3.37.22 \(\int \frac {-12 x^2-10 x^3+e^{2-2 x} (6 x^2+9 x^3+5 x^4)}{1536+3840 x+2400 x^2+e^{2-2 x} (-1536-3840 x-2400 x^2)+e^{4-4 x} (384+960 x+600 x^2)} \, dx\) [3622]

Optimal. Leaf size=27 \[ -\frac {x^3}{48 \left (2-e^{2-2 x}\right ) (4+5 x)} \]

[Out]

-1/48*x^3/(2-exp(2-2*x))/(4+5*x)

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Rubi [F]
time = 3.50, 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 {-12 x^2-10 x^3+e^{2-2 x} \left (6 x^2+9 x^3+5 x^4\right )}{1536+3840 x+2400 x^2+e^{2-2 x} \left (-1536-3840 x-2400 x^2\right )+e^{4-4 x} \left (384+960 x+600 x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-12*x^2 - 10*x^3 + E^(2 - 2*x)*(6*x^2 + 9*x^3 + 5*x^4))/(1536 + 3840*x + 2400*x^2 + E^(2 - 2*x)*(-1536 -
3840*x - 2400*x^2) + E^(4 - 4*x)*(384 + 960*x + 600*x^2)),x]

[Out]

E^2/(750*(E^2 - 2*E^(2*x))) + x/600 - (E^2*x)/(600*(E^2 - 2*E^(2*x))) - x^2/480 + (E^2*x^2)/(480*(E^2 - 2*E^(2
*x))) - (2*E^2)/(375*(E^2 - 2*E^(2*x))*(4 + 5*x)) + (2*E^2*Defer[Int][1/((-E^2 + 2*E^(2*x))*(4 + 5*x)^2), x])/
75 - (4*Defer[Int][E^(2*x)/((-E^2 + 2*E^(2*x))*(4 + 5*x)^2), x])/75

