3.15.32 \(\int \frac {e^{\frac {30-36 x+7 x^2-x^3+e^{16 x} (-45 x+9 x^2)}{30-6 x+x^2}} (-900+360 x-96 x^2+12 x^3-x^4+e^{16 x} (-1350-21060 x+8631 x^2-1584 x^3+144 x^4))}{900-360 x+96 x^2-12 x^3+x^4} \, dx\)

Optimal. Leaf size=33 \[ e^{1-x-\frac {3 e^{16 x} x}{2+\frac {x^2}{3 (5-x)}}} \]

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Rubi [F]  time = 66.40, 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 {\exp \left (\frac {30-36 x+7 x^2-x^3+e^{16 x} \left (-45 x+9 x^2\right )}{30-6 x+x^2}\right ) \left (-900+360 x-96 x^2+12 x^3-x^4+e^{16 x} \left (-1350-21060 x+8631 x^2-1584 x^3+144 x^4\right )\right )}{900-360 x+96 x^2-12 x^3+x^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((30 - 36*x + 7*x^2 - x^3 + E^(16*x)*(-45*x + 9*x^2))/(30 - 6*x + x^2))*(-900 + 360*x - 96*x^2 + 12*x^3
- x^4 + E^(16*x)*(-1350 - 21060*x + 8631*x^2 - 1584*x^3 + 144*x^4)))/(900 - 360*x + 96*x^2 - 12*x^3 + x^4),x]

[Out]

144*Defer[Int][E^((30 + 444*x - 45*E^(16*x)*x - 89*x^2 + 9*E^(16*x)*x^2 + 15*x^3)/(30 - 6*x + x^2)), x] - Defe
r[Int][E^((30 - 36*x + 7*x^2 - x^3 + E^(16*x)*(-45*x + 9*x^2))/(30 - 6*x + x^2)), x] + (360*Defer[Int][E^((30
+ 444*x - 45*E^(16*x)*x - 89*x^2 + 9*E^(16*x)*x^2 + 15*x^3)/(30 - 6*x + x^2))/(6 + (2*I)*Sqrt[21] - 2*x)^2, x]
)/7 - (162*(3 + I*Sqrt[21])*Defer[Int][E^((30 + 444*x - 45*E^(16*x)*x - 89*x^2 + 9*E^(16*x)*x^2 + 15*x^3)/(30
- 6*x + x^2))/(6 + (2*I)*Sqrt[21] - 2*x)^2, x])/7 + (3*I)*Sqrt[3/7]*Defer[Int][E^((30 + 444*x - 45*E^(16*x)*x
- 89*x^2 + 9*E^(16*x)*x^2 + 15*x^3)/(30 - 6*x + x^2))/(6 + (2*I)*Sqrt[21] - 2*x), x] - (32*I)*Sqrt[3/7]*Defer[
Int][E^((30 - 36*x + 7*x^2 - x^3 + E^(16*x)*(-45*x + 9*x^2))/(30 - 6*x + x^2))/(6 + (2*I)*Sqrt[21] - 2*x), x]
+ (3*(336 + (433*I)*Sqrt[21])*Defer[Int][E^((30 + 444*x - 45*E^(16*x)*x - 89*x^2 + 9*E^(16*x)*x^2 + 15*x^3)/(3
0 - 6*x + x^2))/(-6 - (2*I)*Sqrt[21] + 2*x), x])/7 - (4*(21 - I*Sqrt[21])*Defer[Int][E^((30 - 36*x + 7*x^2 - x
^3 + E^(16*x)*(-45*x + 9*x^2))/(30 - 6*x + x^2))/(-6 - (2*I)*Sqrt[21] + 2*x), x])/7 + (12*(7 - (3*I)*Sqrt[21])
*Defer[Int][E^((30 - 36*x + 7*x^2 - x^3 + E^(16*x)*(-45*x + 9*x^2))/(30 - 6*x + x^2))/(-6 - (2*I)*Sqrt[21] + 2
*x), x])/7 + (360*Defer[Int][E^((30 + 444*x - 45*E^(16*x)*x - 89*x^2 + 9*E^(16*x)*x^2 + 15*x^3)/(30 - 6*x + x^
2))/(-6 + (2*I)*Sqrt[21] + 2*x)^2, x])/7 - (162*(3 - I*Sqrt[21])*Defer[Int][E^((30 + 444*x - 45*E^(16*x)*x - 8
9*x^2 + 9*E^(16*x)*x^2 + 15*x^3)/(30 - 6*x + x^2))/(-6 + (2*I)*Sqrt[21] + 2*x)^2, x])/7 + (3*I)*Sqrt[3/7]*Defe
r[Int][E^((30 + 444*x - 45*E^(16*x)*x - 89*x^2 + 9*E^(16*x)*x^2 + 15*x^3)/(30 - 6*x + x^2))/(-6 + (2*I)*Sqrt[2
1] + 2*x), x] + (3*(336 - (433*I)*Sqrt[21])*Defer[Int][E^((30 + 444*x - 45*E^(16*x)*x - 89*x^2 + 9*E^(16*x)*x^
2 + 15*x^3)/(30 - 6*x + x^2))/(-6 + (2*I)*Sqrt[21] + 2*x), x])/7 - (32*I)*Sqrt[3/7]*Defer[Int][E^((30 - 36*x +
7*x^2 - x^3 + E^(16*x)*(-45*x + 9*x^2))/(30 - 6*x + x^2))/(-6 + (2*I)*Sqrt[21] + 2*x), x] - (4*(21 + I*Sqrt[2
1])*Defer[Int][E^((30 - 36*x + 7*x^2 - x^3 + E^(16*x)*(-45*x + 9*x^2))/(30 - 6*x + x^2))/(-6 + (2*I)*Sqrt[21]
+ 2*x), x])/7 + (12*(7 + (3*I)*Sqrt[21])*Defer[Int][E^((30 - 36*x + 7*x^2 - x^3 + E^(16*x)*(-45*x + 9*x^2))/(3
0 - 6*x + x^2))/(-6 + (2*I)*Sqrt[21] + 2*x), x])/7

