3.8.36 \(\int \frac {-6 x-3 x^2+e^{5-e^{\frac {1}{6} (18-\log (4 x+2 x^2))}-x} (-6 x-3 x^2+e^{\frac {1}{6} (18-\log (4 x+2 x^2))} (1+x))}{6 x+3 x^2} \, dx\)

Optimal. Leaf size=27 \[ e^{5-x-\frac {e^3}{\sqrt [6]{x (4+2 x)}}}-x \]

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Rubi [F]  time = 10.81, 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 {-6 x-3 x^2+e^{5-e^{\frac {1}{6} \left (18-\log \left (4 x+2 x^2\right )\right )}-x} \left (-6 x-3 x^2+e^{\frac {1}{6} \left (18-\log \left (4 x+2 x^2\right )\right )} (1+x)\right )}{6 x+3 x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-6*x - 3*x^2 + E^(5 - E^((18 - Log[4*x + 2*x^2])/6) - x)*(-6*x - 3*x^2 + E^((18 - Log[4*x + 2*x^2])/6)*(1
+ x)))/(6*x + 3*x^2),x]

[Out]

-x - Defer[Int][E^(5 - x - E^3/(2^(1/6)*(x*(2 + x))^(1/6))), x] + (2^(5/6)*x^(1/6)*(2 + x)^(1/6)*Defer[Subst][
Defer[Int][E^(8 - x^6 - E^3/(2^(1/6)*(x^6*(2 + x^6))^(1/6)))/(x^2*(2 + x^6)^(7/6)), x], x, x^(1/6)])/(2*x + x^
2)^(1/6) + (2^(5/6)*x^(1/6)*(2 + x)^(1/6)*Defer[Subst][Defer[Int][(E^(8 - x^6 - E^3/(2^(1/6)*(x^6*(2 + x^6))^(
1/6)))*x^4)/(2 + x^6)^(7/6), x], x, x^(1/6)])/(2*x + x^2)^(1/6)

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-6 x-3 x^2+e^{5-e^{\frac {1}{6} \left (18-\log \left (4 x+2 x^2\right )\right )}-x} \left (-6 x-3 x^2+e^{\frac {1}{6} \left (18-\log \left (4 x+2 x^2\right )\right )} (1+x)\right )}{x (6+3 x)} \, dx\\ &=\int \left (-e^{-x-\frac {e^3}{\sqrt [6]{2} \sqrt [6]{x (2+x)}}} \left (e^5+e^{x+\frac {e^3}{\sqrt [6]{2} \sqrt [6]{x (2+x)}}}\right )+\frac {e^{8-x-\frac {e^3}{\sqrt [6]{2} \sqrt [6]{x (2+x)}}} (1+x)}{3 \sqrt [6]{2} x (2+x) \sqrt [6]{2 x+x^2}}\right ) \, dx\\ &=\frac {\int \frac {e^{8-x-\frac {e^3}{\sqrt [6]{2} \sqrt [6]{x (2+x)}}} (1+x)}{x (2+x) \sqrt [6]{2 x+x^2}} \, dx}{3 \sqrt [6]{2}}-\int e^{-x-\frac {e^3}{\sqrt [6]{2} \sqrt [6]{x (2+x)}}} \left (e^5+e^{x+\frac {e^3}{\sqrt [6]{2} \sqrt [6]{x (2+x)}}}\right ) \, dx\\ &=\frac {\int \frac {e^{8-x-\frac {e^3}{\sqrt [6]{2} \sqrt [6]{x (2+x)}}} (1+x)}{\left (2 x+x^2\right )^{7/6}} \, dx}{3 \sqrt [6]{2}}-\int \left (1+e^{5-x-\frac {e^3}{\sqrt [6]{2} \sqrt [6]{x (2+x)}}}\right ) \, dx\\ &=-x+\frac {\left (\sqrt [6]{x} \sqrt [6]{2+x}\right ) \int \frac {e^{8-x-\frac {e^3}{\sqrt [6]{2} \sqrt [6]{x (2+x)}}} (1+x)}{x^{7/6} (2+x)^{7/6}} \, dx}{3 \sqrt [6]{2} \sqrt [6]{2 x+x^2}}-\int e^{5-x-\frac {e^3}{\sqrt [6]{2} \sqrt [6]{x (2+x)}}} \, dx\\ &=-x+\frac {\left (2^{5/6} \sqrt [6]{x} \sqrt [6]{2+x}\right ) \operatorname {Subst}\left (\int \frac {e^{8-x^6-\frac {e^3}{\sqrt [6]{2} \sqrt [6]{x^6 \left (2+x^6\right )}}} \left (1+x^6\right )}{x^2 \left (2+x^6\right )^{7/6}} \, dx,x,\sqrt [6]{x}\right )}{\sqrt [6]{2 x+x^2}}-\int e^{5-x-\frac {e^3}{\sqrt [6]{2} \sqrt [6]{x (2+x)}}} \, dx\\ &=-x+\frac {\left (2^{5/6} \sqrt [6]{x} \sqrt [6]{2+x}\right ) \operatorname {Subst}\left (\int \left (\frac {e^{8-x^6-\frac {e^3}{\sqrt [6]{2} \sqrt [6]{x^6 \left (2+x^6\right )}}}}{x^2 \left (2+x^6\right )^{7/6}}+\frac {e^{8-x^6-\frac {e^3}{\sqrt [6]{2} \sqrt [6]{x^6 \left (2+x^6\right )}}} x^4}{\left (2+x^6\right )^{7/6}}\right ) \, dx,x,\sqrt [6]{x}\right )}{\sqrt [6]{2 x+x^2}}-\int e^{5-x-\frac {e^3}{\sqrt [6]{2} \sqrt [6]{x (2+x)}}} \, dx\\ &=-x+\frac {\left (2^{5/6} \sqrt [6]{x} \sqrt [6]{2+x}\right ) \operatorname {Subst}\left (\int \frac {e^{8-x^6-\frac {e^3}{\sqrt [6]{2} \sqrt [6]{x^6 \left (2+x^6\right )}}}}{x^2 \left (2+x^6\right )^{7/6}} \, dx,x,\sqrt [6]{x}\right )}{\sqrt [6]{2 x+x^2}}+\frac {\left (2^{5/6} \sqrt [6]{x} \sqrt [6]{2+x}\right ) \operatorname {Subst}\left (\int \frac {e^{8-x^6-\frac {e^3}{\sqrt [6]{2} \sqrt [6]{x^6 \left (2+x^6\right )}}} x^4}{\left (2+x^6\right )^{7/6}} \, dx,x,\sqrt [6]{x}\right )}{\sqrt [6]{2 x+x^2}}-\int e^{5-x-\frac {e^3}{\sqrt [6]{2} \sqrt [6]{x (2+x)}}} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [F]  time = 2.15, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-6 x-3 x^2+e^{5-e^{\frac {1}{6} \left (18-\log \left (4 x+2 x^2\right )\right )}-x} \left (-6 x-3 x^2+e^{\frac {1}{6} \left (18-\log \left (4 x+2 x^2\right )\right )} (1+x)\right )}{6 x+3 x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(-6*x - 3*x^2 + E^(5 - E^((18 - Log[4*x + 2*x^2])/6) - x)*(-6*x - 3*x^2 + E^((18 - Log[4*x + 2*x^2])
/6)*(1 + x)))/(6*x + 3*x^2),x]

