3.90.4 \(\int \frac {27 e^{8 x}+e^{4 x} (64-256 x-144 x^2)}{9 e^{8 x}-24 e^{4 x} x+16 x^2} \, dx\)

Optimal. Leaf size=22 \[ \frac {x+4 (4+2 x)}{3-4 e^{-4 x} x} \]

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Rubi [F]  time = 0.99, 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 {27 e^{8 x}+e^{4 x} \left (64-256 x-144 x^2\right )}{9 e^{8 x}-24 e^{4 x} x+16 x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(27*E^(8*x) + E^(4*x)*(64 - 256*x - 144*x^2))/(9*E^(8*x) - 24*E^(4*x)*x + 16*x^2),x]

[Out]

-16/(3*(3*E^(4*x) - 4*x)) + (55*x)/(3*(3*E^(4*x) - 4*x)) + (12*x^2)/(3*E^(4*x) - 4*x) - (220*Defer[Int][x/(3*E
^(4*x) - 4*x)^2, x])/3 - 24*Defer[Int][x/(3*E^(4*x) - 4*x), x] - 48*Defer[Int][x^2/(3*E^(4*x) - 4*x)^2, x] + (
16*Defer[Subst][Defer[Int][(3*E^x - x)^(-2), x], x, 4*x])/3 - (55*Defer[Subst][Defer[Int][(3*E^x - x)^(-1), x]
, x, 4*x])/12 + (9*Defer[Subst][Defer[Int][E^x/(3*E^x - x), x], x, 4*x])/4

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{4 x} \left (64+27 e^{4 x}-256 x-144 x^2\right )}{\left (3 e^{4 x}-4 x\right )^2} \, dx\\ &=\int \left (\frac {9 e^{4 x}}{3 e^{4 x}-4 x}-\frac {4 e^{4 x} \left (-16+55 x+36 x^2\right )}{\left (3 e^{4 x}-4 x\right )^2}\right ) \, dx\\ &=-\left (4 \int \frac {e^{4 x} \left (-16+55 x+36 x^2\right )}{\left (3 e^{4 x}-4 x\right )^2} \, dx\right )+9 \int \frac {e^{4 x}}{3 e^{4 x}-4 x} \, dx\\ &=\frac {9}{4} \operatorname {Subst}\left (\int \frac {e^x}{3 e^x-x} \, dx,x,4 x\right )-4 \int \left (-\frac {16 e^{4 x}}{\left (3 e^{4 x}-4 x\right )^2}+\frac {55 e^{4 x} x}{\left (3 e^{4 x}-4 x\right )^2}+\frac {36 e^{4 x} x^2}{\left (3 e^{4 x}-4 x\right )^2}\right ) \, dx\\ &=\frac {9}{4} \operatorname {Subst}\left (\int \frac {e^x}{3 e^x-x} \, dx,x,4 x\right )+64 \int \frac {e^{4 x}}{\left (3 e^{4 x}-4 x\right )^2} \, dx-144 \int \frac {e^{4 x} x^2}{\left (3 e^{4 x}-4 x\right )^2} \, dx-220 \int \frac {e^{4 x} x}{\left (3 e^{4 x}-4 x\right )^2} \, dx\\ &=-\frac {16}{3 \left (3 e^{4 x}-4 x\right )}+\frac {55 x}{3 \left (3 e^{4 x}-4 x\right )}+\frac {12 x^2}{3 e^{4 x}-4 x}+\frac {9}{4} \operatorname {Subst}\left (\int \frac {e^x}{3 e^x-x} \, dx,x,4 x\right )-\frac {55}{3} \int \frac {1}{3 e^{4 x}-4 x} \, dx+\frac {64}{3} \int \frac {1}{\left (3 e^{4 x}-4 x\right )^2} \, dx-24 \int \frac {x}{3 e^{4 x}-4 x} \, dx-48 \int \frac {x^2}{\left (3 e^{4 x}-4 x\right )^2} \, dx-\frac {220}{3} \int \frac {x}{\left (3 e^{4 x}-4 x\right )^2} \, dx\\ &=-\frac {16}{3 \left (3 e^{4 x}-4 x\right )}+\frac {55 x}{3 \left (3 e^{4 x}-4 x\right )}+\frac {12 x^2}{3 e^{4 x}-4 x}+\frac {9}{4} \operatorname {Subst}\left (\int \frac {e^x}{3 e^x-x} \, dx,x,4 x\right )-\frac {55}{12} \operatorname {Subst}\left (\int \frac {1}{3 e^x-x} \, dx,x,4 x\right )+\frac {16}{3} \operatorname {Subst}\left (\int \frac {1}{\left (3 e^x-x\right )^2} \, dx,x,4 x\right )-24 \int \frac {x}{3 e^{4 x}-4 x} \, dx-48 \int \frac {x^2}{\left (3 e^{4 x}-4 x\right )^2} \, dx-\frac {220}{3} \int \frac {x}{\left (3 e^{4 x}-4 x\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.20, size = 24, normalized size = 1.09 \begin {gather*} \frac {\left (64+27 e^{4 x}\right ) x}{9 e^{4 x}-12 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(27*E^(8*x) + E^(4*x)*(64 - 256*x - 144*x^2))/(9*E^(8*x) - 24*E^(4*x)*x + 16*x^2),x]

