\(\int \frac {75+e^x (-28224-39984 x)+460992 e^{2 x} x^2}{576-240 x+25 x^2+153664 e^{2 x} x^4+e^x (18816 x^2-3920 x^3)} \, dx\) [9770]

Optimal result

Integrand size = 59, antiderivative size = 22 \[ \int \frac {75+e^x (-28224-39984 x)+460992 e^{2 x} x^2}{576-240 x+25 x^2+153664 e^{2 x} x^4+e^x \left (18816 x^2-3920 x^3\right )} \, dx=6-\frac {3}{x+\frac {3}{-\frac {5}{8}+49 e^x x}} \]

[Out]

6-3/(x+3/(49*exp(x)*x-5/8))

Rubi [F]

\[ \int \frac {75+e^x (-28224-39984 x)+460992 e^{2 x} x^2}{576-240 x+25 x^2+153664 e^{2 x} x^4+e^x \left (18816 x^2-3920 x^3\right )} \, dx=\int \frac {75+e^x (-28224-39984 x)+460992 e^{2 x} x^2}{576-240 x+25 x^2+153664 e^{2 x} x^4+e^x \left (18816 x^2-3920 x^3\right )} \, dx \]

[In]

Int[(75 + E^x*(-28224 - 39984*x) + 460992*E^(2*x)*x^2)/(576 - 240*x + 25*x^2 + 153664*E^(2*x)*x^4 + E^x*(18816
*x^2 - 3920*x^3)),x]

[Out]

-3/x - 360*Defer[Int][(24 - 5*x + 392*E^x*x^2)^(-2), x] + 3456*Defer[Int][1/(x^2*(24 - 5*x + 392*E^x*x^2)^2),
x] + 1368*Defer[Int][1/(x*(24 - 5*x + 392*E^x*x^2)^2), x] - 216*Defer[Int][1/(x^2*(24 - 5*x + 392*E^x*x^2)), x
] - 72*Defer[Int][1/(x*(24 - 5*x + 392*E^x*x^2)), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {75+460992 e^{2 x} x^2-2352 e^x (12+17 x)}{\left (24-5 x+392 e^x x^2\right )^2} \, dx \\ & = \int \left (\frac {3}{x^2}-\frac {72 \left (-48-19 x+5 x^2\right )}{x^2 \left (24-5 x+392 e^x x^2\right )^2}-\frac {72 (3+x)}{x^2 \left (24-5 x+392 e^x x^2\right )}\right ) \, dx \\ & = -\frac {3}{x}-72 \int \frac {-48-19 x+5 x^2}{x^2 \left (24-5 x+392 e^x x^2\right )^2} \, dx-72 \int \frac {3+x}{x^2 \left (24-5 x+392 e^x x^2\right )} \, dx \\ & = -\frac {3}{x}-72 \int \left (\frac {5}{\left (24-5 x+392 e^x x^2\right )^2}-\frac {48}{x^2 \left (24-5 x+392 e^x x^2\right )^2}-\frac {19}{x \left (24-5 x+392 e^x x^2\right )^2}\right ) \, dx-72 \int \left (\frac {3}{x^2 \left (24-5 x+392 e^x x^2\right )}+\frac {1}{x \left (24-5 x+392 e^x x^2\right )}\right ) \, dx \\ & = -\frac {3}{x}-72 \int \frac {1}{x \left (24-5 x+392 e^x x^2\right )} \, dx-216 \int \frac {1}{x^2 \left (24-5 x+392 e^x x^2\right )} \, dx-360 \int \frac {1}{\left (24-5 x+392 e^x x^2\right )^2} \, dx+1368 \int \frac {1}{x \left (24-5 x+392 e^x x^2\right )^2} \, dx+3456 \int \frac {1}{x^2 \left (24-5 x+392 e^x x^2\right )^2} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 2.07 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.18 \[ \int \frac {75+e^x (-28224-39984 x)+460992 e^{2 x} x^2}{576-240 x+25 x^2+153664 e^{2 x} x^4+e^x \left (18816 x^2-3920 x^3\right )} \, dx=-\frac {3}{x}+\frac {72}{x \left (24-5 x+392 e^x x^2\right )} \]

[In]

Integrate[(75 + E^x*(-28224 - 39984*x) + 460992*E^(2*x)*x^2)/(576 - 240*x + 25*x^2 + 153664*E^(2*x)*x^4 + E^x*
(18816*x^2 - 3920*x^3)),x]

[Out]

-3/x + 72/(x*(24 - 5*x + 392*E^x*x^2))

Maple [A] (verified)

