3.82.24 \(\int \frac {-11025-2940 x+1274 x^2+196 x^3-49 x^4+e^5 (63+42 x+7 x^2)+e^{2 x} (-2500 x^2+1000 x^3-100 x^4)+e^x (-10500 x+700 x^2+980 x^3-140 x^4+e^5 (-150-90 x-10 x^2+10 x^3))}{11025+2940 x-1274 x^2-196 x^3+49 x^4+e^{2 x} (2500 x^2-1000 x^3+100 x^4)+e^x (10500 x-700 x^2-980 x^3+140 x^4)} \, dx\)

Optimal. Leaf size=30 \[ -x+\frac {e^5}{(5-x) \left (7+\frac {10 e^x x}{3+x}\right )} \]

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Rubi [F]  time = 2.33, 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 {-11025-2940 x+1274 x^2+196 x^3-49 x^4+e^5 \left (63+42 x+7 x^2\right )+e^{2 x} \left (-2500 x^2+1000 x^3-100 x^4\right )+e^x \left (-10500 x+700 x^2+980 x^3-140 x^4+e^5 \left (-150-90 x-10 x^2+10 x^3\right )\right )}{11025+2940 x-1274 x^2-196 x^3+49 x^4+e^{2 x} \left (2500 x^2-1000 x^3+100 x^4\right )+e^x \left (10500 x-700 x^2-980 x^3+140 x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-11025 - 2940*x + 1274*x^2 + 196*x^3 - 49*x^4 + E^5*(63 + 42*x + 7*x^2) + E^(2*x)*(-2500*x^2 + 1000*x^3 -
100*x^4) + E^x*(-10500*x + 700*x^2 + 980*x^3 - 140*x^4 + E^5*(-150 - 90*x - 10*x^2 + 10*x^3)))/(11025 + 2940*
x - 1274*x^2 - 196*x^3 + 49*x^4 + E^(2*x)*(2500*x^2 - 1000*x^3 + 100*x^4) + E^x*(10500*x - 700*x^2 - 980*x^3 +
140*x^4)),x]

[Out]

-x - 77*E^5*Defer[Int][(21 + 7*x + 10*E^x*x)^(-2), x] - (2408*E^5*Defer[Int][1/((-5 + x)*(21 + 7*x + 10*E^x*x)
^2), x])/5 + (63*E^5*Defer[Int][1/(x*(21 + 7*x + 10*E^x*x)^2), x])/5 - 7*E^5*Defer[Int][x/(21 + 7*x + 10*E^x*x
)^2, x] + E^5*Defer[Int][(21 + 7*x + 10*E^x*x)^(-1), x] + 8*E^5*Defer[Int][1/((-5 + x)^2*(21 + 7*x + 10*E^x*x)
), x] + (48*E^5*Defer[Int][1/((-5 + x)*(21 + 7*x + 10*E^x*x)), x])/5 - (3*E^5*Defer[Int][1/(x*(21 + 7*x + 10*E
^x*x)), x])/5

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-100 e^{2 x} (-5+x)^2 x^2-140 e^x (-5+x)^2 x (3+x)+7 e^5 (3+x)^2-49 \left (-15-2 x+x^2\right )^2+10 e^{5+x} \left (-15-9 x-x^2+x^3\right )}{(5-x)^2 \left (21+\left (7+10 e^x\right ) x\right )^2} \, dx\\ &=\int \left (-1+\frac {e^5 \left (-15-9 x-x^2+x^3\right )}{(-5+x)^2 x \left (21+7 x+10 e^x x\right )}-\frac {7 e^5 \left (9+12 x+6 x^2+x^3\right )}{(-5+x) x \left (21+7 x+10 e^x x\right )^2}\right ) \, dx\\ &=-x+e^5 \int \frac {-15-9 x-x^2+x^3}{(-5+x)^2 x \left (21+7 x+10 e^x x\right )} \, dx-\left (7 e^5\right ) \int \frac {9+12 x+6 x^2+x^3}{(-5+x) x \left (21+7 x+10 e^x x\right )^2} \, dx\\ &=-x+e^5 \int \left (\frac {1}{21+7 x+10 e^x x}+\frac {8}{(-5+x)^2 \left (21+7 x+10 e^x x\right )}+\frac {48}{5 (-5+x) \left (21+7 x+10 e^x x\right )}-\frac {3}{5 x \left (21+7 x+10 e^x x\right )}\right ) \, dx-\left (7 e^5\right ) \int \left (\frac {11}{\left (21+7 x+10 e^x x\right )^2}+\frac {344}{5 (-5+x) \left (21+7 x+10 e^x x\right )^2}-\frac {9}{5 x \left (21+7 x+10 e^x x\right )^2}+\frac {x}{\left (21+7 x+10 e^x x\right )^2}\right ) \, dx\\ &=-x-\frac {1}{5} \left (3 e^5\right ) \int \frac {1}{x \left (21+7 x+10 e^x x\right )} \, dx+e^5 \int \frac {1}{21+7 x+10 e^x x} \, dx-\left (7 e^5\right ) \int \frac {x}{\left (21+7 x+10 e^x x\right )^2} \, dx+\left (8 e^5\right ) \int \frac {1}{(-5+x)^2 \left (21+7 x+10 e^x x\right )} \, dx+\frac {1}{5} \left (48 e^5\right ) \int \frac {1}{(-5+x) \left (21+7 x+10 e^x x\right )} \, dx+\frac {1}{5} \left (63 e^5\right ) \int \frac {1}{x \left (21+7 x+10 e^x x\right )^2} \, dx-\left (77 e^5\right ) \int \frac {1}{\left (21+7 x+10 e^x x\right )^2} \, dx-\frac {1}{5} \left (2408 e^5\right ) \int \frac {1}{(-5+x) \left (21+7 x+10 e^x x\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.13, size = 30, normalized size = 1.00 \begin {gather*} -x-\frac {e^5 (3+x)}{(-5+x) \left (21+7 x+10 e^x x\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-11025 - 2940*x + 1274*x^2 + 196*x^3 - 49*x^4 + E^5*(63 + 42*x + 7*x^2) + E^(2*x)*(-2500*x^2 + 1000
*x^3 - 100*x^4) + E^x*(-10500*x + 700*x^2 + 980*x^3 - 140*x^4 + E^5*(-150 - 90*x - 10*x^2 + 10*x^3)))/(11025 +
2940*x - 1274*x^2 - 196*x^3 + 49*x^4 + E^(2*x)*(2500*x^2 - 1000*x^3 + 100*x^4) + E^x*(10500*x - 700*x^2 - 980
*x^3 + 140*x^4)),x]

