3.70.72 \(\int \frac {6250 x+1500 x^2-100 x^3-125 x^4+10 e^{2 x} x^5+10 e^{2 x} x^7+e^x x (-500 x^3+21 x^5+20 e^x x^5+x^6-x^7)+e^x (-500 x^3-20 x^4+22 x^5+2 x^6-x^7)}{3125-250 e^x x^2+5 e^{2 x} x^4+5 e^{2 x} x^6+e^x x (-250 x^2+10 e^x x^4)} \, dx\)

Optimal. Leaf size=30 \[ x \left (x+\frac {(-4+x) (5+x)}{5 \left (e^x-\frac {25}{x^2}+e^x x\right )}\right ) \]

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Rubi [F]  time = 2.73, 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 {6250 x+1500 x^2-100 x^3-125 x^4+10 e^{2 x} x^5+10 e^{2 x} x^7+e^x x \left (-500 x^3+21 x^5+20 e^x x^5+x^6-x^7\right )+e^x \left (-500 x^3-20 x^4+22 x^5+2 x^6-x^7\right )}{3125-250 e^x x^2+5 e^{2 x} x^4+5 e^{2 x} x^6+e^x x \left (-250 x^2+10 e^x x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(6250*x + 1500*x^2 - 100*x^3 - 125*x^4 + 10*E^(2*x)*x^5 + 10*E^(2*x)*x^7 + E^x*x*(-500*x^3 + 21*x^5 + 20*E
^x*x^5 + x^6 - x^7) + E^x*(-500*x^3 - 20*x^4 + 22*x^5 + 2*x^6 - x^7))/(3125 - 250*E^x*x^2 + 5*E^(2*x)*x^4 + 5*
E^(2*x)*x^6 + E^x*x*(-250*x^2 + 10*E^x*x^4)),x]

[Out]

