3.58.31 \(\int \frac {64-32 x+e^{4+x^3} (16-8 x+x^2-48 x^3+24 x^4-3 x^5)}{256+e^{4+x^3} (128-32 x)+e^{8+2 x^3} (16-8 x+x^2)} \, dx\)

Optimal. Leaf size=19 \[ \frac {x}{e^{4+x^3}-\frac {16}{-4+x}} \]

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Rubi [F]  time = 1.77, 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 {64-32 x+e^{4+x^3} \left (16-8 x+x^2-48 x^3+24 x^4-3 x^5\right )}{256+e^{4+x^3} (128-32 x)+e^{8+2 x^3} \left (16-8 x+x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(64 - 32*x + E^(4 + x^3)*(16 - 8*x + x^2 - 48*x^3 + 24*x^4 - 3*x^5))/(256 + E^(4 + x^3)*(128 - 32*x) + E^(
8 + 2*x^3)*(16 - 8*x + x^2)),x]

[Out]

-4*Defer[Int][(-16 + E^(4 + x^3)*(-4 + x))^(-1), x] - 16*Defer[Int][x/(16 - E^(4 + x^3)*(-4 + x))^2, x] + Defe
r[Int][x/(-16 + E^(4 + x^3)*(-4 + x)), x] + 192*Defer[Int][x^3/(16 - E^(4 + x^3)*(-4 + x))^2, x] + 12*Defer[In
t][x^3/(-16 + E^(4 + x^3)*(-4 + x)), x] - 48*Defer[Int][x^4/(16 - E^(4 + x^3)*(-4 + x))^2, x] - 3*Defer[Int][x
^4/(-16 + E^(4 + x^3)*(-4 + x)), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-32 (-2+x)-e^{4+x^3} (-4+x)^2 \left (-1+3 x^3\right )}{\left (16-e^{4+x^3} (-4+x)\right )^2} \, dx\\ &=\int \left (-\frac {16 x \left (1-12 x^2+3 x^3\right )}{\left (-16-4 e^{4+x^3}+e^{4+x^3} x\right )^2}-\frac {4-x-12 x^3+3 x^4}{-16-4 e^{4+x^3}+e^{4+x^3} x}\right ) \, dx\\ &=-\left (16 \int \frac {x \left (1-12 x^2+3 x^3\right )}{\left (-16-4 e^{4+x^3}+e^{4+x^3} x\right )^2} \, dx\right )-\int \frac {4-x-12 x^3+3 x^4}{-16-4 e^{4+x^3}+e^{4+x^3} x} \, dx\\ &=-\left (16 \int \frac {x \left (1-12 x^2+3 x^3\right )}{\left (16-e^{4+x^3} (-4+x)\right )^2} \, dx\right )-\int \left (\frac {4}{-16-4 e^{4+x^3}+e^{4+x^3} x}-\frac {x}{-16-4 e^{4+x^3}+e^{4+x^3} x}-\frac {12 x^3}{-16-4 e^{4+x^3}+e^{4+x^3} x}+\frac {3 x^4}{-16-4 e^{4+x^3}+e^{4+x^3} x}\right ) \, dx\\ &=-\left (3 \int \frac {x^4}{-16-4 e^{4+x^3}+e^{4+x^3} x} \, dx\right )-4 \int \frac {1}{-16-4 e^{4+x^3}+e^{4+x^3} x} \, dx+12 \int \frac {x^3}{-16-4 e^{4+x^3}+e^{4+x^3} x} \, dx-16 \int \left (\frac {x}{\left (-16-4 e^{4+x^3}+e^{4+x^3} x\right )^2}-\frac {12 x^3}{\left (-16-4 e^{4+x^3}+e^{4+x^3} x\right )^2}+\frac {3 x^4}{\left (-16-4 e^{4+x^3}+e^{4+x^3} x\right )^2}\right ) \, dx+\int \frac {x}{-16-4 e^{4+x^3}+e^{4+x^3} x} \, dx\\ &=-\left (3 \int \frac {x^4}{-16+e^{4+x^3} (-4+x)} \, dx\right )-4 \int \frac {1}{-16+e^{4+x^3} (-4+x)} \, dx+12 \int \frac {x^3}{-16+e^{4+x^3} (-4+x)} \, dx-16 \int \frac {x}{\left (-16-4 e^{4+x^3}+e^{4+x^3} x\right )^2} \, dx-48 \int \frac {x^4}{\left (-16-4 e^{4+x^3}+e^{4+x^3} x\right )^2} \, dx+192 \int \frac {x^3}{\left (-16-4 e^{4+x^3}+e^{4+x^3} x\right )^2} \, dx+\int \frac {x}{-16+e^{4+x^3} (-4+x)} \, dx\\ &=-\left (3 \int \frac {x^4}{-16+e^{4+x^3} (-4+x)} \, dx\right )-4 \int \frac {1}{-16+e^{4+x^3} (-4+x)} \, dx+12 \int \frac {x^3}{-16+e^{4+x^3} (-4+x)} \, dx-16 \int \frac {x}{\left (16-e^{4+x^3} (-4+x)\right )^2} \, dx-48 \int \frac {x^4}{\left (16-e^{4+x^3} (-4+x)\right )^2} \, dx+192 \int \frac {x^3}{\left (16-e^{4+x^3} (-4+x)\right )^2} \, dx+\int \frac {x}{-16+e^{4+x^3} (-4+x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.67, size = 20, normalized size = 1.05 \begin {gather*} \frac {(-4+x) x}{-16+e^{4+x^3} (-4+x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(64 - 32*x + E^(4 + x^3)*(16 - 8*x + x^2 - 48*x^3 + 24*x^4 - 3*x^5))/(256 + E^(4 + x^3)*(128 - 32*x)
+ E^(8 + 2*x^3)*(16 - 8*x + x^2)),x]

