3.41.21 $\int \frac{(\frac{6+2x}{x+x^2}{)}^{-x+x^2}(3+3x-5x^2-x^3+(-3+2x+7x^2+2x^3)\log (\frac{6+2x}{x+x^2}))}{3+4x+x^2}dx$

Optimal. Leaf size=22 $${\left(\frac{6+2x}{x(1+x)}\right)}^{-x+x^2}$$

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Rubi [F]  time = 2.93, 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 = {} $$\begin{array}{c}\int \frac{{\left(\frac{6+2x}{x+x^2}\right)}^{-x+x^2}(3+3x-5x^2-x^3+(-3+2x+7x^2+2x^3)\log \left(\frac{6+2x}{x+x^2}\right))}{3+4x+x^2}dx\end{array}$$

Verification is not applicable to the result.

[In]

Int[(((6 + 2*x)/(x + x^2))^(-x + x^2)*(3 + 3*x - 5*x^2 - x^3 + (-3 + 2*x + 7*x^2 + 2*x^3)*Log[(6 + 2*x)/(x + x
^2)]))/(3 + 4*x + x^2),x]

[Out]

-Defer[Int][((6 + 2*x)/(x*(1 + x)))^((-1 + x)*x), x] - Log[(2*(3 + x))/(x*(1 + x))]*Defer[Int][((6 + 2*x)/(x*(
1 + x)))^((-1 + x)*x), x] - 3*Defer[Int][((6 + 2*x)/(x*(1 + x)))^((-1 + x)*x)/(-2 - 2*x), x] - Defer[Int][x*((
6 + 2*x)/(x*(1 + x)))^((-1 + x)*x), x] + 2*Log[(2*(3 + x))/(x*(1 + x))]*Defer[Int][x*((6 + 2*x)/(x*(1 + x)))^(
(-1 + x)*x), x] - 7*Defer[Int][((6 + 2*x)/(x*(1 + x)))^((-1 + x)*x)/(2 + 2*x), x] + 24*Defer[Int][((6 + 2*x)/(
x*(1 + x)))^((-1 + x)*x)/(6 + 2*x), x] - Defer[Int][Defer[Int][((6 + 2*x)/(x*(1 + x)))^((-1 + x)*x), x]/(-3 -
x), x] - Defer[Int][Defer[Int][((6 + 2*x)/(x*(1 + x)))^((-1 + x)*x), x]/x, x] - Defer[Int][Defer[Int][((6 + 2*
x)/(x*(1 + x)))^((-1 + x)*x), x]/(1 + x), x] - 2*Defer[Int][Defer[Int][x*((6 + 2*x)/(x*(1 + x)))^((-1 + x)*x),
x]/(-1 - x), x] + 2*Defer[Int][Defer[Int][x*((6 + 2*x)/(x*(1 + x)))^((-1 + x)*x), x]/x, x] - 2*Defer[Int][Def
er[Int][x*((6 + 2*x)/(x*(1 + x)))^((-1 + x)*x), x]/(3 + x), x]