x^2 + 100*Defer[Int][(-25 + E^x*x^2 + E^x*x^3)^(-2), x] - 100*Defer[Int][x/(-25 + E^x*x^2 + E^x*x^3)^2, x] + 3
00*Defer[Int][x^2/(-25 + E^x*x^2 + E^x*x^3)^2, x] + 90*Defer[Int][x^3/(-25 + E^x*x^2 + E^x*x^3)^2, x] - 20*Def
er[Int][x^4/(-25 + E^x*x^2 + E^x*x^3)^2, x] - 5*Defer[Int][x^5/(-25 + E^x*x^2 + E^x*x^3)^2, x] - 100*Defer[Int
][1/((1 + x)*(-25 + E^x*x^2 + E^x*x^3)^2), x] + 4*Defer[Int][(-25 + E^x*x^2 + E^x*x^3)^(-1), x] - 4*Defer[Int]
[x/(-25 + E^x*x^2 + E^x*x^3), x] + (22*Defer[Int][x^3/(-25 + E^x*x^2 + E^x*x^3), x])/5 + Defer[Int][x^4/(-25 +
E^x*x^2 + E^x*x^3), x]/5 - Defer[Int][x^5/(-25 + E^x*x^2 + E^x*x^3), x]/5 - 4*Defer[Int][1/((1 + x)*(-25 + E^
x*x^2 + E^x*x^3)), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {x \left (6250+1500 x-100 \left (1+5 e^x\right ) x^2-5 \left (25+104 e^x\right ) x^3+2 e^x \left (11+5 e^x\right ) x^4+e^x \left (23+20 e^x\right ) x^5+10 e^{2 x} x^6-e^x x^7\right )}{5 \left (25-e^x x^2 (1+x)\right )^2} \, dx\\ &=\frac {1}{5} \int \frac {x \left (6250+1500 x-100 \left (1+5 e^x\right ) x^2-5 \left (25+104 e^x\right ) x^3+2 e^x \left (11+5 e^x\right ) x^4+e^x \left (23+20 e^x\right ) x^5+10 e^{2 x} x^6-e^x x^7\right )}{\left (25-e^x x^2 (1+x)\right )^2} \, dx\\ &=\frac {1}{5} \int \left (10 x-\frac {x^2 \left (20-22 x-23 x^2+x^4\right )}{(1+x) \left (-25+e^x x^2+e^x x^3\right )}-\frac {25 x^2 \left (-40-78 x-14 x^2+5 x^3+x^4\right )}{(1+x) \left (-25+e^x x^2+e^x x^3\right )^2}\right ) \, dx\\ &=x^2-\frac {1}{5} \int \frac {x^2 \left (20-22 x-23 x^2+x^4\right )}{(1+x) \left (-25+e^x x^2+e^x x^3\right )} \, dx-5 \int \frac {x^2 \left (-40-78 x-14 x^2+5 x^3+x^4\right )}{(1+x) \left (-25+e^x x^2+e^x x^3\right )^2} \, dx\\ &=x^2-\frac {1}{5} \int \left (-\frac {20}{-25+e^x x^2+e^x x^3}+\frac {20 x}{-25+e^x x^2+e^x x^3}-\frac {22 x^3}{-25+e^x x^2+e^x x^3}-\frac {x^4}{-25+e^x x^2+e^x x^3}+\frac {x^5}{-25+e^x x^2+e^x x^3}+\frac {20}{(1+x) \left (-25+e^x x^2+e^x x^3\right )}\right ) \, dx-5 \int \left (-\frac {20}{\left (-25+e^x x^2+e^x x^3\right )^2}+\frac {20 x}{\left (-25+e^x x^2+e^x x^3\right )^2}-\frac {60 x^2}{\left (-25+e^x x^2+e^x x^3\right )^2}-\frac {18 x^3}{\left (-25+e^x x^2+e^x x^3\right )^2}+\frac {4 x^4}{\left (-25+e^x x^2+e^x x^3\right )^2}+\frac {x^5}{\left (-25+e^x x^2+e^x x^3\right )^2}+\frac {20}{(1+x) \left (-25+e^x x^2+e^x x^3\right )^2}\right ) \, dx\\ &=x^2+\frac {1}{5} \int \frac {x^4}{-25+e^x x^2+e^x x^3} \, dx-\frac {1}{5} \int \frac {x^5}{-25+e^x x^2+e^x x^3} \, dx+4 \int \frac {1}{-25+e^x x^2+e^x x^3} \, dx-4 \int \frac {x}{-25+e^x x^2+e^x x^3} \, dx-4 \int \frac {1}{(1+x) \left (-25+e^x x^2+e^x x^3\right )} \, dx+\frac {22}{5} \int \frac {x^3}{-25+e^x x^2+e^x x^3} \, dx-5 \int \frac {x^5}{\left (-25+e^x x^2+e^x x^3\right )^2} \, dx-20 \int \frac {x^4}{\left (-25+e^x x^2+e^x x^3\right )^2} \, dx+90 \int \frac {x^3}{\left (-25+e^x x^2+e^x x^3\right )^2} \, dx+100 \int \frac {1}{\left (-25+e^x x^2+e^x x^3\right )^2} \, dx-100 \int \frac {x}{\left (-25+e^x x^2+e^x x^3\right )^2} \, dx-100 \int \frac {1}{(1+x) \left (-25+e^x x^2+e^x x^3\right )^2} \, dx+300 \int \frac {x^2}{\left (-25+e^x x^2+e^x x^3\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.06, size = 31, normalized size = 1.03 \begin {gather*} \frac {1}{5} x^2 \left (5+\frac {x \left (-20+x+x^2\right )}{-25+e^x x^2 (1+x)}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(6250*x + 1500*x^2 - 100*x^3 - 125*x^4 + 10*E^(2*x)*x^5 + 10*E^(2*x)*x^7 + E^x*x*(-500*x^3 + 21*x^5
+ 20*E^x*x^5 + x^6 - x^7) + E^x*(-500*x^3 - 20*x^4 + 22*x^5 + 2*x^6 - x^7))/(3125 - 250*E^x*x^2 + 5*E^(2*x)*x^
4 + 5*E^(2*x)*x^6 + E^x*x*(-250*x^2 + 10*E^x*x^4)),x]

[Out]

