3.41.73 \(\int \frac {e^{3/x} (240-95 x)-2240 x^2+310 x^3-10 x^4+e^5 (1280 x-160 x^2+5 x^3)}{3 e^{6/x} x^3+588 x^5-84 x^6+3 x^7+e^{10} (768 x^3-96 x^4+3 x^5)+e^5 (-1344 x^4+180 x^5-6 x^6)+e^{3/x} (84 x^4-6 x^5+e^5 (-96 x^3+6 x^4))} \, dx\) [4073]

Optimal. Leaf size=35 \[ \frac {5}{3 x \left (-e^5+\frac {e^{3/x}-2 x}{16-x}+x\right )} \]

[Out]

5/3/(x+(exp(3/x)-2*x)/(16-x)-exp(5))/x

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Rubi [F]
time = 3.15, 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^{3/x} (240-95 x)-2240 x^2+310 x^3-10 x^4+e^5 \left (1280 x-160 x^2+5 x^3\right )}{3 e^{6/x} x^3+588 x^5-84 x^6+3 x^7+e^{10} \left (768 x^3-96 x^4+3 x^5\right )+e^5 \left (-1344 x^4+180 x^5-6 x^6\right )+e^{3/x} \left (84 x^4-6 x^5+e^5 \left (-96 x^3+6 x^4\right )\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(3/x)*(240 - 95*x) - 2240*x^2 + 310*x^3 - 10*x^4 + E^5*(1280*x - 160*x^2 + 5*x^3))/(3*E^(6/x)*x^3 + 588
*x^5 - 84*x^6 + 3*x^7 + E^10*(768*x^3 - 96*x^4 + 3*x^5) + E^5*(-1344*x^4 + 180*x^5 - 6*x^6) + E^(3/x)*(84*x^4
- 6*x^5 + E^5*(-96*x^3 + 6*x^4))),x]

[Out]

(5*(43 + E^5)*Defer[Int][(E^(3/x) + E^5*(-16 + x) - (-14 + x)*x)^(-2), x])/3 + 1280*E^5*Defer[Int][1/(x^3*(E^(
3/x) + E^5*(-16 + x) - (-14 + x)*x)^2), x] - 160*(7 + E^5)*Defer[Int][1/(x^2*(E^(3/x) + E^5*(-16 + x) - (-14 +
x)*x)^2), x] - (5*(134 + 13*E^5)*Defer[Int][1/(x*(E^(3/x) + E^5*(-16 + x) - (-14 + x)*x)^2), x])/3 - (10*Defe
r[Int][x/(E^(3/x) + E^5*(-16 + x) - (-14 + x)*x)^2, x])/3 + 80*Defer[Int][1/(x^3*(E^(3/x) + E^5*(-16 + x) - (-
14 + x)*x)), x] + (95*Defer[Int][1/(x^2*(16*E^5 - E^(3/x) - 14*(1 + E^5/14)*x + x^2)), x])/3