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {900 \exp \left (\frac {30-36 x+7 x^2-x^3+e^{16 x} \left (-45 x+9 x^2\right )}{30-6 x+x^2}\right )}{\left (30-6 x+x^2\right )^2}+\frac {360 \exp \left (\frac {30-36 x+7 x^2-x^3+e^{16 x} \left (-45 x+9 x^2\right )}{30-6 x+x^2}\right ) x}{\left (30-6 x+x^2\right )^2}-\frac {96 \exp \left (\frac {30-36 x+7 x^2-x^3+e^{16 x} \left (-45 x+9 x^2\right )}{30-6 x+x^2}\right ) x^2}{\left (30-6 x+x^2\right )^2}+\frac {12 \exp \left (\frac {30-36 x+7 x^2-x^3+e^{16 x} \left (-45 x+9 x^2\right )}{30-6 x+x^2}\right ) x^3}{\left (30-6 x+x^2\right )^2}-\frac {\exp \left (\frac {30-36 x+7 x^2-x^3+e^{16 x} \left (-45 x+9 x^2\right )}{30-6 x+x^2}\right ) x^4}{\left (30-6 x+x^2\right )^2}+\frac {9 \exp \left (16 x+\frac {30-36 x+7 x^2-x^3+e^{16 x} \left (-45 x+9 x^2\right )}{30-6 x+x^2}\right ) \left (-150-2340 x+959 x^2-176 x^3+16 x^4\right )}{\left (30-6 x+x^2\right )^2}\right ) \, dx\\ &=9 \int \frac {\exp \left (16 x+\frac {30-36 x+7 x^2-x^3+e^{16 x} \left (-45 x+9 x^2\right )}{30-6 x+x^2}\right ) \left (-150-2340 x+959 x^2-176 x^3+16 x^4\right )}{\left (30-6 x+x^2\right )^2} \, dx+12 \int \frac {\exp \left (\frac {30-36 x+7 x^2-x^3+e^{16 x} \left (-45 x+9 x^2\right )}{30-6 x+x^2}\right ) x^3}{\left (30-6 x+x^2\right )^2} \, dx-96 \int \frac {\exp \left (\frac {30-36 x+7 x^2-x^3+e^{16 x} \left (-45 x+9 x^2\right )}{30-6 x+x^2}\right ) x^2}{\left (30-6 x+x^2\right )^2} \, dx+360 \int \frac {\exp \left (\frac {30-36 x+7 x^2-x^3+e^{16 x} \left (-45 x+9 x^2\right )}{30-6 x+x^2}\right ) x}{\left (30-6 x+x^2\right )^2} \, dx-900 \int \frac {\exp \left (\frac {30-36 x+7 x^2-x^3+e^{16 x} \left (-45 x+9 x^2\right )}{30-6 x+x^2}\right )}{\left (30-6 x+x^2\right )^2} \, dx-\int \frac {\exp \left (\frac {30-36 x+7 x^2-x^3+e^{16 x} \left (-45 x+9 x^2\right )}{30-6 x+x^2}\right ) x^4}{\left (30-6 x+x^2\right )^2} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.15, size = 45, normalized size = 1.36 \begin {gather*} e^{\frac {30-9 \left (4+5 e^{16 x}\right ) x+\left (7+9 e^{16 x}\right ) x^2-x^3}{30-6 x+x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((30 - 36*x + 7*x^2 - x^3 + E^(16*x)*(-45*x + 9*x^2))/(30 - 6*x + x^2))*(-900 + 360*x - 96*x^2 +
12*x^3 - x^4 + E^(16*x)*(-1350 - 21060*x + 8631*x^2 - 1584*x^3 + 144*x^4)))/(900 - 360*x + 96*x^2 - 12*x^3 + x
^4),x]

