3.69.96 $\int \frac{3+e+e^x(9+e^2+36x+36x^2+e(6+12x))}{-27-3e^2+e(-18-35x)-105x-102x^2+e^x(9+e^2+36x+36x^2+e(6+12x))}dx$

Optimal. Leaf size=18 $$4+\log (-3+e^x+\frac{x}{3+e+6x})$$

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Rubi [F]  time = 2.04, 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 = {} $$\begin{array}{c}\int \frac{3+e+e^x(9+e^2+36x+36x^2+e(6+12x))}{-27-3e^2+e(-18-35x)-105x-102x^2+e^x(9+e^2+36x+36x^2+e(6+12x))}dx\end{array}$$

Verification is not applicable to the result.

[In]

Int[(3 + E + E^x*(9 + E^2 + 36*x + 36*x^2 + E*(6 + 12*x)))/(-27 - 3*E^2 + E*(-18 - 35*x) - 105*x - 102*x^2 + E
^x*(9 + E^2 + 36*x + 36*x^2 + E*(6 + 12*x))),x]

[Out]

x - 3*(3 + E)*Defer[Int][(9*(1 + E/3) - 3*(1 + E/3)*E^x + 17*x - 6*E^x*x)^(-1), x] - (3 + E)*Defer[Int][1/((3
+ E + 6*x)*(9*(1 + E/3) - 3*(1 + E/3)*E^x + 17*x - 6*E^x*x)), x] + 17*Defer[Int][x/(-9*(1 + E/3) + 3*(1 + E/3)
*E^x - 17*x + 6*E^x*x), x]

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int \frac{-3(1+\frac{e}{3})-e^x(9+e^2+36x+36x^2+e(6+12x))}{(3+e+6x)(9(1+\frac{e}{3})-3(1+\frac{e}{3})e^x+17x-6e^{x}x)}dx\\ & =\int (1+\frac{-((3+e)(10+3e))-35(3+e)x-102x^2}{(3+e+6x)(9(1+\frac{e}{3})-3(1+\frac{e}{3})e^x+17x-6e^{x}x)})dx\\ & =x+\int \frac{-((3+e)(10+3e))-35(3+e)x-102x^2}{(3+e+6x)(9(1+\frac{e}{3})-3(1+\frac{e}{3})e^x+17x-6e^{x}x)}dx\\ & =x+\int (\frac{3(-3-e)}{9(1+\frac{e}{3})-3(1+\frac{e}{3})e^x+17x-6e^{x}x}+\frac{-3-e}{(3+e+6x)(9(1+\frac{e}{3})-3(1+\frac{e}{3})e^x+17x-6e^{x}x)}+\frac{17x}{-9(1+\frac{e}{3})+3(1+\frac{e}{3})e^x-17x+6e^{x}x})dx\\ & =x+17\int \frac{x}{-9(1+\frac{e}{3})+3(1+\frac{e}{3})e^x-17x+6e^{x}x}dx+(-3-e)\int \frac{1}{(3+e+6x)(9(1+\frac{e}{3})-3(1+\frac{e}{3})e^x+17x-6e^{x}x)}dx-(3(3+e))\int \frac{1}{9(1+\frac{e}{3})-3(1+\frac{e}{3})e^x+17x-6e^{x}x}dx\end{array}\end{array}$$

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Mathematica [B]  time = 0.42, size = 37, normalized size = 2.06 $$\begin{array}{c}-\log (3+e+6x)+\log (9+3e-3e^x-e^{1+x}+17x-6e^{x}x)\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(3 + E + E^x*(9 + E^2 + 36*x + 36*x^2 + E*(6 + 12*x)))/(-27 - 3*E^2 + E*(-18 - 35*x) - 105*x - 102*x
^2 + E^x*(9 + E^2 + 36*x + 36*x^2 + E*(6 + 12*x))),x]

[Out]

-Log[3 + E + 6*x] + Log[9 + 3*E - 3*E^x - E^(1 + x) + 17*x - 6*E^x*x]