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {5 \left (e^5 (-16+x)^2 x-e^{3/x} (-48+19 x)-2 x^2 \left (224-31 x+x^2\right )\right )}{3 x^3 \left (e^{3/x}+e^5 (-16+x)-(-14+x) x\right )^2} \, dx\\ &=\frac {5}{3} \int \frac {e^5 (-16+x)^2 x-e^{3/x} (-48+19 x)-2 x^2 \left (224-31 x+x^2\right )}{x^3 \left (e^{3/x}+e^5 (-16+x)-(-14+x) x\right )^2} \, dx\\ &=\frac {5}{3} \int \left (\frac {-48+19 x}{x^3 \left (16 e^5-e^{3/x}-14 \left (1+\frac {e^5}{14}\right ) x+x^2\right )}+\frac {768 e^5-96 \left (7+e^5\right ) x-\left (134+13 e^5\right ) x^2+\left (43+e^5\right ) x^3-2 x^4}{x^3 \left (16 e^5-e^{3/x}-14 \left (1+\frac {e^5}{14}\right ) x+x^2\right )^2}\right ) \, dx\\ &=\frac {5}{3} \int \frac {-48+19 x}{x^3 \left (16 e^5-e^{3/x}-14 \left (1+\frac {e^5}{14}\right ) x+x^2\right )} \, dx+\frac {5}{3} \int \frac {768 e^5-96 \left (7+e^5\right ) x-\left (134+13 e^5\right ) x^2+\left (43+e^5\right ) x^3-2 x^4}{x^3 \left (16 e^5-e^{3/x}-14 \left (1+\frac {e^5}{14}\right ) x+x^2\right )^2} \, dx\\ &=\frac {5}{3} \int \frac {768 e^5-96 \left (7+e^5\right ) x-\left (134+13 e^5\right ) x^2+\left (43+e^5\right ) x^3-2 x^4}{x^3 \left (e^{3/x}+e^5 (-16+x)-(-14+x) x\right )^2} \, dx+\frac {5}{3} \int \left (\frac {48}{x^3 \left (-16 e^5+e^{3/x}+14 \left (1+\frac {e^5}{14}\right ) x-x^2\right )}+\frac {19}{x^2 \left (16 e^5-e^{3/x}-14 \left (1+\frac {e^5}{14}\right ) x+x^2\right )}\right ) \, dx\\ &=\frac {5}{3} \int \left (\frac {43 \left (1+\frac {e^5}{43}\right )}{\left (16 e^5-e^{3/x}-14 \left (1+\frac {e^5}{14}\right ) x+x^2\right )^2}+\frac {768 e^5}{x^3 \left (16 e^5-e^{3/x}-14 \left (1+\frac {e^5}{14}\right ) x+x^2\right )^2}+\frac {96 \left (-7-e^5\right )}{x^2 \left (16 e^5-e^{3/x}-14 \left (1+\frac {e^5}{14}\right ) x+x^2\right )^2}+\frac {-134-13 e^5}{x \left (16 e^5-e^{3/x}-14 \left (1+\frac {e^5}{14}\right ) x+x^2\right )^2}-\frac {2 x}{\left (16 e^5-e^{3/x}-14 \left (1+\frac {e^5}{14}\right ) x+x^2\right )^2}\right ) \, dx+\frac {95}{3} \int \frac {1}{x^2 \left (16 e^5-e^{3/x}-14 \left (1+\frac {e^5}{14}\right ) x+x^2\right )} \, dx+80 \int \frac {1}{x^3 \left (-16 e^5+e^{3/x}+14 \left (1+\frac {e^5}{14}\right ) x-x^2\right )} \, dx\\ &=-\left (\frac {10}{3} \int \frac {x}{\left (16 e^5-e^{3/x}-14 \left (1+\frac {e^5}{14}\right ) x+x^2\right )^2} \, dx\right )+\frac {95}{3} \int \frac {1}{x^2 \left (16 e^5-e^{3/x}-14 \left (1+\frac {e^5}{14}\right ) x+x^2\right )} \, dx+80 \int \frac {1}{x^3 \left (e^{3/x}+e^5 (-16+x)-(-14+x) x\right )} \, dx+\left (1280 e^5\right ) \int \frac {1}{x^3 \left (16 e^5-e^{3/x}-14 \left (1+\frac {e^5}{14}\right ) x+x^2\right )^2} \, dx-\left (160 \left (7+e^5\right )\right ) \int \frac {1}{x^2 \left (16 e^5-e^{3/x}-14 \left (1+\frac {e^5}{14}\right ) x+x^2\right )^2} \, dx+\frac {1}{3} \left (5 \left (43+e^5\right )\right ) \int \frac {1}{\left (16 e^5-e^{3/x}-14 \left (1+\frac {e^5}{14}\right ) x+x^2\right )^2} \, dx-\frac {1}{3} \left (5 \left (134+13 e^5\right )\right ) \int \frac {1}{x \left (16 e^5-e^{3/x}-14 \left (1+\frac {e^5}{14}\right ) x+x^2\right )^2} \, dx\\ &=-\left (\frac {10}{3} \int \frac {x}{\left (e^{3/x}+e^5 (-16+x)-(-14+x) x\right )^2} \, dx\right )+\frac {95}{3} \int \frac {1}{x^2 \left (16 e^5-e^{3/x}-14 \left (1+\frac {e^5}{14}\right ) x+x^2\right )} \, dx+80 \int \frac {1}{x^3 \left (e^{3/x}+e^5 (-16+x)-(-14+x) x\right )} \, dx+\left (1280 e^5\right ) \int \frac {1}{x^3 \left (e^{3/x}+e^5 (-16+x)-(-14+x) x\right )^2} \, dx-\left (160 \left (7+e^5\right )\right ) \int \frac {1}{x^2 \left (e^{3/x}+e^5 (-16+x)-(-14+x) x\right )^2} \, dx+\frac {1}{3} \left (5 \left (43+e^5\right )\right ) \int \frac {1}{\left (e^{3/x}+e^5 (-16+x)-(-14+x) x\right )^2} \, dx-\frac {1}{3} \left (5 \left (134+13 e^5\right )\right ) \int \frac {1}{x \left (e^{3/x}+e^5 (-16+x)-(-14+x) x\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.15, size = 33, normalized size = 0.94 \begin {gather*} -\frac {5 (-16+x)}{3 x \left (e^{3/x}+e^5 (-16+x)-(-14+x) x\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(3/x)*(240 - 95*x) - 2240*x^2 + 310*x^3 - 10*x^4 + E^5*(1280*x - 160*x^2 + 5*x^3))/(3*E^(6/x)*x^3
+ 588*x^5 - 84*x^6 + 3*x^7 + E^10*(768*x^3 - 96*x^4 + 3*x^5) + E^5*(-1344*x^4 + 180*x^5 - 6*x^6) + E^(3/x)*(8
4*x^4 - 6*x^5 + E^5*(-96*x^3 + 6*x^4))),x]

