3.11.26 \(\int \frac {960 x^2+656 x^3+12 x^4+e^x (160 x^3-77 x^4-x^5)}{64+128 x+64 x^2+4 e^{2 x} x^2+e^x (32 x+32 x^2)} \, dx\)

Optimal. Leaf size=26 \[ \frac {4 x \left (5 x+\frac {x^2}{16}\right )}{4+e^x+\frac {4}{x}} \]

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Rubi [F]  time = 1.54, 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 {960 x^2+656 x^3+12 x^4+e^x \left (160 x^3-77 x^4-x^5\right )}{64+128 x+64 x^2+4 e^{2 x} x^2+e^x \left (32 x+32 x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(960*x^2 + 656*x^3 + 12*x^4 + E^x*(160*x^3 - 77*x^4 - x^5))/(64 + 128*x + 64*x^2 + 4*E^(2*x)*x^2 + E^x*(32
*x + 32*x^2)),x]

[Out]

80*Defer[Int][x^2/(4 + 4*x + E^x*x)^2, x] + 81*Defer[Int][x^3/(4 + 4*x + E^x*x)^2, x] + 81*Defer[Int][x^4/(4 +
4*x + E^x*x)^2, x] + Defer[Int][x^5/(4 + 4*x + E^x*x)^2, x] + 40*Defer[Int][x^2/(4 + 4*x + E^x*x), x] - (77*D
efer[Int][x^3/(4 + 4*x + E^x*x), x])/4 - Defer[Int][x^4/(4 + 4*x + E^x*x), x]/4

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {x^2 \left (960+16 \left (41+10 e^x\right ) x-\left (-12+77 e^x\right ) x^2-e^x x^3\right )}{4 \left (4+\left (4+e^x\right ) x\right )^2} \, dx\\ &=\frac {1}{4} \int \frac {x^2 \left (960+16 \left (41+10 e^x\right ) x-\left (-12+77 e^x\right ) x^2-e^x x^3\right )}{\left (4+\left (4+e^x\right ) x\right )^2} \, dx\\ &=\frac {1}{4} \int \left (-\frac {x^2 \left (-160+77 x+x^2\right )}{4+4 x+e^x x}+\frac {4 x^2 \left (80+81 x+81 x^2+x^3\right )}{\left (4+4 x+e^x x\right )^2}\right ) \, dx\\ &=-\left (\frac {1}{4} \int \frac {x^2 \left (-160+77 x+x^2\right )}{4+4 x+e^x x} \, dx\right )+\int \frac {x^2 \left (80+81 x+81 x^2+x^3\right )}{\left (4+4 x+e^x x\right )^2} \, dx\\ &=-\left (\frac {1}{4} \int \left (-\frac {160 x^2}{4+4 x+e^x x}+\frac {77 x^3}{4+4 x+e^x x}+\frac {x^4}{4+4 x+e^x x}\right ) \, dx\right )+\int \left (\frac {80 x^2}{\left (4+4 x+e^x x\right )^2}+\frac {81 x^3}{\left (4+4 x+e^x x\right )^2}+\frac {81 x^4}{\left (4+4 x+e^x x\right )^2}+\frac {x^5}{\left (4+4 x+e^x x\right )^2}\right ) \, dx\\ &=-\left (\frac {1}{4} \int \frac {x^4}{4+4 x+e^x x} \, dx\right )-\frac {77}{4} \int \frac {x^3}{4+4 x+e^x x} \, dx+40 \int \frac {x^2}{4+4 x+e^x x} \, dx+80 \int \frac {x^2}{\left (4+4 x+e^x x\right )^2} \, dx+81 \int \frac {x^3}{\left (4+4 x+e^x x\right )^2} \, dx+81 \int \frac {x^4}{\left (4+4 x+e^x x\right )^2} \, dx+\int \frac {x^5}{\left (4+4 x+e^x x\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.28, size = 22, normalized size = 0.85 \begin {gather*} \frac {x^3 (80+x)}{4 \left (4+4 x+e^x x\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(960*x^2 + 656*x^3 + 12*x^4 + E^x*(160*x^3 - 77*x^4 - x^5))/(64 + 128*x + 64*x^2 + 4*E^(2*x)*x^2 + E
^x*(32*x + 32*x^2)),x]

