3.47.27 \(\int \frac {e^{\frac {9+24 x^2+72 x^3+70 x^4+96 x^5+216 x^6+216 x^7+81 x^8}{3 x+x^2}} (-27-18 x+72 x^2+432 x^3+702 x^4+1292 x^5+3528 x^6+4752 x^7+2781 x^8+486 x^9)}{9 x^2+6 x^3+x^4} \, dx\)

Optimal. Leaf size=30 \[ -2+e^{\frac {\left (3+\left (-x+3 \left (x+x^2\right )\right )^2\right )^2}{x (3+x)}} \]

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Rubi [F]  time = 14.23, 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 {\exp \left (\frac {9+24 x^2+72 x^3+70 x^4+96 x^5+216 x^6+216 x^7+81 x^8}{3 x+x^2}\right ) \left (-27-18 x+72 x^2+432 x^3+702 x^4+1292 x^5+3528 x^6+4752 x^7+2781 x^8+486 x^9\right )}{9 x^2+6 x^3+x^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((9 + 24*x^2 + 72*x^3 + 70*x^4 + 96*x^5 + 216*x^6 + 216*x^7 + 81*x^8)/(3*x + x^2))*(-27 - 18*x + 72*x^2
+ 432*x^3 + 702*x^4 + 1292*x^5 + 3528*x^6 + 4752*x^7 + 2781*x^8 + 486*x^9))/(9*x^2 + 6*x^3 + x^4),x]

[Out]

-7293*Defer[Int][E^((3 + 4*x^2 + 12*x^3 + 9*x^4)^2/(x*(3 + x))), x] - 3*Defer[Int][E^((3 + 4*x^2 + 12*x^3 + 9*
x^4)^2/(x*(3 + x)))/x^2, x] + 4910*Defer[Int][E^((3 + 4*x^2 + 12*x^3 + 9*x^4)^2/(x*(3 + x)))*x, x] - 2385*Defe
r[Int][E^((3 + 4*x^2 + 12*x^3 + 9*x^4)^2/(x*(3 + x)))*x^2, x] + 1188*Defer[Int][E^((3 + 4*x^2 + 12*x^3 + 9*x^4
)^2/(x*(3 + x)))*x^3, x] - 135*Defer[Int][E^((3 + 4*x^2 + 12*x^3 + 9*x^4)^2/(x*(3 + x)))*x^4, x] + 486*Defer[I
nt][E^((3 + 4*x^2 + 12*x^3 + 9*x^4)^2/(x*(3 + x)))*x^5, x] + 65712*Defer[Int][E^((3 + 4*x^2 + 12*x^3 + 9*x^4)^
2/(x*(3 + x)))/(3 + x)^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\exp \left (\frac {9+24 x^2+72 x^3+70 x^4+96 x^5+216 x^6+216 x^7+81 x^8}{3 x+x^2}\right ) \left (-27-18 x+72 x^2+432 x^3+702 x^4+1292 x^5+3528 x^6+4752 x^7+2781 x^8+486 x^9\right )}{x^2 \left (9+6 x+x^2\right )} \, dx\\ &=\int \frac {\exp \left (\frac {9+24 x^2+72 x^3+70 x^4+96 x^5+216 x^6+216 x^7+81 x^8}{3 x+x^2}\right ) \left (-27-18 x+72 x^2+432 x^3+702 x^4+1292 x^5+3528 x^6+4752 x^7+2781 x^8+486 x^9\right )}{x^2 (3+x)^2} \, dx\\ &=\int \frac {e^{\frac {\left (3+4 x^2+12 x^3+9 x^4\right )^2}{x (3+x)}} \left (-27-18 x+72 x^2+432 x^3+702 x^4+1292 x^5+3528 x^6+4752 x^7+2781 x^8+486 x^9\right )}{x^2 (3+x)^2} \, dx\\ &=\int \left (-7293 e^{\frac {\left (3+4 x^2+12 x^3+9 x^4\right )^2}{x (3+x)}}-\frac {3 e^{\frac {\left (3+4 x^2+12 x^3+9 x^4\right )^2}{x (3+x)}}}{x^2}+4910 e^{\frac {\left (3+4 x^2+12 x^3+9 x^4\right )^2}{x (3+x)}} x-2385 e^{\frac {\left (3+4 x^2+12 x^3+9 x^4\right )^2}{x (3+x)}} x^2+1188 e^{\frac {\left (3+4 x^2+12 x^3+9 x^4\right )^2}{x (3+x)}} x^3-135 e^{\frac {\left (3+4 x^2+12 x^3+9 x^4\right )^2}{x (3+x)}} x^4+486 e^{\frac {\left (3+4 x^2+12 x^3+9 x^4\right )^2}{x (3+x)}} x^5+\frac {65712 e^{\frac {\left (3+4 x^2+12 x^3+9 x^4\right )^2}{x (3+x)}}}{(3+x)^2}\right ) \, dx\\ &=-\left (3 \int \frac {e^{\frac {\left (3+4 x^2+12 x^3+9 x^4\right )^2}{x (3+x)}}}{x^2} \, dx\right )-135 \int e^{\frac {\left (3+4 x^2+12 x^3+9 x^4\right )^2}{x (3+x)}} x^4 \, dx+486 \int e^{\frac {\left (3+4 x^2+12 x^3+9 x^4\right )^2}{x (3+x)}} x^5 \, dx+1188 \int e^{\frac {\left (3+4 x^2+12 x^3+9 x^4\right )^2}{x (3+x)}} x^3 \, dx-2385 \int e^{\frac {\left (3+4 x^2+12 x^3+9 x^4\right )^2}{x (3+x)}} x^2 \, dx+4910 \int e^{\frac {\left (3+4 x^2+12 x^3+9 x^4\right )^2}{x (3+x)}} x \, dx-7293 \int e^{\frac {\left (3+4 x^2+12 x^3+9 x^4\right )^2}{x (3+x)}} \, dx+65712 \int \frac {e^{\frac {\left (3+4 x^2+12 x^3+9 x^4\right )^2}{x (3+x)}}}{(3+x)^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.10, size = 30, normalized size = 1.00 \begin {gather*} e^{\frac {\left (3+4 x^2+12 x^3+9 x^4\right )^2}{x (3+x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((9 + 24*x^2 + 72*x^3 + 70*x^4 + 96*x^5 + 216*x^6 + 216*x^7 + 81*x^8)/(3*x + x^2))*(-27 - 18*x +
72*x^2 + 432*x^3 + 702*x^4 + 1292*x^5 + 3528*x^6 + 4752*x^7 + 2781*x^8 + 486*x^9))/(9*x^2 + 6*x^3 + x^4),x]

