3.64.45 \(\int \frac {48 x^2+12 x^3+e^{-1+x} (-12 x^2+8 x^3)+e^x (-64+e^{-3+3 x}+e^{-2+2 x} (-12-9 x)-144 x-108 x^2-27 x^3+e^{-1+x} (48+72 x+27 x^2))}{-256+4 e^{-3+3 x}+e^{-2+2 x} (-48-36 x)-576 x-432 x^2-108 x^3+e^{-1+x} (192+288 x+108 x^2)} \, dx\)

Optimal. Leaf size=26 \[ 3+\frac {e^x}{4}-\frac {x^3}{\left (-4+e^{-1+x}-3 x\right )^2} \]

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Rubi [F]  time = 1.45, 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 {48 x^2+12 x^3+e^{-1+x} \left (-12 x^2+8 x^3\right )+e^x \left (-64+e^{-3+3 x}+e^{-2+2 x} (-12-9 x)-144 x-108 x^2-27 x^3+e^{-1+x} \left (48+72 x+27 x^2\right )\right )}{-256+4 e^{-3+3 x}+e^{-2+2 x} (-48-36 x)-576 x-432 x^2-108 x^3+e^{-1+x} \left (192+288 x+108 x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(48*x^2 + 12*x^3 + E^(-1 + x)*(-12*x^2 + 8*x^3) + E^x*(-64 + E^(-3 + 3*x) + E^(-2 + 2*x)*(-12 - 9*x) - 144
*x - 108*x^2 - 27*x^3 + E^(-1 + x)*(48 + 72*x + 27*x^2)))/(-256 + 4*E^(-3 + 3*x) + E^(-2 + 2*x)*(-48 - 36*x) -
576*x - 432*x^2 - 108*x^3 + E^(-1 + x)*(192 + 288*x + 108*x^2)),x]

[Out]

E^x/4 - 2*E^3*Defer[Int][x^3/(4*E - E^x + 3*E*x)^3, x] - 6*E^3*Defer[Int][x^4/(4*E - E^x + 3*E*x)^3, x] - 3*E^
2*Defer[Int][x^2/(4*E - E^x + 3*E*x)^2, x] + 2*E^2*Defer[Int][x^3/(4*E - E^x + 3*E*x)^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{4 x}+12 e^3 x^2 (4+x)+4 e^{2+x} x^2 (-3+2 x)-3 e^{1+3 x} (4+3 x)+3 e^{2+2 x} (4+3 x)^2-e^{3+x} (4+3 x)^3}{4 \left (e^x-e (4+3 x)\right )^3} \, dx\\ &=\frac {1}{4} \int \frac {e^{4 x}+12 e^3 x^2 (4+x)+4 e^{2+x} x^2 (-3+2 x)-3 e^{1+3 x} (4+3 x)+3 e^{2+2 x} (4+3 x)^2-e^{3+x} (4+3 x)^3}{\left (e^x-e (4+3 x)\right )^3} \, dx\\ &=\frac {1}{4} \int \left (e^x-\frac {8 e^3 x^3 (1+3 x)}{\left (4 e-e^x+3 e x\right )^3}+\frac {4 e^2 x^2 (-3+2 x)}{\left (4 e-e^x+3 e x\right )^2}\right ) \, dx\\ &=\frac {\int e^x \, dx}{4}+e^2 \int \frac {x^2 (-3+2 x)}{\left (4 e-e^x+3 e x\right )^2} \, dx-\left (2 e^3\right ) \int \frac {x^3 (1+3 x)}{\left (4 e-e^x+3 e x\right )^3} \, dx\\ &=\frac {e^x}{4}+e^2 \int \left (-\frac {3 x^2}{\left (4 e-e^x+3 e x\right )^2}+\frac {2 x^3}{\left (4 e-e^x+3 e x\right )^2}\right ) \, dx-\left (2 e^3\right ) \int \left (\frac {x^3}{\left (4 e-e^x+3 e x\right )^3}+\frac {3 x^4}{\left (4 e-e^x+3 e x\right )^3}\right ) \, dx\\ &=\frac {e^x}{4}+\left (2 e^2\right ) \int \frac {x^3}{\left (4 e-e^x+3 e x\right )^2} \, dx-\left (3 e^2\right ) \int \frac {x^2}{\left (4 e-e^x+3 e x\right )^2} \, dx-\left (2 e^3\right ) \int \frac {x^3}{\left (4 e-e^x+3 e x\right )^3} \, dx-\left (6 e^3\right ) \int \frac {x^4}{\left (4 e-e^x+3 e x\right )^3} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.08, size = 29, normalized size = 1.12 \begin {gather*} \frac {1}{4} \left (e^x-\frac {4 e^2 x^3}{\left (-4 e+e^x-3 e x\right )^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(48*x^2 + 12*x^3 + E^(-1 + x)*(-12*x^2 + 8*x^3) + E^x*(-64 + E^(-3 + 3*x) + E^(-2 + 2*x)*(-12 - 9*x)
- 144*x - 108*x^2 - 27*x^3 + E^(-1 + x)*(48 + 72*x + 27*x^2)))/(-256 + 4*E^(-3 + 3*x) + E^(-2 + 2*x)*(-48 - 3
6*x) - 576*x - 432*x^2 - 108*x^3 + E^(-1 + x)*(192 + 288*x + 108*x^2)),x]

