3.75.98 \(\int \frac {2205-13230 x+31080 x^2-36120 x^3+20900 x^4-4840 x^5+e^x (-105+430 x-645 x^2+430 x^3-110 x^4)}{441 x^2-2646 x^3+6657 x^4-8988 x^5+6868 x^6-2816 x^7+484 x^8+e^{2 x} (1-4 x+6 x^2-4 x^3+x^4)+e^x (-42 x+210 x^2-422 x^3+426 x^4-216 x^5+44 x^6)} \, dx\)

Optimal. Leaf size=30 \[ \frac {5}{-x+x^2+\frac {e^x}{21+\frac {x^2}{(1-x)^2}}} \]

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Rubi [F]  time = 2.37, 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 {2205-13230 x+31080 x^2-36120 x^3+20900 x^4-4840 x^5+e^x \left (-105+430 x-645 x^2+430 x^3-110 x^4\right )}{441 x^2-2646 x^3+6657 x^4-8988 x^5+6868 x^6-2816 x^7+484 x^8+e^{2 x} \left (1-4 x+6 x^2-4 x^3+x^4\right )+e^x \left (-42 x+210 x^2-422 x^3+426 x^4-216 x^5+44 x^6\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(2205 - 13230*x + 31080*x^2 - 36120*x^3 + 20900*x^4 - 4840*x^5 + E^x*(-105 + 430*x - 645*x^2 + 430*x^3 - 1
10*x^4))/(441*x^2 - 2646*x^3 + 6657*x^4 - 8988*x^5 + 6868*x^6 - 2816*x^7 + 484*x^8 + E^(2*x)*(1 - 4*x + 6*x^2
- 4*x^3 + x^4) + E^x*(-42*x + 210*x^2 - 422*x^3 + 426*x^4 - 216*x^5 + 44*x^6)),x]

[Out]

