3.28.60 \(\int \frac {23-2 x+529 x^2+108 e^{24+3 x} x^2+9 e^{32+4 x} x^2-46 x^3+x^4+e^{8+x} (18+18 x+828 x^2-36 x^3)+e^{16+2 x} (3+6 x+462 x^2-6 x^3)}{529 x^2+108 e^{24+3 x} x^2+9 e^{32+4 x} x^2-46 x^3+x^4+e^{8+x} (828 x^2-36 x^3)+e^{16+2 x} (462 x^2-6 x^3)} \, dx\)

Optimal. Leaf size=22 \[ x+\frac {1}{x \left (4-3 \left (3+e^{8+x}\right )^2+x\right )} \]

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Rubi [F]  time = 2.61, 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 {23-2 x+529 x^2+108 e^{24+3 x} x^2+9 e^{32+4 x} x^2-46 x^3+x^4+e^{8+x} \left (18+18 x+828 x^2-36 x^3\right )+e^{16+2 x} \left (3+6 x+462 x^2-6 x^3\right )}{529 x^2+108 e^{24+3 x} x^2+9 e^{32+4 x} x^2-46 x^3+x^4+e^{8+x} \left (828 x^2-36 x^3\right )+e^{16+2 x} \left (462 x^2-6 x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(23 - 2*x + 529*x^2 + 108*E^(24 + 3*x)*x^2 + 9*E^(32 + 4*x)*x^2 - 46*x^3 + x^4 + E^(8 + x)*(18 + 18*x + 82
8*x^2 - 36*x^3) + E^(16 + 2*x)*(3 + 6*x + 462*x^2 - 6*x^3))/(529*x^2 + 108*E^(24 + 3*x)*x^2 + 9*E^(32 + 4*x)*x
^2 - 46*x^3 + x^4 + E^(8 + x)*(828*x^2 - 36*x^3) + E^(16 + 2*x)*(462*x^2 - 6*x^3)),x]

[Out]

x + 2*Defer[Int][(23 + 18*E^(8 + x) + 3*E^(16 + 2*x) - x)^(-2), x] + Defer[Int][1/((23 + 18*E^(8 + x) + 3*E^(1
6 + 2*x) - x)*x^2), x] - 47*Defer[Int][1/((23 + 18*E^(8 + x) + 3*E^(16 + 2*x) - x)^2*x), x] - 18*Defer[Int][E^
(8 + x)/((23 + 18*E^(8 + x) + 3*E^(16 + 2*x) - x)^2*x), x] + 2*Defer[Int][1/((23 + 18*E^(8 + x) + 3*E^(16 + 2*
x) - x)*x), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {23-2 x+529 x^2+108 e^{3 (8+x)} x^2+9 e^{4 (8+x)} x^2-46 x^3+x^4+e^{2 (8+x)} \left (3+6 x+462 x^2-6 x^3\right )+18 e^{8+x} \left (1+x+46 x^2-2 x^3\right )}{\left (23+18 e^{8+x}+3 e^{2 (8+x)}-x\right )^2 x^2} \, dx\\ &=\int \left (1-\frac {47+18 e^{8+x}-2 x}{\left (23+18 e^{8+x}+3 e^{16+2 x}-x\right )^2 x}+\frac {1+2 x}{\left (23+18 e^{8+x}+3 e^{16+2 x}-x\right ) x^2}\right ) \, dx\\ &=x-\int \frac {47+18 e^{8+x}-2 x}{\left (23+18 e^{8+x}+3 e^{16+2 x}-x\right )^2 x} \, dx+\int \frac {1+2 x}{\left (23+18 e^{8+x}+3 e^{16+2 x}-x\right ) x^2} \, dx\\ &=x-\int \left (-\frac {2}{\left (23+18 e^{8+x}+3 e^{16+2 x}-x\right )^2}+\frac {47}{\left (23+18 e^{8+x}+3 e^{16+2 x}-x\right )^2 x}+\frac {18 e^{8+x}}{\left (23+18 e^{8+x}+3 e^{16+2 x}-x\right )^2 x}\right ) \, dx+\int \left (\frac {1}{\left (23+18 e^{8+x}+3 e^{16+2 x}-x\right ) x^2}+\frac {2}{\left (23+18 e^{8+x}+3 e^{16+2 x}-x\right ) x}\right ) \, dx\\ &=x+2 \int \frac {1}{\left (23+18 e^{8+x}+3 e^{16+2 x}-x\right )^2} \, dx+2 \int \frac {1}{\left (23+18 e^{8+x}+3 e^{16+2 x}-x\right ) x} \, dx-18 \int \frac {e^{8+x}}{\left (23+18 e^{8+x}+3 e^{16+2 x}-x\right )^2 x} \, dx-47 \int \frac {1}{\left (23+18 e^{8+x}+3 e^{16+2 x}-x\right )^2 x} \, dx+\int \frac {1}{\left (23+18 e^{8+x}+3 e^{16+2 x}-x\right ) x^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.09, size = 30, normalized size = 1.36 \begin {gather*} -\frac {1}{\left (23+18 e^{8+x}+3 e^{16+2 x}-x\right ) x}+x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(23 - 2*x + 529*x^2 + 108*E^(24 + 3*x)*x^2 + 9*E^(32 + 4*x)*x^2 - 46*x^3 + x^4 + E^(8 + x)*(18 + 18*
x + 828*x^2 - 36*x^3) + E^(16 + 2*x)*(3 + 6*x + 462*x^2 - 6*x^3))/(529*x^2 + 108*E^(24 + 3*x)*x^2 + 9*E^(32 +
4*x)*x^2 - 46*x^3 + x^4 + E^(8 + x)*(828*x^2 - 36*x^3) + E^(16 + 2*x)*(462*x^2 - 6*x^3)),x]