[Out]

(-5*(-16 + x))/(3*x*(E^(3/x) + E^5*(-16 + x) - (-14 + x)*x))

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Maple [A]
time = 1.68, size = 34, normalized size = 0.97

method	result	size
risch	\(-\frac {5 \left (x -16\right )}{3 x \left (x \,{\mathrm e}^{5}-x^{2}-16 \,{\mathrm e}^{5}+{\mathrm e}^{\frac {3}{x}}+14 x \right )}\)	\(34\)
norman	\(\frac {-\frac {5}{3} x^{2}+\frac {80}{3} x}{x^{2} \left (x \,{\mathrm e}^{5}-x^{2}-16 \,{\mathrm e}^{5}+{\mathrm e}^{\frac {3}{x}}+14 x \right )}\)	\(39\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-95*x+240)*exp(3/x)+(5*x^3-160*x^2+1280*x)*exp(5)-10*x^4+310*x^3-2240*x^2)/(3*x^3*exp(3/x)^2+((6*x^4-96*
x^3)*exp(5)-6*x^5+84*x^4)*exp(3/x)+(3*x^5-96*x^4+768*x^3)*exp(5)^2+(-6*x^6+180*x^5-1344*x^4)*exp(5)+3*x^7-84*x
^6+588*x^5),x,method=_RETURNVERBOSE)

[Out]

-5/3*(x-16)/x/(x*exp(5)-x^2-16*exp(5)+exp(3/x)+14*x)

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Maxima [A]
time = 0.33, size = 34, normalized size = 0.97 \begin {gather*} \frac {5 \, {\left (x - 16\right )}}{3 \, {\left (x^{3} - x^{2} {\left (e^{5} + 14\right )} + 16 \, x e^{5} - x e^{\frac {3}{x}}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-95*x+240)*exp(3/x)+(5*x^3-160*x^2+1280*x)*exp(5)-10*x^4+310*x^3-2240*x^2)/(3*x^3*exp(3/x)^2+((6*x
^4-96*x^3)*exp(5)-6*x^5+84*x^4)*exp(3/x)+(3*x^5-96*x^4+768*x^3)*exp(5)^2+(-6*x^6+180*x^5-1344*x^4)*exp(5)+3*x^
7-84*x^6+588*x^5),x, algorithm="maxima")

[Out]

5/3*(x - 16)/(x^3 - x^2*(e^5 + 14) + 16*x*e^5 - x*e^(3/x))

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Fricas [A]
time = 0.35, size = 36, normalized size = 1.03 \begin {gather*} \frac {5 \, {\left (x - 16\right )}}{3 \, {\left (x^{3} - 14 \, x^{2} - {\left (x^{2} - 16 \, x\right )} e^{5} - x e^{\frac {3}{x}}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-95*x+240)*exp(3/x)+(5*x^3-160*x^2+1280*x)*exp(5)-10*x^4+310*x^3-2240*x^2)/(3*x^3*exp(3/x)^2+((6*x
^4-96*x^3)*exp(5)-6*x^5+84*x^4)*exp(3/x)+(3*x^5-96*x^4+768*x^3)*exp(5)^2+(-6*x^6+180*x^5-1344*x^4)*exp(5)+3*x^
7-84*x^6+588*x^5),x, algorithm="fricas")

