3.73.78 $\int \frac{-1250-1250e^x}{e^{3+2x}+625x+e^3{x}^2+e^x(625+2e^{3}x)}dx$

Optimal. Leaf size=18 $$2+\log \left({(e^3+\frac{625}{e^x+x})}^2\right)$$

________________________________________________________________________________________

Rubi [F]  time = 0.43, 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 = {} $$\begin{array}{c}\int \frac{-1250-1250e^x}{e^{3+2x}+625x+e^3{x}^2+e^x(625+2e^{3}x)}dx\end{array}$$

Verification is not applicable to the result.

[In]

Int[(-1250 - 1250*E^x)/(E^(3 + 2*x) + 625*x + E^3*x^2 + E^x*(625 + 2*E^3*x)),x]

[Out]

-2*Defer[Int][(E^x + x)^(-1), x] + 2*Defer[Int][x/(E^x + x), x] - 2*(625 - E^3)*Defer[Int][(625 + E^(3 + x) +
E^3*x)^(-1), x] - 2*E^3*Defer[Int][x/(625 + E^(3 + x) + E^3*x), x]

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int \frac{1250(-1-e^x)}{e^{3+2x}+625x+e^3{x}^2+e^x(625+2e^{3}x)}dx\\ & =1250\int \frac{-1-e^x}{e^{3+2x}+625x+e^3{x}^2+e^x(625+2e^{3}x)}dx\\ & =1250\int (\frac{-1+x}{625(e^x+x)}-\frac{625-e^3+e^{3}x}{625(625+e^{3+x}+e^{3}x)})dx\\ & =2\int \frac{-1+x}{e^x+x}dx-2\int \frac{625-e^3+e^{3}x}{625+e^{3+x}+e^{3}x}dx\\ & =2\int (-\frac{1}{e^x+x}+\frac{x}{e^x+x})dx-2\int (\frac{625(1-\frac{e^3}{625})}{625+e^{3+x}+e^{3}x}+\frac{e^{3}x}{625+e^{3+x}+e^{3}x})dx\\ & =-(2\int \frac{1}{e^x+x}dx)+2\int \frac{x}{e^x+x}dx-\left(2e^3\right)\int \frac{x}{625+e^{3+x}+e^{3}x}dx-\left(2(625-e^3)\right)\int \frac{1}{625+e^{3+x}+e^{3}x}dx\end{array}\end{array}$$

________________________________________________________________________________________

Mathematica [A]  time = 0.14, size = 30, normalized size = 1.67 $$\begin{array}{c}-1250(\frac{1}{625}\log (e^x+x)-\frac{1}{625}\log (625+e^{3+x}+e^{3}x))\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(-1250 - 1250*E^x)/(E^(3 + 2*x) + 625*x + E^3*x^2 + E^x*(625 + 2*E^3*x)),x]

[Out]

-1250*(Log[E^x + x]/625 - Log[625 + E^(3 + x) + E^3*x]/625)

________________________________________________________________________________________

fricas [A]  time = 0.65, size = 21, normalized size = 1.17 $$\begin{array}{c}2\log (x{e}^3+e^{(x+3)}+625)-2\log (x+e^x)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1250*exp(x)-1250)/(exp(3)*exp(x)^2+(2*x*exp(3)+625)*exp(x)+x^2*exp(3)+625*x),x, algorithm="fricas"
)

[Out]

2*log(x*e^3 + e^(x + 3) + 625) - 2*log(x + e^x)

________________________________________________________________________________________

giac [A]  time = 0.21, size = 24, normalized size = 1.33 $$\begin{array}{c}2\log (-x{e}^3-e^{(x+3)}-625)-2\log (x+e^x)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1250*exp(x)-1250)/(exp(3)*exp(x)^2+(2*x*exp(3)+625)*exp(x)+x^2*exp(3)+625*x),x, algorithm="giac")

[Out]

2*log(-x*e^3 - e^(x + 3) - 625) - 2*log(x + e^x)

________________________________________________________________________________________

maple [A]  time = 0.11, size = 23, normalized size = 1.28

method	result	size
norman	$-2\ln ({\mathrm{e}}^x+x)+2\ln ({\mathrm{e}}^x{\mathrm{e}}^3+x{\mathrm{e}}^3+625)$	$23$
risch	$2\ln ({\mathrm{e}}^x+(x{\mathrm{e}}^3+625){\mathrm{e}}^{-3})-2\ln ({\mathrm{e}}^x+x)$	$24$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-1250*exp(x)-1250)/(exp(3)*exp(x)^2+(2*x*exp(3)+625)*exp(x)+x^2*exp(3)+625*x),x,method=_RETURNVERBOSE)

[Out]

-2*ln(exp(x)+x)+2*ln(exp(x)*exp(3)+x*exp(3)+625)

________________________________________________________________________________________

maxima [A]  time = 0.40, size = 24, normalized size = 1.33 $$\begin{array}{c}2\log \left((x{e}^3+e^{(x+3)}+625)e^{(-3)}\right)-2\log (x+e^x)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1250*exp(x)-1250)/(exp(3)*exp(x)^2+(2*x*exp(3)+625)*exp(x)+x^2*exp(3)+625*x),x, algorithm="maxima"
)

[Out]

2*log((x*e^3 + e^(x + 3) + 625)*e^(-3)) - 2*log(x + e^x)

________________________________________________________________________________________

mupad [B]  time = 0.20, size = 19, normalized size = 1.06 $$\begin{array}{c}2\ln (x+625{\mathrm{e}}^{-3}+{\mathrm{e}}^x)-2\ln (x+{\mathrm{e}}^x)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(1250*exp(x) + 1250)/(625*x + exp(2*x)*exp(3) + x^2*exp(3) + exp(x)*(2*x*exp(3) + 625)),x)

[Out]

2*log(x + 625*exp(-3) + exp(x)) - 2*log(x + exp(x))

________________________________________________________________________________________

sympy [A]  time = 0.27, size = 27, normalized size = 1.50 $$\begin{array}{c}-2\log (x+e^x)+2\log (\frac{4x{e}^3+2500}{4e^3}+e^x)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1250*exp(x)-1250)/(exp(3)*exp(x)**2+(2*x*exp(3)+625)*exp(x)+x**2*exp(3)+625*x),x)

[Out]

-2*log(x + exp(x)) + 2*log((4*x*exp(3) + 2500)*exp(-3)/4 + exp(x))

________________________________________________________________________________________