3.11.29 \(\int \frac {518 x+256 x^2-16 x^4-2 x^5}{67081+132608 x+112674 x^2+54880 x^3+16991 x^4+3424 x^5+438 x^6+32 x^7+x^8} \, dx\)

Optimal. Leaf size=23 \[ e^3+\frac {x}{-5 x+\frac {3+(4+x)^4}{x}} \]

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Rubi [F]  time = 0.47, 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 {518 x+256 x^2-16 x^4-2 x^5}{67081+132608 x+112674 x^2+54880 x^3+16991 x^4+3424 x^5+438 x^6+32 x^7+x^8} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(518*x + 256*x^2 - 16*x^4 - 2*x^5)/(67081 + 132608*x + 112674*x^2 + 54880*x^3 + 16991*x^4 + 3424*x^5 + 438
*x^6 + 32*x^7 + x^8),x]

[Out]

37/(2*(259 + 256*x + 91*x^2 + 16*x^3 + x^4)) + 592*Defer[Int][(259 + 256*x + 91*x^2 + 16*x^3 + x^4)^(-2), x] +
307*Defer[Int][x/(259 + 256*x + 91*x^2 + 16*x^3 + x^4)^2, x] + 200*Defer[Int][x^2/(259 + 256*x + 91*x^2 + 16*
x^3 + x^4)^2, x] + 16*Defer[Int][(259 + 256*x + 91*x^2 + 16*x^3 + x^4)^(-1), x] - 2*Defer[Int][x/(259 + 256*x
+ 91*x^2 + 16*x^3 + x^4), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {2 \left (2072+1530 x+344 x^2+37 x^3\right )}{\left (259+256 x+91 x^2+16 x^3+x^4\right )^2}-\frac {2 (-8+x)}{259+256 x+91 x^2+16 x^3+x^4}\right ) \, dx\\ &=-\left (2 \int \frac {2072+1530 x+344 x^2+37 x^3}{\left (259+256 x+91 x^2+16 x^3+x^4\right )^2} \, dx\right )-2 \int \frac {-8+x}{259+256 x+91 x^2+16 x^3+x^4} \, dx\\ &=\frac {37}{2 \left (259+256 x+91 x^2+16 x^3+x^4\right )}-\frac {1}{2} \int \frac {-1184-614 x-400 x^2}{\left (259+256 x+91 x^2+16 x^3+x^4\right )^2} \, dx-2 \int \left (-\frac {8}{259+256 x+91 x^2+16 x^3+x^4}+\frac {x}{259+256 x+91 x^2+16 x^3+x^4}\right ) \, dx\\ &=\frac {37}{2 \left (259+256 x+91 x^2+16 x^3+x^4\right )}-\frac {1}{2} \int \left (-\frac {1184}{\left (259+256 x+91 x^2+16 x^3+x^4\right )^2}-\frac {614 x}{\left (259+256 x+91 x^2+16 x^3+x^4\right )^2}-\frac {400 x^2}{\left (259+256 x+91 x^2+16 x^3+x^4\right )^2}\right ) \, dx-2 \int \frac {x}{259+256 x+91 x^2+16 x^3+x^4} \, dx+16 \int \frac {1}{259+256 x+91 x^2+16 x^3+x^4} \, dx\\ &=\frac {37}{2 \left (259+256 x+91 x^2+16 x^3+x^4\right )}-2 \int \frac {x}{259+256 x+91 x^2+16 x^3+x^4} \, dx+16 \int \frac {1}{259+256 x+91 x^2+16 x^3+x^4} \, dx+200 \int \frac {x^2}{\left (259+256 x+91 x^2+16 x^3+x^4\right )^2} \, dx+307 \int \frac {x}{\left (259+256 x+91 x^2+16 x^3+x^4\right )^2} \, dx+592 \int \frac {1}{\left (259+256 x+91 x^2+16 x^3+x^4\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 24, normalized size = 1.04 \begin {gather*} \frac {x^2}{259+256 x+91 x^2+16 x^3+x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(518*x + 256*x^2 - 16*x^4 - 2*x^5)/(67081 + 132608*x + 112674*x^2 + 54880*x^3 + 16991*x^4 + 3424*x^5
+ 438*x^6 + 32*x^7 + x^8),x]