[Out]

Integrate[(-6*x - 3*x^2 + E^(5 - E^((18 - Log[4*x + 2*x^2])/6) - x)*(-6*x - 3*x^2 + E^((18 - Log[4*x + 2*x^2])
/6)*(1 + x)))/(6*x + 3*x^2), x]

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fricas [A]  time = 0.62, size = 44, normalized size = 1.63 \begin {gather*} -x + e^{\left (-\frac {2 \, x^{3} - 6 \, x^{2} + {\left (2 \, x^{2} + 4 \, x\right )}^{\frac {5}{6}} e^{3} - 20 \, x}{2 \, {\left (x^{2} + 2 \, x\right )}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x + e^(-1/2*(2*x^3 - 6*x^2 + (2*x^2 + 4*x)^(5/6)*e^3 - 20*x)/(x^2 + 2*x))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {3 \, x^{2} + {\left (3 \, x^{2} - {\left (x + 1\right )} e^{\left (-\frac {1}{6} \, \log \left (2 \, x^{2} + 4 \, x\right ) + 3\right )} + 6 \, x\right )} e^{\left (-x - e^{\left (-\frac {1}{6} \, \log \left (2 \, x^{2} + 4 \, x\right ) + 3\right )} + 5\right )} + 6 \, x}{3 \, {\left (x^{2} + 2 \, x\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {\left (\left (x +1\right ) {\mathrm e}^{-\frac {\ln \left (2 x^{2}+4 x \right )}{6}+3}-3 x^{2}-6 x \right ) {\mathrm e}^{-{\mathrm e}^{-\frac {\ln \left (2 x^{2}+4 x \right )}{6}+3}+5-x}-3 x^{2}-6 x}{3 x^{2}+6 x}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.09, size = 25, normalized size = 0.93 \begin {gather*} -x + e^{\left (-\frac {2^{\frac {5}{6}} e^{3}}{2 \, {\left (x + 2\right )}^{\frac {1}{6}} x^{\frac {1}{6}}} - x + 5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int -\frac {6\,x+3\,x^2+{\mathrm {e}}^{5-{\mathrm {e}}^{3-\frac {\ln \left (2\,x^2+4\,x\right )}{6}}-x}\,\left (6\,x-{\mathrm {e}}^{3-\frac {\ln \left (2\,x^2+4\,x\right )}{6}}\,\left (x+1\right )+3\,x^2\right )}{3\,x^2+6\,x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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