[Out]

-x - (E^5*(3 + x))/((-5 + x)*(21 + 7*x + 10*E^x*x))

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fricas [B]  time = 0.94, size = 58, normalized size = 1.93 \begin {gather*} -\frac {7 \, x^{3} - 14 \, x^{2} + {\left (x + 3\right )} e^{5} + 10 \, {\left (x^{3} - 5 \, x^{2}\right )} e^{x} - 105 \, x}{7 \, x^{2} + 10 \, {\left (x^{2} - 5 \, x\right )} e^{x} - 14 \, x - 105} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-100*x^4+1000*x^3-2500*x^2)*exp(x)^2+((10*x^3-10*x^2-90*x-150)*exp(5)-140*x^4+980*x^3+700*x^2-1050
0*x)*exp(x)+(7*x^2+42*x+63)*exp(5)-49*x^4+196*x^3+1274*x^2-2940*x-11025)/((100*x^4-1000*x^3+2500*x^2)*exp(x)^2
+(140*x^4-980*x^3-700*x^2+10500*x)*exp(x)+49*x^4-196*x^3-1274*x^2+2940*x+11025),x, algorithm="fricas")

[Out]

-(7*x^3 - 14*x^2 + (x + 3)*e^5 + 10*(x^3 - 5*x^2)*e^x - 105*x)/(7*x^2 + 10*(x^2 - 5*x)*e^x - 14*x - 105)

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giac [B]  time = 0.35, size = 62, normalized size = 2.07 \begin {gather*} -\frac {10 \, x^{3} e^{x} + 7 \, x^{3} - 50 \, x^{2} e^{x} - 14 \, x^{2} + x e^{5} - 105 \, x + 3 \, e^{5}}{10 \, x^{2} e^{x} + 7 \, x^{2} - 50 \, x e^{x} - 14 \, x - 105} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-100*x^4+1000*x^3-2500*x^2)*exp(x)^2+((10*x^3-10*x^2-90*x-150)*exp(5)-140*x^4+980*x^3+700*x^2-1050
0*x)*exp(x)+(7*x^2+42*x+63)*exp(5)-49*x^4+196*x^3+1274*x^2-2940*x-11025)/((100*x^4-1000*x^3+2500*x^2)*exp(x)^2
+(140*x^4-980*x^3-700*x^2+10500*x)*exp(x)+49*x^4-196*x^3-1274*x^2+2940*x+11025),x, algorithm="giac")

[Out]

-(10*x^3*e^x + 7*x^3 - 50*x^2*e^x - 14*x^2 + x*e^5 - 105*x + 3*e^5)/(10*x^2*e^x + 7*x^2 - 50*x*e^x - 14*x - 10
5)

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maple [A]  time = 0.17, size = 29, normalized size = 0.97

method	result	size
risch	\(-x -\frac {{\mathrm e}^{5} \left (3+x \right )}{\left (x -5\right ) \left (10 \,{\mathrm e}^{x} x +7 x +21\right )}\)	\(29\)
norman	\(\frac {-21 x^{2}+\left (175-{\mathrm e}^{5}\right ) x +250 \,{\mathrm e}^{x} x -7 x^{3}-10 \,{\mathrm e}^{x} x^{3}+525-3 \,{\mathrm e}^{5}}{10 \,{\mathrm e}^{x} x^{2}-50 \,{\mathrm e}^{x} x +7 x^{2}-14 x -105}\)	\(62\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-100*x^4+1000*x^3-2500*x^2)*exp(x)^2+((10*x^3-10*x^2-90*x-150)*exp(5)-140*x^4+980*x^3+700*x^2-10500*x)*e
xp(x)+(7*x^2+42*x+63)*exp(5)-49*x^4+196*x^3+1274*x^2-2940*x-11025)/((100*x^4-1000*x^3+2500*x^2)*exp(x)^2+(140*
x^4-980*x^3-700*x^2+10500*x)*exp(x)+49*x^4-196*x^3-1274*x^2+2940*x+11025),x,method=_RETURNVERBOSE)

