3.77.69 \(\int \frac {-16 e^{3 x}+625 x-375 x^2-1000 x^3+640 x^4+384 x^5-256 x^6+e^{2 x} (80+124 x-162 x^2+8 x^3)+e^x (-100-560 x+547 x^2+437 x^3-392 x^4+16 x^5)}{8 e^{3 x} x^9-1000 x^{12}+2400 x^{13}-1920 x^{14}+512 x^{15}+e^{2 x} (-120 x^{10}+96 x^{11})+e^x (600 x^{11}-960 x^{12}+384 x^{13})} \, dx\)

Optimal. Leaf size=32 \[ \frac {\left (2-\frac {-1+x}{-\frac {e^x}{5-4 x}+x}\right )^2}{16 x^8} \]

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Rubi [F]  time = 3.18, 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 {-16 e^{3 x}+625 x-375 x^2-1000 x^3+640 x^4+384 x^5-256 x^6+e^{2 x} \left (80+124 x-162 x^2+8 x^3\right )+e^x \left (-100-560 x+547 x^2+437 x^3-392 x^4+16 x^5\right )}{8 e^{3 x} x^9-1000 x^{12}+2400 x^{13}-1920 x^{14}+512 x^{15}+e^{2 x} \left (-120 x^{10}+96 x^{11}\right )+e^x \left (600 x^{11}-960 x^{12}+384 x^{13}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-16*E^(3*x) + 625*x - 375*x^2 - 1000*x^3 + 640*x^4 + 384*x^5 - 256*x^6 + E^(2*x)*(80 + 124*x - 162*x^2 +
8*x^3) + E^x*(-100 - 560*x + 547*x^2 + 437*x^3 - 392*x^4 + 16*x^5))/(8*E^(3*x)*x^9 - 1000*x^12 + 2400*x^13 - 1
920*x^14 + 512*x^15 + E^(2*x)*(-120*x^10 + 96*x^11) + E^x*(600*x^11 - 960*x^12 + 384*x^13)),x]

[Out]