[Out]

-(1/((23 + 18*E^(8 + x) + 3*E^(16 + 2*x) - x)*x)) + x

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fricas [B]  time = 0.49, size = 56, normalized size = 2.55 \begin {gather*} \frac {x^{3} - 3 \, x^{2} e^{\left (2 \, x + 16\right )} - 18 \, x^{2} e^{\left (x + 8\right )} - 23 \, x^{2} + 1}{x^{2} - 3 \, x e^{\left (2 \, x + 16\right )} - 18 \, x e^{\left (x + 8\right )} - 23 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((9*x^2*exp(4)^8*exp(x)^4+108*x^2*exp(4)^6*exp(x)^3+(-6*x^3+462*x^2+6*x+3)*exp(4)^4*exp(x)^2+(-36*x^3
+828*x^2+18*x+18)*exp(4)^2*exp(x)+x^4-46*x^3+529*x^2-2*x+23)/(9*x^2*exp(4)^8*exp(x)^4+108*x^2*exp(4)^6*exp(x)^
3+(-6*x^3+462*x^2)*exp(4)^4*exp(x)^2+(-36*x^3+828*x^2)*exp(4)^2*exp(x)+x^4-46*x^3+529*x^2),x, algorithm="frica
s")

[Out]

(x^3 - 3*x^2*e^(2*x + 16) - 18*x^2*e^(x + 8) - 23*x^2 + 1)/(x^2 - 3*x*e^(2*x + 16) - 18*x*e^(x + 8) - 23*x)

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giac [B]  time = 0.85, size = 56, normalized size = 2.55 \begin {gather*} \frac {x^{3} - 3 \, x^{2} e^{\left (2 \, x + 16\right )} - 18 \, x^{2} e^{\left (x + 8\right )} - 23 \, x^{2} + 2}{x^{2} - 3 \, x e^{\left (2 \, x + 16\right )} - 18 \, x e^{\left (x + 8\right )} - 23 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((9*x^2*exp(4)^8*exp(x)^4+108*x^2*exp(4)^6*exp(x)^3+(-6*x^3+462*x^2+6*x+3)*exp(4)^4*exp(x)^2+(-36*x^3
+828*x^2+18*x+18)*exp(4)^2*exp(x)+x^4-46*x^3+529*x^2-2*x+23)/(9*x^2*exp(4)^8*exp(x)^4+108*x^2*exp(4)^6*exp(x)^
3+(-6*x^3+462*x^2)*exp(4)^4*exp(x)^2+(-36*x^3+828*x^2)*exp(4)^2*exp(x)+x^4-46*x^3+529*x^2),x, algorithm="giac"
)

[Out]

(x^3 - 3*x^2*e^(2*x + 16) - 18*x^2*e^(x + 8) - 23*x^2 + 2)/(x^2 - 3*x*e^(2*x + 16) - 18*x*e^(x + 8) - 23*x)

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maple [A]  time = 0.34, size = 29, normalized size = 1.32

method	result	size
risch	\(x -\frac {1}{x \left (3 \,{\mathrm e}^{2 x +16}+18 \,{\mathrm e}^{x +8}-x +23\right )}\)	\(29\)
norman	\(\frac {-1+529 x +414 \,{\mathrm e}^{8} {\mathrm e}^{x} x +69 x \,{\mathrm e}^{16} {\mathrm e}^{2 x}-x^{3}+18 \,{\mathrm e}^{8} {\mathrm e}^{x} x^{2}+3 \,{\mathrm e}^{16} {\mathrm e}^{2 x} x^{2}}{x \left (3 \,{\mathrm e}^{16} {\mathrm e}^{2 x}+18 \,{\mathrm e}^{8} {\mathrm e}^{x}-x +23\right )}\)	\(84\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((9*x^2*exp(4)^8*exp(x)^4+108*x^2*exp(4)^6*exp(x)^3+(-6*x^3+462*x^2+6*x+3)*exp(4)^4*exp(x)^2+(-36*x^3+828*x
^2+18*x+18)*exp(4)^2*exp(x)+x^4-46*x^3+529*x^2-2*x+23)/(9*x^2*exp(4)^8*exp(x)^4+108*x^2*exp(4)^6*exp(x)^3+(-6*
x^3+462*x^2)*exp(4)^4*exp(x)^2+(-36*x^3+828*x^2)*exp(4)^2*exp(x)+x^4-46*x^3+529*x^2),x,method=_RETURNVERBOSE)

