3.103.94 \(\int \frac {-560+840 x+e^x (112-56 x-84 x^2)}{400 x^2-600 x^3+225 x^4+e^x (-160 x^2+240 x^3-90 x^4)+e^{2 x} (16 x^2-24 x^3+9 x^4)} \, dx\)

Optimal. Leaf size=25 \[ \frac {28}{3 \left (5-e^x\right ) \left (\frac {4}{3}-x\right ) x} \]

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Rubi [F]  time = 1.34, 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 {-560+840 x+e^x \left (112-56 x-84 x^2\right )}{400 x^2-600 x^3+225 x^4+e^x \left (-160 x^2+240 x^3-90 x^4\right )+e^{2 x} \left (16 x^2-24 x^3+9 x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-560 + 840*x + E^x*(112 - 56*x - 84*x^2))/(400*x^2 - 600*x^3 + 225*x^4 + E^x*(-160*x^2 + 240*x^3 - 90*x^4
) + E^(2*x)*(16*x^2 - 24*x^3 + 9*x^4)),x]

[Out]

7*Defer[Int][1/((-5 + E^x)*x^2), x] + 35*Defer[Int][1/((-5 + E^x)^2*x), x] + 7*Defer[Int][1/((-5 + E^x)*x), x]
- 63*Defer[Int][1/((-5 + E^x)*(-4 + 3*x)^2), x] - 105*Defer[Int][1/((-5 + E^x)^2*(-4 + 3*x)), x] - 21*Defer[I
nt][1/((-5 + E^x)*(-4 + 3*x)), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {28 \left (-20+30 x-e^x \left (-4+2 x+3 x^2\right )\right )}{\left (5-e^x\right )^2 (4-3 x)^2 x^2} \, dx\\ &=28 \int \frac {-20+30 x-e^x \left (-4+2 x+3 x^2\right )}{\left (5-e^x\right )^2 (4-3 x)^2 x^2} \, dx\\ &=28 \int \left (-\frac {5}{\left (-5+e^x\right )^2 x (-4+3 x)}-\frac {-4+2 x+3 x^2}{\left (-5+e^x\right ) x^2 (-4+3 x)^2}\right ) \, dx\\ &=-\left (28 \int \frac {-4+2 x+3 x^2}{\left (-5+e^x\right ) x^2 (-4+3 x)^2} \, dx\right )-140 \int \frac {1}{\left (-5+e^x\right )^2 x (-4+3 x)} \, dx\\ &=-\left (28 \int \left (-\frac {1}{4 \left (-5+e^x\right ) x^2}-\frac {1}{4 \left (-5+e^x\right ) x}+\frac {9}{4 \left (-5+e^x\right ) (-4+3 x)^2}+\frac {3}{4 \left (-5+e^x\right ) (-4+3 x)}\right ) \, dx\right )-140 \int \left (-\frac {1}{4 \left (-5+e^x\right )^2 x}+\frac {3}{4 \left (-5+e^x\right )^2 (-4+3 x)}\right ) \, dx\\ &=7 \int \frac {1}{\left (-5+e^x\right ) x^2} \, dx+7 \int \frac {1}{\left (-5+e^x\right ) x} \, dx-21 \int \frac {1}{\left (-5+e^x\right ) (-4+3 x)} \, dx+35 \int \frac {1}{\left (-5+e^x\right )^2 x} \, dx-63 \int \frac {1}{\left (-5+e^x\right ) (-4+3 x)^2} \, dx-105 \int \frac {1}{\left (-5+e^x\right )^2 (-4+3 x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.22, size = 19, normalized size = 0.76 \begin {gather*} \frac {28}{\left (-5+e^x\right ) x (-4+3 x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-560 + 840*x + E^x*(112 - 56*x - 84*x^2))/(400*x^2 - 600*x^3 + 225*x^4 + E^x*(-160*x^2 + 240*x^3 -
90*x^4) + E^(2*x)*(16*x^2 - 24*x^3 + 9*x^4)),x]