1/(4*x^8) + (125*Defer[Int][1/(x^8*(E^x - 5*x + 4*x^2)^3), x])/8 - (775*Defer[Int][1/(x^7*(E^x - 5*x + 4*x^2)^
3), x])/8 + (1875*Defer[Int][1/(x^6*(E^x - 5*x + 4*x^2)^3), x])/8 - (2293*Defer[Int][1/(x^5*(E^x - 5*x + 4*x^2
)^3), x])/8 + (375*Defer[Int][1/(x^4*(E^x - 5*x + 4*x^2)^3), x])/2 - 62*Defer[Int][1/(x^3*(E^x - 5*x + 4*x^2)^
3), x] + 8*Defer[Int][1/(x^2*(E^x - 5*x + 4*x^2)^3), x] - (25*Defer[Int][1/(x^9*(E^x - 5*x + 4*x^2)^2), x])/2
+ 30*Defer[Int][1/(x^8*(E^x - 5*x + 4*x^2)^2), x] - (53*Defer[Int][1/(x^7*(E^x - 5*x + 4*x^2)^2), x])/8 - (255
*Defer[Int][1/(x^6*(E^x - 5*x + 4*x^2)^2), x])/8 + 27*Defer[Int][1/(x^5*(E^x - 5*x + 4*x^2)^2), x] - 6*Defer[I
nt][1/(x^4*(E^x - 5*x + 4*x^2)^2), x] + 10*Defer[Int][1/(x^9*(E^x - 5*x + 4*x^2)), x] - (29*Defer[Int][1/(x^8*
(E^x - 5*x + 4*x^2)), x])/2 + (15*Defer[Int][1/(x^7*(E^x - 5*x + 4*x^2)), x])/4 + Defer[Int][1/(x^6*(E^x - 5*x
+ 4*x^2)), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-16 e^{3 x}-x (-5+4 x)^3 \left (5+9 x+4 x^2\right )+2 e^{2 x} \left (40+62 x-81 x^2+4 x^3\right )+e^x \left (-100-560 x+547 x^2+437 x^3-392 x^4+16 x^5\right )}{8 x^9 \left (e^x+x (-5+4 x)\right )^3} \, dx\\ &=\frac {1}{8} \int \frac {-16 e^{3 x}-x (-5+4 x)^3 \left (5+9 x+4 x^2\right )+2 e^{2 x} \left (40+62 x-81 x^2+4 x^3\right )+e^x \left (-100-560 x+547 x^2+437 x^3-392 x^4+16 x^5\right )}{x^9 \left (e^x+x (-5+4 x)\right )^3} \, dx\\ &=\frac {1}{8} \int \left (-\frac {16}{x^9}+\frac {\left (5-13 x+4 x^2\right ) \left (5-9 x+4 x^2\right )^2}{x^8 \left (e^x-5 x+4 x^2\right )^3}+\frac {2 \left (40-58 x+15 x^2+4 x^3\right )}{x^9 \left (e^x-5 x+4 x^2\right )}-\frac {100-240 x+53 x^2+255 x^3-216 x^4+48 x^5}{x^9 \left (e^x-5 x+4 x^2\right )^2}\right ) \, dx\\ &=\frac {1}{4 x^8}+\frac {1}{8} \int \frac {\left (5-13 x+4 x^2\right ) \left (5-9 x+4 x^2\right )^2}{x^8 \left (e^x-5 x+4 x^2\right )^3} \, dx-\frac {1}{8} \int \frac {100-240 x+53 x^2+255 x^3-216 x^4+48 x^5}{x^9 \left (e^x-5 x+4 x^2\right )^2} \, dx+\frac {1}{4} \int \frac {40-58 x+15 x^2+4 x^3}{x^9 \left (e^x-5 x+4 x^2\right )} \, dx\\ &=\frac {1}{4 x^8}+\frac {1}{8} \int \left (\frac {125}{x^8 \left (e^x-5 x+4 x^2\right )^3}-\frac {775}{x^7 \left (e^x-5 x+4 x^2\right )^3}+\frac {1875}{x^6 \left (e^x-5 x+4 x^2\right )^3}-\frac {2293}{x^5 \left (e^x-5 x+4 x^2\right )^3}+\frac {1500}{x^4 \left (e^x-5 x+4 x^2\right )^3}-\frac {496}{x^3 \left (e^x-5 x+4 x^2\right )^3}+\frac {64}{x^2 \left (e^x-5 x+4 x^2\right )^3}\right ) \, dx-\frac {1}{8} \int \left (\frac {100}{x^9 \left (e^x-5 x+4 x^2\right )^2}-\frac {240}{x^8 \left (e^x-5 x+4 x^2\right )^2}+\frac {53}{x^7 \left (e^x-5 x+4 x^2\right )^2}+\frac {255}{x^6 \left (e^x-5 x+4 x^2\right )^2}-\frac {216}{x^5 \left (e^x-5 x+4 x^2\right )^2}+\frac {48}{x^4 \left (e^x-5 x+4 x^2\right )^2}\right ) \, dx+\frac {1}{4} \int \left (\frac {40}{x^9 \left (e^x-5 x+4 x^2\right )}-\frac {58}{x^8 \left (e^x-5 x+4 x^2\right )}+\frac {15}{x^7 \left (e^x-5 x+4 x^2\right )}+\frac {4}{x^6 \left (e^x-5 x+4 x^2\right )}\right ) \, dx\\ &=\frac {1}{4 x^8}+\frac {15}{4} \int \frac {1}{x^7 \left (e^x-5 x+4 x^2\right )} \, dx-6 \int \frac {1}{x^4 \left (e^x-5 x+4 x^2\right )^2} \, dx-\frac {53}{8} \int \frac {1}{x^7 \left (e^x-5 x+4 x^2\right )^2} \, dx+8 \int \frac {1}{x^2 \left (e^x-5 x+4 x^2\right )^3} \, dx+10 \int \frac {1}{x^9 \left (e^x-5 x+4 x^2\right )} \, dx-\frac {25}{2} \int \frac {1}{x^9 \left (e^x-5 x+4 x^2\right )^2} \, dx-\frac {29}{2} \int \frac {1}{x^8 \left (e^x-5 x+4 x^2\right )} \, dx+\frac {125}{8} \int \frac {1}{x^8 \left (e^x-5 x+4 x^2\right )^3} \, dx+27 \int \frac {1}{x^5 \left (e^x-5 x+4 x^2\right )^2} \, dx+30 \int \frac {1}{x^8 \left (e^x-5 x+4 x^2\right )^2} \, dx-\frac {255}{8} \int \frac {1}{x^6 \left (e^x-5 x+4 x^2\right )^2} \, dx-62 \int \frac {1}{x^3 \left (e^x-5 x+4 x^2\right )^3} \, dx-\frac {775}{8} \int \frac {1}{x^7 \left (e^x-5 x+4 x^2\right )^3} \, dx+\frac {375}{2} \int \frac {1}{x^4 \left (e^x-5 x+4 x^2\right )^3} \, dx+\frac {1875}{8} \int \frac {1}{x^6 \left (e^x-5 x+4 x^2\right )^3} \, dx-\frac {2293}{8} \int \frac {1}{x^5 \left (e^x-5 x+4 x^2\right )^3} \, dx+\int \frac {1}{x^6 \left (e^x-5 x+4 x^2\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.07, size = 35, normalized size = 1.09 \begin {gather*} \frac {\left (5-2 e^x+x-4 x^2\right )^2}{16 x^8 \left (e^x+x (-5+4 x)\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-16*E^(3*x) + 625*x - 375*x^2 - 1000*x^3 + 640*x^4 + 384*x^5 - 256*x^6 + E^(2*x)*(80 + 124*x - 162*
x^2 + 8*x^3) + E^x*(-100 - 560*x + 547*x^2 + 437*x^3 - 392*x^4 + 16*x^5))/(8*E^(3*x)*x^9 - 1000*x^12 + 2400*x^
13 - 1920*x^14 + 512*x^15 + E^(2*x)*(-120*x^10 + 96*x^11) + E^x*(600*x^11 - 960*x^12 + 384*x^13)),x]

