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

Optimal. Leaf size=22 \[ 6-\frac {3}{x+\frac {3}{-\frac {5}{8}+49 e^x x}} \]

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Rubi [F]  time = 1.01, 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 {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 \end {gather*}

Verification is not applicable to the result.

[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 {gather*} \begin {aligned} \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 {aligned} \end {gather*}

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Mathematica [A]  time = 0.22, size = 26, normalized size = 1.18 \begin {gather*} -\frac {3}{x}+\frac {72}{x \left (24-5 x+392 e^x x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[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))

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fricas [A]  time = 0.82, size = 23, normalized size = 1.05 \begin {gather*} -\frac {3 \, {\left (392 \, x e^{x} - 5\right )}}{392 \, x^{2} e^{x} - 5 \, x + 24} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [A]  time = 0.24, size = 23, normalized size = 1.05 \begin {gather*} -\frac {3 \, {\left (392 \, x e^{x} - 5\right )}}{392 \, x^{2} e^{x} - 5 \, x + 24} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [A]  time = 0.05, size = 23, normalized size = 1.05

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maxima [A]  time = 0.39, size = 23, normalized size = 1.05 \begin {gather*} -\frac {3 \, {\left (392 \, x e^{x} - 5\right )}}{392 \, x^{2} e^{x} - 5 \, x + 24} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [B]  time = 0.14, size = 23, normalized size = 1.05 \begin {gather*} -\frac {1176\,x\,{\mathrm {e}}^x-15}{392\,x^2\,{\mathrm {e}}^x-5\,x+24} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [A]  time = 0.18, size = 20, normalized size = 0.91 \begin {gather*} \frac {72}{392 x^{3} e^{x} - 5 x^{2} + 24 x} - \frac {3}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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