[Out]

28/((-5 + E^x)*x*(-4 + 3*x))

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fricas [A]  time = 0.56, size = 26, normalized size = 1.04 \begin {gather*} -\frac {28}{15 \, x^{2} - {\left (3 \, x^{2} - 4 \, x\right )} e^{x} - 20 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-84*x^2-56*x+112)*exp(x)+840*x-560)/((9*x^4-24*x^3+16*x^2)*exp(x)^2+(-90*x^4+240*x^3-160*x^2)*exp(
x)+225*x^4-600*x^3+400*x^2),x, algorithm="fricas")

[Out]

-28/(15*x^2 - (3*x^2 - 4*x)*e^x - 20*x)

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giac [A]  time = 0.20, size = 25, normalized size = 1.00 \begin {gather*} \frac {28}{3 \, x^{2} e^{x} - 15 \, x^{2} - 4 \, x e^{x} + 20 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-84*x^2-56*x+112)*exp(x)+840*x-560)/((9*x^4-24*x^3+16*x^2)*exp(x)^2+(-90*x^4+240*x^3-160*x^2)*exp(
x)+225*x^4-600*x^3+400*x^2),x, algorithm="giac")

[Out]

28/(3*x^2*e^x - 15*x^2 - 4*x*e^x + 20*x)

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maple [A]  time = 0.07, size = 19, normalized size = 0.76

method	result	size
norman	\(\frac {28}{x \left ({\mathrm e}^{x}-5\right ) \left (3 x -4\right )}\)	\(19\)
risch	\(\frac {28}{x \left ({\mathrm e}^{x}-5\right ) \left (3 x -4\right )}\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-84*x^2-56*x+112)*exp(x)+840*x-560)/((9*x^4-24*x^3+16*x^2)*exp(x)^2+(-90*x^4+240*x^3-160*x^2)*exp(x)+225
*x^4-600*x^3+400*x^2),x,method=_RETURNVERBOSE)

[Out]

28/x/(exp(x)-5)/(3*x-4)

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maxima [A]  time = 0.41, size = 26, normalized size = 1.04 \begin {gather*} -\frac {28}{15 \, x^{2} - {\left (3 \, x^{2} - 4 \, x\right )} e^{x} - 20 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-84*x^2-56*x+112)*exp(x)+840*x-560)/((9*x^4-24*x^3+16*x^2)*exp(x)^2+(-90*x^4+240*x^3-160*x^2)*exp(
x)+225*x^4-600*x^3+400*x^2),x, algorithm="maxima")

[Out]

-28/(15*x^2 - (3*x^2 - 4*x)*e^x - 20*x)

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mupad [B]  time = 6.70, size = 27, normalized size = 1.08 \begin {gather*} -\frac {28\,\left (4\,x-3\,x^2\right )}{x^2\,{\left (3\,x-4\right )}^2\,\left ({\mathrm {e}}^x-5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(x)*(56*x + 84*x^2 - 112) - 840*x + 560)/(exp(2*x)*(16*x^2 - 24*x^3 + 9*x^4) - exp(x)*(160*x^2 - 240*
x^3 + 90*x^4) + 400*x^2 - 600*x^3 + 225*x^4),x)

[Out]

-(28*(4*x - 3*x^2))/(x^2*(3*x - 4)^2*(exp(x) - 5))

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sympy [A]  time = 0.14, size = 20, normalized size = 0.80 \begin {gather*} \frac {28}{- 15 x^{2} + 20 x + \left (3 x^{2} - 4 x\right ) e^{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-84*x**2-56*x+112)*exp(x)+840*x-560)/((9*x**4-24*x**3+16*x**2)*exp(x)**2+(-90*x**4+240*x**3-160*x*
*2)*exp(x)+225*x**4-600*x**3+400*x**2),x)

[Out]

28/(-15*x**2 + 20*x + (3*x**2 - 4*x)*exp(x))

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