[Out]

x^2/(259 + 256*x + 91*x^2 + 16*x^3 + x^4)

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fricas [A]  time = 0.60, size = 24, normalized size = 1.04 \begin {gather*} \frac {x^{2}}{x^{4} + 16 \, x^{3} + 91 \, x^{2} + 256 \, x + 259} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x^5-16*x^4+256*x^2+518*x)/(x^8+32*x^7+438*x^6+3424*x^5+16991*x^4+54880*x^3+112674*x^2+132608*x+6
7081),x, algorithm="fricas")

[Out]

x^2/(x^4 + 16*x^3 + 91*x^2 + 256*x + 259)

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giac [A]  time = 0.25, size = 24, normalized size = 1.04 \begin {gather*} \frac {x^{2}}{x^{4} + 16 \, x^{3} + 91 \, x^{2} + 256 \, x + 259} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x^5-16*x^4+256*x^2+518*x)/(x^8+32*x^7+438*x^6+3424*x^5+16991*x^4+54880*x^3+112674*x^2+132608*x+6
7081),x, algorithm="giac")

[Out]

x^2/(x^4 + 16*x^3 + 91*x^2 + 256*x + 259)

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maple [A]  time = 0.03, size = 25, normalized size = 1.09

method	result	size
gosper	\(\frac {x^{2}}{x^{4}+16 x^{3}+91 x^{2}+256 x +259}\)	\(25\)
default	\(\frac {x^{2}}{x^{4}+16 x^{3}+91 x^{2}+256 x +259}\)	\(25\)
norman	\(\frac {x^{2}}{x^{4}+16 x^{3}+91 x^{2}+256 x +259}\)	\(25\)
risch	\(\frac {x^{2}}{x^{4}+16 x^{3}+91 x^{2}+256 x +259}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2*x^5-16*x^4+256*x^2+518*x)/(x^8+32*x^7+438*x^6+3424*x^5+16991*x^4+54880*x^3+112674*x^2+132608*x+67081),
x,method=_RETURNVERBOSE)

[Out]

x^2/(x^4+16*x^3+91*x^2+256*x+259)

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maxima [A]  time = 0.37, size = 24, normalized size = 1.04 \begin {gather*} \frac {x^{2}}{x^{4} + 16 \, x^{3} + 91 \, x^{2} + 256 \, x + 259} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x^5-16*x^4+256*x^2+518*x)/(x^8+32*x^7+438*x^6+3424*x^5+16991*x^4+54880*x^3+112674*x^2+132608*x+6
7081),x, algorithm="maxima")

[Out]

x^2/(x^4 + 16*x^3 + 91*x^2 + 256*x + 259)

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mupad [B]  time = 0.77, size = 24, normalized size = 1.04 \begin {gather*} \frac {x^2}{x^4+16\,x^3+91\,x^2+256\,x+259} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((518*x + 256*x^2 - 16*x^4 - 2*x^5)/(132608*x + 112674*x^2 + 54880*x^3 + 16991*x^4 + 3424*x^5 + 438*x^6 + 3
2*x^7 + x^8 + 67081),x)

[Out]

x^2/(256*x + 91*x^2 + 16*x^3 + x^4 + 259)

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sympy [A]  time = 0.12, size = 20, normalized size = 0.87 \begin {gather*} \frac {x^{2}}{x^{4} + 16 x^{3} + 91 x^{2} + 256 x + 259} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x**5-16*x**4+256*x**2+518*x)/(x**8+32*x**7+438*x**6+3424*x**5+16991*x**4+54880*x**3+112674*x**2+
132608*x+67081),x)

[Out]

x**2/(x**4 + 16*x**3 + 91*x**2 + 256*x + 259)

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