3.78.9 \(\int \frac {108 x+198 x^2+96 x^3+e^{2 x} (-54 x^2-441 x^3-696 x^4-192 x^5)}{288+1536 x+2048 x^2+e^{2 x} (720 x+3840 x^2+5120 x^3)+e^{4 x} (720 x^2+3840 x^3+5120 x^4)+e^{6 x} (360 x^3+1920 x^4+2560 x^5)+e^{8 x} (90 x^4+480 x^5+640 x^6)+e^{10 x} (9 x^5+48 x^6+64 x^7)} \, dx\)

Optimal. Leaf size=26 \[ \frac {3 x (3+x)}{\left (8+\frac {3}{x}\right ) \left (2+e^{2 x} x\right )^4} \]

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Rubi [F]  time = 1.88, 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 {108 x+198 x^2+96 x^3+e^{2 x} \left (-54 x^2-441 x^3-696 x^4-192 x^5\right )}{288+1536 x+2048 x^2+e^{2 x} \left (720 x+3840 x^2+5120 x^3\right )+e^{4 x} \left (720 x^2+3840 x^3+5120 x^4\right )+e^{6 x} \left (360 x^3+1920 x^4+2560 x^5\right )+e^{8 x} \left (90 x^4+480 x^5+640 x^6\right )+e^{10 x} \left (9 x^5+48 x^6+64 x^7\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(108*x + 198*x^2 + 96*x^3 + E^(2*x)*(-54*x^2 - 441*x^3 - 696*x^4 - 192*x^5))/(288 + 1536*x + 2048*x^2 + E^
(2*x)*(720*x + 3840*x^2 + 5120*x^3) + E^(4*x)*(720*x^2 + 3840*x^3 + 5120*x^4) + E^(6*x)*(360*x^3 + 1920*x^4 +
2560*x^5) + E^(8*x)*(90*x^4 + 480*x^5 + 640*x^6) + E^(10*x)*(9*x^5 + 48*x^6 + 64*x^7)),x]

[Out]

(63*Defer[Int][(2 + E^(2*x)*x)^(-5), x])/32 + (75*Defer[Int][x/(2 + E^(2*x)*x)^5, x])/4 + 6*Defer[Int][x^2/(2
+ E^(2*x)*x)^5, x] - (189*Defer[Int][1/((3 + 8*x)*(2 + E^(2*x)*x)^5), x])/32 - (69*Defer[Int][x/(2 + E^(2*x)*x
)^4, x])/8 - 3*Defer[Int][x^2/(2 + E^(2*x)*x)^4, x] - (567*Defer[Int][1/((3 + 8*x)^2*(2 + E^(2*x)*x)^4), x])/6
4 + (189*Defer[Int][1/((3 + 8*x)*(2 + E^(2*x)*x)^4), x])/64

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {3 x \left (36-6 \left (-11+3 e^{2 x}\right ) x-\left (-32+147 e^{2 x}\right ) x^2-232 e^{2 x} x^3-64 e^{2 x} x^4\right )}{(3+8 x)^2 \left (2+e^{2 x} x\right )^5} \, dx\\ &=3 \int \frac {x \left (36-6 \left (-11+3 e^{2 x}\right ) x-\left (-32+147 e^{2 x}\right ) x^2-232 e^{2 x} x^3-64 e^{2 x} x^4\right )}{(3+8 x)^2 \left (2+e^{2 x} x\right )^5} \, dx\\ &=3 \int \left (\frac {8 x \left (3+7 x+2 x^2\right )}{(3+8 x) \left (2+e^{2 x} x\right )^5}-\frac {x \left (18+147 x+232 x^2+64 x^3\right )}{(3+8 x)^2 \left (2+e^{2 x} x\right )^4}\right ) \, dx\\ &=-\left (3 \int \frac {x \left (18+147 x+232 x^2+64 x^3\right )}{(3+8 x)^2 \left (2+e^{2 x} x\right )^4} \, dx\right )+24 \int \frac {x \left (3+7 x+2 x^2\right )}{(3+8 x) \left (2+e^{2 x} x\right )^5} \, dx\\ &=-\left (3 \int \left (\frac {23 x}{8 \left (2+e^{2 x} x\right )^4}+\frac {x^2}{\left (2+e^{2 x} x\right )^4}+\frac {189}{64 (3+8 x)^2 \left (2+e^{2 x} x\right )^4}-\frac {63}{64 (3+8 x) \left (2+e^{2 x} x\right )^4}\right ) \, dx\right )+24 \int \left (\frac {21}{256 \left (2+e^{2 x} x\right )^5}+\frac {25 x}{32 \left (2+e^{2 x} x\right )^5}+\frac {x^2}{4 \left (2+e^{2 x} x\right )^5}-\frac {63}{256 (3+8 x) \left (2+e^{2 x} x\right )^5}\right ) \, dx\\ &=\frac {63}{32} \int \frac {1}{\left (2+e^{2 x} x\right )^5} \, dx+\frac {189}{64} \int \frac {1}{(3+8 x) \left (2+e^{2 x} x\right )^4} \, dx-3 \int \frac {x^2}{\left (2+e^{2 x} x\right )^4} \, dx-\frac {189}{32} \int \frac {1}{(3+8 x) \left (2+e^{2 x} x\right )^5} \, dx+6 \int \frac {x^2}{\left (2+e^{2 x} x\right )^5} \, dx-\frac {69}{8} \int \frac {x}{\left (2+e^{2 x} x\right )^4} \, dx-\frac {567}{64} \int \frac {1}{(3+8 x)^2 \left (2+e^{2 x} x\right )^4} \, dx+\frac {75}{4} \int \frac {x}{\left (2+e^{2 x} x\right )^5} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.36, size = 26, normalized size = 1.00 \begin {gather*} \frac {3 x^2 (3+x)}{(3+8 x) \left (2+e^{2 x} x\right )^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(108*x + 198*x^2 + 96*x^3 + E^(2*x)*(-54*x^2 - 441*x^3 - 696*x^4 - 192*x^5))/(288 + 1536*x + 2048*x^
2 + E^(2*x)*(720*x + 3840*x^2 + 5120*x^3) + E^(4*x)*(720*x^2 + 3840*x^3 + 5120*x^4) + E^(6*x)*(360*x^3 + 1920*
x^4 + 2560*x^5) + E^(8*x)*(90*x^4 + 480*x^5 + 640*x^6) + E^(10*x)*(9*x^5 + 48*x^6 + 64*x^7)),x]

