3.31.34 \(\int \frac {e^{\frac {1+x-3 x^3-6 x^4-3 x^5}{5 e^x x^2-5 x^3}} (3 x+2 x^2+6 x^5+6 x^6+e^x (-2-2 x-x^2-3 x^3-9 x^4-3 x^5+3 x^6))}{5 e^{2 x} x^3-10 e^x x^4+5 x^5} \, dx\)

Optimal. Leaf size=29 \[ e^{\frac {(1+x) \left (\frac {1}{x^2}-3 \left (x+x^2\right )\right )}{5 \left (e^x-x\right )}} \]

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Rubi [F]  time = 24.47, 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 {1+x-3 x^3-6 x^4-3 x^5}{5 e^x x^2-5 x^3}\right ) \left (3 x+2 x^2+6 x^5+6 x^6+e^x \left (-2-2 x-x^2-3 x^3-9 x^4-3 x^5+3 x^6\right )\right )}{5 e^{2 x} x^3-10 e^x x^4+5 x^5} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((1 + x - 3*x^3 - 6*x^4 - 3*x^5)/(5*E^x*x^2 - 5*x^3))*(3*x + 2*x^2 + 6*x^5 + 6*x^6 + E^x*(-2 - 2*x - x^
2 - 3*x^3 - 9*x^4 - 3*x^5 + 3*x^6)))/(5*E^(2*x)*x^3 - 10*E^x*x^4 + 5*x^5),x]

[Out]

-1/5*Defer[Int][E^((1 + x - 3*x^3 - 6*x^4 - 3*x^5)/(5*(E^x - x)*x^2))/(E^x - x)^2, x] - (3*Defer[Int][E^((1 +
x - 3*x^3 - 6*x^4 - 3*x^5)/(5*(E^x - x)*x^2))/(E^x - x), x])/5 - (2*Defer[Int][E^((1 + x - 3*x^3 - 6*x^4 - 3*x
^5)/(5*(E^x - x)*x^2))/((E^x - x)*x^3), x])/5 + Defer[Int][E^((1 + x - 3*x^3 - 6*x^4 - 3*x^5)/(5*(E^x - x)*x^2
))/((E^x - x)^2*x^2), x]/5 - (2*Defer[Int][E^((1 + x - 3*x^3 - 6*x^4 - 3*x^5)/(5*(E^x - x)*x^2))/((E^x - x)*x^
2), x])/5 - Defer[Int][E^((1 + x - 3*x^3 - 6*x^4 - 3*x^5)/(5*(E^x - x)*x^2))/((E^x - x)*x), x]/5 - (3*Defer[In
t][(E^((1 + x - 3*x^3 - 6*x^4 - 3*x^5)/(5*(E^x - x)*x^2))*x)/(E^x - x)^2, x])/5 - (9*Defer[Int][(E^((1 + x - 3
*x^3 - 6*x^4 - 3*x^5)/(5*(E^x - x)*x^2))*x)/(E^x - x), x])/5 - (3*Defer[Int][(E^((1 + x - 3*x^3 - 6*x^4 - 3*x^
5)/(5*(E^x - x)*x^2))*x^2)/(E^x - x)^2, x])/5 - (3*Defer[Int][(E^((1 + x - 3*x^3 - 6*x^4 - 3*x^5)/(5*(E^x - x)
*x^2))*x^2)/(E^x - x), x])/5 + (3*Defer[Int][(E^((1 + x - 3*x^3 - 6*x^4 - 3*x^5)/(5*(E^x - x)*x^2))*x^3)/(E^x
- x)^2, x])/5 + (3*Defer[Int][(E^((1 + x - 3*x^3 - 6*x^4 - 3*x^5)/(5*(E^x - x)*x^2))*x^3)/(E^x - x), x])/5 + (
3*Defer[Int][(E^((1 + x - 3*x^3 - 6*x^4 - 3*x^5)/(5*(E^x - x)*x^2))*x^4)/(E^x - x)^2, x])/5