[Out]

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

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fricas [B]  time = 0.70, size = 76, normalized size = 2.92 \begin {gather*} -\frac {4 \, x^{3} e^{2} + 2 \, {\left (3 \, x + 4\right )} e^{\left (2 \, x + 1\right )} - {\left (9 \, x^{2} + 24 \, x + 16\right )} e^{\left (x + 2\right )} - e^{\left (3 \, x\right )}}{4 \, {\left ({\left (9 \, x^{2} + 24 \, x + 16\right )} e^{2} - 2 \, {\left (3 \, x + 4\right )} e^{\left (x + 1\right )} + e^{\left (2 \, x\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((exp(x-1)^3+(-9*x-12)*exp(x-1)^2+(27*x^2+72*x+48)*exp(x-1)-27*x^3-108*x^2-144*x-64)*exp(x)+(8*x^3-1
2*x^2)*exp(x-1)+12*x^3+48*x^2)/(4*exp(x-1)^3+(-36*x-48)*exp(x-1)^2+(108*x^2+288*x+192)*exp(x-1)-108*x^3-432*x^
2-576*x-256),x, algorithm="fricas")

[Out]

-1/4*(4*x^3*e^2 + 2*(3*x + 4)*e^(2*x + 1) - (9*x^2 + 24*x + 16)*e^(x + 2) - e^(3*x))/((9*x^2 + 24*x + 16)*e^2
- 2*(3*x + 4)*e^(x + 1) + e^(2*x))

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giac [B]  time = 0.27, size = 91, normalized size = 3.50 \begin {gather*} -\frac {4 \, x^{3} e^{2} - 9 \, x^{2} e^{\left (x + 2\right )} + 6 \, x e^{\left (2 \, x + 1\right )} - 24 \, x e^{\left (x + 2\right )} - e^{\left (3 \, x\right )} + 8 \, e^{\left (2 \, x + 1\right )} - 16 \, e^{\left (x + 2\right )}}{4 \, {\left (9 \, x^{2} e^{2} + 24 \, x e^{2} - 6 \, x e^{\left (x + 1\right )} + 16 \, e^{2} + e^{\left (2 \, x\right )} - 8 \, e^{\left (x + 1\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((exp(x-1)^3+(-9*x-12)*exp(x-1)^2+(27*x^2+72*x+48)*exp(x-1)-27*x^3-108*x^2-144*x-64)*exp(x)+(8*x^3-1
2*x^2)*exp(x-1)+12*x^3+48*x^2)/(4*exp(x-1)^3+(-36*x-48)*exp(x-1)^2+(108*x^2+288*x+192)*exp(x-1)-108*x^3-432*x^
2-576*x-256),x, algorithm="giac")

[Out]

-1/4*(4*x^3*e^2 - 9*x^2*e^(x + 2) + 6*x*e^(2*x + 1) - 24*x*e^(x + 2) - e^(3*x) + 8*e^(2*x + 1) - 16*e^(x + 2))
/(9*x^2*e^2 + 24*x*e^2 - 6*x*e^(x + 1) + 16*e^2 + e^(2*x) - 8*e^(x + 1))

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maple [A]  time = 0.48, size = 22, normalized size = 0.85