[Out]

-x-exp(5)*(3+x)/(x-5)/(10*exp(x)*x+7*x+21)

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maxima [B]  time = 0.44, size = 59, normalized size = 1.97 \begin {gather*} -\frac {7 \, x^{3} - 14 \, x^{2} + x {\left (e^{5} - 105\right )} + 10 \, {\left (x^{3} - 5 \, x^{2}\right )} e^{x} + 3 \, e^{5}}{7 \, x^{2} + 10 \, {\left (x^{2} - 5 \, x\right )} e^{x} - 14 \, x - 105} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-100*x^4+1000*x^3-2500*x^2)*exp(x)^2+((10*x^3-10*x^2-90*x-150)*exp(5)-140*x^4+980*x^3+700*x^2-1050
0*x)*exp(x)+(7*x^2+42*x+63)*exp(5)-49*x^4+196*x^3+1274*x^2-2940*x-11025)/((100*x^4-1000*x^3+2500*x^2)*exp(x)^2
+(140*x^4-980*x^3-700*x^2+10500*x)*exp(x)+49*x^4-196*x^3-1274*x^2+2940*x+11025),x, algorithm="maxima")

[Out]

-(7*x^3 - 14*x^2 + x*(e^5 - 105) + 10*(x^3 - 5*x^2)*e^x + 3*e^5)/(7*x^2 + 10*(x^2 - 5*x)*e^x - 14*x - 105)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int -\frac {2940\,x-{\mathrm {e}}^5\,\left (7\,x^2+42\,x+63\right )+{\mathrm {e}}^x\,\left (10500\,x+{\mathrm {e}}^5\,\left (-10\,x^3+10\,x^2+90\,x+150\right )-700\,x^2-980\,x^3+140\,x^4\right )+{\mathrm {e}}^{2\,x}\,\left (100\,x^4-1000\,x^3+2500\,x^2\right )-1274\,x^2-196\,x^3+49\,x^4+11025}{2940\,x+{\mathrm {e}}^x\,\left (140\,x^4-980\,x^3-700\,x^2+10500\,x\right )+{\mathrm {e}}^{2\,x}\,\left (100\,x^4-1000\,x^3+2500\,x^2\right )-1274\,x^2-196\,x^3+49\,x^4+11025} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(2940*x - exp(5)*(42*x + 7*x^2 + 63) + exp(x)*(10500*x + exp(5)*(90*x + 10*x^2 - 10*x^3 + 150) - 700*x^2
- 980*x^3 + 140*x^4) + exp(2*x)*(2500*x^2 - 1000*x^3 + 100*x^4) - 1274*x^2 - 196*x^3 + 49*x^4 + 11025)/(2940*x
+ exp(x)*(10500*x - 700*x^2 - 980*x^3 + 140*x^4) + exp(2*x)*(2500*x^2 - 1000*x^3 + 100*x^4) - 1274*x^2 - 196*
x^3 + 49*x^4 + 11025),x)

[Out]

int(-(2940*x - exp(5)*(42*x + 7*x^2 + 63) + exp(x)*(10500*x + exp(5)*(90*x + 10*x^2 - 10*x^3 + 150) - 700*x^2
- 980*x^3 + 140*x^4) + exp(2*x)*(2500*x^2 - 1000*x^3 + 100*x^4) - 1274*x^2 - 196*x^3 + 49*x^4 + 11025)/(2940*x
+ exp(x)*(10500*x - 700*x^2 - 980*x^3 + 140*x^4) + exp(2*x)*(2500*x^2 - 1000*x^3 + 100*x^4) - 1274*x^2 - 196*
x^3 + 49*x^4 + 11025), x)

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sympy [A]  time = 0.26, size = 34, normalized size = 1.13 \begin {gather*} - x + \frac {- x e^{5} - 3 e^{5}}{7 x^{2} - 14 x + \left (10 x^{2} - 50 x\right ) e^{x} - 105} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-100*x**4+1000*x**3-2500*x**2)*exp(x)**2+((10*x**3-10*x**2-90*x-150)*exp(5)-140*x**4+980*x**3+700*
x**2-10500*x)*exp(x)+(7*x**2+42*x+63)*exp(5)-49*x**4+196*x**3+1274*x**2-2940*x-11025)/((100*x**4-1000*x**3+250
0*x**2)*exp(x)**2+(140*x**4-980*x**3-700*x**2+10500*x)*exp(x)+49*x**4-196*x**3-1274*x**2+2940*x+11025),x)

[Out]

-x + (-x*exp(5) - 3*exp(5))/(7*x**2 - 14*x + (10*x**2 - 50*x)*exp(x) - 105)

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