3.17.33 \(\int \frac {e^{12+4 x} (5+10 x)+e^{6+2 x} (-10 x+10 x^2)}{x^4+e^{6+2 x} (18 x^2-2 x^3-2 x^4)+e^{12+4 x} (81-18 x-17 x^2+2 x^3+x^4)} \, dx\)

Optimal. Leaf size=25 \[ \frac {5}{9-x-x^2+e^{-6-2 x} x^2} \]

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Rubi [F]  time = 9.51, 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 {e^{12+4 x} (5+10 x)+e^{6+2 x} \left (-10 x+10 x^2\right )}{x^4+e^{6+2 x} \left (18 x^2-2 x^3-2 x^4\right )+e^{12+4 x} \left (81-18 x-17 x^2+2 x^3+x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(12 + 4*x)*(5 + 10*x) + E^(6 + 2*x)*(-10*x + 10*x^2))/(x^4 + E^(6 + 2*x)*(18*x^2 - 2*x^3 - 2*x^4) + E^(
12 + 4*x)*(81 - 18*x - 17*x^2 + 2*x^3 + x^4)),x]

[Out]

-5*Defer[Int][E^(6 + 2*x)/(x^2 - E^(6 + 2*x)*(-9 + x + x^2))^2, x] + (90*Defer[Int][E^(6 + 2*x)/((-1 + Sqrt[37
] - 2*x)*(x^2 - E^(6 + 2*x)*(-9 + x + x^2))^2), x])/Sqrt[37] + 10*Defer[Int][(E^(6 + 2*x)*x^2)/(x^2 - E^(6 + 2
*x)*(-9 + x + x^2))^2, x] + (95*(37 - Sqrt[37])*Defer[Int][E^(6 + 2*x)/((1 - Sqrt[37] + 2*x)*(x^2 - E^(6 + 2*x
)*(-9 + x + x^2))^2), x])/37 + (90*Defer[Int][E^(6 + 2*x)/((1 + Sqrt[37] + 2*x)*(x^2 - E^(6 + 2*x)*(-9 + x + x
^2))^2), x])/Sqrt[37] + (95*(37 + Sqrt[37])*Defer[Int][E^(6 + 2*x)/((1 + Sqrt[37] + 2*x)*(x^2 - E^(6 + 2*x)*(-
9 + x + x^2))^2), x])/37 + (10*(37 + Sqrt[37])*Defer[Int][E^(6 + 2*x)/((-1 - Sqrt[37] - 2*x)*(x^2 - E^(6 + 2*x
)*(-9 + x + x^2))), x])/37 + (10*Defer[Int][E^(6 + 2*x)/((-1 + Sqrt[37] - 2*x)*(x^2 - E^(6 + 2*x)*(-9 + x + x^
2))), x])/Sqrt[37] + (10*(37 - Sqrt[37])*Defer[Int][E^(6 + 2*x)/((-1 + Sqrt[37] - 2*x)*(x^2 - E^(6 + 2*x)*(-9
+ x + x^2))), x])/37 + (10*Defer[Int][E^(6 + 2*x)/((1 + Sqrt[37] + 2*x)*(x^2 - E^(6 + 2*x)*(-9 + x + x^2))), x
])/Sqrt[37]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {5 e^{6+2 x} \left (2 (-1+x) x+e^{6+2 x} (1+2 x)\right )}{\left (x^2-e^{6+2 x} \left (-9+x+x^2\right )\right )^2} \, dx\\ &=5 \int \frac {e^{6+2 x} \left (2 (-1+x) x+e^{6+2 x} (1+2 x)\right )}{\left (x^2-e^{6+2 x} \left (-9+x+x^2\right )\right )^2} \, dx\\ &=5 \int \left (\frac {e^{6+2 x} (1+2 x)}{\left (-9+x+x^2\right ) \left (-9 e^{6+2 x}+e^{6+2 x} x-x^2+e^{6+2 x} x^2\right )}+\frac {e^{6+2 x} x \left (18-19 x+2 x^2+2 x^3\right )}{\left (-9+x+x^2\right ) \left (-9 e^{6+2 x}+e^{6+2 x} x-x^2+e^{6+2 x} x^2\right )^2}\right ) \, dx\\ &=5 \int \frac {e^{6+2 x} (1+2 x)}{\left (-9+x+x^2\right ) \left (-9 e^{6+2 x}+e^{6+2 x} x-x^2+e^{6+2 x} x^2\right )} \, dx+5 \int \frac {e^{6+2 x} x \left (18-19 x+2 x^2+2 x^3\right )}{\left (-9+x+x^2\right ) \left (-9 e^{6+2 x}+e^{6+2 x} x-x^2+e^{6+2 x} x^2\right )^2} \, dx\\ &=5 \int \frac {e^{6+2 x} (1+2 x)}{\left (9-x-x^2\right ) \left (x^2-e^{6+2 x} \left (-9+x+x^2\right )\right )} \, dx+5 \int \left (-\frac {e^{6+2 x}}{\left (-9 e^{6+2 x}+e^{6+2 x} x-x^2+e^{6+2 x} x^2\right )^2}+\frac {2 e^{6+2 x} x^2}{\left (-9 e^{6+2 x}+e^{6+2 x} x-x^2+e^{6+2 x} x^2\right )^2}+\frac {e^{6+2 x} (-9+19 x)}{\left (-9+x+x^2\right ) \left (-9 e^{6+2 x}+e^{6+2 x} x-x^2+e^{6+2 x} x^2\right )^2}\right ) \, dx\\ &=-\left (5 \int \frac {e^{6+2 