3.8.34 \(\int \frac {896 x+2 e^{2 x} x+68 x^2-324 x^3+5 x^4+50 x^5+e^x (-88 x+7 x^2+23 x^3)}{768+3 e^{2 x}-96 x-477 x^2+30 x^3+75 x^4+e^x (-96+6 x+30 x^2)} \, dx\)

Optimal. Leaf size=33 \[ \frac {1}{3} \left (x^2+\frac {3 x^2 (4+x)}{16-e^x-x-5 x^2}\right ) \]

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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 {896 x+2 e^{2 x} x+68 x^2-324 x^3+5 x^4+50 x^5+e^x \left (-88 x+7 x^2+23 x^3\right )}{768+3 e^{2 x}-96 x-477 x^2+30 x^3+75 x^4+e^x \left (-96+6 x+30 x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(896*x + 2*E^(2*x)*x + 68*x^2 - 324*x^3 + 5*x^4 + 50*x^5 + E^x*(-88*x + 7*x^2 + 23*x^3))/(768 + 3*E^(2*x)
- 96*x - 477*x^2 + 30*x^3 + 75*x^4 + E^x*(-96 + 6*x + 30*x^2)),x]

[Out]

x^2/3 + 68*Defer[Int][x^2/(-16 + E^x + x + 5*x^2)^2, x] + 53*Defer[Int][x^3/(-16 + E^x + x + 5*x^2)^2, x] - 11
*Defer[Int][x^4/(-16 + E^x + x + 5*x^2)^2, x] - 5*Defer[Int][x^5/(-16 + E^x + x + 5*x^2)^2, x] - 8*Defer[Int][
x/(-16 + E^x + x + 5*x^2), x] + Defer[Int][x^2/(-16 + E^x + x + 5*x^2), x] + Defer[Int][x^3/(-16 + E^x + x + 5
*x^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {x \left (896+2 e^{2 x}+68 x-324 x^2+5 x^3+50 x^4+e^x \left (-88+7 x+23 x^2\right )\right )}{3 \left (16-e^x-x-5 x^2\right )^2} \, dx\\ &=\frac {1}{3} \int \frac {x \left (896+2 e^{2 x}+68 x-324 x^2+5 x^3+50 x^4+e^x \left (-88+7 x+23 x^2\right )\right )}{\left (16-e^x-x-5 x^2\right )^2} \, dx\\ &=\frac {1}{3} \int \left (2 x+\frac {3 x \left (-8+x+x^2\right )}{-16+e^x+x+5 x^2}-\frac {3 x^2 \left (-68-53 x+11 x^2+5 x^3\right )}{\left (-16+e^x+x+5 x^2\right )^2}\right ) \, dx\\ &=\frac {x^2}{3}+\int \frac {x \left (-8+x+x^2\right )}{-16+e^x+x+5 x^2} \, dx-\int \frac {x^2 \left (-68-53 x+11 x^2+5 x^3\right )}{\left (-16+e^x+x+5 x^2\right )^2} \, dx\\ &=\frac {x^2}{3}-\int \left (-\frac {68 x^2}{\left (-16+e^x+x+5 x^2\right )^2}-\frac {53 x^3}{\left (-16+e^x+x+5 x^2\right )^2}+\frac {11 x^4}{\left (-16+e^x+x+5 x^2\right )^2}+\frac {5 x^5}{\left (-16+e^x+x+5 x^2\right )^2}\right ) \, dx+\int \left (-\frac {8 x}{-16+e^x+x+5 x^2}+\frac {x^2}{-16+e^x+x+5 x^2}+\frac {x^3}{-16+e^x+x+5 x^2}\right ) \, dx\\ &=\frac {x^2}{3}-5 \int \frac {x^5}{\left (-16+e^x+x+5 x^2\right )^2} \, dx-8 \int \frac {x}{-16+e^x+x+5 x^2} \, dx-11 \int \frac {x^4}{\left (-16+e^x+x+5 x^2\right )^2} \, dx+53 \int \frac {x^3}{\left (-16+e^x+x+5 x^2\right )^2} \, dx+68 \int \frac {x^2}{\left (-16+e^x+x+5 x^2\right )^2} \, dx+\int \frac {x^2}{-16+e^x+x+5 x^2} \, dx+\int \frac {x^3}{-16+e^x+x+5 x^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.27, size = 27, normalized size = 0.82 \begin {gather*} \frac {1}{3} x^2 \left (1-\frac {3 (4+x)}{-16+e^x+x+5 x^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(896*x + 2*E^(2*x)*x + 68*x^2 - 324*x^3 + 5*x^4 + 50*x^5 + E^x*(-88*x + 7*x^2 + 23*x^3))/(768 + 3*E^
(2*x) - 96*x - 477*x^2 + 30*x^3 + 75*x^4 + E^x*(-96 + 6*x + 30*x^2)),x]

[Out]

