3.69.88 $\int \frac{3x^2+x^3+e^{x+x^2}(-6x+6x^2+21x^3+6x^4)+(6x+2x^2+e^{x+x^2}(9x+21x^2+6x^3))\log (\frac{2}{e^{5/3}(3+x)})+(3+x){\log }^2(\frac{2}{e^{5/3}(3+x)})}{3x^3+x^4+(6x^2+2x^3)\log (\frac{2}{e^{5/3}(3+x)})+(3x+x^2){\log }^2(\frac{2}{e^{5/3}(3+x)})}dx$

Optimal. Leaf size=30 $$2+\log (x)+\frac{3e^{x+x^2}}{x+\log \left(\frac{2}{e^{5/3}(3+x)}\right)}$$

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Rubi [F]  time = 13.39, 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 = {} $$\begin{array}{c}\int \frac{3x^2+x^3+e^{x+x^2}(-6x+6x^2+21x^3+6x^4)+(6x+2x^2+e^{x+x^2}(9x+21x^2+6x^3))\log \left(\frac{2}{e^{5/3}(3+x)}\right)+(3+x){\log }^2\left(\frac{2}{e^{5/3}(3+x)}\right)}{3x^3+x^4+(6x^2+2x^3)\log \left(\frac{2}{e^{5/3}(3+x)}\right)+(3x+x^2){\log }^2\left(\frac{2}{e^{5/3}(3+x)}\right)}dx\end{array}$$

Verification is not applicable to the result.

[In]

Int[(3*x^2 + x^3 + E^(x + x^2)*(-6*x + 6*x^2 + 21*x^3 + 6*x^4) + (6*x + 2*x^2 + E^(x + x^2)*(9*x + 21*x^2 + 6*
x^3))*Log[2/(E^(5/3)*(3 + x))] + (3 + x)*Log[2/(E^(5/3)*(3 + x))]^2)/(3*x^3 + x^4 + (6*x^2 + 2*x^3)*Log[2/(E^(
5/3)*(3 + x))] + (3*x + x^2)*Log[2/(E^(5/3)*(3 + x))]^2),x]

[Out]

Log[x] + 6*Defer[Int][(-3*x + 5*(1 - (3*Log[2])/5) - 3*Log[(3 + x)^(-1)])^(-1), x] - 27*Defer[Int][E^(x + x^2)
/(3*x - 5*(1 - (3*Log[2])/5) + 3*Log[(3 + x)^(-1)])^2, x] + 81*Defer[Int][1/((-3 - x)*(3*x - 5*(1 - (3*Log[2])
/5) + 3*Log[(3 + x)^(-1)])^2), x] + 81*Defer[Int][1/((3 + x)*(3*x - 5*(1 - (3*Log[2])/5) + 3*Log[(3 + x)^(-1)]
)^2), x] + 27*Defer[Int][E^(x + x^2)/((3 + x)*(3*x - 5*(1 - (3*Log[2])/5) + 3*Log[(3 + x)^(-1)])^2), x] + 6*De
fer[Int][(3*x - 5*(1 - (3*Log[2])/5) + 3*Log[(3 + x)^(-1)])^(-1), x] + 9*Defer[Int][E^(x + x^2)/(3*x - 5*(1 -
(3*Log[2])/5) + 3*Log[(3 + x)^(-1)]), x] + 18*Defer[Int][(E^(x + x^2)*x)/(3*x - 5*(1 - (3*Log[2])/5) + 3*Log[(
3 + x)^(-1)]), x]