[Out]

x-1/x/(3*exp(2*x+16)+18*exp(x+8)-x+23)

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maxima [B]  time = 0.56, size = 56, normalized size = 2.55 \begin {gather*} \frac {x^{3} - 3 \, x^{2} e^{\left (2 \, x + 16\right )} - 18 \, x^{2} e^{\left (x + 8\right )} - 23 \, x^{2} + 1}{x^{2} - 3 \, x e^{\left (2 \, x + 16\right )} - 18 \, x e^{\left (x + 8\right )} - 23 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((9*x^2*exp(4)^8*exp(x)^4+108*x^2*exp(4)^6*exp(x)^3+(-6*x^3+462*x^2+6*x+3)*exp(4)^4*exp(x)^2+(-36*x^3
+828*x^2+18*x+18)*exp(4)^2*exp(x)+x^4-46*x^3+529*x^2-2*x+23)/(9*x^2*exp(4)^8*exp(x)^4+108*x^2*exp(4)^6*exp(x)^
3+(-6*x^3+462*x^2)*exp(4)^4*exp(x)^2+(-36*x^3+828*x^2)*exp(4)^2*exp(x)+x^4-46*x^3+529*x^2),x, algorithm="maxim
a")

[Out]

(x^3 - 3*x^2*e^(2*x + 16) - 18*x^2*e^(x + 8) - 23*x^2 + 1)/(x^2 - 3*x*e^(2*x + 16) - 18*x*e^(x + 8) - 23*x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} \int \frac {{\mathrm {e}}^{x+8}\,\left (-36\,x^3+828\,x^2+18\,x+18\right )-2\,x+{\mathrm {e}}^{2\,x+16}\,\left (-6\,x^3+462\,x^2+6\,x+3\right )+108\,x^2\,{\mathrm {e}}^{3\,x+24}+9\,x^2\,{\mathrm {e}}^{4\,x+32}+529\,x^2-46\,x^3+x^4+23}{{\mathrm {e}}^{x+8}\,\left (828\,x^2-36\,x^3\right )+{\mathrm {e}}^{2\,x+16}\,\left (462\,x^2-6\,x^3\right )+108\,x^2\,{\mathrm {e}}^{3\,x+24}+9\,x^2\,{\mathrm {e}}^{4\,x+32}+529\,x^2-46\,x^3+x^4} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((529*x^2 - 2*x - 46*x^3 + x^4 + exp(8)*exp(x)*(18*x + 828*x^2 - 36*x^3 + 18) + exp(2*x)*exp(16)*(6*x + 462
*x^2 - 6*x^3 + 3) + 108*x^2*exp(3*x)*exp(24) + 9*x^2*exp(4*x)*exp(32) + 23)/(529*x^2 - 46*x^3 + x^4 + exp(8)*e
xp(x)*(828*x^2 - 36*x^3) + exp(2*x)*exp(16)*(462*x^2 - 6*x^3) + 108*x^2*exp(3*x)*exp(24) + 9*x^2*exp(4*x)*exp(
32)),x)

[Out]

int((exp(x + 8)*(18*x + 828*x^2 - 36*x^3 + 18) - 2*x + exp(2*x + 16)*(6*x + 462*x^2 - 6*x^3 + 3) + 108*x^2*exp
(3*x + 24) + 9*x^2*exp(4*x + 32) + 529*x^2 - 46*x^3 + x^4 + 23)/(exp(x + 8)*(828*x^2 - 36*x^3) + exp(2*x + 16)
*(462*x^2 - 6*x^3) + 108*x^2*exp(3*x + 24) + 9*x^2*exp(4*x + 32) + 529*x^2 - 46*x^3 + x^4), x)

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sympy [A]  time = 0.22, size = 31, normalized size = 1.41 \begin {gather*} x - \frac {1}{- x^{2} + 3 x e^{16} e^{2 x} + 18 x e^{8} e^{x} + 23 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((9*x**2*exp(4)**8*exp(x)**4+108*x**2*exp(4)**6*exp(x)**3+(-6*x**3+462*x**2+6*x+3)*exp(4)**4*exp(x)**
2+(-36*x**3+828*x**2+18*x+18)*exp(4)**2*exp(x)+x**4-46*x**3+529*x**2-2*x+23)/(9*x**2*exp(4)**8*exp(x)**4+108*x
**2*exp(4)**6*exp(x)**3+(-6*x**3+462*x**2)*exp(4)**4*exp(x)**2+(-36*x**3+828*x**2)*exp(4)**2*exp(x)+x**4-46*x*
*3+529*x**2),x)

[Out]

x - 1/(-x**2 + 3*x*exp(16)*exp(2*x) + 18*x*exp(8)*exp(x) + 23*x)

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