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{2 x} x^2 \left (-2 e^{2 x} (6+5 x)+e^2 \left (6+9 x+5 x^2\right )\right )}{24 \left (e^2-2 e^{2 x}\right )^2 (4+5 x)^2} \, dx\\ &=\frac {1}{24} \int \frac {e^{2 x} x^2 \left (-2 e^{2 x} (6+5 x)+e^2 \left (6+9 x+5 x^2\right )\right )}{\left (e^2-2 e^{2 x}\right )^2 (4+5 x)^2} \, dx\\ &=\frac {1}{24} \int \left (\frac {e^{2+2 x} x^3}{\left (e^2-2 e^{2 x}\right )^2 (4+5 x)}-\frac {e^{2 x} x^2 (6+5 x)}{\left (-e^2+2 e^{2 x}\right ) (4+5 x)^2}\right ) \, dx\\ &=\frac {1}{24} \int \frac {e^{2+2 x} x^3}{\left (e^2-2 e^{2 x}\right )^2 (4+5 x)} \, dx-\frac {1}{24} \int \frac {e^{2 x} x^2 (6+5 x)}{\left (-e^2+2 e^{2 x}\right ) (4+5 x)^2} \, dx\\ &=-\left (\frac {1}{24} \int \left (-\frac {2 e^{2 x}}{25 \left (-e^2+2 e^{2 x}\right )}+\frac {e^{2 x} x}{5 \left (-e^2+2 e^{2 x}\right )}+\frac {32 e^{2 x}}{25 \left (-e^2+2 e^{2 x}\right ) (4+5 x)^2}\right ) \, dx\right )+\frac {1}{24} \int \left (\frac {16 e^{2+2 x}}{125 \left (-e^2+2 e^{2 x}\right )^2}-\frac {4 e^{2+2 x} x}{25 \left (-e^2+2 e^{2 x}\right )^2}+\frac {e^{2+2 x} x^2}{5 \left (-e^2+2 e^{2 x}\right )^2}-\frac {64 e^{2+2 x}}{125 \left (-e^2+2 e^{2 x}\right )^2 (4+5 x)}\right ) \, dx\\ &=\frac {1}{300} \int \frac {e^{2 x}}{-e^2+2 e^{2 x}} \, dx+\frac {2}{375} \int \frac {e^{2+2 x}}{\left (-e^2+2 e^{2 x}\right )^2} \, dx-\frac {1}{150} \int \frac {e^{2+2 x} x}{\left (-e^2+2 e^{2 x}\right )^2} \, dx-\frac {1}{120} \int \frac {e^{2 x} x}{-e^2+2 e^{2 x}} \, dx+\frac {1}{120} \int \frac {e^{2+2 x} x^2}{\left (-e^2+2 e^{2 x}\right )^2} \, dx-\frac {8}{375} \int \frac {e^{2+2 x}}{\left (-e^2+2 e^{2 x}\right )^2 (4+5 x)} \, dx-\frac {4}{75} \int \frac {e^{2 x}}{\left (-e^2+2 e^{2 x}\right ) (4+5 x)^2} \, dx\\ &=-\frac {1}{480} x \log \left (1-2 e^{-2+2 x}\right )+\frac {1}{600} \text {Subst}\left (\int \frac {1}{-e^2+2 x} \, dx,x,e^{2 x}\right )+\frac {1}{480} \int \log \left (1-2 e^{-2+2 x}\right ) \, dx-\frac {4}{75} \int \frac {e^{2 x}}{\left (-e^2+2 e^{2 x}\right ) (4+5 x)^2} \, dx+\frac {1}{375} \left (2 e^2\right ) \int \frac {e^{2 x}}{\left (-e^2+2 e^{2 x}\right )^2} \, dx-\frac {1}{150} e^2 \int \frac {e^{2 x} x}{\left (-e^2+2 e^{2 x}\right )^2} \, dx+\frac {1}{120} e^2 \int \frac {e^{2 x} x^2}{\left (-e^2+2 e^{2 x}\right )^2} \, dx-\frac {1}{375} \left (8 e^2\right ) \int \frac {e^{2 x}}{\left (-e^2+2 e^{2 x}\right )^2 (4+5 x)} \, dx\\ &=-\frac {e^2 x}{600 \left (e^2-2 e^{2 x}\right )}+\frac {e^2 x^2}{480 \left (e^2-2 e^{2 x}\right )}-\frac {2 e^2}{375 \left (e^2-2 e^{2 x}\right ) (4+5 x)}+\frac {\log \left (e^2-2 e^{2 x}\right )}{1200}-\frac {1}{480} x \log \left (1-2 e^{-2+2 x}\right )+\frac {1}{960} \text {Subst}\left (\int \frac {\log (1-2 x)}{x} \, dx,x,e^{-2+2 x}\right )-\frac {4}{75} \int \frac {e^{2 x}}{\left (-e^2+2 e^{2 x}\right ) (4+5 x)^2} \, dx-\frac {1}{600} e^2 \int \frac {1}{-e^2+2 e^{2 x}} \, dx+\frac {1}{375} e^2 \text {Subst}\left (\int \frac {1}{\left (-e^2+2 x\right )^2} \, dx,x,e^{2 x}\right )+\frac {1}{240} e^2 \int \frac {x}{-e^2+2 e^{2 x}} \, dx+\frac {1}{75} \left (2 e^2\right ) \int \frac {1}{\left (-e^2+2 e^{2 x}\right ) (4+5 x)^2} \, dx\\ &=\frac {e^2}{750 \left (e^2-2 e^{2 x}\right )}-\frac {e^2 x}{600 \left (e^2-2 e^{2 x}\right )}-\frac {x^2}{480}+\frac {e^2 x^2}{480 \left (e^2-2 e^{2 x}\right )}-\frac {2 e^2}{375 \left (e^2-2 e^{2 x}\right ) (4+5 x)}+\frac {\log \left (e^2-2 e^{2 x}\right )}{1200}-\frac {1}{480} x \log \left (1-2 e^{-2+2 x}\right )-\frac {1}{960} \text {Li}_2\left (2 e^{-2+2 x}\right )+\frac {1}{120} \int \frac {e^{2 x} x}{-e^2+2 e^{2 x}} \, dx-\frac {4}{75} \int \frac {e^{2 x}}{\left (-e^2+2 e^{2 x}\right ) (4+5 x)^2} \, dx-\frac {e^2 \text {Subst}\left (\int \frac {1}{x \left (-e^2+2 x\right )} \, dx,x,e^{2 x}\right )}{1200}+\frac {1}{75} \left (2 e^2\right ) \int \frac {1}{\left (-e^2+2 e^{2 x}\right ) (4+5 x)^2} \, dx\\ &=\frac {e^2}{750 \left (e^2-2 e^{2 x}\right )}-\frac {e^2 x}{600 \left (e^2-2 e^{2 x}\right )}-\frac {x^2}{480}+\frac {e^2 x^2}{480 \left (e^2-2 e^{2 x}\right )}-\frac {2 e^2}{375 \left (e^2-2 e^{2 x}\right ) (4+5 x)}+\frac {\log \left (e^2-2 e^{2 x}\right )}{1200}-\frac {1}{960} \text {Li}_2\left (2 e^{-2+2 x}\right )+\frac {\text {Subst}\left (\int \frac {1}{x} \, dx,x,e^{2 x}\right )}{1200}-\frac {1}{600} \text {Subst}\left (\int \frac {1}{-e^2+2 x} \, dx,x,e^{2 x}\right )-\frac {1}{480} \int \log \left (1-2 e^{-2+2 x}\right ) \, dx-\frac {4}{75} \int \frac {e^{2 x}}{\left (-e^2+2 e^{2 x}\right ) (4+5 x)^2} \, dx+\frac {1}{75} \left (2 e^2\right ) \int \frac {1}{\left (-e^2+2 e^{2 x}\right ) (4+5 x)^2} \, dx\\ &=\frac {e^2}{750 \left (e^2-2 e^{2 x}\right )}+\frac {x}{600}-\frac {e^2 x}{600 \left (e^2-2 e^{2 x}\right )}-\frac {x^2}{480}+\frac {e^2 x^2}{480 \left (e^2-2 e^{2 x}\right )}-\frac {2 e^2}{375 \left (e^2-2 e^{2 x}\right ) (4+5 x)}-\frac {1}{960} \text {Li}_2\left (2 e^{-2+2 x}\right )-\frac {1}{960} \text {Subst}\left (\int \frac {\log (1-2 x)}{x} \, dx,x,e^{-2+2 x}\right )-\frac {4}{75} \int \frac {e^{2 x}}{\left (-e^2+2 e^{2 x}\right ) (4+5 x)^2} \, dx+\frac {1}{75} \left (2 e^2\right ) \int \frac {1}{\left (-e^2+2 e^{2 x}\right ) (4+5 x)^2} \, dx\\ &=\frac {e^2}{750 \left (e^2-2 e^{2 x}\right )}+\frac {x}{600}-\frac {e^2 x}{600 \left (e^2-2 e^{2 x}\right )}-\frac {x^2}{480}+\frac {e^2 x^2}{480 \left (e^2-2 e^{2 x}\right )}-\frac {2 e^2}{375 \left (e^2-2 e^{2 x}\right ) (4+5 x)}-\frac {4}{75} \int \frac {e^{2 x}}{\left (-e^2+2 e^{2 x}\right ) (4+5 x)^2} \, dx+\frac {1}{75} \left (2 e^2\right ) \int \frac {1}{\left (-e^2+2 e^{2 x}\right ) (4+5 x)^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 9.60, size = 53, normalized size = 1.96 \begin {gather*} \frac {-24 e^2 (4+5 x)+e^{2 x} \left (192+240 x-125 x^3\right )}{6000 \left (-e^2+2 e^{2 x}\right ) (4+5 x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-12*x^2 - 10*x^3 + E^(2 - 2*x)*(6*x^2 + 9*x^3 + 5*x^4))/(1536 + 3840*x + 2400*x^2 + E^(2 - 2*x)*(-1
536 - 3840*x - 2400*x^2) + E^(4 - 4*x)*(384 + 960*x + 600*x^2)),x]