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int \frac{3x^2+x^3+3e^{x+x^2}x(-2+2x+7x^2+2x^3)+x(3+x)(2+e^{x+x^2}(3+6x))(-\frac{5}{3}+\log (2)+\log \left(\frac{1}{3+x}\right))+(3+x){(-\frac{5}{3}+\log (2)+\log \left(\frac{1}{3+x}\right))}^2}{x(3+x){(x-\frac{5}{3}(1-\frac{3\log (2)}{5})+\log \left(\frac{1}{3+x}\right))}^2}dx\\ & =\int \frac{9(3x^2+x^3+3e^{x+x^2}x(-2+2x+7x^2+2x^3)+x(3+x)(2+e^{x+x^2}(3+6x))(-\frac{5}{3}+\log (2)+\log \left(\frac{1}{3+x}\right))+(3+x){(-\frac{5}{3}+\log (2)+\log \left(\frac{1}{3+x}\right))}^2)}{x(3+x){(3x-5(1-\frac{3\log (2)}{5})+3\log \left(\frac{1}{3+x}\right))}^2}dx\\ & =9\int \frac{3x^2+x^3+3e^{x+x^2}x(-2+2x+7x^2+2x^3)+x(3+x)(2+e^{x+x^2}(3+6x))(-\frac{5}{3}+\log (2)+\log \left(\frac{1}{3+x}\right))+(3+x){(-\frac{5}{3}+\log (2)+\log \left(\frac{1}{3+x}\right))}^2}{x(3+x){(3x-5(1-\frac{3\log (2)}{5})+3\log \left(\frac{1}{3+x}\right))}^2}dx\\ & =9\int (\frac{3x}{(3+x){(3x-5(1-\frac{3\log (2)}{5})+3\log \left(\frac{1}{3+x}\right))}^2}+\frac{x^2}{(3+x){(3x-5(1-\frac{3\log (2)}{5})+3\log \left(\frac{1}{3+x}\right))}^2}+\frac{{(5(1-\frac{3\log (2)}{5})-3\log \left(\frac{1}{3+x}\right))}^2}{9x{(3x-5(1-\frac{3\log (2)}{5})+3\log \left(\frac{1}{3+x}\right))}^2}+\frac{-10(1-\frac{3\log (2)}{5})+6\log \left(\frac{1}{3+x}\right)}{3{(3x-5(1-\frac{3\log (2)}{5})+3\log \left(\frac{1}{3+x}\right))}^2}+\frac{e^{x+x^2}(6x^3-29x(1-\frac{21\log (2)}{29})-21(1-\frac{3\log (2)}{7})+11x^2(1+\frac{6\log (2)}{11})+9\log \left(\frac{1}{3+x}\right)+21x\log \left(\frac{1}{3+x}\right)+6x^2\log \left(\frac{1}{3+x}\right))}{(3+x){(3x-5(1-\frac{3\log (2)}{5})+3\log \left(\frac{1}{3+x}\right))}^2})dx\\ & =3\int \frac{-10(1-\frac{3\log (2)}{5})+6\log \left(\frac{1}{3+x}\right)}{{(3x-5(1-\frac{3\log (2)}{5})+3\log \left(\frac{1}{3+x}\right))}^2}dx+9\int \frac{x^2}{(3+x){(3x-5(1-\frac{3\log (2)}{5})+3\log \left(\frac{1}{3+x}\right))}^2}dx+9\int \frac{e^{x+x^2}(6x^3-29x(1-\frac{21\log (2)}{29})-21(1-\frac{3\log (2)}{7})+11x^2(1+\frac{6\log (2)}{11})+9\log \left(\frac{1}{3+x}\right)+21x\log \left(\frac{1}{3+x}\right)+6x^2\log \left(\frac{1}{3+x}\right))}{(3+x){(3x-5(1-\frac{3\log (2)}{5})+3\log \left(\frac{1}{3+x}\right))}^2}dx+27\int \frac{x}{(3+x){(3x-5(1-\frac{3\log (2)}{5})+3\log \left(\frac{1}{3+x}\right))}^2}dx+\int \frac{{(5(1-\frac{3\log (2)}{5})-3\log \left(\frac{1}{3+x}\right))}^2}{x{(3x-5(1-\frac{3\log (2)}{5})+3\log \left(\frac{1}{3+x}\right))}^2}dx\\ & =3\int (-\frac{6x}{{(3x-5(1-\frac{3\log (2)}{5})+3\log \left(\frac{1}{3+x}\right))}^2}+\frac{2}{3x-5(1-\frac{3\log (2)}{5})+3\log \left(\frac{1}{3+x}\right)})dx+9\int \frac{e^{x+x^2}(6x^3-21(1-\frac{3\log (2)}{7})+x(-29+21\log (2))+x^2(11+\log (64))+3(3+7x+2x^2)\log \left(\frac{1}{3+x}\right))}{(3+x){(3x-5(1-\frac{3\log (2)}{5})+3\log \left(\frac{1}{3+x}\right))}^2}dx+9\int (-\frac{3}{{(3x-5(1-\frac{3\log (2)}{5})+3\log \left(\frac{1}{3+x}\right))}^2}+\frac{x}{{(3x-5(1-\frac{3\log (2)}{5})+3\log \left(\frac{1}{3+x}\right))}^2}+\frac{9}{(3+x){(3x-5(1-\frac{3\log (2)}{5})+3\log \left(\frac{1}{3+x}\right))}^2})dx+27\int (\frac{1}{{(3x-5(1-\frac{3\log (2)}{5})+3\log \left(\frac{1}{3+x}\right))}^2}+\frac{3}{(-3-x){(3x-5(1-\frac{3\log (2)}{5})+3\log \left(\frac{1}{3+x}\right))}^2})dx+\int (\frac{1}{x}+\frac{6}{-3x+5(1-\frac{3\log (2)}{5})-3\log \left(\frac{1}{3+x}\right)}+\frac{9x}{{(3x-5(1-\frac{3\log (2)}{5})+3\log \left(\frac{1}{3+x}\right))}^2})dx\\ & =\log (x)+6\int \frac{1}{-3x+5(1-\frac{3\log (2)}{5})-3\log \left(\frac{1}{3+x}\right)}dx+6\int \frac{1}{3x-5(1-\frac{3\log (2)}{5})+3\log \left(\frac{1}{3+x}\right)}dx+2(9\int \frac{x}{{(3x-5(1-\frac{3\log (2)}{5})+3\log \left(\frac{1}{3+x}\right))}^2}dx)+9\int (\frac{3e^{x+x^2}(-2-x)}{(3+x){(3x-5(1-\frac{3\log (2)}{5})+3\log \left(\frac{1}{3+x}\right))}^2}+\frac{e^{x+x^2}(1+2x)}{3x-5(1-\frac{3\log (2)}{5})+3\log \left(\frac{1}{3+x}\right)})dx-18\int \frac{x}{{(3x-5(1-\frac{3\log (2)}{5})+3\log \left(\frac{1}{3+x}\right))}^2}dx+81\int \frac{1}{(-3-x){(3x-5(1-\frac{3\log (2)}{5})+3\log \left(\frac{1}{3+x}\right))}^2}dx+81\int \frac{1}{(3+x){(3x-5(1-\frac{3\log (2)}{5})+3\log \left(\frac{1}{3+x}\right))}^2}dx\\ & =\log (x)+6\int \frac{1}{-3x+5(1-\frac{3\log (2)}{5})-3\log \left(\frac{1}{3+x}\right)}dx+6\int \frac{1}{3x-5(1-\frac{3\log (2)}{5})+3\log \left(\frac{1}{3+x}\right)}dx+2(9\int \frac{x}{{(3x-5(1-\frac{3\log (2)}{5})+3\log \left(\frac{1}{3+x}\right))}^2}dx)+9\int \frac{e^{x+x^2}(1+2x)}{3x-5(1-\frac{3\log (2)}{5})+3\log \left(\frac{1}{3+x}\right)}dx-18\int \frac{x}{{(3x-5(1-\frac{3\log (2)}{5})+3\log \left(\frac{1}{3+x}\right))}^2}dx+27\int \frac{e^{x+x^2}(-2-x)}{(3+x){(3x-5(1-\frac{3\log (2)}{5})+3\log \left(\frac{1}{3+x}\right))}^2}dx+81\int \frac{1}{(-3-x){(3x-5(1-\frac{3\log (2)}{5})+3\log \left(\frac{1}{3+x}\right))}^2}dx+81\int \frac{1}{(3+x){(3x-5(1-\frac{3\log (2)}{5})+3\log \left(\frac{1}{3+x}\right))}^2}dx\\ & =\log (x)+6\int \frac{1}{-3x+5(1-\frac{3\log (2)}{5})-3\log \left(\frac{1}{3+x}\right)}dx+6\int \frac{1}{3x-5(1-\frac{3\log (2)}{5})+3\log \left(\frac{1}{3+x}\right)}dx+2(9\int \frac{x}{{(3x-5(1-\frac{3\log (2)}{5})+3\log \left(\frac{1}{3+x}\right))}^2}dx)+9\int (\frac{e^{x+x^2}}{3x-5(1-\frac{3\log (2)}{5})+3\log \left(\frac{1}{3+x}\right)}+\frac{2e^{x+x^2}x}{3x-5(1-\frac{3\log (2)}{5})+3\log \left(\frac{1}{3+x}\right)})dx-18\int \frac{x}{{(3x-5(1-\frac{3\log (2)}{5})+3\log \left(\frac{1}{3+x}\right))}^2}dx+27\int (-\frac{e^{x+x^2}}{{(3x-5(1-\frac{3\log (2)}{5})+3\log \left(\frac{1}{3+x}\right))}^2}+\frac{e^{x+x^2}}{(3+x){(3x-5(1-\frac{3\log (2)}{5})+3\log \left(\frac{1}{3+x}\right))}^2})dx+81\int \frac{1}{(-3-x){(3x-5(1-\frac{3\log (2)}{5})+3\log \left(\frac{1}{3+x}\right))}^2}dx+81\int \frac{1}{(3+x){(3x-5(1-\frac{3\log (2)}{5})+3\log \left(\frac{1}{3+x}\right))}^2}dx\\ & =\log (x)+6\int \frac{1}{-3x+5(1-\frac{3\log (2)}{5})-3\log \left(\frac{1}{3+x}\right)}dx+6\int \frac{1}{3x-5(1-\frac{3\log (2)}{5})+3\log \left(\frac{1}{3+x}\right)}dx+2(9\int \frac{x}{{(3x-5(1-\frac{3\log (2)}{5})+3\log \left(\frac{1}{3+x}\right))}^2}dx)+9\int \frac{e^{x+x^2}}{3x-5(1-\frac{3\log (2)}{5})+3\log \left(\frac{1}{3+x}\right)}dx-18\int \frac{x}{{(3x-5(1-\frac{3\log (2)}{5})+3\log \left(\frac{1}{3+x}\right))}^2}dx+18\int \frac{e^{x+x^2}x}{3x-5(1-\frac{3\log (2)}{5})+3\log \left(\frac{1}{3+x}\right)}dx-27\int \frac{e^{x+x^2}}{{(3x-5(1-\frac{3\log (2)}{5})+3\log \left(\frac{1}{3+x}\right))}^2}dx+27\int \frac{e^{x+x^2}}{(3+x){(3x-5(1-\frac{3\log (2)}{5})+3\log \left(\frac{1}{3+x}\right))}^2}dx+81\int \frac{1}{(-3-x){(3x-5(1-\frac{3\log (2)}{5})+3\log \left(\frac{1}{3+x}\right))}^2}dx+81\int \frac{1}{(3+x){(3x-5(1-\frac{3\log (2)}{5})+3\log \left(\frac{1}{3+x}\right))}^2}dx\end{array}\end{array}$$