2215*Defer[Int][(-E^x + 21*x + E^x*x - 42*x^2 + 22*x^3)^(-2), x] + 5*Defer[Int][1/((-1 + x)^2*(-E^x + 21*x + E
^x*x - 42*x^2 + 22*x^3)^2), x] + 15*Defer[Int][1/((-1 + x)*(-E^x + 21*x + E^x*x - 42*x^2 + 22*x^3)^2), x] - 11
020*Defer[Int][x/(-E^x + 21*x + E^x*x - 42*x^2 + 22*x^3)^2, x] + 18060*Defer[Int][x^2/(-E^x + 21*x + E^x*x - 4
2*x^2 + 22*x^3)^2, x] - 11660*Defer[Int][x^3/(-E^x + 21*x + E^x*x - 42*x^2 + 22*x^3)^2, x] + 2420*Defer[Int][x
^4/(-E^x + 21*x + E^x*x - 42*x^2 + 22*x^3)^2, x] + 100*Defer[Int][(-E^x + 21*x + E^x*x - 42*x^2 + 22*x^3)^(-1)
, x] - 10*Defer[Int][1/((-1 + x)^2*(-E^x + 21*x + E^x*x - 42*x^2 + 22*x^3)), x] - 15*Defer[Int][1/((-1 + x)*(-
E^x + 21*x + E^x*x - 42*x^2 + 22*x^3)), x] - 110*Defer[Int][x/(-E^x + 21*x + E^x*x - 42*x^2 + 22*x^3), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {5 \left (-\left ((-1+2 x) \left (21-42 x+22 x^2\right )^2\right )-e^x \left (21-86 x+129 x^2-86 x^3+22 x^4\right )\right )}{(1-x)^2 \left (e^x (-1+x)+x \left (21-42 x+22 x^2\right )\right )^2} \, dx\\ &=5 \int \frac {-\left ((-1+2 x) \left (21-42 x+22 x^2\right )^2\right )-e^x \left (21-86 x+129 x^2-86 x^3+22 x^4\right )}{(1-x)^2 \left (e^x (-1+x)+x \left (21-42 x+22 x^2\right )\right )^2} \, dx\\ &=5 \int \left (-\frac {-21+65 x-64 x^2+22 x^3}{(-1+x)^2 \left (-e^x+21 x+e^x x-42 x^2+22 x^3\right )}+\frac {441-3087 x+8463 x^2-11760 x^3+8760 x^4-3300 x^5+484 x^6}{(-1+x)^2 \left (-e^x+21 x+e^x x-42 x^2+22 x^3\right )^2}\right ) \, dx\\ &=-\left (5 \int \frac {-21+65 x-64 x^2+22 x^3}{(-1+x)^2 \left (-e^x+21 x+e^x x-42 x^2+22 x^3\right )} \, dx\right )+5 \int \frac {441-3087 x+8463 x^2-11760 x^3+8760 x^4-3300 x^5+484 x^6}{(-1+x)^2 \left (-e^x+21 x+e^x x-42 x^2+22 x^3\right )^2} \, dx\\ &=5 \int \left (\frac {443}{\left (-e^x+21 x+e^x x-42 x^2+22 x^3\right )^2}+\frac {1}{(-1+x)^2 \left (-e^x+21 x+e^x x-42 x^2+22 x^3\right )^2}+\frac {3}{(-1+x) \left (-e^x+21 x+e^x x-42 x^2+22 x^3\right )^2}-\frac {2204 x}{\left (-e^x+21 x+e^x x-42 x^2+22 x^3\right )^2}+\frac {3612 x^2}{\left (-e^x+21 x+e^x x-42 x^2+22 x^3\right )^2}-\frac {2332 x^3}{\left (-e^x+21 x+e^x x-42 x^2+22 x^3\right )^2}+\frac {484 x^4}{\left (-e^x+21 x+e^x x-42 x^2+22 x^3\right )^2}\right ) \, dx-5 \int \left (-\frac {20}{-e^x+21 x+e^x x-42 x^2+22 x^3}+\frac {2}{(-1+x)^2 \left (-e^x+21 x+e^x x-42 x^2+22 x^3\right )}+\frac {3}{(-1+x) \left (-e^x+21 x+e^x x-42 x^2+22 x^3\right )}+\frac {22 x}{-e^x+21 x+e^x x-42 x^2+22 x^3}\right ) \, dx\\ &=5 \int \frac {1}{(-1+x)^2 \left (-e^x+21 x+e^x x-42 x^2+22 x^3\right )^2} \, dx-10 \int \frac {1}{(-1+x)^2 \left (-e^x+21 x+e^x x-42 x^2+22 x^3\right )} \, dx+15 \int \frac {1}{(-1+x) \left (-e^x+21 x+e^x x-42 x^2+22 x^3\right )^2} \, dx-15 \int \frac {1}{(-1+x) \left (-e^x+21 x+e^x x-42 x^2+22 x^3\right )} \, dx+100 \int \frac {1}{-e^x+21 x+e^x x-42 x^2+22 x^3} \, dx-110 \int \frac {x}{-e^x+21 x+e^x x-42 x^2+22 x^3} \, dx+2215 \int \frac {1}{\left (-e^x+21 x+e^x x-42 x^2+22 x^3\right )^2} \, dx+2420 \int \frac {x^4}{\left (-e^x+21 x+e^x x-42 x^2+22 x^3\right )^2} \, dx-11020 \int \frac {x}{\left (-e^x+21 x+e^x x-42 x^2+22 x^3\right )^2} \, dx-11660 \int \frac {x^3}{\left (-e^x+21 x+e^x x-42 x^2+22 x^3\right )^2} \, dx+18060 \int \frac {x^2}{\left (-e^x+21 x+e^x x-42 x^2+22 x^3\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.55, size = 39, normalized size = 1.30 \begin {gather*} -\frac {5 \left (-21+42 x-22 x^2\right )}{(-1+x) \left (e^x (-1+x)+x \left (21-42 x+22 x^2\right )\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2205 - 13230*x + 31080*x^2 - 36120*x^3 + 20900*x^4 - 4840*x^5 + E^x*(-105 + 430*x - 645*x^2 + 430*x
^3 - 110*x^4))/(441*x^2 - 2646*x^3 + 6657*x^4 - 8988*x^5 + 6868*x^6 - 2816*x^7 + 484*x^8 + E^(2*x)*(1 - 4*x +
6*x^2 - 4*x^3 + x^4) + E^x*(-42*x + 210*x^2 - 422*x^3 + 426*x^4 - 216*x^5 + 44*x^6)),x]

