3.16.8 \(\int \frac {-25 x+e^{\frac {1}{5} (-26-5 x^2+5 \log (x^2))} (-250+250 x^2)}{25 e^{\frac {2}{5} (-26-5 x^2+5 \log (x^2))} x+10 e^{\frac {1}{5} (-26-5 x^2+5 \log (x^2))} x^2+x^3} \, dx\)

Optimal. Leaf size=25 \[ \frac {5}{\frac {x}{5}+e^{-\frac {26}{5}-x^2} x^2} \]

________________________________________________________________________________________

Rubi [F]  time = 2.32, 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 {-25 x+e^{\frac {1}{5} \left (-26-5 x^2+5 \log \left (x^2\right )\right )} \left (-250+250 x^2\right )}{25 e^{\frac {2}{5} \left (-26-5 x^2+5 \log \left (x^2\right )\right )} x+10 e^{\frac {1}{5} \left (-26-5 x^2+5 \log \left (x^2\right )\right )} x^2+x^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-25*x + E^((-26 - 5*x^2 + 5*Log[x^2])/5)*(-250 + 250*x^2))/(25*E^((2*(-26 - 5*x^2 + 5*Log[x^2]))/5)*x + 1
0*E^((-26 - 5*x^2 + 5*Log[x^2])/5)*x^2 + x^3),x]

[Out]

-125*Defer[Int][E^(26/5 + x^2)/(x*(E^(26/5 + x^2) + 5*x)^2), x] + 250*Defer[Int][(E^(26/5 + x^2)*x)/(E^(26/5 +
x^2) + 5*x)^2, x] - 25*Defer[Int][E^(26/5 + x^2)/(x^2*(E^(26/5 + x^2) + 5*x)), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {25 e^{\frac {26}{5}+x^2} \left (-e^{\frac {26}{5}+x^2}+10 x \left (-1+x^2\right )\right )}{x^2 \left (e^{\frac {26}{5}+x^2}+5 x\right )^2} \, dx\\ &=25 \int \frac {e^{\frac {26}{5}+x^2} \left (-e^{\frac {26}{5}+x^2}+10 x \left (-1+x^2\right )\right )}{x^2 \left (e^{\frac {26}{5}+x^2}+5 x\right )^2} \, dx\\ &=25 \int \left (-\frac {e^{\frac {26}{5}+x^2}}{x^2 \left (e^{\frac {26}{5}+x^2}+5 x\right )}+\frac {5 e^{\frac {26}{5}+x^2} \left (-1+2 x^2\right )}{x \left (e^{\frac {26}{5}+x^2}+5 x\right )^2}\right ) \, dx\\ &=-\left (25 \int \frac {e^{\frac {26}{5}+x^2}}{x^2 \left (e^{\frac {26}{5}+x^2}+5 x\right )} \, dx\right )+125 \int \frac {e^{\frac {26}{5}+x^2} \left (-1+2 x^2\right )}{x \left (e^{\frac {26}{5}+x^2}+5 x\right )^2} \, dx\\ &=-\left (25 \int \frac {e^{\frac {26}{5}+x^2}}{x^2 \left (e^{\frac {26}{5}+x^2}+5 x\right )} \, dx\right )+125 \int \left (-\frac {e^{\frac {26}{5}+x^2}}{x \left (e^{\frac {26}{5}+x^2}+5 x\right )^2}+\frac {2 e^{\frac {26}{5}+x^2} x}{\left (e^{\frac {26}{5}+x^2}+5 x\right )^2}\right ) \, dx\\ &=-\left (25 \int \frac {e^{\frac {26}{5}+x^2}}{x^2 \left (e^{\frac {26}{5}+x^2}+5 x\right )} \, dx\right )-125 \int \frac {e^{\frac {26}{5}+x^2}}{x \left (e^{\frac {26}{5}+x^2}+5 x\right )^2} \, dx+250 \int \frac {e^{\frac {26}{5}+x^2} x}{\left (e^{\frac {26}{5}+x^2}+5 x\right )^2} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.31, size = 25, normalized size = 1.00 \begin {gather*} -25 \left (-\frac {1}{x}+\frac {5}{e^{\frac {26}{5}+x^2}+5 x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-25*x + E^((-26 - 5*x^2 + 5*Log[x^2])/5)*(-250 + 250*x^2))/(25*E^((2*(-26 - 5*x^2 + 5*Log[x^2]))/5)
*x + 10*E^((-26 - 5*x^2 + 5*Log[x^2])/5)*x^2 + x^3),x]