[Out]

((64 + 27*E^(4*x))*x)/(9*E^(4*x) - 12*x)

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fricas [A]  time = 0.58, size = 25, normalized size = 1.14 \begin {gather*} -\frac {27 \, x e^{\left (4 \, x\right )} + 64 \, x}{3 \, {\left (4 \, x - 3 \, e^{\left (4 \, x\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((27*exp(x)^8+(-144*x^2-256*x+64)*exp(x)^4)/(9*exp(x)^8-24*x*exp(x)^4+16*x^2),x, algorithm="fricas")

[Out]

-1/3*(27*x*e^(4*x) + 64*x)/(4*x - 3*e^(4*x))

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giac [A]  time = 0.17, size = 25, normalized size = 1.14 \begin {gather*} -\frac {27 \, x e^{\left (4 \, x\right )} + 64 \, x}{3 \, {\left (4 \, x - 3 \, e^{\left (4 \, x\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((27*exp(x)^8+(-144*x^2-256*x+64)*exp(x)^4)/(9*exp(x)^8-24*x*exp(x)^4+16*x^2),x, algorithm="giac")

[Out]

-1/3*(27*x*e^(4*x) + 64*x)/(4*x - 3*e^(4*x))

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maple [A]  time = 0.06, size = 25, normalized size = 1.14

method	result	size
norman	\(\frac {-\frac {64 x}{3}-9 x \,{\mathrm e}^{4 x}}{-3 \,{\mathrm e}^{4 x}+4 x}\)	\(25\)
risch	\(3 x -\frac {4 \left (9 x +16\right ) x}{3 \left (-3 \,{\mathrm e}^{4 x}+4 x \right )}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((27*exp(x)^8+(-144*x^2-256*x+64)*exp(x)^4)/(9*exp(x)^8-24*x*exp(x)^4+16*x^2),x,method=_RETURNVERBOSE)

[Out]

(-64/3*x-9*x*exp(x)^4)/(-3*exp(x)^4+4*x)

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maxima [A]  time = 0.41, size = 25, normalized size = 1.14 \begin {gather*} -\frac {27 \, x e^{\left (4 \, x\right )} + 64 \, x}{3 \, {\left (4 \, x - 3 \, e^{\left (4 \, x\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((27*exp(x)^8+(-144*x^2-256*x+64)*exp(x)^4)/(9*exp(x)^8-24*x*exp(x)^4+16*x^2),x, algorithm="maxima")

[Out]

-1/3*(27*x*e^(4*x) + 64*x)/(4*x - 3*e^(4*x))

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mupad [B]  time = 0.09, size = 23, normalized size = 1.05 \begin {gather*} -\frac {x\,\left (27\,{\mathrm {e}}^{4\,x}+64\right )}{3\,\left (4\,x-3\,{\mathrm {e}}^{4\,x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((27*exp(8*x) - exp(4*x)*(256*x + 144*x^2 - 64))/(9*exp(8*x) - 24*x*exp(4*x) + 16*x^2),x)

[Out]

-(x*(27*exp(4*x) + 64))/(3*(4*x - 3*exp(4*x)))

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sympy [A]  time = 0.15, size = 20, normalized size = 0.91 \begin {gather*} 3 x + \frac {36 x^{2} + 64 x}{- 12 x + 9 e^{4 x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((27*exp(x)**8+(-144*x**2-256*x+64)*exp(x)**4)/(9*exp(x)**8-24*x*exp(x)**4+16*x**2),x)

[Out]

3*x + (36*x**2 + 64*x)/(-12*x + 9*exp(4*x))

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