5 \,{\mathrm e}^{x}-5 x +6\right )}\)	\(40\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-16*x^2-80*x-100)*exp(x)^2+(8*x^2+116*x+220)*exp(x)-120)/((4*x^4+12*x^3-11*x^2-30*x+25)*exp(x)^2+(-20*x^
3-6*x^2+86*x-60)*exp(x)+25*x^2-60*x+36),x,method=_RETURNVERBOSE)

[Out]

(20*exp(x)+8*exp(x)*x-24)/(2*exp(x)*x^2+3*exp(x)*x-5*exp(x)-5*x+6)

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maxima [A]  time = 0.40, size = 32, normalized size = 1.03 \begin {gather*} \frac {4 \, {\left ({\left (2 \, x + 5\right )} e^{x} - 6\right )}}{{\left (2 \, x^{2} + 3 \, x - 5\right )} e^{x} - 5 \, x + 6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-16*x^2-80*x-100)*exp(x)^2+(8*x^2+116*x+220)*exp(x)-120)/((4*x^4+12*x^3-11*x^2-30*x+25)*exp(x)^2+(
-20*x^3-6*x^2+86*x-60)*exp(x)+25*x^2-60*x+36),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 7.48, size = 36, normalized size = 1.16 \begin {gather*} \frac {4\,\left (5\,{\mathrm {e}}^x+2\,x\,{\mathrm {e}}^x-6\right )}{2\,x^2\,{\mathrm {e}}^x-5\,{\mathrm {e}}^x-5\,x+3\,x\,{\mathrm {e}}^x+6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(2*x)*(80*x + 16*x^2 + 100) - exp(x)*(116*x + 8*x^2 + 220) + 120)/(exp(2*x)*(12*x^3 - 11*x^2 - 30*x +
4*x^4 + 25) - 60*x + 25*x^2 - exp(x)*(6*x^2 - 86*x + 20*x^3 + 60) + 36),x)

[Out]

(4*(5*exp(x) + 2*x*exp(x) - 6))/(2*x^2*exp(x) - 5*exp(x) - 5*x + 3*x*exp(x) + 6)

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sympy [A]  time = 0.31, size = 34, normalized size = 1.10 \begin {gather*} - \frac {4 x}{- 5 x^{2} + 11 x + \left (2 x^{3} + x^{2} - 8 x + 5\right ) e^{x} - 6} + \frac {4}{x - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-16*x**2-80*x-100)*exp(x)**2+(8*x**2+116*x+220)*exp(x)-120)/((4*x**4+12*x**3-11*x**2-30*x+25)*exp(
x)**2+(-20*x**3-6*x**2+86*x-60)*exp(x)+25*x**2-60*x+36),x)

[Out]

-4*x/(-5*x**2 + 11*x + (2*x**3 + x**2 - 8*x + 5)*exp(x) - 6) + 4/(x - 1)

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