[Out]

(5 - 2*E^x + x - 4*x^2)^2/(16*x^8*(E^x + x*(-5 + 4*x))^2)

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fricas [B]  time = 0.80, size = 83, normalized size = 2.59 \begin {gather*} \frac {16 \, x^{4} - 8 \, x^{3} - 39 \, x^{2} + 4 \, {\left (4 \, x^{2} - x - 5\right )} e^{x} + 10 \, x + 4 \, e^{\left (2 \, x\right )} + 25}{16 \, {\left (16 \, x^{12} - 40 \, x^{11} + 25 \, x^{10} + x^{8} e^{\left (2 \, x\right )} + 2 \, {\left (4 \, x^{10} - 5 \, x^{9}\right )} e^{x}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-16*exp(x)^3+(8*x^3-162*x^2+124*x+80)*exp(x)^2+(16*x^5-392*x^4+437*x^3+547*x^2-560*x-100)*exp(x)-25
6*x^6+384*x^5+640*x^4-1000*x^3-375*x^2+625*x)/(8*x^9*exp(x)^3+(96*x^11-120*x^10)*exp(x)^2+(384*x^13-960*x^12+6
00*x^11)*exp(x)+512*x^15-1920*x^14+2400*x^13-1000*x^12),x, algorithm="fricas")

[Out]

1/16*(16*x^4 - 8*x^3 - 39*x^2 + 4*(4*x^2 - x - 5)*e^x + 10*x + 4*e^(2*x) + 25)/(16*x^12 - 40*x^11 + 25*x^10 +
x^8*e^(2*x) + 2*(4*x^10 - 5*x^9)*e^x)