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fricas [A]  time = 0.63, size = 30, normalized size = 1.67 $$\begin{array}{c}\log \left(\frac{(6x+e+3)e^x-17x-3e-9}{6x+e+3}\right)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((exp(1)^2+(12*x+6)*exp(1)+36*x^2+36*x+9)*exp(x)+3+exp(1))/((exp(1)^2+(12*x+6)*exp(1)+36*x^2+36*x+9)
*exp(x)-3*exp(1)^2+(-35*x-18)*exp(1)-102*x^2-105*x-27),x, algorithm="fricas")

[Out]

log(((6*x + e + 3)*e^x - 17*x - 3*e - 9)/(6*x + e + 3))

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 $$\begin{array}{c}\text{Timed out}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((exp(1)^2+(12*x+6)*exp(1)+36*x^2+36*x+9)*exp(x)+3+exp(1))/((exp(1)^2+(12*x+6)*exp(1)+36*x^2+36*x+9)
*exp(x)-3*exp(1)^2+(-35*x-18)*exp(1)-102*x^2-105*x-27),x, algorithm="giac")

[Out]

Timed out

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maple [A]  time = 0.11, size = 25, normalized size = 1.39

method	result	size
risch	$\ln ({\mathrm{e}}^x-\frac{3\mathrm{e}+17x+9}{6x+3+\mathrm{e}})$	$25$
norman	$-\ln (6x+3+\mathrm{e})+\ln (\mathrm{e}{\mathrm{e}}^x+6{\mathrm{e}}^{x}x-3\mathrm{e}-17x+3{\mathrm{e}}^x-9)$	$36$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((exp(1)^2+(12*x+6)*exp(1)+36*x^2+36*x+9)*exp(x)+3+exp(1))/((exp(1)^2+(12*x+6)*exp(1)+36*x^2+36*x+9)*exp(x
)-3*exp(1)^2+(-35*x-18)*exp(1)-102*x^2-105*x-27),x,method=_RETURNVERBOSE)

[Out]

ln(exp(x)-(3*exp(1)+17*x+9)/(6*x+3+exp(1)))

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maxima [A]  time = 0.43, size = 30, normalized size = 1.67 $$\begin{array}{c}\log \left(\frac{(6x+e+3)e^x-17x-3e-9}{6x+e+3}\right)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((exp(1)^2+(12*x+6)*exp(1)+36*x^2+36*x+9)*exp(x)+3+exp(1))/((exp(1)^2+(12*x+6)*exp(1)+36*x^2+36*x+9)
*exp(x)-3*exp(1)^2+(-35*x-18)*exp(1)-102*x^2-105*x-27),x, algorithm="maxima")

[Out]

log(((6*x + e + 3)*e^x - 17*x - 3*e - 9)/(6*x + e + 3))

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mupad [B]  time = 4.35, size = 35, normalized size = 1.94 $$\begin{array}{c}\ln (3{\mathrm{e}}^x-3\mathrm{e}-17x+\mathrm{e}{\mathrm{e}}^x+6x{\mathrm{e}}^x-9)-\ln (6x+\mathrm{e}+3)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(1) + exp(x)*(36*x + exp(2) + 36*x^2 + exp(1)*(12*x + 6) + 9) + 3)/(105*x + 3*exp(2) - exp(x)*(36*x +
exp(2) + 36*x^2 + exp(1)*(12*x + 6) + 9) + 102*x^2 + exp(1)*(35*x + 18) + 27),x)

[Out]

log(3*exp(x) - 3*exp(1) - 17*x + exp(1)*exp(x) + 6*x*exp(x) - 9) - log(6*x + exp(1) + 3)

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sympy [A]  time = 0.50, size = 24, normalized size = 1.33 $$\begin{array}{c}\log (\frac{-17x-9-3e}{6x+e+3}+e^x)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((exp(1)**2+(12*x+6)*exp(1)+36*x**2+36*x+9)*exp(x)+3+exp(1))/((exp(1)**2+(12*x+6)*exp(1)+36*x**2+36*
x+9)*exp(x)-3*exp(1)**2+(-35*x-18)*exp(1)-102*x**2-105*x-27),x)

[Out]

log((-17*x - 9 - 3*E)/(6*x + E + 3) + exp(x))

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