[Out]

E^((3 + 4*x^2 + 12*x^3 + 9*x^4)^2/(x*(3 + x)))

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fricas [A]  time = 0.68, size = 48, normalized size = 1.60 \begin {gather*} e^{\left (\frac {81 \, x^{8} + 216 \, x^{7} + 216 \, x^{6} + 96 \, x^{5} + 70 \, x^{4} + 72 \, x^{3} + 24 \, x^{2} + 9}{x^{2} + 3 \, x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((486*x^9+2781*x^8+4752*x^7+3528*x^6+1292*x^5+702*x^4+432*x^3+72*x^2-18*x-27)*exp((81*x^8+216*x^7+216
*x^6+96*x^5+70*x^4+72*x^3+24*x^2+9)/(x^2+3*x))/(x^4+6*x^3+9*x^2),x, algorithm="fricas")

[Out]

e^((81*x^8 + 216*x^7 + 216*x^6 + 96*x^5 + 70*x^4 + 72*x^3 + 24*x^2 + 9)/(x^2 + 3*x))

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giac [B]  time = 0.15, size = 111, normalized size = 3.70 \begin {gather*} e^{\left (\frac {81 \, x^{8}}{x^{2} + 3 \, x} + \frac {216 \, x^{7}}{x^{2} + 3 \, x} + \frac {216 \, x^{6}}{x^{2} + 3 \, x} + \frac {96 \, x^{5}}{x^{2} + 3 \, x} + \frac {70 \, x^{4}}{x^{2} + 3 \, x} + \frac {72 \, x^{3}}{x^{2} + 3 \, x} + \frac {24 \, x^{2}}{x^{2} + 3 \, x} + \frac {9}{x^{2} + 3 \, x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((486*x^9+2781*x^8+4752*x^7+3528*x^6+1292*x^5+702*x^4+432*x^3+72*x^2-18*x-27)*exp((81*x^8+216*x^7+216
*x^6+96*x^5+70*x^4+72*x^3+24*x^2+9)/(x^2+3*x))/(x^4+6*x^3+9*x^2),x, algorithm="giac")

[Out]

e^(81*x^8/(x^2 + 3*x) + 216*x^7/(x^2 + 3*x) + 216*x^6/(x^2 + 3*x) + 96*x^5/(x^2 + 3*x) + 70*x^4/(x^2 + 3*x) +
72*x^3/(x^2 + 3*x) + 24*x^2/(x^2 + 3*x) + 9/(x^2 + 3*x))

