3.100.21 \(\int \frac {-7500+e^{4 x}+4600 x+2600 x^2-19900 x^3+2501 x^4+2500 x^5+50 x^6+625 x^8+e^{2 x} (-300+100 x+2 x^2+50 x^4)}{2500+e^{4 x}+5000 x+2600 x^2+100 x^3+2501 x^4+2500 x^5+50 x^6+625 x^8+e^{2 x} (100+100 x+2 x^2+50 x^4)} \, dx\)

Optimal. Leaf size=27 \[ x+\frac {8}{2+2 x+x^4+\frac {1}{25} \left (e^{2 x}+x^2\right )} \]

________________________________________________________________________________________

Rubi [F]  time = 1.04, 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 {-7500+e^{4 x}+4600 x+2600 x^2-19900 x^3+2501 x^4+2500 x^5+50 x^6+625 x^8+e^{2 x} \left (-300+100 x+2 x^2+50 x^4\right )}{2500+e^{4 x}+5000 x+2600 x^2+100 x^3+2501 x^4+2500 x^5+50 x^6+625 x^8+e^{2 x} \left (100+100 x+2 x^2+50 x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-7500 + E^(4*x) + 4600*x + 2600*x^2 - 19900*x^3 + 2501*x^4 + 2500*x^5 + 50*x^6 + 625*x^8 + E^(2*x)*(-300
+ 100*x + 2*x^2 + 50*x^4))/(2500 + E^(4*x) + 5000*x + 2600*x^2 + 100*x^3 + 2501*x^4 + 2500*x^5 + 50*x^6 + 625*
x^8 + E^(2*x)*(100 + 100*x + 2*x^2 + 50*x^4)),x]

[Out]

x + 10000*Defer[Int][(50 + E^(2*x) + 50*x + x^2 + 25*x^4)^(-2), x] + 19600*Defer[Int][x/(50 + E^(2*x) + 50*x +
x^2 + 25*x^4)^2, x] + 400*Defer[Int][x^2/(50 + E^(2*x) + 50*x + x^2 + 25*x^4)^2, x] - 20000*Defer[Int][x^3/(5
0 + E^(2*x) + 50*x + x^2 + 25*x^4)^2, x] + 10000*Defer[Int][x^4/(50 + E^(2*x) + 50*x + x^2 + 25*x^4)^2, x] - 4
00*Defer[Int][(50 + E^(2*x) + 50*x + x^2 + 25*x^4)^(-1), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-7500+e^{4 x}+4600 x+2600 x^2-19900 x^3+2501 x^4+2500 x^5+50 x^6+625 x^8+2 e^{2 x} \left (-150+50 x+x^2+25 x^4\right )}{\left (50+e^{2 x}+50 x+x^2+25 x^4\right )^2} \, dx\\ &=\int \left (1-\frac {400}{50+e^{2 x}+50 x+x^2+25 x^4}+\frac {400 \left (25+49 x+x^2-50 x^3+25 x^4\right )}{\left (50+e^{2 x}+50 x+x^2+25 x^4\right )^2}\right ) \, dx\\ &=x-400 \int \frac {1}{50+e^{2 x}+50 x+x^2+25 x^4} \, dx+400 \int \frac {25+49 x+x^2-50 x^3+25 x^4}{\left (50+e^{2 x}+50 x+x^2+25 x^4\right )^2} \, dx\\ &=x-400 \int \frac {1}{50+e^{2 x}+50 x+x^2+25 x^4} \, dx+400 \int \left (\frac {25}{\left (50+e^{2 x}+50 x+x^2+25 x^4\right )^2}+\frac {49 x}{\left (50+e^{2 x}+50 x+x^2+25 x^4\right )^2}+\frac {x^2}{\left (50+e^{2 x}+50 x+x^2+25 x^4\right )^2}-\frac {50 x^3}{\left (50+e^{2 x}+50 x+x^2+25 x^4\right )^2}+\frac {25 x^4}{\left (50+e^{2 x}+50 x+x^2+25 x^4\right )^2}\right ) \, dx\\ &=x+400 \int \frac {x^2}{\left (50+e^{2 x}+50 x+x^2+25 x^4\right )^2} \, dx-400 \int \frac {1}{50+e^{2 x}+50 x+x^2+25 x^4} \, dx+10000 \int \frac {1}{\left (50+e^{2 x}+50 x+x^2+25 x^4\right )^2} \, dx+10000 \int \frac {x^4}{\left (50+e^{2 x}+50 x+x^2+25 x^4\right )^2} \, dx+19600 \int \frac {x}{\left (50+e^{2 x}+50 x+x^2+25 x^4\right )^2} \, dx-20000 \int \frac {x^3}{\left (50+e^{2 x}+50 x+x^2+25 x^4\right )^2} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.03, size = 24, normalized size = 0.89 \begin {gather*} x+\frac {200}{50+e^{2 x}+50 x+x^2+25 x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-7500 + E^(4*x) + 4600*x + 2600*x^2 - 19900*x^3 + 2501*x^4 + 2500*x^5 + 50*x^6 + 625*x^8 + E^(2*x)*
(-300 + 100*x + 2*x^2 + 50*x^4))/(2500 + E^(4*x) + 5000*x + 2600*x^2 + 100*x^3 + 2501*x^4 + 2500*x^5 + 50*x^6
+ 625*x^8 + E^(2*x)*(100 + 100*x + 2*x^2 + 50*x^4)),x]

[Out]

x + 200/(50 + E^(2*x) + 50*x + x^2 + 25*x^4)

