3.39.62 \(\int \frac {6340+144 e^{2 x}-640 x+16 x^2+e^x (-1911+87 x)}{6400+144 e^{2 x}-640 x+16 x^2+e^x (-1920+96 x)} \, dx\)

Optimal. Leaf size=21 \[ x+\frac {x}{16 \left (-5+e^x+\frac {1}{3} (-5+x)\right )} \]

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Rubi [F]  time = 0.56, 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 {6340+144 e^{2 x}-640 x+16 x^2+e^x (-1911+87 x)}{6400+144 e^{2 x}-640 x+16 x^2+e^x (-1920+96 x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(6340 + 144*E^(2*x) - 640*x + 16*x^2 + E^x*(-1911 + 87*x))/(6400 + 144*E^(2*x) - 640*x + 16*x^2 + E^x*(-19
20 + 96*x)),x]

[Out]

x - (63*Defer[Int][x/(-20 + 3*E^x + x)^2, x])/16 + (3*Defer[Int][x^2/(-20 + 3*E^x + x)^2, x])/16 + (3*Defer[In
t][(-20 + 3*E^x + x)^(-1), x])/16 - (3*Defer[Int][x/(-20 + 3*E^x + x), x])/16

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {6340+144 e^{2 x}-640 x+16 x^2+e^x (-1911+87 x)}{16 \left (20-3 e^x-x\right )^2} \, dx\\ &=\frac {1}{16} \int \frac {6340+144 e^{2 x}-640 x+16 x^2+e^x (-1911+87 x)}{\left (20-3 e^x-x\right )^2} \, dx\\ &=\frac {1}{16} \int \left (16+\frac {3 (-21+x) x}{\left (-20+3 e^x+x\right )^2}-\frac {3 (-1+x)}{-20+3 e^x+x}\right ) \, dx\\ &=x+\frac {3}{16} \int \frac {(-21+x) x}{\left (-20+3 e^x+x\right )^2} \, dx-\frac {3}{16} \int \frac {-1+x}{-20+3 e^x+x} \, dx\\ &=x+\frac {3}{16} \int \left (-\frac {21 x}{\left (-20+3 e^x+x\right )^2}+\frac {x^2}{\left (-20+3 e^x+x\right )^2}\right ) \, dx-\frac {3}{16} \int \left (-\frac {1}{-20+3 e^x+x}+\frac {x}{-20+3 e^x+x}\right ) \, dx\\ &=x+\frac {3}{16} \int \frac {x^2}{\left (-20+3 e^x+x\right )^2} \, dx+\frac {3}{16} \int \frac {1}{-20+3 e^x+x} \, dx-\frac {3}{16} \int \frac {x}{-20+3 e^x+x} \, dx-\frac {63}{16} \int \frac {x}{\left (-20+3 e^x+x\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.15, size = 21, normalized size = 1.00 \begin {gather*} \frac {1}{16} \left (16 x+\frac {3 x}{-20+3 e^x+x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(6340 + 144*E^(2*x) - 640*x + 16*x^2 + E^x*(-1911 + 87*x))/(6400 + 144*E^(2*x) - 640*x + 16*x^2 + E^
x*(-1920 + 96*x)),x]

[Out]

(16*x + (3*x)/(-20 + 3*E^x + x))/16

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fricas [A]  time = 0.54, size = 25, normalized size = 1.19 \begin {gather*} \frac {16 \, x^{2} + 48 \, x e^{x} - 317 \, x}{16 \, {\left (x + 3 \, e^{x} - 20\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((144*exp(x)^2+(87*x-1911)*exp(x)+16*x^2-640*x+6340)/(144*exp(x)^2+(96*x-1920)*exp(x)+16*x^2-640*x+64
00),x, algorithm="fricas")

[Out]

1/16*(16*x^2 + 48*x*e^x - 317*x)/(x + 3*e^x - 20)

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giac [A]  time = 0.18, size = 25, normalized size = 1.19 \begin {gather*} \frac {16 \, x^{2} + 48 \, x e^{x} - 317 \, x}{16 \, {\left (x + 3 \, e^{x} - 20\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((144*exp(x)^2+(87*x-1911)*exp(x)+16*x^2-640*x+6340)/(144*exp(x)^2+(96*x-1920)*exp(x)+16*x^2-640*x+64
00),x, algorithm="giac")

[Out]

1/16*(16*x^2 + 48*x*e^x - 317*x)/(x + 3*e^x - 20)

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maple [A]  time = 0.04, size = 15, normalized size = 0.71

method	result	size
risch	\(x +\frac {3 x}{16 \left (-20+3 \,{\mathrm e}^{x}+x \right )}\)	\(15\)
norman	\(\frac {x^{2}+\frac {951 \,{\mathrm e}^{x}}{16}+3 \,{\mathrm e}^{x} x -\frac {1585}{4}}{-20+3 \,{\mathrm e}^{x}+x}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((144*exp(x)^2+(87*x-1911)*exp(x)+16*x^2-640*x+6340)/(144*exp(x)^2+(96*x-1920)*exp(x)+16*x^2-640*x+6400),x,
method=_RETURNVERBOSE)

[Out]

x+3/16*x/(-20+3*exp(x)+x)

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maxima [A]  time = 0.40, size = 25, normalized size = 1.19 \begin {gather*} \frac {16 \, x^{2} + 48 \, x e^{x} - 317 \, x}{16 \, {\left (x + 3 \, e^{x} - 20\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((144*exp(x)^2+(87*x-1911)*exp(x)+16*x^2-640*x+6340)/(144*exp(x)^2+(96*x-1920)*exp(x)+16*x^2-640*x+64
00),x, algorithm="maxima")

[Out]

1/16*(16*x^2 + 48*x*e^x - 317*x)/(x + 3*e^x - 20)

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mupad [B]  time = 2.49, size = 16, normalized size = 0.76 \begin {gather*} x+\frac {3\,x}{16\,x+48\,{\mathrm {e}}^x-320} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((144*exp(2*x) - 640*x + exp(x)*(87*x - 1911) + 16*x^2 + 6340)/(144*exp(2*x) - 640*x + exp(x)*(96*x - 1920)
+ 16*x^2 + 6400),x)

[Out]

x + (3*x)/(16*x + 48*exp(x) - 320)

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sympy [A]  time = 0.12, size = 14, normalized size = 0.67 \begin {gather*} x + \frac {3 x}{16 x + 48 e^{x} - 320} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((144*exp(x)**2+(87*x-1911)*exp(x)+16*x**2-640*x+6340)/(144*exp(x)**2+(96*x-1920)*exp(x)+16*x**2-640*
x+6400),x)

[Out]

x + 3*x/(16*x + 48*exp(x) - 320)

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