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Mathematica [A]  time = 0.23, size = 31, normalized size = 1.03 $$\begin{array}{c}\log (x)+\frac{9e^{x+x^2}}{-5+3x+3\log (2)+3\log \left(\frac{1}{3+x}\right)}\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(3*x^2 + x^3 + E^(x + x^2)*(-6*x + 6*x^2 + 21*x^3 + 6*x^4) + (6*x + 2*x^2 + E^(x + x^2)*(9*x + 21*x^
2 + 6*x^3))*Log[2/(E^(5/3)*(3 + x))] + (3 + x)*Log[2/(E^(5/3)*(3 + x))]^2)/(3*x^3 + x^4 + (6*x^2 + 2*x^3)*Log[
2/(E^(5/3)*(3 + x))] + (3*x + x^2)*Log[2/(E^(5/3)*(3 + x))]^2),x]

[Out]

Log[x] + (9*E^(x + x^2))/(-5 + 3*x + 3*Log[2] + 3*Log[(3 + x)^(-1)])

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fricas [A]  time = 1.02, size = 39, normalized size = 1.30 $$\begin{array}{c}\frac{(x+\log \left(\frac{2e^{(-\frac{5}{3})}}{x+3}\right))\log (x)+3e^{(x^2+x)}}{x+\log \left(\frac{2e^{(-\frac{5}{3})}}{x+3}\right)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3+x)*log(2/(3+x)/exp(5/3))^2+((6*x^3+21*x^2+9*x)*exp(x^2+x)+2*x^2+6*x)*log(2/(3+x)/exp(5/3))+(6*x^
4+21*x^3+6*x^2-6*x)*exp(x^2+x)+x^3+3*x^2)/((x^2+3*x)*log(2/(3+x)/exp(5/3))^2+(2*x^3+6*x^2)*log(2/(3+x)/exp(5/3
))+x^4+3*x^3),x, algorithm="fricas")

[Out]

((x + log(2*e^(-5/3)/(x + 3)))*log(x) + 3*e^(x^2 + x))/(x + log(2*e^(-5/3)/(x + 3)))

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giac [A]  time = 0.57, size = 48, normalized size = 1.60 $$\begin{array}{c}\frac{3x\log (x)+3\log (x)\log \left(\frac{2}{x+3}\right)+9e^{(x^2+x)}-5\log (x)}{3x+3\log \left(\frac{2}{x+3}\right)-5}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3+x)*log(2/(3+x)/exp(5/3))^2+((6*x^3+21*x^2+9*x)*exp(x^2+x)+2*x^2+6*x)*log(2/(3+x)/exp(5/3))+(6*x^
4+21*x^3+6*x^2-6*x)*exp(x^2+x)+x^3+3*x^2)/((x^2+3*x)*log(2/(3+x)/exp(5/3))^2+(2*x^3+6*x^2)*log(2/(3+x)/exp(5/3
))+x^4+3*x^3),x, algorithm="giac")