(x^2*(5 + (x*(-20 + x + x^2))/(-25 + E^x*x^2*(1 + x))))/5

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fricas [A]  time = 0.60, size = 48, normalized size = 1.60 \begin {gather*} \frac {x^{5} + x^{4} - 20 \, x^{3} - 125 \, x^{2} + 5 \, {\left (x^{4} + x^{3}\right )} e^{\left (x + \log \relax (x)\right )}}{5 \, {\left ({\left (x^{2} + x\right )} e^{\left (x + \log \relax (x)\right )} - 25\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((10*x^5*exp(x+log(x))^2+(20*x^5*exp(x)-x^7+x^6+21*x^5-500*x^3)*exp(x+log(x))+10*x^5*exp(x)^2+(-x^7+2
*x^6+22*x^5-20*x^4-500*x^3)*exp(x)-125*x^4-100*x^3+1500*x^2+6250*x)/(5*x^4*exp(x+log(x))^2+(10*exp(x)*x^4-250*
x^2)*exp(x+log(x))+5*exp(x)^2*x^4-250*exp(x)*x^2+3125),x, algorithm="fricas")

[Out]

1/5*(x^5 + x^4 - 20*x^3 - 125*x^2 + 5*(x^4 + x^3)*e^(x + log(x)))/((x^2 + x)*e^(x + log(x)) - 25)

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giac [A]  time = 1.26, size = 49, normalized size = 1.63 \begin {gather*} \frac {5 \, x^{5} e^{x} + x^{5} + 5 \, x^{4} e^{x} + x^{4} - 20 \, x^{3} - 125 \, x^{2}}{5 \, {\left (x^{3} e^{x} + x^{2} e^{x} - 25\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((10*x^5*exp(x+log(x))^2+(20*x^5*exp(x)-x^7+x^6+21*x^5-500*x^3)*exp(x+log(x))+10*x^5*exp(x)^2+(-x^7+2
*x^6+22*x^5-20*x^4-500*x^3)*exp(x)-125*x^4-100*x^3+1500*x^2+6250*x)/(5*x^4*exp(x+log(x))^2+(10*exp(x)*x^4-250*
x^2)*exp(x+log(x))+5*exp(x)^2*x^4-250*exp(x)*x^2+3125),x, algorithm="giac")

[Out]

1/5*(5*x^5*e^x + x^5 + 5*x^4*e^x + x^4 - 20*x^3 - 125*x^2)/(x^3*e^x + x^2*e^x - 25)

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maple [A]  time = 0.08, size = 32, normalized size = 1.07

method	result	size
risch	\(x^{2}+\frac {\left (x^{2}+x -20\right ) x^{3}}{5 \,{\mathrm e}^{x} x^{3}+5 \,{\mathrm e}^{x} x^{2}-125}\)	\(32\)
norman	\(\frac {x^{5} {\mathrm e}^{x}+{\mathrm e}^{x} x^{4}-25 x^{2}-4 x^{3}+\frac {x^{4}}{5}+\frac {x^{5}}{5}}{{\mathrm e}^{x} x^{3}+{\mathrm e}^{x} x^{2}-25}\)	\(51\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((10*x^5*exp(x+ln(x))^2+(20*x^5*exp(x)-x^7+x^6+21*x^5-500*x^3)*exp(x+ln(x))+10*x^5*exp(x)^2+(-x^7+2*x^6+22*
x^5-20*x^4-500*x^3)*exp(x)-125*x^4-100*x^3+1500*x^2+6250*x)/(5*x^4*exp(x+ln(x))^2+(10*exp(x)*x^4-250*x^2)*exp(
x+ln(x))+5*exp(x)^2*x^4-250*exp(x)*x^2+3125),x,method=_RETURNVERBOSE)

[Out]

x^2+1/5*(x^2+x-20)*x^3/(exp(x)*x^3+exp(x)*x^2-25)