Time = 0.07 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.05

method	result	size
norman	\(\frac {15-1176 \,{\mathrm e}^{x} x}{392 \,{\mathrm e}^{x} x^{2}-5 x +24}\)	\(23\)
parallelrisch	\(\frac {5880-460992 \,{\mathrm e}^{x} x}{153664 \,{\mathrm e}^{x} x^{2}-1960 x +9408}\)	\(24\)
risch	\(-\frac {3}{x}+\frac {72}{x \left (392 \,{\mathrm e}^{x} x^{2}-5 x +24\right )}\)	\(26\)

[In]

int((460992*exp(x)^2*x^2+(-39984*x-28224)*exp(x)+75)/(153664*exp(x)^2*x^4+(-3920*x^3+18816*x^2)*exp(x)+25*x^2-
240*x+576),x,method=_RETURNVERBOSE)

[Out]

(15-1176*exp(x)*x)/(392*exp(x)*x^2-5*x+24)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.05 \[ \int \frac {75+e^x (-28224-39984 x)+460992 e^{2 x} x^2}{576-240 x+25 x^2+153664 e^{2 x} x^4+e^x \left (18816 x^2-3920 x^3\right )} \, dx=-\frac {3 \, {\left (392 \, x e^{x} - 5\right )}}{392 \, x^{2} e^{x} - 5 \, x + 24} \]

[In]

integrate((460992*exp(x)^2*x^2+(-39984*x-28224)*exp(x)+75)/(153664*exp(x)^2*x^4+(-3920*x^3+18816*x^2)*exp(x)+2
5*x^2-240*x+576),x, algorithm="fricas")

[Out]

-3*(392*x*e^x - 5)/(392*x^2*e^x - 5*x + 24)

Sympy [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {75+e^x (-28224-39984 x)+460992 e^{2 x} x^2}{576-240 x+25 x^2+153664 e^{2 x} x^4+e^x \left (18816 x^2-3920 x^3\right )} \, dx=\frac {72}{392 x^{3} e^{x} - 5 x^{2} + 24 x} - \frac {3}{x} \]

[In]

integrate((460992*exp(x)**2*x**2+(-39984*x-28224)*exp(x)+75)/(153664*exp(x)**2*x**4+(-3920*x**3+18816*x**2)*ex
p(x)+25*x**2-240*x+576),x)

[Out]

72/(392*x**3*exp(x) - 5*x**2 + 24*x) - 3/x

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.05 \[ \int \frac {75+e^x (-28224-39984 x)+460992 e^{2 x} x^2}{576-240 x+25 x^2+153664 e^{2 x} x^4+e^x \left (18816 x^2-3920 x^3\right )} \, dx=-\frac {3 \, {\left (392 \, x e^{x} - 5\right )}}{392 \, x^{2} e^{x} - 5 \, x + 24} \]

[In]

integrate((460992*exp(x)^2*x^2+(-39984*x-28224)*exp(x)+75)/(153664*exp(x)^2*x^4+(-3920*x^3+18816*x^2)*exp(x)+2
5*x^2-240*x+576),x, algorithm="maxima")

[Out]

-3*(392*x*e^x - 5)/(392*x^2*e^x - 5*x + 24)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.05 \[ \int \frac {75+e^x (-28224-39984 x)+460992 e^{2 x} x^2}{576-240 x+25 x^2+153664 e^{2 x} x^4+e^x \left (18816 x^2-3920 x^3\right )} \, dx=-\frac {3 \, {\left (392 \, x e^{x} - 5\right )}}{392 \, x^{2} e^{x} - 5 \, x + 24} \]

[In]

integrate((460992*exp(x)^2*x^2+(-39984*x-28224)*exp(x)+75)/(153664*exp(x)^2*x^4+(-3920*x^3+18816*x^2)*exp(x)+2
5*x^2-240*x+576),x, algorithm="giac")

[Out]

-3*(392*x*e^x - 5)/(392*x^2*e^x - 5*x + 24)

Mupad [B] (verification not implemented)

Time = 14.51 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.05 \[ \int \frac {75+e^x (-28224-39984 x)+460992 e^{2 x} x^2}{576-240 x+25 x^2+153664 e^{2 x} x^4+e^x \left (18816 x^2-3920 x^3\right )} \, dx=-\frac {1176\,x\,{\mathrm {e}}^x-15}{392\,x^2\,{\mathrm {e}}^x-5\,x+24} \]

[In]

int((460992*x^2*exp(2*x) - exp(x)*(39984*x + 28224) + 75)/(exp(x)*(18816*x^2 - 3920*x^3) - 240*x + 153664*x^4*
exp(2*x) + 25*x^2 + 576),x)

[Out]

-(1176*x*exp(x) - 15)/(392*x^2*exp(x) - 5*x + 24)