[Out]

-25*(-x^(-1) + 5/(E^(26/5 + x^2) + 5*x))

________________________________________________________________________________________

fricas [A]  time = 0.81, size = 20, normalized size = 0.80 \begin {gather*} \frac {25}{x + 5 \, e^{\left (-x^{2} + \log \left (x^{2}\right ) - \frac {26}{5}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((250*x^2-250)*exp(log(x^2)-x^2-26/5)-25*x)/(25*x*exp(log(x^2)-x^2-26/5)^2+10*x^2*exp(log(x^2)-x^2-2
6/5)+x^3),x, algorithm="fricas")

[Out]

25/(x + 5*e^(-x^2 + log(x^2) - 26/5))

________________________________________________________________________________________

giac [A]  time = 0.18, size = 19, normalized size = 0.76 \begin {gather*} \frac {25}{5 \, x^{2} e^{\left (-x^{2} - \frac {26}{5}\right )} + x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((250*x^2-250)*exp(log(x^2)-x^2-26/5)-25*x)/(25*x*exp(log(x^2)-x^2-26/5)^2+10*x^2*exp(log(x^2)-x^2-2
6/5)+x^3),x, algorithm="giac")

[Out]

25/(5*x^2*e^(-x^2 - 26/5) + x)

________________________________________________________________________________________

maple [A]  time = 0.14, size = 20, normalized size = 0.80

method	result	size
risch	\(\frac {25}{5 x^{2} {\mathrm e}^{-\frac {26}{5}-x^{2}}+x}\)	\(20\)
default	\(\frac {25}{5 \,{\mathrm e}^{\ln \left (x^{2}\right )-x^{2}-\frac {26}{5}}+x}\)	\(21\)
norman	\(\frac {25}{5 \,{\mathrm e}^{\ln \left (x^{2}\right )-x^{2}-\frac {26}{5}}+x}\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((250*x^2-250)*exp(ln(x^2)-x^2-26/5)-25*x)/(25*x*exp(ln(x^2)-x^2-26/5)^2+10*x^2*exp(ln(x^2)-x^2-26/5)+x^3)
,x,method=_RETURNVERBOSE)

[Out]

25/(5*x^2*exp(-26/5-x^2)+x)

________________________________________________________________________________________

maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((250*x^2-250)*exp(log(x^2)-x^2-26/5)-25*x)/(25*x*exp(log(x^2)-x^2-26/5)^2+10*x^2*exp(log(x^2)-x^2-2
6/5)+x^3),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is undefined.

________________________________________________________________________________________

mupad [B]  time = 1.11, size = 24, normalized size = 0.96 \begin {gather*} \frac {25\,x^2}{x^3+5\,x^4\,{\mathrm {e}}^{-\frac {26}{5}}\,{\mathrm {e}}^{-x^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(25*x - exp(log(x^2) - x^2 - 26/5)*(250*x^2 - 250))/(25*x*exp(2*log(x^2) - 2*x^2 - 52/5) + 10*x^2*exp(log
(x^2) - x^2 - 26/5) + x^3),x)

[Out]

(25*x^2)/(x^3 + 5*x^4*exp(-26/5)*exp(-x^2))

________________________________________________________________________________________

sympy [A]  time = 0.14, size = 17, normalized size = 0.68 \begin {gather*} \frac {25}{5 x^{2} e^{- x^{2} - \frac {26}{5}} + x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((250*x**2-250)*exp(ln(x**2)-x**2-26/5)-25*x)/(25*x*exp(ln(x**2)-x**2-26/5)**2+10*x**2*exp(ln(x**2)-
x**2-26/5)+x**3),x)

[Out]

25/(5*x**2*exp(-x**2 - 26/5) + x)

________________________________________________________________________________________