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maple [A]  time = 0.18, size = 30, normalized size = 1.00

method	result	size
risch	\({\mathrm e}^{\frac {\left (9 x^{4}+12 x^{3}+4 x^{2}+3\right )^{2}}{\left (3+x \right ) x}}\)	\(30\)
gosper	\({\mathrm e}^{\frac {81 x^{8}+216 x^{7}+216 x^{6}+96 x^{5}+70 x^{4}+72 x^{3}+24 x^{2}+9}{\left (3+x \right ) x}}\)	\(48\)
norman	\(\frac {x^{2} {\mathrm e}^{\frac {81 x^{8}+216 x^{7}+216 x^{6}+96 x^{5}+70 x^{4}+72 x^{3}+24 x^{2}+9}{x^{2}+3 x}}+3 x \,{\mathrm e}^{\frac {81 x^{8}+216 x^{7}+216 x^{6}+96 x^{5}+70 x^{4}+72 x^{3}+24 x^{2}+9}{x^{2}+3 x}}}{\left (3+x \right ) x}\)	\(114\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((486*x^9+2781*x^8+4752*x^7+3528*x^6+1292*x^5+702*x^4+432*x^3+72*x^2-18*x-27)*exp((81*x^8+216*x^7+216*x^6+9
6*x^5+70*x^4+72*x^3+24*x^2+9)/(x^2+3*x))/(x^4+6*x^3+9*x^2),x,method=_RETURNVERBOSE)

[Out]

exp((9*x^4+12*x^3+4*x^2+3)^2/(3+x)/x)

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maxima [A]  time = 0.96, size = 43, normalized size = 1.43 \begin {gather*} e^{\left (81 \, x^{6} - 27 \, x^{5} + 297 \, x^{4} - 795 \, x^{3} + 2455 \, x^{2} - 7293 \, x - \frac {65712}{x + 3} + \frac {3}{x} + 21903\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((486*x^9+2781*x^8+4752*x^7+3528*x^6+1292*x^5+702*x^4+432*x^3+72*x^2-18*x-27)*exp((81*x^8+216*x^7+216
*x^6+96*x^5+70*x^4+72*x^3+24*x^2+9)/(x^2+3*x))/(x^4+6*x^3+9*x^2),x, algorithm="maxima")

[Out]

e^(81*x^6 - 27*x^5 + 297*x^4 - 795*x^3 + 2455*x^2 - 7293*x - 65712/(x + 3) + 3/x + 21903)

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mupad [B]  time = 3.60, size = 88, normalized size = 2.93 \begin {gather*} {\mathrm {e}}^{\frac {9}{x^2+3\,x}}\,{\mathrm {e}}^{\frac {24\,x}{x+3}}\,{\mathrm {e}}^{\frac {70\,x^3}{x+3}}\,{\mathrm {e}}^{\frac {72\,x^2}{x+3}}\,{\mathrm {e}}^{\frac {81\,x^7}{x+3}}\,{\mathrm {e}}^{\frac {96\,x^4}{x+3}}\,{\mathrm {e}}^{\frac {216\,x^5}{x+3}}\,{\mathrm {e}}^{\frac {216\,x^6}{x+3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp((24*x^2 + 72*x^3 + 70*x^4 + 96*x^5 + 216*x^6 + 216*x^7 + 81*x^8 + 9)/(3*x + x^2))*(72*x^2 - 18*x + 43
2*x^3 + 702*x^4 + 1292*x^5 + 3528*x^6 + 4752*x^7 + 2781*x^8 + 486*x^9 - 27))/(9*x^2 + 6*x^3 + x^4),x)

[Out]

exp(9/(3*x + x^2))*exp((24*x)/(x + 3))*exp((70*x^3)/(x + 3))*exp((72*x^2)/(x + 3))*exp((81*x^7)/(x + 3))*exp((
96*x^4)/(x + 3))*exp((216*x^5)/(x + 3))*exp((216*x^6)/(x + 3))

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sympy [B]  time = 0.28, size = 44, normalized size = 1.47 \begin {gather*} e^{\frac {81 x^{8} + 216 x^{7} + 216 x^{6} + 96 x^{5} + 70 x^{4} + 72 x^{3} + 24 x^{2} + 9}{x^{2} + 3 x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((486*x**9+2781*x**8+4752*x**7+3528*x**6+1292*x**5+702*x**4+432*x**3+72*x**2-18*x-27)*exp((81*x**8+21
6*x**7+216*x**6+96*x**5+70*x**4+72*x**3+24*x**2+9)/(x**2+3*x))/(x**4+6*x**3+9*x**2),x)

[Out]

exp((81*x**8 + 216*x**7 + 216*x**6 + 96*x**5 + 70*x**4 + 72*x**3 + 24*x**2 + 9)/(x**2 + 3*x))

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