method	result	size
risch	\(\frac {{\mathrm e}^{x}}{4}-\frac {x^{3}}{\left ({\mathrm e}^{x -1}-4-3 x \right )^{2}}\)	\(22\)
norman	\(\frac {\left (-12 \,{\mathrm e}^{4} {\mathrm e}^{x}+18 x^{2} {\mathrm e}^{5}+48 x \,{\mathrm e}^{5}-6 x \,{\mathrm e}^{4} {\mathrm e}^{x}-x^{3} {\mathrm e}^{4}+\frac {{\mathrm e}^{2} {\mathrm e}^{3 x}}{4}-\frac {3 x \,{\mathrm e}^{3} {\mathrm e}^{2 x}}{2}+\frac {9 x^{2} {\mathrm e}^{4} {\mathrm e}^{x}}{4}+32 \,{\mathrm e}^{5}\right ) {\mathrm e}^{-2}}{\left (3 x \,{\mathrm e}+4 \,{\mathrm e}-{\mathrm e}^{x}\right )^{2}}\)	\(103\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((exp(x-1)^3+(-9*x-12)*exp(x-1)^2+(27*x^2+72*x+48)*exp(x-1)-27*x^3-108*x^2-144*x-64)*exp(x)+(8*x^3-12*x^2)
*exp(x-1)+12*x^3+48*x^2)/(4*exp(x-1)^3+(-36*x-48)*exp(x-1)^2+(108*x^2+288*x+192)*exp(x-1)-108*x^3-432*x^2-576*
x-256),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [B]  time = 0.44, size = 90, normalized size = 3.46 \begin {gather*} -\frac {4 \, x^{3} e^{2} + 2 \, {\left (3 \, x e + 4 \, e\right )} e^{\left (2 \, x\right )} - {\left (9 \, x^{2} e^{2} + 24 \, x e^{2} + 16 \, e^{2}\right )} e^{x} - e^{\left (3 \, x\right )}}{4 \, {\left (9 \, x^{2} e^{2} + 24 \, x e^{2} - 2 \, {\left (3 \, x e + 4 \, e\right )} e^{x} + 16 \, e^{2} + e^{\left (2 \, x\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((exp(x-1)^3+(-9*x-12)*exp(x-1)^2+(27*x^2+72*x+48)*exp(x-1)-27*x^3-108*x^2-144*x-64)*exp(x)+(8*x^3-1
2*x^2)*exp(x-1)+12*x^3+48*x^2)/(4*exp(x-1)^3+(-36*x-48)*exp(x-1)^2+(108*x^2+288*x+192)*exp(x-1)-108*x^3-432*x^
2-576*x-256),x, algorithm="maxima")

[Out]

-1/4*(4*x^3*e^2 + 2*(3*x*e + 4*e)*e^(2*x) - (9*x^2*e^2 + 24*x*e^2 + 16*e^2)*e^x - e^(3*x))/(9*x^2*e^2 + 24*x*e
^2 - 2*(3*x*e + 4*e)*e^x + 16*e^2 + e^(2*x))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {{\mathrm {e}}^x\,\left (144\,x-{\mathrm {e}}^{3\,x-3}-{\mathrm {e}}^{x-1}\,\left (27\,x^2+72\,x+48\right )+{\mathrm {e}}^{2\,x-2}\,\left (9\,x+12\right )+108\,x^2+27\,x^3+64\right )+{\mathrm {e}}^{x-1}\,\left (12\,x^2-8\,x^3\right )-48\,x^2-12\,x^3}{576\,x-4\,{\mathrm {e}}^{3\,x-3}-{\mathrm {e}}^{x-1}\,\left (108\,x^2+288\,x+192\right )+{\mathrm {e}}^{2\,x-2}\,\left (36\,x+48\right )+432\,x^2+108\,x^3+256} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x)*(144*x - exp(3*x - 3) - exp(x - 1)*(72*x + 27*x^2 + 48) + exp(2*x - 2)*(9*x + 12) + 108*x^2 + 27*x
^3 + 64) + exp(x - 1)*(12*x^2 - 8*x^3) - 48*x^2 - 12*x^3)/(576*x - 4*exp(3*x - 3) - exp(x - 1)*(288*x + 108*x^
2 + 192) + exp(2*x - 2)*(36*x + 48) + 432*x^2 + 108*x^3 + 256),x)

[Out]

int((exp(x)*(144*x - exp(3*x - 3) - exp(x - 1)*(72*x + 27*x^2 + 48) + exp(2*x - 2)*(9*x + 12) + 108*x^2 + 27*x
^3 + 64) + exp(x - 1)*(12*x^2 - 8*x^3) - 48*x^2 - 12*x^3)/(576*x - 4*exp(3*x - 3) - exp(x - 1)*(288*x + 108*x^
2 + 192) + exp(2*x - 2)*(36*x + 48) + 432*x^2 + 108*x^3 + 256), 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(((exp(x-1)**3+(-9*x-12)*exp(x-1)**2+(27*x**2+72*x+48)*exp(x-1)-27*x**3-108*x**2-144*x-64)*exp(x)+(8*
x**3-12*x**2)*exp(x-1)+12*x**3+48*x**2)/(4*exp(x-1)**3+(-36*x-48)*exp(x-1)**2+(108*x**2+288*x+192)*exp(x-1)-10
8*x**3-432*x**2-576*x-256),x)

[Out]

Timed out

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