[Out]

(-5*(-21 + 42*x - 22*x^2))/((-1 + x)*(E^x*(-1 + x) + x*(21 - 42*x + 22*x^2)))

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fricas [A]  time = 0.69, size = 44, normalized size = 1.47 \begin {gather*} \frac {5 \, {\left (22 \, x^{2} - 42 \, x + 21\right )}}{22 \, x^{4} - 64 \, x^{3} + 63 \, x^{2} + {\left (x^{2} - 2 \, x + 1\right )} e^{x} - 21 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-110*x^4+430*x^3-645*x^2+430*x-105)*exp(x)-4840*x^5+20900*x^4-36120*x^3+31080*x^2-13230*x+2205)/((
x^4-4*x^3+6*x^2-4*x+1)*exp(x)^2+(44*x^6-216*x^5+426*x^4-422*x^3+210*x^2-42*x)*exp(x)+484*x^8-2816*x^7+6868*x^6
-8988*x^5+6657*x^4-2646*x^3+441*x^2),x, algorithm="fricas")

[Out]

5*(22*x^2 - 42*x + 21)/(22*x^4 - 64*x^3 + 63*x^2 + (x^2 - 2*x + 1)*e^x - 21*x)

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giac [A]  time = 0.21, size = 46, normalized size = 1.53 \begin {gather*} \frac {5 \, {\left (22 \, x^{2} - 42 \, x + 21\right )}}{22 \, x^{4} - 64 \, x^{3} + x^{2} e^{x} + 63 \, x^{2} - 2 \, x e^{x} - 21 \, x + e^{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-110*x^4+430*x^3-645*x^2+430*x-105)*exp(x)-4840*x^5+20900*x^4-36120*x^3+31080*x^2-13230*x+2205)/((
x^4-4*x^3+6*x^2-4*x+1)*exp(x)^2+(44*x^6-216*x^5+426*x^4-422*x^3+210*x^2-42*x)*exp(x)+484*x^8-2816*x^7+6868*x^6
-8988*x^5+6657*x^4-2646*x^3+441*x^2),x, algorithm="giac")

[Out]

5*(22*x^2 - 42*x + 21)/(22*x^4 - 64*x^3 + x^2*e^x + 63*x^2 - 2*x*e^x - 21*x + e^x)

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maple [A]  time = 0.17, size = 42, normalized size = 1.40

method	result	size
risch	\(\frac {110 x^{2}-210 x +105}{\left (x -1\right ) \left (22 x^{3}-42 x^{2}+{\mathrm e}^{x} x +21 x -{\mathrm e}^{x}\right )}\)	\(42\)
norman	\(\frac {110 x^{2}-210 x +105}{22 x^{4}+{\mathrm e}^{x} x^{2}-64 x^{3}-2 \,{\mathrm e}^{x} x +63 x^{2}+{\mathrm e}^{x}-21 x}\)	\(46\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-110*x^4+430*x^3-645*x^2+430*x-105)*exp(x)-4840*x^5+20900*x^4-36120*x^3+31080*x^2-13230*x+2205)/((x^4-4*
x^3+6*x^2-4*x+1)*exp(x)^2+(44*x^6-216*x^5+426*x^4-422*x^3+210*x^2-42*x)*exp(x)+484*x^8-2816*x^7+6868*x^6-8988*
x^5+6657*x^4-2646*x^3+441*x^2),x,method=_RETURNVERBOSE)

[Out]