[Out]

(-24*E^2*(4 + 5*x) + E^(2*x)*(192 + 240*x - 125*x^3))/(6000*(-E^2 + 2*E^(2*x))*(4 + 5*x))

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Maple [A]
time = 0.88, size = 23, normalized size = 0.85

method	result	size
norman	\(\frac {x^{3}}{48 \left ({\mathrm e}^{-2 x +2}-2\right ) \left (4+5 x \right )}\)	\(23\)
risch	\(\frac {x^{3}}{48 \left ({\mathrm e}^{-2 x +2}-2\right ) \left (4+5 x \right )}\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((5*x^4+9*x^3+6*x^2)*exp(-2*x+2)-10*x^3-12*x^2)/((600*x^2+960*x+384)*exp(-2*x+2)^2+(-2400*x^2-3840*x-1536)
*exp(-2*x+2)+2400*x^2+3840*x+1536),x,method=_RETURNVERBOSE)

[Out]

1/48*x^3/(exp(-2*x+2)-2)/(4+5*x)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 50 vs. \(2 (22) = 44\).
time = 0.33, size = 50, normalized size = 1.85 \begin {gather*} \frac {40 \, x e^{2} + {\left (125 \, x^{3} - 80 \, x - 64\right )} e^{\left (2 \, x\right )} + 32 \, e^{2}}{6000 \, {\left (5 \, x e^{2} - 2 \, {\left (5 \, x + 4\right )} e^{\left (2 \, x\right )} + 4 \, e^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*x^4+9*x^3+6*x^2)*exp(2-2*x)-10*x^3-12*x^2)/((600*x^2+960*x+384)*exp(2-2*x)^2+(-2400*x^2-3840*x-1
536)*exp(2-2*x)+2400*x^2+3840*x+1536),x, algorithm="maxima")

[Out]

1/6000*(40*x*e^2 + (125*x^3 - 80*x - 64)*e^(2*x) + 32*e^2)/(5*x*e^2 - 2*(5*x + 4)*e^(2*x) + 4*e^2)

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Fricas [A]
time = 0.36, size = 24, normalized size = 0.89 \begin {gather*} \frac {x^{3}}{48 \, {\left ({\left (5 \, x + 4\right )} e^{\left (-2 \, x + 2\right )} - 10 \, x - 8\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*x^4+9*x^3+6*x^2)*exp(2-2*x)-10*x^3-12*x^2)/((600*x^2+960*x+384)*exp(2-2*x)^2+(-2400*x^2-3840*x-1
536)*exp(2-2*x)+2400*x^2+3840*x+1536),x, algorithm="fricas")

[Out]

1/48*x^3/((5*x + 4)*e^(-2*x + 2) - 10*x - 8)

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Sympy [A]
time = 0.07, size = 20, normalized size = 0.74 \begin {gather*} \frac {25 x^{3}}{- 12000 x + \left (6000 x + 4800\right ) e^{2 - 2 x} - 9600} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*x**4+9*x**3+6*x**2)*exp(2-2*x)-10*x**3-12*x**2)/((600*x**2+960*x+384)*exp(2-2*x)**2+(-2400*x**2-
3840*x-1536)*exp(2-2*x)+2400*x**2+3840*x+1536),x)

[Out]

25*x**3/(-12000*x + (6000*x + 4800)*exp(2 - 2*x) - 9600)

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Giac [A]
time = 0.42, size = 29, normalized size = 1.07 \begin {gather*} \frac {x^{3}}{48 \, {\left (5 \, x e^{\left (-2 \, x + 2\right )} - 10 \, x + 4 \, e^{\left (-2 \, x + 2\right )} - 8\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*x^4+9*x^3+6*x^2)*exp(2-2*x)-10*x^3-12*x^2)/((600*x^2+960*x+384)*exp(2-2*x)^2+(-2400*x^2-3840*x-1
536)*exp(2-2*x)+2400*x^2+3840*x+1536),x, algorithm="giac")

[Out]

1/48*x^3/(5*x*e^(-2*x + 2) - 10*x + 4*e^(-2*x + 2) - 8)

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Mupad [B]
time = 2.20, size = 30, normalized size = 1.11 \begin {gather*} \frac {5\,x^4+4\,x^3}{48\,{\left (5\,x+4\right )}^2\,\left ({\mathrm {e}}^{2-2\,x}-2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(12*x^2 + 10*x^3 - exp(2 - 2*x)*(6*x^2 + 9*x^3 + 5*x^4))/(3840*x + exp(4 - 4*x)*(960*x + 600*x^2 + 384) -
exp(2 - 2*x)*(3840*x + 2400*x^2 + 1536) + 2400*x^2 + 1536),x)

[Out]

(4*x^3 + 5*x^4)/(48*(5*x + 4)^2*(exp(2 - 2*x) - 2))

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