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{\frac {1+x-3 x^3-6 x^4-3 x^5}{5 \left (e^x-x\right ) x^2}} \left (3 x+2 x^2+6 x^5+6 x^6+e^x \left (-2-2 x-x^2-3 x^3-9 x^4-3 x^5+3 x^6\right )\right )}{5 \left (e^x-x\right )^2 x^3} \, dx\\ &=\frac {1}{5} \int \frac {e^{\frac {1+x-3 x^3-6 x^4-3 x^5}{5 \left (e^x-x\right ) x^2}} \left (3 x+2 x^2+6 x^5+6 x^6+e^x \left (-2-2 x-x^2-3 x^3-9 x^4-3 x^5+3 x^6\right )\right )}{\left (e^x-x\right )^2 x^3} \, dx\\ &=\frac {1}{5} \int \left (\frac {e^{\frac {1+x-3 x^3-6 x^4-3 x^5}{5 \left (e^x-x\right ) x^2}} \left (-2-2 x-x^2-3 x^3-9 x^4-3 x^5+3 x^6\right )}{\left (e^x-x\right ) x^3}+\frac {e^{\frac {1+x-3 x^3-6 x^4-3 x^5}{5 \left (e^x-x\right ) x^2}} \left (1-x^2-3 x^3-3 x^4+3 x^5+3 x^6\right )}{\left (e^x-x\right )^2 x^2}\right ) \, dx\\ &=\frac {1}{5} \int \frac {e^{\frac {1+x-3 x^3-6 x^4-3 x^5}{5 \left (e^x-x\right ) x^2}} \left (-2-2 x-x^2-3 x^3-9 x^4-3 x^5+3 x^6\right )}{\left (e^x-x\right ) x^3} \, dx+\frac {1}{5} \int \frac {e^{\frac {1+x-3 x^3-6 x^4-3 x^5}{5 \left (e^x-x\right ) x^2}} \left (1-x^2-3 x^3-3 x^4+3 x^5+3 x^6\right )}{\left (e^x-x\right )^2 x^2} \, dx\\ &=\frac {1}{5} \int \left (-\frac {3 e^{\frac {1+x-3 x^3-6 x^4-3 x^5}{5 \left (e^x-x\right ) x^2}}}{e^x-x}-\frac {2 e^{\frac {1+x-3 x^3-6 x^4-3 x^5}{5 \left (e^x-x\right ) x^2}}}{\left (e^x-x\right ) x^3}-\frac {2 e^{\frac {1+x-3 x^3-6 x^4-3 x^5}{5 \left (e^x-x\right ) x^2}}}{\left (e^x-x\right ) x^2}-\frac {e^{\frac {1+x-3 x^3-6 x^4-3 x^5}{5 \left (e^x-x\right ) x^2}}}{\left (e^x-x\right ) x}-\frac {9 e^{\frac {1+x-3 x^3-6 x^4-3 x^5}{5 \left (e^x-x\right ) x^2}} x}{e^x-x}-\frac {3 e^{\frac {1+x-3 x^3-6 x^4-3 x^5}{5 \left (e^x-x\right ) x^2}} x^2}{e^x-x}+\frac {3 e^{\frac {1+x-3 x^3-6 x^4-3 x^5}{5 \left (e^x-x\right ) x^2}} x^3}{e^x-x}\right ) \, dx+\frac {1}{5} \int \left (-\frac {e^{\frac {1+x-3 x^3-6 x^4-3 x^5}{5 \left (e^x-x\right ) x^2}}}{\left (e^x-x\right )^2}+\frac {e^{\frac {1+x-3 x^3-6 x^4-3 x^5}{5 \left (e^x-x\right ) x^2}}}{\left (e^x-x\right )^2 x^2}-\frac {3 e^{\frac {1+x-3 x^3-6 x^4-3 x^5}{5 \left (e^x-x\right ) x^2}} x}{\left (e^x-x\right )^2}-\frac {3 e^{\frac {1+x-3 x^3-6 x^4-3 x^5}{5 \left (e^x-x\right ) x^2}} x^2}{\left (e^x-x\right )^2}+\frac {3 e^{\frac {1+x-3 x^3-6 x^4-3 x^5}{5 \left (e^x-x\right ) x^2}} x^3}{\left (e^x-x\right )^2}+\frac {3 e^{\frac {1+x-3 x^3-6 x^4-3 x^5}{5 \left (e^x-x\right ) x^2}} x^4}{\left (e^x-x\right )^2}\right ) \, dx\\ &=-\left (\frac {1}{5} \int \frac {e^{\frac {1+x-3 x^3-6 x^4-3 x^5}{5 \left (e^x-x\right ) x^2}}}{\left (e^x-x\right )^2} \, dx\right )+\frac {1}{5} \int \frac {e^{\frac {1+x-3 x^3-6 x^4-3 x^5}{5 \left (e^x-x\right ) x^2}}}{\left (e^x-x\right )^2 x^2} \, dx-\frac {1}{5} \int \frac {e^{\frac {1+x-3 x^3-6 x^4-3 x^5}{5 \left (e^x-x\right ) x^2}}}{\left (e^x-x\right ) x} \, dx-\frac {2}{5} \int \frac {e^{\frac {1+x-3 x^3-6 x^4-3 x^5}{5 \left (e^x-x\right ) x^2}}}{\left (e^x-x\right ) x^3} \, dx-\frac {2}{5} \int \frac {e^{\frac {1+x-3 x^3-6 x^4-3 x^5}{5 \left (e^x-x\right ) x^2}}}{\left (e^x-x\right ) x^2} \, dx-\frac {3}{5} \int \frac {e^{\frac {1+x-3 x^3-6 x^4-3 x^5}{5 \left (e^x-x\right ) x^2}}}{e^x-x} \, dx-\frac {3}{5} \int \frac {e^{\frac {1+x-3 x^3-6 x^4-3 x^5}{5 \left (e^x-x\right ) x^2}} x}{\left (e^x-x\right )^2} \, dx-\frac {3}{5} \int \frac {e^{\frac {1+x-3 x^3-6 x^4-3 x^5}{5 \left (e^x-x\right ) x^2}} x^2}{\left (e^x-x\right )^2} \, dx-\frac {3}{5} \int \frac {e^{\frac {1+x-3 x^3-6 x^4-3 x^5}{5 \left (e^x-x\right ) x^2}} x^2}{e^x-x} \, dx+\frac {3}{5} \int \frac {e^{\frac {1+x-3 x^3-6 x^4-3 x^5}{5 \left (e^x-x\right ) x^2}} x^3}{\left (e^x-x\right )^2} \, dx+\frac {3}{5} \int \frac {e^{\frac {1+x-3 x^3-6 x^4-3 x^5}{5 \left (e^x-x\right ) x^2}} x^3}{e^x-x} \, dx+\frac {3}{5} \int \frac {e^{\frac {1+x-3 x^3-6 x^4-3 x^5}{5 \left (e^x-x\right ) x^2}} x^4}{\left (e^x-x\right )^2} \, dx-\frac {9}{5} \int \frac {e^{\frac {1+x-3 x^3-6 x^4-3 x^5}{5 \left (e^x-x\right ) x^2}} x}{e^x-x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.10, size = 36, normalized size = 1.24 \begin {gather*} e^{\frac {1+x-3 x^3-6 x^4-3 x^5}{5 \left (e^x-x\right ) x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((1 + x - 3*x^3 - 6*x^4 - 3*x^5)/(5*E^x*x^2 - 5*x^3))*(3*x + 2*x^2 + 6*x^5 + 6*x^6 + E^x*(-2 - 2*
x - x^2 - 3*x^3 - 9*x^4 - 3*x^5 + 3*x^6)))/(5*E^(2*x)*x^3 - 10*E^x*x^4 + 5*x^5),x]