x}}{\left (-9 e^{6+2 x}+e^{6+2 x} x-x^2+e^{6+2 x} x^2\right )^2} \, dx\right )+5 \int \frac {e^{6+2 x} (-9+19 x)}{\left (-9+x+x^2\right ) \left (-9 e^{6+2 x}+e^{6+2 x} x-x^2+e^{6+2 x} x^2\right )^2} \, dx+5 \int \left (\frac {e^{6+2 x}}{\left (-9+x+x^2\right ) \left (-9 e^{6+2 x}+e^{6+2 x} x-x^2+e^{6+2 x} x^2\right )}+\frac {2 e^{6+2 x} x}{\left (-9+x+x^2\right ) \left (-9 e^{6+2 x}+e^{6+2 x} x-x^2+e^{6+2 x} x^2\right )}\right ) \, dx+10 \int \frac {e^{6+2 x} x^2}{\left (-9 e^{6+2 x}+e^{6+2 x} x-x^2+e^{6+2 x} x^2\right )^2} \, dx\\ &=5 \int \frac {e^{6+2 x}}{\left (-9+x+x^2\right ) \left (-9 e^{6+2 x}+e^{6+2 x} x-x^2+e^{6+2 x} x^2\right )} \, dx-5 \int \frac {e^{6+2 x}}{\left (x^2-e^{6+2 x} \left (-9+x+x^2\right )\right )^2} \, dx+5 \int \left (-\frac {9 e^{6+2 x}}{\left (-9+x+x^2\right ) \left (-9 e^{6+2 x}+e^{6+2 x} x-x^2+e^{6+2 x} x^2\right )^2}+\frac {19 e^{6+2 x} x}{\left (-9+x+x^2\right ) \left (-9 e^{6+2 x}+e^{6+2 x} x-x^2+e^{6+2 x} x^2\right )^2}\right ) \, dx+10 \int \frac {e^{6+2 x} x}{\left (-9+x+x^2\right ) \left (-9 e^{6+2 x}+e^{6+2 x} x-x^2+e^{6+2 x} x^2\right )} \, dx+10 \int \frac {e^{6+2 x} x^2}{\left (x^2-e^{6+2 x} \left (-9+x+x^2\right )\right )^2} \, dx\\ &=-\left (5 \int \frac {e^{6+2 x}}{\left (x^2-e^{6+2 x} \left (-9+x+x^2\right )\right )^2} \, dx\right )+5 \int \frac {e^{6+2 x}}{\left (9-x-x^2\right ) \left (x^2-e^{6+2 x} \left (-9+x+x^2\right )\right )} \, dx+10 \int \frac {e^{6+2 x} x^2}{\left (x^2-e^{6+2 x} \left (-9+x+x^2\right )\right )^2} \, dx+10 \int \frac {e^{6+2 x} x}{\left (9-x-x^2\right ) \left (x^2-e^{6+2 x} \left (-9+x+x^2\right )\right )} \, dx-45 \int \frac {e^{6+2 x}}{\left (-9+x+x^2\right ) \left (-9 e^{6+2 x}+e^{6+2 x} x-x^2+e^{6+2 x} x^2\right )^2} \, dx+95 \int \frac {e^{6+2 x} x}{\left (-9+x+x^2\right ) \left (-9 e^{6+2 x}+e^{6+2 x} x-x^2+e^{6+2 x} x^2\right )^2} \, dx\\ &=-\left (5 \int \frac {e^{6+2 x}}{\left (x^2-e^{6+2 x} \left (-9+x+x^2\right )\right )^2} \, dx\right )+5 \int \left (\frac {2 e^{6+2 x}}{\sqrt {37} \left (-1+\sqrt {37}-2 x\right ) \left (x^2-e^{6+2 x} \left (-9+x+x^2\right )\right )}+\frac {2 e^{6+2 x}}{\sqrt {37} \left (1+\sqrt {37}+2 x\right ) \left (x^2-e^{6+2 x} \left (-9+x+x^2\right )\right )}\right ) \, dx+10 \int \frac {e^{6+2 x} x^2}{\left (x^2-e^{6+2 x} \left (-9+x+x^2\right )\right )^2} \, dx+10 \int \left (\frac {\left (1+\frac {1}{\sqrt {37}}\right ) e^{6+2 x}}{\left (-1-\sqrt {37}-2 x\right ) \left (x^2-e^{6+2 x} \left (-9+x+x^2\right )\right )}+\frac {\left (1-\frac {1}{\sqrt {37}}\right ) e^{6+2 x}}{\left (-1+\sqrt {37}-2 x\right ) \left (x^2-e^{6+2 x} \left (-9+x+x^2\right )\right )}\right ) \, dx-45 \int \frac {e^{6+2 x}}{\left (-9+x+x^2\right ) \left (x^2-e^{6+2 x} \left (-9+x+x^2\right )\right )^2} \, dx+95 \int \frac {e^{6+2 x} x}{\left (-9+x+x^2\right ) \left (x^2-e^{6+2 x} \left (-9+x+x^2\right )\right )^2} \, dx\\ &=-\left (5 \int \frac {e^{6+2 x}}{\left (x^2-e^{6+2 x} \left (-9+x+x^2\right )\right )^2} \, dx\right )+10 \int \frac {e^{6+2 x} x^2}{\left (x^2-e^{6+2 x} \left (-9+x+x^2\right )\right )^2} \, dx-45 \int \left (-\frac {2 e^{6+2 x}}{\sqrt {37} \left (-1+\sqrt {37}-2 x\right ) \left (x^2-e^{6+2 x} \left (-9+x+x^2\right )\right )^2}-\frac {2 e^{6+2 x}}{\sqrt {37} \left (1+\sqrt {37}+2 x\right ) \left (x^2-e^{6+2 x} \left (-9+x+x^2\right )\right )^2}\right ) \, dx+95 \int \left (\frac {\left (1-\frac {1}{\sqrt {37}}\right ) e^{6+2 x}}{\left (1-\sqrt {37}+2 x\right ) \left (x^2-e^{6+2 x} \left (-9+x+x^2\right )\right )^2}+\frac {\left (1+\frac {1}{\sqrt {37}}\right ) e^{6+2 x}}{\left (1+\sqrt {37}+2 x\right ) \left (x^2-e^{6+2 x} \left (-9+x+x^2\right )\right )^2}\right ) \, dx+\frac {10 \int \frac {e^{6+2 x}}{\left (-1+\sqrt {37}-2 x\right ) \left (x^2-e^{6+2 x} \left (-9+x+x^2\right )\right )} \, dx}{\sqrt {37}}+\frac {10 \int \frac {e^{6+2 x}}{\left (1+\sqrt {37}+2 x\right ) \left (x^2-e^{6+2 x} \left (-9+x+x^2\right )\right )} \, dx}{\sqrt {37}}+\frac {1}{37} \left (10 \left (37-\sqrt {37}\right )\right ) \int \frac {e^{6+2 x}}{\left (-1+\sqrt {37}-2 x\right ) \left (x^2-e^{6+2 x} \left (-9+x+x^2\right )\right )} \, dx+\frac {1}{37} \left (10 \left (37+\sqrt {37}\right )\right ) \int \frac {e^{6+2 x}}{\left (-1-\sqrt {37}-2 x\right ) \left (x^2-e^{6+2 x} \left (-9+x+x^2\right )\right )} \, dx\\ &=-\left (5 \int \frac {e^{6+2 x}}{\left (x^2-e^{6+2 x} \left (-9+x+x^2\right )\right )^2} \, dx\right )+10 \int \frac {e^{6+2 x} x^2}{\left (x^2-e^{6+2 x} \left (-9+x+x^2\right )\right )^2} \, dx+\frac {10 \int \frac {e^{6+2 x}}{\left (-1+\sqrt {37}-2 x\right ) \left (x^2-e^{6+2 x} \left (-9+x+x^2\right )\right )} \, dx}{\sqrt {37}}+\frac {10 \int \frac {e^{6+2 x}}{\left (1+\sqrt {37}+2 x\right ) \left (x^2-e^{6+2 x} \left (-9+x+x^2\right )\right )} \, dx}{\sqrt {37}}+\frac {90 \int \frac {e^{6+2 x}}{\left (-1+\sqrt {37}-2 x\right ) \left (x^2-e^{6+2 x} \left (-9+x+x^2\right )\right )^2} \, dx}{\sqrt {37}}+\frac {90 \int \frac {e^{6+2 x}}{\left (1+\sqrt {37}+2 x\right ) \left (x^2-e^{6+2 x} \left (-9+x+x^2\right )\right )^2} \, dx}{\sqrt {37}}+\frac {1}{37} \left (10 \left (37-\sqrt {37}\right )\right ) \int \frac {e^{6+2 x}}{\left (-1+\sqrt {37}-2 x\right ) \left (x^2-e^{6+2 x} \left (-9+x+x^2\right )\right )} \, dx+\frac {1}{37} \left (95 \left (37-\sqrt {37}\right )\right ) \int \frac {e^{6+2 x}}{\left (1-\sqrt {37}+2 x\right ) \left (x^2-e^{6+2 x} \left (-9+x+x^2\right )\right )^2} \, dx+\frac {1}{37} \left (10 \left (37+\sqrt {37}\right )\right ) \int \frac {e^{6+2 x}}{\left (-1-\sqrt {37}-2 x\right ) \left (x^2-e^{6+2 x} \left (-9+x+x^2\right )\right )} \, dx+\frac {1}{37} \left (95 \left (37+\sqrt {37}\right )\right ) \int \frac {e^{6+2 x}}{\left (1+\sqrt {37}+2 x\right ) \left (x^2-e^{6+2 x} \left (-9+x+x^2\right )\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 1.62, size = 31, normalized size = 1.24 \begin {gather*} -\frac {5 e^{6+2 x}}{-x^2+e^{6+2 x} \left (-9+x+x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(12 + 4*x)*(5 + 10*x) + E^(6 + 2*x)*(-10*x + 10*x^2))/(x^4 + E^(6 + 2*x)*(18*x^2 - 2*x^3 - 2*x^4)
+ E^(12 + 4*x)*(81 - 18*x - 17*x^2 + 2*x^3 + x^4)),x]