[Out]

(3*x^2*(3 + x))/((3 + 8*x)*(2 + E^(2*x)*x)^4)

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fricas [B]  time = 0.55, size = 83, normalized size = 3.19 \begin {gather*} \frac {3 \, {\left (x^{3} + 3 \, x^{2}\right )}}{{\left (8 \, x^{5} + 3 \, x^{4}\right )} e^{\left (8 \, x\right )} + 8 \, {\left (8 \, x^{4} + 3 \, x^{3}\right )} e^{\left (6 \, x\right )} + 24 \, {\left (8 \, x^{3} + 3 \, x^{2}\right )} e^{\left (4 \, x\right )} + 32 \, {\left (8 \, x^{2} + 3 \, x\right )} e^{\left (2 \, x\right )} + 128 \, x + 48} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-192*x^5-696*x^4-441*x^3-54*x^2)*exp(x)^2+96*x^3+198*x^2+108*x)/((64*x^7+48*x^6+9*x^5)*exp(x)^10+(
640*x^6+480*x^5+90*x^4)*exp(x)^8+(2560*x^5+1920*x^4+360*x^3)*exp(x)^6+(5120*x^4+3840*x^3+720*x^2)*exp(x)^4+(51
20*x^3+3840*x^2+720*x)*exp(x)^2+2048*x^2+1536*x+288),x, algorithm="fricas")

[Out]

3*(x^3 + 3*x^2)/((8*x^5 + 3*x^4)*e^(8*x) + 8*(8*x^4 + 3*x^3)*e^(6*x) + 24*(8*x^3 + 3*x^2)*e^(4*x) + 32*(8*x^2
+ 3*x)*e^(2*x) + 128*x + 48)

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giac [B]  time = 0.30, size = 88, normalized size = 3.38 \begin {gather*} \frac {3 \, {\left (x^{3} + 3 \, x^{2}\right )}}{8 \, x^{5} e^{\left (8 \, x\right )} + 3 \, x^{4} e^{\left (8 \, x\right )} + 64 \, x^{4} e^{\left (6 \, x\right )} + 24 \, x^{3} e^{\left (6 \, x\right )} + 192 \, x^{3} e^{\left (4 \, x\right )} + 72 \, x^{2} e^{\left (4 \, x\right )} + 256 \, x^{2} e^{\left (2 \, x\right )} + 96 \, x e^{\left (2 \, x\right )} + 128 \, x + 48} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-192*x^5-696*x^4-441*x^3-54*x^2)*exp(x)^2+96*x^3+198*x^2+108*x)/((64*x^7+48*x^6+9*x^5)*exp(x)^10+(
640*x^6+480*x^5+90*x^4)*exp(x)^8+(2560*x^5+1920*x^4+360*x^3)*exp(x)^6+(5120*x^4+3840*x^3+720*x^2)*exp(x)^4+(51
20*x^3+3840*x^2+720*x)*exp(x)^2+2048*x^2+1536*x+288),x, algorithm="giac")

[Out]

3*(x^3 + 3*x^2)/(8*x^5*e^(8*x) + 3*x^4*e^(8*x) + 64*x^4*e^(6*x) + 24*x^3*e^(6*x) + 192*x^3*e^(4*x) + 72*x^2*e^
(4*x) + 256*x^2*e^(2*x) + 96*x*e^(2*x) + 128*x + 48)