[Out]

E^((1 + x - 3*x^3 - 6*x^4 - 3*x^5)/(5*(E^x - x)*x^2))

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fricas [A]  time = 0.81, size = 36, normalized size = 1.24 \begin {gather*} e^{\left (\frac {3 \, x^{5} + 6 \, x^{4} + 3 \, x^{3} - x - 1}{5 \, {\left (x^{3} - x^{2} e^{x}\right )}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x^6-3*x^5-9*x^4-3*x^3-x^2-2*x-2)*exp(x)+6*x^6+6*x^5+2*x^2+3*x)*exp((-3*x^5-6*x^4-3*x^3+x+1)/(5*e
xp(x)*x^2-5*x^3))/(5*exp(x)^2*x^3-10*exp(x)*x^4+5*x^5),x, algorithm="fricas")

[Out]

e^(1/5*(3*x^5 + 6*x^4 + 3*x^3 - x - 1)/(x^3 - x^2*e^x))

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x^6-3*x^5-9*x^4-3*x^3-x^2-2*x-2)*exp(x)+6*x^6+6*x^5+2*x^2+3*x)*exp((-3*x^5-6*x^4-3*x^3+x+1)/(5*e
xp(x)*x^2-5*x^3))/(5*exp(x)^2*x^3-10*exp(x)*x^4+5*x^5),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Evaluation time: 0.46Unable to divide, perhaps due to rounding error%%%{6075,[1,36]%%%}+%%%{6075,[1,35]%%%}
+%%%{-66825

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maple [A]  time = 0.12, size = 30, normalized size = 1.03

method	result	size
risch	\({\mathrm e}^{-\frac {\left (x +1\right ) \left (3 x^{4}+3 x^{3}-1\right )}{5 x^{2} \left ({\mathrm e}^{x}-x \right )}}\)	\(30\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((3*x^6-3*x^5-9*x^4-3*x^3-x^2-2*x-2)*exp(x)+6*x^6+6*x^5+2*x^2+3*x)*exp((-3*x^5-6*x^4-3*x^3+x+1)/(5*exp(x)*
x^2-5*x^3))/(5*exp(x)^2*x^3-10*exp(x)*x^4+5*x^5),x,method=_RETURNVERBOSE)

[Out]

exp(-1/5*(x+1)*(3*x^4+3*x^3-1)/x^2/(exp(x)-x))

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x^6-3*x^5-9*x^4-3*x^3-x^2-2*x-2)*exp(x)+6*x^6+6*x^5+2*x^2+3*x)*exp((-3*x^5-6*x^4-3*x^3+x+1)/(5*e
xp(x)*x^2-5*x^3))/(5*exp(x)^2*x^3-10*exp(x)*x^4+5*x^5),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is undefined.

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mupad [B]  time = 2.07, size = 77, normalized size = 2.66 \begin {gather*} {\mathrm {e}}^{\frac {3\,x}{5\,x-5\,{\mathrm {e}}^x}}\,{\mathrm {e}}^{\frac {1}{5\,x\,{\mathrm {e}}^x-5\,x^2}}\,{\mathrm {e}}^{\frac {3\,x^3}{5\,x-5\,{\mathrm {e}}^x}}\,{\mathrm {e}}^{\frac {6\,x^2}{5\,x-5\,{\mathrm {e}}^x}}\,{\mathrm {e}}^{\frac {1}{5\,x^2\,{\mathrm {e}}^x-5\,x^3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-(3*x^3 - x + 6*x^4 + 3*x^5 - 1)/(5*x^2*exp(x) - 5*x^3))*(3*x - exp(x)*(2*x + x^2 + 3*x^3 + 9*x^4 + 3
*x^5 - 3*x^6 + 2) + 2*x^2 + 6*x^5 + 6*x^6))/(5*x^3*exp(2*x) - 10*x^4*exp(x) + 5*x^5),x)

[Out]

exp((3*x)/(5*x - 5*exp(x)))*exp(1/(5*x*exp(x) - 5*x^2))*exp((3*x^3)/(5*x - 5*exp(x)))*exp((6*x^2)/(5*x - 5*exp
(x)))*exp(1/(5*x^2*exp(x) - 5*x^3))

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sympy [A]  time = 0.52, size = 32, normalized size = 1.10 \begin {gather*} e^{\frac {- 3 x^{5} - 6 x^{4} - 3 x^{3} + x + 1}{- 5 x^{3} + 5 x^{2} e^{x}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x**6-3*x**5-9*x**4-3*x**3-x**2-2*x-2)*exp(x)+6*x**6+6*x**5+2*x**2+3*x)*exp((-3*x**5-6*x**4-3*x**
3+x+1)/(5*exp(x)*x**2-5*x**3))/(5*exp(x)**2*x**3-10*exp(x)*x**4+5*x**5),x)

[Out]

exp((-3*x**5 - 6*x**4 - 3*x**3 + x + 1)/(-5*x**3 + 5*x**2*exp(x)))

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