[Out]

E^((30 - 9*(4 + 5*E^(16*x))*x + (7 + 9*E^(16*x))*x^2 - x^3)/(30 - 6*x + x^2))

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fricas [A]  time = 0.77, size = 39, normalized size = 1.18 \begin {gather*} e^{\left (-\frac {x^{3} - 7 \, x^{2} - 9 \, {\left (x^{2} - 5 \, x\right )} e^{\left (16 \, x\right )} + 36 \, x - 30}{x^{2} - 6 \, x + 30}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((144*x^4-1584*x^3+8631*x^2-21060*x-1350)*exp(16*x)-x^4+12*x^3-96*x^2+360*x-900)*exp(((9*x^2-45*x)*e
xp(16*x)-x^3+7*x^2-36*x+30)/(x^2-6*x+30))/(x^4-12*x^3+96*x^2-360*x+900),x, algorithm="fricas")

[Out]

e^(-(x^3 - 7*x^2 - 9*(x^2 - 5*x)*e^(16*x) + 36*x - 30)/(x^2 - 6*x + 30))

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giac [B]  time = 0.90, size = 93, normalized size = 2.82 \begin {gather*} e^{\left (-\frac {x^{3}}{x^{2} - 6 \, x + 30} + \frac {9 \, x^{2} e^{\left (16 \, x\right )}}{x^{2} - 6 \, x + 30} + \frac {7 \, x^{2}}{x^{2} - 6 \, x + 30} - \frac {45 \, x e^{\left (16 \, x\right )}}{x^{2} - 6 \, x + 30} - \frac {36 \, x}{x^{2} - 6 \, x + 30} + \frac {30}{x^{2} - 6 \, x + 30}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((144*x^4-1584*x^3+8631*x^2-21060*x-1350)*exp(16*x)-x^4+12*x^3-96*x^2+360*x-900)*exp(((9*x^2-45*x)*e
xp(16*x)-x^3+7*x^2-36*x+30)/(x^2-6*x+30))/(x^4-12*x^3+96*x^2-360*x+900),x, algorithm="giac")

[Out]

e^(-x^3/(x^2 - 6*x + 30) + 9*x^2*e^(16*x)/(x^2 - 6*x + 30) + 7*x^2/(x^2 - 6*x + 30) - 45*x*e^(16*x)/(x^2 - 6*x
+ 30) - 36*x/(x^2 - 6*x + 30) + 30/(x^2 - 6*x + 30))

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maple [A]  time = 0.17, size = 43, normalized size = 1.30