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giac [B]  time = 0.15, size = 84, normalized size = 2.62 \begin {gather*} \frac {16 \, x^{4} - 8 \, x^{3} + 16 \, x^{2} e^{x} - 39 \, x^{2} - 4 \, x e^{x} + 10 \, x + 4 \, e^{\left (2 \, x\right )} - 20 \, e^{x} + 25}{16 \, {\left (16 \, x^{12} - 40 \, x^{11} + 8 \, x^{10} e^{x} + 25 \, x^{10} - 10 \, x^{9} e^{x} + x^{8} e^{\left (2 \, x\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-16*exp(x)^3+(8*x^3-162*x^2+124*x+80)*exp(x)^2+(16*x^5-392*x^4+437*x^3+547*x^2-560*x-100)*exp(x)-25
6*x^6+384*x^5+640*x^4-1000*x^3-375*x^2+625*x)/(8*x^9*exp(x)^3+(96*x^11-120*x^10)*exp(x)^2+(384*x^13-960*x^12+6
00*x^11)*exp(x)+512*x^15-1920*x^14+2400*x^13-1000*x^12),x, algorithm="giac")

[Out]

1/16*(16*x^4 - 8*x^3 + 16*x^2*e^x - 39*x^2 - 4*x*e^x + 10*x + 4*e^(2*x) - 20*e^x + 25)/(16*x^12 - 40*x^11 + 8*
x^10*e^x + 25*x^10 - 10*x^9*e^x + x^8*e^(2*x))

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maple [B]  time = 0.04, size = 61, normalized size = 1.91

method	result	size
risch	\(\frac {1}{4 x^{8}}-\frac {48 x^{4}-152 x^{3}+16 \,{\mathrm e}^{x} x^{2}+139 x^{2}-36 \,{\mathrm e}^{x} x -10 x +20 \,{\mathrm e}^{x}-25}{16 x^{8} \left (4 x^{2}+{\mathrm e}^{x}-5 x \right )^{2}}\)	\(61\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-16*exp(x)^3+(8*x^3-162*x^2+124*x+80)*exp(x)^2+(16*x^5-392*x^4+437*x^3+547*x^2-560*x-100)*exp(x)-256*x^6+
384*x^5+640*x^4-1000*x^3-375*x^2+625*x)/(8*x^9*exp(x)^3+(96*x^11-120*x^10)*exp(x)^2+(384*x^13-960*x^12+600*x^1
1)*exp(x)+512*x^15-1920*x^14+2400*x^13-1000*x^12),x,method=_RETURNVERBOSE)

[Out]

1/4/x^8-1/16*(48*x^4-152*x^3+16*exp(x)*x^2+139*x^2-36*exp(x)*x-10*x+20*exp(x)-25)/x^8/(4*x^2+exp(x)-5*x)^2

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maxima [B]  time = 0.44, size = 83, normalized size = 2.59 \begin {gather*} \frac {16 \, x^{4} - 8 \, x^{3} - 39 \, x^{2} + 4 \, {\left (4 \, x^{2} - x - 5\right )} e^{x} + 10 \, x + 4 \, e^{\left (2 \, x\right )} + 25}{16 \, {\left (16 \, x^{12} - 40 \, x^{11} + 25 \, x^{10} + x^{8} e^{\left (2 \, x\right )} + 2 \, {\left (4 \, x^{10} - 5 \, x^{9}\right )} e^{x}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-16*exp(x)^3+(8*x^3-162*x^2+124*x+80)*exp(x)^2+(16*x^5-392*x^4+437*x^3+547*x^2-560*x-100)*exp(x)-25
6*x^6+384*x^5+640*x^4-1000*x^3-375*x^2+625*x)/(8*x^9*exp(x)^3+(96*x^11-120*x^10)*exp(x)^2+(384*x^13-960*x^12+6
00*x^11)*exp(x)+512*x^15-1920*x^14+2400*x^13-1000*x^12),x, algorithm="maxima")

