\(\int \frac {518 x+256 x^2-16 x^4-2 x^5}{67081+132608 x+112674 x^2+54880 x^3+16991 x^4+3424 x^5+438 x^6+32 x^7+x^8} \, dx\) [1029]

Optimal result

Integrand size = 60, antiderivative size = 23 \[ \int \frac {518 x+256 x^2-16 x^4-2 x^5}{67081+132608 x+112674 x^2+54880 x^3+16991 x^4+3424 x^5+438 x^6+32 x^7+x^8} \, dx=e^3+\frac {x}{-5 x+\frac {3+(4+x)^4}{x}} \]

[Out]

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

Rubi [F]

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

[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{align*} \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{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.04 \[ \int \frac {518 x+256 x^2-16 x^4-2 x^5}{67081+132608 x+112674 x^2+54880 x^3+16991 x^4+3424 x^5+438 x^6+32 x^7+x^8} \, dx=\frac {x^2}{259+256 x+91 x^2+16 x^3+x^4} \]

[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)

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 25, normalized size of antiderivative = 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	\(\frac {x^{2}}{x^{4}+16 x^{3}+91 x^{2}+256 x +259}\)	\(25\)
parallelrisch	\(\frac {x^{2}}{x^{4}+16 x^{3}+91 x^{2}+256 x +259}\)	\(25\)

[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)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.04 \[ \int \frac {518 x+256 x^2-16 x^4-2 x^5}{67081+132608 x+112674 x^2+54880 x^3+16991 x^4+3424 x^5+438 x^6+32 x^7+x^8} \, dx=\frac {x^{2}}{x^{4} + 16 \, x^{3} + 91 \, x^{2} + 256 \, x + 259} \]

[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)

Sympy [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87 \[ \int \frac {518 x+256 x^2-16 x^4-2 x^5}{67081+132608 x+112674 x^2+54880 x^3+16991 x^4+3424 x^5+438 x^6+32 x^7+x^8} \, dx=\frac {x^{2}}{x^{4} + 16 x^{3} + 91 x^{2} + 256 x + 259} \]

[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)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.04 \[ \int \frac {518 x+256 x^2-16 x^4-2 x^5}{67081+132608 x+112674 x^2+54880 x^3+16991 x^4+3424 x^5+438 x^6+32 x^7+x^8} \, dx=\frac {x^{2}}{x^{4} + 16 \, x^{3} + 91 \, x^{2} + 256 \, x + 259} \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.04 \[ \int \frac {518 x+256 x^2-16 x^4-2 x^5}{67081+132608 x+112674 x^2+54880 x^3+16991 x^4+3424 x^5+438 x^6+32 x^7+x^8} \, dx=\frac {x^{2}}{x^{4} + 16 \, x^{3} + 91 \, x^{2} + 256 \, x + 259} \]

[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)

Mupad [B] (verification not implemented)

Time = 9.18 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.04 \[ \int \frac {518 x+256 x^2-16 x^4-2 x^5}{67081+132608 x+112674 x^2+54880 x^3+16991 x^4+3424 x^5+438 x^6+32 x^7+x^8} \, dx=\frac {x^2}{x^4+16\,x^3+91\,x^2+256\,x+259} \]

[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)