[Out]

(-5*E^(6 + 2*x))/(-x^2 + E^(6 + 2*x)*(-9 + x + x^2))

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fricas [A]  time = 0.84, size = 28, normalized size = 1.12 \begin {gather*} \frac {5 \, e^{\left (2 \, x + 6\right )}}{x^{2} - {\left (x^{2} + x - 9\right )} e^{\left (2 \, x + 6\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((10*x+5)*exp(3)^4*exp(x)^4+(10*x^2-10*x)*exp(3)^2*exp(x)^2)/((x^4+2*x^3-17*x^2-18*x+81)*exp(3)^4*ex
p(x)^4+(-2*x^4-2*x^3+18*x^2)*exp(3)^2*exp(x)^2+x^4),x, algorithm="fricas")

[Out]

5*e^(2*x + 6)/(x^2 - (x^2 + x - 9)*e^(2*x + 6))

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giac [A]  time = 0.53, size = 42, normalized size = 1.68 \begin {gather*} -\frac {5 \, e^{\left (2 \, x + 6\right )}}{x^{2} e^{\left (2 \, x + 6\right )} - x^{2} + x e^{\left (2 \, x + 6\right )} - 9 \, e^{\left (2 \, x + 6\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((10*x+5)*exp(3)^4*exp(x)^4+(10*x^2-10*x)*exp(3)^2*exp(x)^2)/((x^4+2*x^3-17*x^2-18*x+81)*exp(3)^4*ex
p(x)^4+(-2*x^4-2*x^3+18*x^2)*exp(3)^2*exp(x)^2+x^4),x, algorithm="giac")

[Out]