[Out]

1/16*(16*x^4 - 8*x^3 - 39*x^2 + 4*(4*x^2 - x - 5)*e^x + 10*x + 4*e^(2*x) + 25)/(16*x^12 - 40*x^11 + 25*x^10 +
x^8*e^(2*x) + 2*(4*x^10 - 5*x^9)*e^x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} -\int \frac {16\,{\mathrm {e}}^{3\,x}-625\,x-{\mathrm {e}}^{2\,x}\,\left (8\,x^3-162\,x^2+124\,x+80\right )+{\mathrm {e}}^x\,\left (-16\,x^5+392\,x^4-437\,x^3-547\,x^2+560\,x+100\right )+375\,x^2+1000\,x^3-640\,x^4-384\,x^5+256\,x^6}{{\mathrm {e}}^x\,\left (384\,x^{13}-960\,x^{12}+600\,x^{11}\right )-{\mathrm {e}}^{2\,x}\,\left (120\,x^{10}-96\,x^{11}\right )+8\,x^9\,{\mathrm {e}}^{3\,x}-1000\,x^{12}+2400\,x^{13}-1920\,x^{14}+512\,x^{15}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(16*exp(3*x) - 625*x - exp(2*x)*(124*x - 162*x^2 + 8*x^3 + 80) + exp(x)*(560*x - 547*x^2 - 437*x^3 + 392*
x^4 - 16*x^5 + 100) + 375*x^2 + 1000*x^3 - 640*x^4 - 384*x^5 + 256*x^6)/(exp(x)*(600*x^11 - 960*x^12 + 384*x^1
3) - exp(2*x)*(120*x^10 - 96*x^11) + 8*x^9*exp(3*x) - 1000*x^12 + 2400*x^13 - 1920*x^14 + 512*x^15),x)

[Out]

-int((16*exp(3*x) - 625*x - exp(2*x)*(124*x - 162*x^2 + 8*x^3 + 80) + exp(x)*(560*x - 547*x^2 - 437*x^3 + 392*
x^4 - 16*x^5 + 100) + 375*x^2 + 1000*x^3 - 640*x^4 - 384*x^5 + 256*x^6)/(exp(x)*(600*x^11 - 960*x^12 + 384*x^1
3) - exp(2*x)*(120*x^10 - 96*x^11) + 8*x^9*exp(3*x) - 1000*x^12 + 2400*x^13 - 1920*x^14 + 512*x^15), x)

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sympy [B]  time = 0.26, size = 78, normalized size = 2.44 \begin {gather*} \frac {- 48 x^{4} + 152 x^{3} - 139 x^{2} + 10 x + \left (- 16 x^{2} + 36 x - 20\right ) e^{x} + 25}{256 x^{12} - 640 x^{11} + 400 x^{10} + 16 x^{8} e^{2 x} + \left (128 x^{10} - 160 x^{9}\right ) e^{x}} + \frac {1}{4 x^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-16*exp(x)**3+(8*x**3-162*x**2+124*x+80)*exp(x)**2+(16*x**5-392*x**4+437*x**3+547*x**2-560*x-100)*e
xp(x)-256*x**6+384*x**5+640*x**4-1000*x**3-375*x**2+625*x)/(8*x**9*exp(x)**3+(96*x**11-120*x**10)*exp(x)**2+(3
84*x**13-960*x**12+600*x**11)*exp(x)+512*x**15-1920*x**14+2400*x**13-1000*x**12),x)

[Out]

(-48*x**4 + 152*x**3 - 139*x**2 + 10*x + (-16*x**2 + 36*x - 20)*exp(x) + 25)/(256*x**12 - 640*x**11 + 400*x**1
0 + 16*x**8*exp(2*x) + (128*x**10 - 160*x**9)*exp(x)) + 1/(4*x**8)

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