[Out]

(3*x*log(x) + 3*log(x)*log(2/(x + 3)) + 9*e^(x^2 + x) - 5*log(x))/(3*x + 3*log(2/(x + 3)) - 5)

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maple [C]  time = 0.54, size = 34, normalized size = 1.13

method	result	size
risch	$\ln (x)+\frac{18i{\mathrm{e}}^{(x+1)x}}{6i\ln (2)+6ix-6i\ln (3+x)-10i}$	$34$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((3+x)*ln(2/(3+x)/exp(5/3))^2+((6*x^3+21*x^2+9*x)*exp(x^2+x)+2*x^2+6*x)*ln(2/(3+x)/exp(5/3))+(6*x^4+21*x^3
+6*x^2-6*x)*exp(x^2+x)+x^3+3*x^2)/((x^2+3*x)*ln(2/(3+x)/exp(5/3))^2+(2*x^3+6*x^2)*ln(2/(3+x)/exp(5/3))+x^4+3*x
^3),x,method=_RETURNVERBOSE)

[Out]

ln(x)+18*I*exp((x+1)*x)/(6*I*ln(2)+6*I*x-6*I*ln(3+x)-10*I)

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maxima [A]  time = 0.49, size = 28, normalized size = 0.93 $$\begin{array}{c}\frac{9e^{(x^2+x)}}{3x+3\log (2)-3\log (x+3)-5}+\log (x)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3+x)*log(2/(3+x)/exp(5/3))^2+((6*x^3+21*x^2+9*x)*exp(x^2+x)+2*x^2+6*x)*log(2/(3+x)/exp(5/3))+(6*x^
4+21*x^3+6*x^2-6*x)*exp(x^2+x)+x^3+3*x^2)/((x^2+3*x)*log(2/(3+x)/exp(5/3))^2+(2*x^3+6*x^2)*log(2/(3+x)/exp(5/3
))+x^4+3*x^3),x, algorithm="maxima")

[Out]

9*e^(x^2 + x)/(3*x + 3*log(2) - 3*log(x + 3) - 5) + log(x)

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mupad [B]  time = 0.44, size = 41, normalized size = 1.37 $$\begin{array}{c}\frac{3{\mathrm{e}}^{x^2+x}+\ln \left(\frac{2{\mathrm{e}}^{-\frac{5}{3}}}{x+3}\right)\ln (x)+x\ln (x)}{x+\ln \left(\frac{2{\mathrm{e}}^{-\frac{5}{3}}}{x+3}\right)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log((2*exp(-5/3))/(x + 3))*(6*x + exp(x + x^2)*(9*x + 21*x^2 + 6*x^3) + 2*x^2) + exp(x + x^2)*(6*x^2 - 6*
x + 21*x^3 + 6*x^4) + 3*x^2 + x^3 + log((2*exp(-5/3))/(x + 3))^2*(x + 3))/(log((2*exp(-5/3))/(x + 3))^2*(3*x +
x^2) + log((2*exp(-5/3))/(x + 3))*(6*x^2 + 2*x^3) + 3*x^3 + x^4),x)

[Out]

(3*exp(x + x^2) + log((2*exp(-5/3))/(x + 3))*log(x) + x*log(x))/(x + log((2*exp(-5/3))/(x + 3)))

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sympy [A]  time = 0.52, size = 24, normalized size = 0.80 $$\begin{array}{c}\log (x)+\frac{3e^{x^2+x}}{x+\log \left(\frac{2}{(x+3)e^{\frac{5}{3}}}\right)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3+x)*ln(2/(3+x)/exp(5/3))**2+((6*x**3+21*x**2+9*x)*exp(x**2+x)+2*x**2+6*x)*ln(2/(3+x)/exp(5/3))+(6
*x**4+21*x**3+6*x**2-6*x)*exp(x**2+x)+x**3+3*x**2)/((x**2+3*x)*ln(2/(3+x)/exp(5/3))**2+(2*x**3+6*x**2)*ln(2/(3
+x)/exp(5/3))+x**4+3*x**3),x)

[Out]

log(x) + 3*exp(x**2 + x)/(x + log(2*exp(-5/3)/(x + 3)))

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