-5*e^(2*x + 6)/(x^2*e^(2*x + 6) - x^2 + x*e^(2*x + 6) - 9*e^(2*x + 6))

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maple [A]  time = 0.35, size = 51, normalized size = 2.04

method	result	size
norman	\(-\frac {5 \,{\mathrm e}^{6} {\mathrm e}^{2 x}}{x^{2} {\mathrm e}^{6} {\mathrm e}^{2 x}+{\mathrm e}^{6} {\mathrm e}^{2 x} x -9 \,{\mathrm e}^{6} {\mathrm e}^{2 x}-x^{2}}\)	\(51\)
risch	\(-\frac {5}{x^{2}+x -9}-\frac {5 x^{2}}{\left (x^{2}+x -9\right ) \left (x^{2} {\mathrm e}^{2 x +6}+x \,{\mathrm e}^{2 x +6}-9 \,{\mathrm e}^{2 x +6}-x^{2}\right )}\)	\(59\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((10*x+5)*exp(3)^4*exp(x)^4+(10*x^2-10*x)*exp(3)^2*exp(x)^2)/((x^4+2*x^3-17*x^2-18*x+81)*exp(3)^4*exp(x)^4
+(-2*x^4-2*x^3+18*x^2)*exp(3)^2*exp(x)^2+x^4),x,method=_RETURNVERBOSE)

[Out]

-5*exp(3)^2*exp(x)^2/(x^2*exp(3)^2*exp(x)^2+exp(3)^2*exp(x)^2*x-9*exp(3)^2*exp(x)^2-x^2)

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maxima [A]  time = 0.45, size = 35, normalized size = 1.40 \begin {gather*} \frac {5 \, e^{\left (2 \, x + 6\right )}}{x^{2} - {\left (x^{2} e^{6} + x e^{6} - 9 \, e^{6}\right )} e^{\left (2 \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((10*x+5)*exp(3)^4*exp(x)^4+(10*x^2-10*x)*exp(3)^2*exp(x)^2)/((x^4+2*x^3-17*x^2-18*x+81)*exp(3)^4*ex
p(x)^4+(-2*x^4-2*x^3+18*x^2)*exp(3)^2*exp(x)^2+x^4),x, algorithm="maxima")

[Out]

5*e^(2*x + 6)/(x^2 - (x^2*e^6 + x*e^6 - 9*e^6)*e^(2*x))

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mupad [B]  time = 0.19, size = 57, normalized size = 2.28 \begin {gather*} \frac {5\,x\,\left ({\mathrm {e}}^{2\,x+6}-x+x\,{\mathrm {e}}^{2\,x+6}\right )}{9\,\left (9\,{\mathrm {e}}^{2\,x+6}-x\,{\mathrm {e}}^{2\,x+6}-x^2\,{\mathrm {e}}^{2\,x+6}+x^2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(4*x)*exp(12)*(10*x + 5) - exp(2*x)*exp(6)*(10*x - 10*x^2))/(x^4 - exp(2*x)*exp(6)*(2*x^3 - 18*x^2 + 2
*x^4) + exp(4*x)*exp(12)*(2*x^3 - 17*x^2 - 18*x + x^4 + 81)),x)

[Out]

(5*x*(exp(2*x + 6) - x + x*exp(2*x + 6)))/(9*(9*exp(2*x + 6) - x*exp(2*x + 6) - x^2*exp(2*x + 6) + x^2))

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sympy [B]  time = 0.34, size = 66, normalized size = 2.64 \begin {gather*} - \frac {5 x^{2}}{- x^{4} - x^{3} + 9 x^{2} + \left (x^{4} e^{6} + 2 x^{3} e^{6} - 17 x^{2} e^{6} - 18 x e^{6} + 81 e^{6}\right ) e^{2 x}} - \frac {5}{x^{2} + x - 9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((10*x+5)*exp(3)**4*exp(x)**4+(10*x**2-10*x)*exp(3)**2*exp(x)**2)/((x**4+2*x**3-17*x**2-18*x+81)*exp
(3)**4*exp(x)**4+(-2*x**4-2*x**3+18*x**2)*exp(3)**2*exp(x)**2+x**4),x)

[Out]

-5*x**2/(-x**4 - x**3 + 9*x**2 + (x**4*exp(6) + 2*x**3*exp(6) - 17*x**2*exp(6) - 18*x*exp(6) + 81*exp(6))*exp(
2*x)) - 5/(x**2 + x - 9)

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