[Out]

((-4 + x)*x)/(-16 + E^(4 + x^3)*(-4 + x))

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fricas [A]  time = 1.13, size = 22, normalized size = 1.16 \begin {gather*} \frac {x^{2} - 4 \, x}{{\left (x - 4\right )} e^{\left (x^{3} + 4\right )} - 16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*x^5+24*x^4-48*x^3+x^2-8*x+16)*exp(x^3+4)-32*x+64)/((x^2-8*x+16)*exp(x^3+4)^2+(-32*x+128)*exp(x^
3+4)+256),x, algorithm="fricas")

[Out]

(x^2 - 4*x)/((x - 4)*e^(x^3 + 4) - 16)

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giac [A]  time = 1.14, size = 28, normalized size = 1.47 \begin {gather*} \frac {x^{2} - 4 \, x}{x e^{\left (x^{3} + 4\right )} - 4 \, e^{\left (x^{3} + 4\right )} - 16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*x^5+24*x^4-48*x^3+x^2-8*x+16)*exp(x^3+4)-32*x+64)/((x^2-8*x+16)*exp(x^3+4)^2+(-32*x+128)*exp(x^
3+4)+256),x, algorithm="giac")

[Out]

(x^2 - 4*x)/(x*e^(x^3 + 4) - 4*e^(x^3 + 4) - 16)

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maple [A]  time = 0.12, size = 26, normalized size = 1.37

method	result	size
risch	\(\frac {\left (x -4\right ) x}{{\mathrm e}^{x^{3}+4} x -4 \,{\mathrm e}^{x^{3}+4}-16}\)	\(26\)
norman	\(\frac {x^{2}-4 x}{{\mathrm e}^{x^{3}+4} x -4 \,{\mathrm e}^{x^{3}+4}-16}\)	\(29\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-3*x^5+24*x^4-48*x^3+x^2-8*x+16)*exp(x^3+4)-32*x+64)/((x^2-8*x+16)*exp(x^3+4)^2+(-32*x+128)*exp(x^3+4)+2
56),x,method=_RETURNVERBOSE)

[Out]

(x-4)*x/(exp(x^3+4)*x-4*exp(x^3+4)-16)

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maxima [A]  time = 0.82, size = 26, normalized size = 1.37 \begin {gather*} \frac {x^{2} - 4 \, x}{{\left (x e^{4} - 4 \, e^{4}\right )} e^{\left (x^{3}\right )} - 16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*x^5+24*x^4-48*x^3+x^2-8*x+16)*exp(x^3+4)-32*x+64)/((x^2-8*x+16)*exp(x^3+4)^2+(-32*x+128)*exp(x^
3+4)+256),x, algorithm="maxima")

[Out]

(x^2 - 4*x)/((x*e^4 - 4*e^4)*e^(x^3) - 16)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} \int -\frac {32\,x+{\mathrm {e}}^{x^3+4}\,\left (3\,x^5-24\,x^4+48\,x^3-x^2+8\,x-16\right )-64}{{\mathrm {e}}^{2\,x^3+8}\,\left (x^2-8\,x+16\right )-{\mathrm {e}}^{x^3+4}\,\left (32\,x-128\right )+256} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(32*x + exp(x^3 + 4)*(8*x - x^2 + 48*x^3 - 24*x^4 + 3*x^5 - 16) - 64)/(exp(2*x^3 + 8)*(x^2 - 8*x + 16) -
exp(x^3 + 4)*(32*x - 128) + 256),x)

[Out]

int(-(32*x + exp(x^3 + 4)*(8*x - x^2 + 48*x^3 - 24*x^4 + 3*x^5 - 16) - 64)/(exp(2*x^3 + 8)*(x^2 - 8*x + 16) -
exp(x^3 + 4)*(32*x - 128) + 256), x)

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sympy [A]  time = 0.16, size = 17, normalized size = 0.89 \begin {gather*} \frac {x^{2} - 4 x}{\left (x - 4\right ) e^{x^{3} + 4} - 16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*x**5+24*x**4-48*x**3+x**2-8*x+16)*exp(x**3+4)-32*x+64)/((x**2-8*x+16)*exp(x**3+4)**2+(-32*x+128
)*exp(x**3+4)+256),x)

[Out]

(x**2 - 4*x)/((x - 4)*exp(x**3 + 4) - 16)

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