[Out]

5/3*(x - 16)/(x^3 - 14*x^2 - (x^2 - 16*x)*e^5 - x*e^(3/x))

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Sympy [A]
time = 0.11, size = 37, normalized size = 1.06 \begin {gather*} \frac {80 - 5 x}{- 3 x^{3} + 42 x^{2} + 3 x^{2} e^{5} + 3 x e^{\frac {3}{x}} - 48 x e^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-95*x+240)*exp(3/x)+(5*x**3-160*x**2+1280*x)*exp(5)-10*x**4+310*x**3-2240*x**2)/(3*x**3*exp(3/x)**
2+((6*x**4-96*x**3)*exp(5)-6*x**5+84*x**4)*exp(3/x)+(3*x**5-96*x**4+768*x**3)*exp(5)**2+(-6*x**6+180*x**5-1344
*x**4)*exp(5)+3*x**7-84*x**6+588*x**5),x)

[Out]

(80 - 5*x)/(-3*x**3 + 42*x**2 + 3*x**2*exp(5) + 3*x*exp(3/x) - 48*x*exp(5))

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Giac [A]
time = 0.61, size = 43, normalized size = 1.23 \begin {gather*} -\frac {5 \, {\left (\frac {1}{x^{2}} - \frac {16}{x^{3}}\right )}}{3 \, {\left (\frac {e^{5}}{x} + \frac {14}{x} - \frac {16 \, e^{5}}{x^{2}} + \frac {e^{\frac {3}{x}}}{x^{2}} - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-95*x+240)*exp(3/x)+(5*x^3-160*x^2+1280*x)*exp(5)-10*x^4+310*x^3-2240*x^2)/(3*x^3*exp(3/x)^2+((6*x
^4-96*x^3)*exp(5)-6*x^5+84*x^4)*exp(3/x)+(3*x^5-96*x^4+768*x^3)*exp(5)^2+(-6*x^6+180*x^5-1344*x^4)*exp(5)+3*x^
7-84*x^6+588*x^5),x, algorithm="giac")

[Out]

-5/3*(1/x^2 - 16/x^3)/(e^5/x + 14/x - 16*e^5/x^2 + e^(3/x)/x^2 - 1)

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Mupad [B]
time = 3.28, size = 106, normalized size = 3.03 \begin {gather*} \frac {\frac {10\,x^5}{3}+\left (-\frac {5\,{\mathrm {e}}^5}{3}-\frac {215}{3}\right )\,x^4+\left (\frac {65\,{\mathrm {e}}^5}{3}+\frac {670}{3}\right )\,x^3+\left (160\,{\mathrm {e}}^5+1120\right )\,x^2-1280\,{\mathrm {e}}^5\,x}{\left (14\,x-16\,{\mathrm {e}}^5+{\mathrm {e}}^{3/x}+x\,{\mathrm {e}}^5-x^2\right )\,\left (3\,x^3\,{\mathrm {e}}^5-48\,x^2\,{\mathrm {e}}^5+x^4\,{\mathrm {e}}^5+42\,x^3+11\,x^4-2\,x^5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(3/x)*(95*x - 240) - exp(5)*(1280*x - 160*x^2 + 5*x^3) + 2240*x^2 - 310*x^3 + 10*x^4)/(exp(10)*(768*x
^3 - 96*x^4 + 3*x^5) - exp(3/x)*(exp(5)*(96*x^3 - 6*x^4) - 84*x^4 + 6*x^5) - exp(5)*(1344*x^4 - 180*x^5 + 6*x^
6) + 3*x^3*exp(6/x) + 588*x^5 - 84*x^6 + 3*x^7),x)

[Out]

(x^3*((65*exp(5))/3 + 670/3) - x^4*((5*exp(5))/3 + 215/3) - 1280*x*exp(5) + x^2*(160*exp(5) + 1120) + (10*x^5)
/3)/((14*x - 16*exp(5) + exp(3/x) + x*exp(5) - x^2)*(3*x^3*exp(5) - 48*x^2*exp(5) + x^4*exp(5) + 42*x^3 + 11*x
^4 - 2*x^5))

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