(x^2*(1 - (3*(4 + x))/(-16 + E^x + x + 5*x^2)))/3

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fricas [A]  time = 0.58, size = 36, normalized size = 1.09 \begin {gather*} \frac {5 \, x^{4} - 2 \, x^{3} + x^{2} e^{x} - 28 \, x^{2}}{3 \, {\left (5 \, x^{2} + x + e^{x} - 16\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x*exp(x)^2+(23*x^3+7*x^2-88*x)*exp(x)+50*x^5+5*x^4-324*x^3+68*x^2+896*x)/(3*exp(x)^2+(30*x^2+6*x-
96)*exp(x)+75*x^4+30*x^3-477*x^2-96*x+768),x, algorithm="fricas")

[Out]

1/3*(5*x^4 - 2*x^3 + x^2*e^x - 28*x^2)/(5*x^2 + x + e^x - 16)

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giac [A]  time = 0.29, size = 36, normalized size = 1.09 \begin {gather*} \frac {5 \, x^{4} - 2 \, x^{3} + x^{2} e^{x} - 28 \, x^{2}}{3 \, {\left (5 \, x^{2} + x + e^{x} - 16\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x*exp(x)^2+(23*x^3+7*x^2-88*x)*exp(x)+50*x^5+5*x^4-324*x^3+68*x^2+896*x)/(3*exp(x)^2+(30*x^2+6*x-
96)*exp(x)+75*x^4+30*x^3-477*x^2-96*x+768),x, algorithm="giac")

[Out]

1/3*(5*x^4 - 2*x^3 + x^2*e^x - 28*x^2)/(5*x^2 + x + e^x - 16)

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maple [A]  time = 0.07, size = 27, normalized size = 0.82

method	result	size
risch	\(\frac {x^{2}}{3}-\frac {x^{2} \left (4+x \right )}{5 x^{2}+{\mathrm e}^{x}+x -16}\)	\(27\)
norman	\(\frac {\frac {28 x}{15}+\frac {28 \,{\mathrm e}^{x}}{15}-\frac {2 x^{3}}{3}+\frac {5 x^{4}}{3}+\frac {{\mathrm e}^{x} x^{2}}{3}-\frac {448}{15}}{5 x^{2}+{\mathrm e}^{x}+x -16}\)	\(40\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x*exp(x)^2+(23*x^3+7*x^2-88*x)*exp(x)+50*x^5+5*x^4-324*x^3+68*x^2+896*x)/(3*exp(x)^2+(30*x^2+6*x-96)*ex
p(x)+75*x^4+30*x^3-477*x^2-96*x+768),x,method=_RETURNVERBOSE)

[Out]

1/3*x^2-x^2*(4+x)/(5*x^2+exp(x)+x-16)

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maxima [A]  time = 0.52, size = 36, normalized size = 1.09 \begin {gather*} \frac {5 \, x^{4} - 2 \, x^{3} + x^{2} e^{x} - 28 \, x^{2}}{3 \, {\left (5 \, x^{2} + x + e^{x} - 16\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x*exp(x)^2+(23*x^3+7*x^2-88*x)*exp(x)+50*x^5+5*x^4-324*x^3+68*x^2+896*x)/(3*exp(x)^2+(30*x^2+6*x-
96)*exp(x)+75*x^4+30*x^3-477*x^2-96*x+768),x, algorithm="maxima")

[Out]

1/3*(5*x^4 - 2*x^3 + x^2*e^x - 28*x^2)/(5*x^2 + x + e^x - 16)

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mupad [B]  time = 0.69, size = 35, normalized size = 1.06 \begin {gather*} -\frac {x^2\,\left (2\,x-{\mathrm {e}}^x-5\,x^2+28\right )}{3\,\left (x+{\mathrm {e}}^x+5\,x^2-16\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((896*x + 2*x*exp(2*x) + 68*x^2 - 324*x^3 + 5*x^4 + 50*x^5 + exp(x)*(7*x^2 - 88*x + 23*x^3))/(3*exp(2*x) -
96*x + exp(x)*(6*x + 30*x^2 - 96) - 477*x^2 + 30*x^3 + 75*x^4 + 768),x)

[Out]

-(x^2*(2*x - exp(x) - 5*x^2 + 28))/(3*(x + exp(x) + 5*x^2 - 16))

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sympy [A]  time = 0.14, size = 26, normalized size = 0.79 \begin {gather*} \frac {x^{2}}{3} + \frac {- x^{3} - 4 x^{2}}{5 x^{2} + x + e^{x} - 16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x*exp(x)**2+(23*x**3+7*x**2-88*x)*exp(x)+50*x**5+5*x**4-324*x**3+68*x**2+896*x)/(3*exp(x)**2+(30*
x**2+6*x-96)*exp(x)+75*x**4+30*x**3-477*x**2-96*x+768),x)

[Out]

x**2/3 + (-x**3 - 4*x**2)/(5*x**2 + x + exp(x) - 16)

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