[Out]

(x^3*(80 + x))/(4*(4 + 4*x + E^x*x))

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fricas [A]  time = 0.94, size = 22, normalized size = 0.85 \begin {gather*} \frac {x^{4} + 80 \, x^{3}}{4 \, {\left (x e^{x} + 4 \, x + 4\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^5-77*x^4+160*x^3)*exp(x)+12*x^4+656*x^3+960*x^2)/(4*exp(x)^2*x^2+(32*x^2+32*x)*exp(x)+64*x^2+12
8*x+64),x, algorithm="fricas")

[Out]

1/4*(x^4 + 80*x^3)/(x*e^x + 4*x + 4)

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giac [A]  time = 0.44, size = 22, normalized size = 0.85 \begin {gather*} \frac {x^{4} + 80 \, x^{3}}{4 \, {\left (x e^{x} + 4 \, x + 4\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^5-77*x^4+160*x^3)*exp(x)+12*x^4+656*x^3+960*x^2)/(4*exp(x)^2*x^2+(32*x^2+32*x)*exp(x)+64*x^2+12
8*x+64),x, algorithm="giac")

[Out]

1/4*(x^4 + 80*x^3)/(x*e^x + 4*x + 4)

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maple [A]  time = 0.04, size = 20, normalized size = 0.77

method	result	size
risch	\(\frac {\left (x +80\right ) x^{3}}{4 \,{\mathrm e}^{x} x +16 x +16}\)	\(20\)
norman	\(\frac {20 x^{3}+\frac {1}{4} x^{4}}{{\mathrm e}^{x} x +4 x +4}\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-x^5-77*x^4+160*x^3)*exp(x)+12*x^4+656*x^3+960*x^2)/(4*exp(x)^2*x^2+(32*x^2+32*x)*exp(x)+64*x^2+128*x+64
),x,method=_RETURNVERBOSE)

[Out]

1/4*(x+80)*x^3/(exp(x)*x+4*x+4)

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maxima [A]  time = 0.44, size = 22, normalized size = 0.85 \begin {gather*} \frac {x^{4} + 80 \, x^{3}}{4 \, {\left (x e^{x} + 4 \, x + 4\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^5-77*x^4+160*x^3)*exp(x)+12*x^4+656*x^3+960*x^2)/(4*exp(x)^2*x^2+(32*x^2+32*x)*exp(x)+64*x^2+12
8*x+64),x, algorithm="maxima")

[Out]

1/4*(x^4 + 80*x^3)/(x*e^x + 4*x + 4)

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mupad [B]  time = 0.15, size = 19, normalized size = 0.73 \begin {gather*} \frac {x^3\,\left (x+80\right )}{4\,\left (4\,x+x\,{\mathrm {e}}^x+4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((960*x^2 + 656*x^3 + 12*x^4 - exp(x)*(77*x^4 - 160*x^3 + x^5))/(128*x + 4*x^2*exp(2*x) + exp(x)*(32*x + 32
*x^2) + 64*x^2 + 64),x)

[Out]

(x^3*(x + 80))/(4*(4*x + x*exp(x) + 4))

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sympy [A]  time = 0.15, size = 19, normalized size = 0.73 \begin {gather*} \frac {x^{4} + 80 x^{3}}{4 x e^{x} + 16 x + 16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x**5-77*x**4+160*x**3)*exp(x)+12*x**4+656*x**3+960*x**2)/(4*exp(x)**2*x**2+(32*x**2+32*x)*exp(x)+
64*x**2+128*x+64),x)

[Out]

(x**4 + 80*x**3)/(4*x*exp(x) + 16*x + 16)

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