________________________________________________________________________________________

fricas [A]  time = 1.03, size = 44, normalized size = 1.63 \begin {gather*} \frac {25 \, x^{5} + x^{3} + 50 \, x^{2} + x e^{\left (2 \, x\right )} + 50 \, x + 200}{25 \, x^{4} + x^{2} + 50 \, x + e^{\left (2 \, x\right )} + 50} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(x)^4+(50*x^4+2*x^2+100*x-300)*exp(x)^2+625*x^8+50*x^6+2500*x^5+2501*x^4-19900*x^3+2600*x^2+4600
*x-7500)/(exp(x)^4+(50*x^4+2*x^2+100*x+100)*exp(x)^2+625*x^8+50*x^6+2500*x^5+2501*x^4+100*x^3+2600*x^2+5000*x+
2500),x, algorithm="fricas")

[Out]

(25*x^5 + x^3 + 50*x^2 + x*e^(2*x) + 50*x + 200)/(25*x^4 + x^2 + 50*x + e^(2*x) + 50)

________________________________________________________________________________________

giac [A]  time = 0.18, size = 44, normalized size = 1.63 \begin {gather*} \frac {25 \, x^{5} + x^{3} + 50 \, x^{2} + x e^{\left (2 \, x\right )} + 50 \, x + 200}{25 \, x^{4} + x^{2} + 50 \, x + e^{\left (2 \, x\right )} + 50} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(x)^4+(50*x^4+2*x^2+100*x-300)*exp(x)^2+625*x^8+50*x^6+2500*x^5+2501*x^4-19900*x^3+2600*x^2+4600
*x-7500)/(exp(x)^4+(50*x^4+2*x^2+100*x+100)*exp(x)^2+625*x^8+50*x^6+2500*x^5+2501*x^4+100*x^3+2600*x^2+5000*x+
2500),x, algorithm="giac")

[Out]

(25*x^5 + x^3 + 50*x^2 + x*e^(2*x) + 50*x + 200)/(25*x^4 + x^2 + 50*x + e^(2*x) + 50)

________________________________________________________________________________________

maple [A]  time = 0.04, size = 24, normalized size = 0.89

method	result	size
risch	\(x +\frac {200}{25 x^{4}+{\mathrm e}^{2 x}+x^{2}+50 x +50}\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x)^4+(50*x^4+2*x^2+100*x-300)*exp(x)^2+625*x^8+50*x^6+2500*x^5+2501*x^4-19900*x^3+2600*x^2+4600*x-750
0)/(exp(x)^4+(50*x^4+2*x^2+100*x+100)*exp(x)^2+625*x^8+50*x^6+2500*x^5+2501*x^4+100*x^3+2600*x^2+5000*x+2500),
x,method=_RETURNVERBOSE)

[Out]

x+200/(25*x^4+exp(2*x)+x^2+50*x+50)

________________________________________________________________________________________

maxima [A]  time = 0.39, size = 44, normalized size = 1.63 \begin {gather*} \frac {25 \, x^{5} + x^{3} + 50 \, x^{2} + x e^{\left (2 \, x\right )} + 50 \, x + 200}{25 \, x^{4} + x^{2} + 50 \, x + e^{\left (2 \, x\right )} + 50} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(x)^4+(50*x^4+2*x^2+100*x-300)*exp(x)^2+625*x^8+50*x^6+2500*x^5+2501*x^4-19900*x^3+2600*x^2+4600
*x-7500)/(exp(x)^4+(50*x^4+2*x^2+100*x+100)*exp(x)^2+625*x^8+50*x^6+2500*x^5+2501*x^4+100*x^3+2600*x^2+5000*x+
2500),x, algorithm="maxima")

[Out]

(25*x^5 + x^3 + 50*x^2 + x*e^(2*x) + 50*x + 200)/(25*x^4 + x^2 + 50*x + e^(2*x) + 50)

________________________________________________________________________________________

mupad [B]  time = 6.54, size = 23, normalized size = 0.85 \begin {gather*} x+\frac {200}{50\,x+{\mathrm {e}}^{2\,x}+x^2+25\,x^4+50} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4600*x + exp(4*x) + exp(2*x)*(100*x + 2*x^2 + 50*x^4 - 300) + 2600*x^2 - 19900*x^3 + 2501*x^4 + 2500*x^5
+ 50*x^6 + 625*x^8 - 7500)/(5000*x + exp(4*x) + exp(2*x)*(100*x + 2*x^2 + 50*x^4 + 100) + 2600*x^2 + 100*x^3 +
2501*x^4 + 2500*x^5 + 50*x^6 + 625*x^8 + 2500),x)

[Out]

x + 200/(50*x + exp(2*x) + x^2 + 25*x^4 + 50)

________________________________________________________________________________________

sympy [A]  time = 0.17, size = 20, normalized size = 0.74 \begin {gather*} x + \frac {200}{25 x^{4} + x^{2} + 50 x + e^{2 x} + 50} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(x)**4+(50*x**4+2*x**2+100*x-300)*exp(x)**2+625*x**8+50*x**6+2500*x**5+2501*x**4-19900*x**3+2600
*x**2+4600*x-7500)/(exp(x)**4+(50*x**4+2*x**2+100*x+100)*exp(x)**2+625*x**8+50*x**6+2500*x**5+2501*x**4+100*x*
*3+2600*x**2+5000*x+2500),x)

[Out]

x + 200/(25*x**4 + x**2 + 50*x + exp(2*x) + 50)

________________________________________________________________________________________