5*(22*x^2-42*x+21)/(x-1)/(22*x^3-42*x^2+exp(x)*x+21*x-exp(x))

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maxima [A]  time = 0.45, size = 44, normalized size = 1.47 \begin {gather*} \frac {5 \, {\left (22 \, x^{2} - 42 \, x + 21\right )}}{22 \, x^{4} - 64 \, x^{3} + 63 \, x^{2} + {\left (x^{2} - 2 \, x + 1\right )} e^{x} - 21 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-110*x^4+430*x^3-645*x^2+430*x-105)*exp(x)-4840*x^5+20900*x^4-36120*x^3+31080*x^2-13230*x+2205)/((
x^4-4*x^3+6*x^2-4*x+1)*exp(x)^2+(44*x^6-216*x^5+426*x^4-422*x^3+210*x^2-42*x)*exp(x)+484*x^8-2816*x^7+6868*x^6
-8988*x^5+6657*x^4-2646*x^3+441*x^2),x, algorithm="maxima")

[Out]

5*(22*x^2 - 42*x + 21)/(22*x^4 - 64*x^3 + 63*x^2 + (x^2 - 2*x + 1)*e^x - 21*x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int -\frac {13230\,x+{\mathrm {e}}^x\,\left (110\,x^4-430\,x^3+645\,x^2-430\,x+105\right )-31080\,x^2+36120\,x^3-20900\,x^4+4840\,x^5-2205}{{\mathrm {e}}^{2\,x}\,\left (x^4-4\,x^3+6\,x^2-4\,x+1\right )-{\mathrm {e}}^x\,\left (-44\,x^6+216\,x^5-426\,x^4+422\,x^3-210\,x^2+42\,x\right )+441\,x^2-2646\,x^3+6657\,x^4-8988\,x^5+6868\,x^6-2816\,x^7+484\,x^8} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(13230*x + exp(x)*(645*x^2 - 430*x - 430*x^3 + 110*x^4 + 105) - 31080*x^2 + 36120*x^3 - 20900*x^4 + 4840*
x^5 - 2205)/(exp(2*x)*(6*x^2 - 4*x - 4*x^3 + x^4 + 1) - exp(x)*(42*x - 210*x^2 + 422*x^3 - 426*x^4 + 216*x^5 -
44*x^6) + 441*x^2 - 2646*x^3 + 6657*x^4 - 8988*x^5 + 6868*x^6 - 2816*x^7 + 484*x^8),x)

[Out]

int(-(13230*x + exp(x)*(645*x^2 - 430*x - 430*x^3 + 110*x^4 + 105) - 31080*x^2 + 36120*x^3 - 20900*x^4 + 4840*
x^5 - 2205)/(exp(2*x)*(6*x^2 - 4*x - 4*x^3 + x^4 + 1) - exp(x)*(42*x - 210*x^2 + 422*x^3 - 426*x^4 + 216*x^5 -
44*x^6) + 441*x^2 - 2646*x^3 + 6657*x^4 - 8988*x^5 + 6868*x^6 - 2816*x^7 + 484*x^8), x)

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sympy [B]  time = 0.32, size = 39, normalized size = 1.30 \begin {gather*} \frac {110 x^{2} - 210 x + 105}{22 x^{4} - 64 x^{3} + 63 x^{2} - 21 x + \left (x^{2} - 2 x + 1\right ) e^{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-110*x**4+430*x**3-645*x**2+430*x-105)*exp(x)-4840*x**5+20900*x**4-36120*x**3+31080*x**2-13230*x+2
205)/((x**4-4*x**3+6*x**2-4*x+1)*exp(x)**2+(44*x**6-216*x**5+426*x**4-422*x**3+210*x**2-42*x)*exp(x)+484*x**8-
2816*x**7+6868*x**6-8988*x**5+6657*x**4-2646*x**3+441*x**2),x)

[Out]

(110*x**2 - 210*x + 105)/(22*x**4 - 64*x**3 + 63*x**2 - 21*x + (x**2 - 2*x + 1)*exp(x))

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