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maple [A]  time = 0.05, size = 26, normalized size = 1.00

method	result	size
risch	\(\frac {3 x^{2} \left (3+x \right )}{\left (8 x +3\right ) \left (x \,{\mathrm e}^{2 x}+2\right )^{4}}\)	\(26\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-192*x^5-696*x^4-441*x^3-54*x^2)*exp(x)^2+96*x^3+198*x^2+108*x)/((64*x^7+48*x^6+9*x^5)*exp(x)^10+(640*x^
6+480*x^5+90*x^4)*exp(x)^8+(2560*x^5+1920*x^4+360*x^3)*exp(x)^6+(5120*x^4+3840*x^3+720*x^2)*exp(x)^4+(5120*x^3
+3840*x^2+720*x)*exp(x)^2+2048*x^2+1536*x+288),x,method=_RETURNVERBOSE)

[Out]

3*x^2*(3+x)/(8*x+3)/(x*exp(2*x)+2)^4

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maxima [B]  time = 0.46, size = 83, normalized size = 3.19 \begin {gather*} \frac {3 \, {\left (x^{3} + 3 \, x^{2}\right )}}{{\left (8 \, x^{5} + 3 \, x^{4}\right )} e^{\left (8 \, x\right )} + 8 \, {\left (8 \, x^{4} + 3 \, x^{3}\right )} e^{\left (6 \, x\right )} + 24 \, {\left (8 \, x^{3} + 3 \, x^{2}\right )} e^{\left (4 \, x\right )} + 32 \, {\left (8 \, x^{2} + 3 \, x\right )} e^{\left (2 \, x\right )} + 128 \, x + 48} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-192*x^5-696*x^4-441*x^3-54*x^2)*exp(x)^2+96*x^3+198*x^2+108*x)/((64*x^7+48*x^6+9*x^5)*exp(x)^10+(
640*x^6+480*x^5+90*x^4)*exp(x)^8+(2560*x^5+1920*x^4+360*x^3)*exp(x)^6+(5120*x^4+3840*x^3+720*x^2)*exp(x)^4+(51
20*x^3+3840*x^2+720*x)*exp(x)^2+2048*x^2+1536*x+288),x, algorithm="maxima")

[Out]

3*(x^3 + 3*x^2)/((8*x^5 + 3*x^4)*e^(8*x) + 8*(8*x^4 + 3*x^3)*e^(6*x) + 24*(8*x^3 + 3*x^2)*e^(4*x) + 32*(8*x^2
+ 3*x)*e^(2*x) + 128*x + 48)

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mupad [B]  time = 5.39, size = 74, normalized size = 2.85 \begin {gather*} \frac {3\,\left (16\,x^5+62\,x^4+45\,x^3+9\,x^2\right )}{\left (2\,x+1\right )\,{\left (8\,x+3\right )}^2\,\left (32\,x\,{\mathrm {e}}^{2\,x}+24\,x^2\,{\mathrm {e}}^{4\,x}+8\,x^3\,{\mathrm {e}}^{6\,x}+x^4\,{\mathrm {e}}^{8\,x}+16\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((108*x - exp(2*x)*(54*x^2 + 441*x^3 + 696*x^4 + 192*x^5) + 198*x^2 + 96*x^3)/(1536*x + exp(2*x)*(720*x + 3
840*x^2 + 5120*x^3) + exp(10*x)*(9*x^5 + 48*x^6 + 64*x^7) + exp(8*x)*(90*x^4 + 480*x^5 + 640*x^6) + exp(6*x)*(
360*x^3 + 1920*x^4 + 2560*x^5) + exp(4*x)*(720*x^2 + 3840*x^3 + 5120*x^4) + 2048*x^2 + 288),x)

[Out]

(3*(9*x^2 + 45*x^3 + 62*x^4 + 16*x^5))/((2*x + 1)*(8*x + 3)^2*(32*x*exp(2*x) + 24*x^2*exp(4*x) + 8*x^3*exp(6*x
) + x^4*exp(8*x) + 16))

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sympy [B]  time = 0.32, size = 73, normalized size = 2.81 \begin {gather*} \frac {3 x^{3} + 9 x^{2}}{128 x + \left (256 x^{2} + 96 x\right ) e^{2 x} + \left (192 x^{3} + 72 x^{2}\right ) e^{4 x} + \left (64 x^{4} + 24 x^{3}\right ) e^{6 x} + \left (8 x^{5} + 3 x^{4}\right ) e^{8 x} + 48} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-192*x**5-696*x**4-441*x**3-54*x**2)*exp(x)**2+96*x**3+198*x**2+108*x)/((64*x**7+48*x**6+9*x**5)*e
xp(x)**10+(640*x**6+480*x**5+90*x**4)*exp(x)**8+(2560*x**5+1920*x**4+360*x**3)*exp(x)**6+(5120*x**4+3840*x**3+
720*x**2)*exp(x)**4+(5120*x**3+3840*x**2+720*x)*exp(x)**2+2048*x**2+1536*x+288),x)

[Out]

(3*x**3 + 9*x**2)/(128*x + (256*x**2 + 96*x)*exp(2*x) + (192*x**3 + 72*x**2)*exp(4*x) + (64*x**4 + 24*x**3)*ex
p(6*x) + (8*x**5 + 3*x**4)*exp(8*x) + 48)

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