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maxima [A]  time = 0.43, size = 44, normalized size = 1.47 \begin {gather*} \frac {x^{5} + x^{4} - 20 \, x^{3} - 125 \, x^{2} + 5 \, {\left (x^{5} + x^{4}\right )} e^{x}}{5 \, {\left ({\left (x^{3} + x^{2}\right )} e^{x} - 25\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((10*x^5*exp(x+log(x))^2+(20*x^5*exp(x)-x^7+x^6+21*x^5-500*x^3)*exp(x+log(x))+10*x^5*exp(x)^2+(-x^7+2
*x^6+22*x^5-20*x^4-500*x^3)*exp(x)-125*x^4-100*x^3+1500*x^2+6250*x)/(5*x^4*exp(x+log(x))^2+(10*exp(x)*x^4-250*
x^2)*exp(x+log(x))+5*exp(x)^2*x^4-250*exp(x)*x^2+3125),x, algorithm="maxima")

[Out]

1/5*(x^5 + x^4 - 20*x^3 - 125*x^2 + 5*(x^5 + x^4)*e^x)/((x^3 + x^2)*e^x - 25)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {6250\,x+10\,x^5\,{\mathrm {e}}^{2\,x+2\,\ln \relax (x)}-{\mathrm {e}}^x\,\left (x^7-2\,x^6-22\,x^5+20\,x^4+500\,x^3\right )+{\mathrm {e}}^{x+\ln \relax (x)}\,\left (20\,x^5\,{\mathrm {e}}^x-500\,x^3+21\,x^5+x^6-x^7\right )+10\,x^5\,{\mathrm {e}}^{2\,x}+1500\,x^2-100\,x^3-125\,x^4}{5\,x^4\,{\mathrm {e}}^{2\,x+2\,\ln \relax (x)}-250\,x^2\,{\mathrm {e}}^x+{\mathrm {e}}^{x+\ln \relax (x)}\,\left (10\,x^4\,{\mathrm {e}}^x-250\,x^2\right )+5\,x^4\,{\mathrm {e}}^{2\,x}+3125} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((6250*x + 10*x^5*exp(2*x + 2*log(x)) - exp(x)*(500*x^3 + 20*x^4 - 22*x^5 - 2*x^6 + x^7) + exp(x + log(x))*
(20*x^5*exp(x) - 500*x^3 + 21*x^5 + x^6 - x^7) + 10*x^5*exp(2*x) + 1500*x^2 - 100*x^3 - 125*x^4)/(5*x^4*exp(2*
x + 2*log(x)) - 250*x^2*exp(x) + exp(x + log(x))*(10*x^4*exp(x) - 250*x^2) + 5*x^4*exp(2*x) + 3125),x)

[Out]

int((6250*x + 10*x^5*exp(2*x + 2*log(x)) - exp(x)*(500*x^3 + 20*x^4 - 22*x^5 - 2*x^6 + x^7) + exp(x + log(x))*
(20*x^5*exp(x) - 500*x^3 + 21*x^5 + x^6 - x^7) + 10*x^5*exp(2*x) + 1500*x^2 - 100*x^3 - 125*x^4)/(5*x^4*exp(2*
x + 2*log(x)) - 250*x^2*exp(x) + exp(x + log(x))*(10*x^4*exp(x) - 250*x^2) + 5*x^4*exp(2*x) + 3125), x)

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sympy [A]  time = 0.22, size = 29, normalized size = 0.97 \begin {gather*} x^{2} + \frac {x^{5} + x^{4} - 20 x^{3}}{\left (5 x^{3} + 5 x^{2}\right ) e^{x} - 125} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((10*x**5*exp(x+ln(x))**2+(20*x**5*exp(x)-x**7+x**6+21*x**5-500*x**3)*exp(x+ln(x))+10*x**5*exp(x)**2+
(-x**7+2*x**6+22*x**5-20*x**4-500*x**3)*exp(x)-125*x**4-100*x**3+1500*x**2+6250*x)/(5*x**4*exp(x+ln(x))**2+(10
*exp(x)*x**4-250*x**2)*exp(x+ln(x))+5*exp(x)**2*x**4-250*exp(x)*x**2+3125),x)

[Out]

x**2 + (x**5 + x**4 - 20*x**3)/((5*x**3 + 5*x**2)*exp(x) - 125)

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