method	result	size
risch	\({\mathrm e}^{-\frac {-9 \,{\mathrm e}^{16 x} x^{2}+x^{3}+45 \,{\mathrm e}^{16 x} x -7 x^{2}+36 x -30}{x^{2}-6 x +30}}\)	\(43\)
norman	\(\frac {x^{2} {\mathrm e}^{\frac {\left (9 x^{2}-45 x \right ) {\mathrm e}^{16 x}-x^{3}+7 x^{2}-36 x +30}{x^{2}-6 x +30}}-6 x \,{\mathrm e}^{\frac {\left (9 x^{2}-45 x \right ) {\mathrm e}^{16 x}-x^{3}+7 x^{2}-36 x +30}{x^{2}-6 x +30}}+30 \,{\mathrm e}^{\frac {\left (9 x^{2}-45 x \right ) {\mathrm e}^{16 x}-x^{3}+7 x^{2}-36 x +30}{x^{2}-6 x +30}}}{x^{2}-6 x +30}\)	\(145\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((144*x^4-1584*x^3+8631*x^2-21060*x-1350)*exp(16*x)-x^4+12*x^3-96*x^2+360*x-900)*exp(((9*x^2-45*x)*exp(16*
x)-x^3+7*x^2-36*x+30)/(x^2-6*x+30))/(x^4-12*x^3+96*x^2-360*x+900),x,method=_RETURNVERBOSE)

[Out]

exp(-(-9*exp(16*x)*x^2+x^3+45*exp(16*x)*x-7*x^2+36*x-30)/(x^2-6*x+30))

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maxima [A]  time = 1.32, size = 45, normalized size = 1.36 \begin {gather*} e^{\left (-x + \frac {9 \, x e^{\left (16 \, x\right )}}{x^{2} - 6 \, x + 30} - \frac {270 \, e^{\left (16 \, x\right )}}{x^{2} - 6 \, x + 30} + 9 \, e^{\left (16 \, x\right )} + 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((144*x^4-1584*x^3+8631*x^2-21060*x-1350)*exp(16*x)-x^4+12*x^3-96*x^2+360*x-900)*exp(((9*x^2-45*x)*e
xp(16*x)-x^3+7*x^2-36*x+30)/(x^2-6*x+30))/(x^4-12*x^3+96*x^2-360*x+900),x, algorithm="maxima")

[Out]

e^(-x + 9*x*e^(16*x)/(x^2 - 6*x + 30) - 270*e^(16*x)/(x^2 - 6*x + 30) + 9*e^(16*x) + 1)

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mupad [B]  time = 1.18, size = 98, normalized size = 2.97 \begin {gather*} {\mathrm {e}}^{-\frac {x^3}{x^2-6\,x+30}}\,{\mathrm {e}}^{\frac {7\,x^2}{x^2-6\,x+30}}\,{\mathrm {e}}^{\frac {30}{x^2-6\,x+30}}\,{\mathrm {e}}^{\frac {9\,x^2\,{\mathrm {e}}^{16\,x}}{x^2-6\,x+30}}\,{\mathrm {e}}^{-\frac {36\,x}{x^2-6\,x+30}}\,{\mathrm {e}}^{-\frac {45\,x\,{\mathrm {e}}^{16\,x}}{x^2-6\,x+30}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(-(36*x + exp(16*x)*(45*x - 9*x^2) - 7*x^2 + x^3 - 30)/(x^2 - 6*x + 30))*(exp(16*x)*(21060*x - 8631*x
^2 + 1584*x^3 - 144*x^4 + 1350) - 360*x + 96*x^2 - 12*x^3 + x^4 + 900))/(96*x^2 - 360*x - 12*x^3 + x^4 + 900),
x)

[Out]

exp(-x^3/(x^2 - 6*x + 30))*exp((7*x^2)/(x^2 - 6*x + 30))*exp(30/(x^2 - 6*x + 30))*exp((9*x^2*exp(16*x))/(x^2 -
6*x + 30))*exp(-(36*x)/(x^2 - 6*x + 30))*exp(-(45*x*exp(16*x))/(x^2 - 6*x + 30))

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sympy [A]  time = 0.50, size = 36, normalized size = 1.09 \begin {gather*} e^{\frac {- x^{3} + 7 x^{2} - 36 x + \left (9 x^{2} - 45 x\right ) e^{16 x} + 30}{x^{2} - 6 x + 30}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((144*x**4-1584*x**3+8631*x**2-21060*x-1350)*exp(16*x)-x**4+12*x**3-96*x**2+360*x-900)*exp(((9*x**2-
45*x)*exp(16*x)-x**3+7*x**2-36*x+30)/(x**2-6*x+30))/(x**4-12*x**3+96*x**2-360*x+900),x)

[Out]

exp((-x**3 + 7*x**2 - 36*x + (9*x**2 - 45*x)*exp(16*x) + 30)/(x**2 - 6*x + 30))

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