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int (\frac{3{\left(\frac{6+2x}{x(1+x)}\right)}^{(-1+x)x}}{3+4x+x^2}+\frac{3x{\left(\frac{6+2x}{x(1+x)}\right)}^{(-1+x)x}}{3+4x+x^2}-\frac{5x^2{\left(\frac{6+2x}{x(1+x)}\right)}^{(-1+x)x}}{3+4x+x^2}-\frac{x^3{\left(\frac{6+2x}{x(1+x)}\right)}^{(-1+x)x}}{3+4x+x^2}+(-1+2x){\left(\frac{6+2x}{x(1+x)}\right)}^{(-1+x)x}\log \left(\frac{6+2x}{x(1+x)}\right))dx\\ & =3\int \frac{{\left(\frac{6+2x}{x(1+x)}\right)}^{(-1+x)x}}{3+4x+x^2}dx+3\int \frac{x{\left(\frac{6+2x}{x(1+x)}\right)}^{(-1+x)x}}{3+4x+x^2}dx-5\int \frac{x^2{\left(\frac{6+2x}{x(1+x)}\right)}^{(-1+x)x}}{3+4x+x^2}dx-\int \frac{x^3{\left(\frac{6+2x}{x(1+x)}\right)}^{(-1+x)x}}{3+4x+x^2}dx+\int (-1+2x){\left(\frac{6+2x}{x(1+x)}\right)}^{(-1+x)x}\log \left(\frac{6+2x}{x(1+x)}\right)dx\\ & =3\int (-\frac{{\left(\frac{6+2x}{x(1+x)}\right)}^{(-1+x)x}}{-2-2x}-\frac{{\left(\frac{6+2x}{x(1+x)}\right)}^{(-1+x)x}}{6+2x})dx+3\int (-\frac{{\left(\frac{6+2x}{x(1+x)}\right)}^{(-1+x)x}}{2+2x}+\frac{3{\left(\frac{6+2x}{x(1+x)}\right)}^{(-1+x)x}}{6+2x})dx-5\int ({\left(\frac{6+2x}{x(1+x)}\right)}^{(-1+x)x}-\frac{{\left(\frac{6+2x}{x(1+x)}\right)}^{(-1+x)x}(3+4x)}{3+4x+x^2})dx-\log \left(\frac{6+2x}{x(1+x)}\right)\int {\left(\frac{6+2x}{x(1+x)}\right)}^{(-1+x)x}dx+(2\log \left(\frac{6+2x}{x(1+x)}\right))\int x{\left(\frac{6+2x}{x(1+x)}\right)}^{(-1+x)x}dx-\int (-4{\left(\frac{6+2x}{x(1+x)}\right)}^{(-1+x)x}+x{\left(\frac{6+2x}{x(1+x)}\right)}^{(-1+x)x}+\frac{{\left(\frac{6+2x}{x(1+x)}\right)}^{(-1+x)x}(12+13x)}{3+4x+x^2})dx-\int \frac{(3+6x+x^2)(\int {\left(\frac{6+2x}{x+x^2}\right)}^{(-1+x)x}dx-2\int x{\left(\frac{6+2x}{x+x^2}\right)}^{(-1+x)x}dx)}{x(1+x)(3+x)}dx\\ & =-(3\int \frac{{\left(\frac{6+2x}{x(1+x)}\right)}^{(-1+x)x}}{-2-2x}dx)-3\int \frac{{\left(\frac{6+2x}{x(1+x)}\right)}^{(-1+x)x}}{2+2x}dx-3\int \frac{{\left(\frac{6+2x}{x(1+x)}\right)}^{(-1+x)x}}{6+2x}dx+4\int {\left(\frac{6+2x}{x(1+x)}\right)}^{(-1+x)x}dx-5\int {\left(\frac{6+2x}{x(1+x)}\right)}^{(-1+x)x}dx+5\int \frac{{\left(\frac{6+2x}{x(1+x)}\right)}^{(-1+x)x}(3+4x)}{3+4x+x^2}dx+9\int \frac{{\left(\frac{6+2x}{x(1+x)}\right)}^{(-1+x)x}}{6+2x}dx-\log \left(\frac{6+2x}{x(1+x)}\right)\int {\left(\frac{6+2x}{x(1+x)}\right)}^{(-1+x)x}dx+(2\log \left(\frac{6+2x}{x(1+x)}\right))\int x{\left(\frac{6+2x}{x(1+x)}\right)}^{(-1+x)x}dx-\int x{\left(\frac{6+2x}{x(1+x)}\right)}^{(-1+x)x}dx-\int \frac{{\left(\frac{6+2x}{x(1+x)}\right)}^{(-1+x)x}(12+13x)}{3+4x+x^2}dx-\int (\frac{(3+6x+x^2)\int {\left(\frac{6+2x}{x(1+x)}\right)}^{(-1+x)x}dx}{x(1+x)(3+x)}+\frac{2(-3-6x-x^2)\int x{\left(\frac{6+2x}{x(1+x)}\right)}^{(-1+x)x}dx}{x(1+x)(3+x)})dx\\ & =-(2\int \frac{(-3-6x-x^2)\int x{\left(\frac{6+2x}{x(1+x)}\right)}^{(-1+x)x}dx}{x(1+x)(3+x)}dx)-3\int \frac{{\left(\frac{6+2x}{x(1+x)}\right)}^{(-1+x)x}}{-2-2x}dx-3\int \frac{{\left(\frac{6+2x}{x(1+x)}\right)}^{(-1+x)x}}{2+2x}dx-3\int \frac{{\left(\frac{6+2x}{x(1+x)}\right)}^{(-1+x)x}}{6+2x}dx+4\int {\left(\frac{6+2x}{x(1+x)}\right)}^{(-1+x)x}dx-5\int {\left(\frac{6+2x}{x(1+x)}\right)}^{(-1+x)x}dx+5\int (-\frac{{\left(\frac{6+2x}{x(1+x)}\right)}^{(-1+x)x}}{2+2x}+\frac{9{\left(\frac{6+2x}{x(1+x)}\right)}^{(-1+x)x}}{6+2x})dx+9\int \frac{{\left(\frac{6+2x}{x(1+x)}\right)}^{(-1+x)x}}{6+2x}dx-\log \left(\frac{6+2x}{x(1+x)}\right)\int {\left(\frac{6+2x}{x(1+x)}\right)}^{(-1+x)x}dx+(2\log \left(\frac{6+2x}{x(1+x)}\right))\int x{\left(\frac{6+2x}{x(1+x)}\right)}^{(-1+x)x}dx-\int x{\left(\frac{6+2x}{x(1+x)}\right)}^{(-1+x)x}dx-\int (-\frac{{\left(\frac{6+2x}{x(1+x)}\right)}^{(-1+x)x}}{2+2x}+\frac{27{\left(\frac{6+2x}{x(1+x)}\right)}^{(-1+x)x}}{6+2x})dx-\int \frac{(3+6x+x^2)\int {\left(\frac{6+2x}{x(1+x)}\right)}^{(-1+x)x}dx}{x(1+x)(3+x)}dx\\ & =-(2\int (\frac{\int x{\left(\frac{6+2x}{x(1+x)}\right)}^{(-1+x)x}dx}{-1-x}-\frac{\int x{\left(\frac{6+2x}{x(1+x)}\right)}^{(-1+x)x}dx}{x}+\frac{\int x{\left(\frac{6+2x}{x(1+x)}\right)}^{(-1+x)x}dx}{3+x})dx)-3\int \frac{{\left(\frac{6+2x}{x(1+x)}\right)}^{(-1+x)x}}{-2-2x}dx-3\int \frac{{\left(\frac{6+2x}{x(1+x)}\right)}^{(-1+x)x}}{2+2x}dx-3\int \frac{{\left(\frac{6+2x}{x(1+x)}\right)}^{(-1+x)x}}{6+2x}dx+4\int {\left(\frac{6+2x}{x(1+x)}\right)}^{(-1+x)x}dx-5\int {\left(\frac{6+2x}{x(1+x)}\right)}^{(-1+x)x}dx-5\int \frac{{\left(\frac{6+2x}{x(1+x)}\right)}^{(-1+x)x}}{2+2x}dx+9\int \frac{{\left(\frac{6+2x}{x(1+x)}\right)}^{(-1+x)x}}{6+2x}dx-27\int \frac{{\left(\frac{6+2x}{x(1+x)}\right)}^{(-1+x)x}}{6+2x}dx+45\int \frac{{\left(\frac{6+2x}{x(1+x)}\right)}^{(-1+x)x}}{6+2x}dx-\log \left(\frac{6+2x}{x(1+x)}\right)\int {\left(\frac{6+2x}{x(1+x)}\right)}^{(-1+x)x}dx+(2\log \left(\frac{6+2x}{x(1+x)}\right))\int x{\left(\frac{6+2x}{x(1+x)}\right)}^{(-1+x)x}dx-\int x{\left(\frac{6+2x}{x(1+x)}\right)}^{(-1+x)x}dx+\int \frac{{\left(\frac{6+2x}{x(1+x)}\right)}^{(-1+x)x}}{2+2x}dx-\int (\frac{\int {\left(\frac{6+2x}{x(1+x)}\right)}^{(-1+x)x}dx}{-3-x}+\frac{\int {\left(\frac{6+2x}{x(1+x)}\right)}^{(-1+x)x}dx}{x}+\frac{\int {\left(\frac{6+2x}{x(1+x)}\right)}^{(-1+x)x}dx}{1+x})dx\\ & =-(2\int \frac{\int x{\left(\frac{6+2x}{x(1+x)}\right)}^{(-1+x)x}dx}{-1-x}dx)+2\int \frac{\int x{\left(\frac{6+2x}{x(1+x)}\right)}^{(-1+x)x}dx}{x}dx-2\int \frac{\int x{\left(\frac{6+2x}{x(1+x)}\right)}^{(-1+x)x}dx}{3+x}dx-3\int \frac{{\left(\frac{6+2x}{x(1+x)}\right)}^{(-1+x)x}}{-2-2x}dx-3\int \frac{{\left(\frac{6+2x}{x(1+x)}\right)}^{(-1+x)x}}{2+2x}dx-3\int \frac{{\left(\frac{6+2x}{x(1+x)}\right)}^{(-1+x)x}}{6+2x}dx+4\int {\left(\frac{6+2x}{x(1+x)}\right)}^{(-1+x)x}dx-5\int {\left(\frac{6+2x}{x(1+x)}\right)}^{(-1+x)x}dx-5\int \frac{{\left(\frac{6+2x}{x(1+x)}\right)}^{(-1+x)x}}{2+2x}dx+9\int \frac{{\left(\frac{6+2x}{x(1+x)}\right)}^{(-1+x)x}}{6+2x}dx-27\int \frac{{\left(\frac{6+2x}{x(1+x)}\right)}^{(-1+x)x}}{6+2x}dx+45\int \frac{{\left(\frac{6+2x}{x(1+x)}\right)}^{(-1+x)x}}{6+2x}dx-\log \left(\frac{6+2x}{x(1+x)}\right)\int {\left(\frac{6+2x}{x(1+x)}\right)}^{(-1+x)x}dx+(2\log \left(\frac{6+2x}{x(1+x)}\right))\int x{\left(\frac{6+2x}{x(1+x)}\right)}^{(-1+x)x}dx-\int x{\left(\frac{6+2x}{x(1+x)}\right)}^{(-1+x)x}dx+\int \frac{{\left(\frac{6+2x}{x(1+x)}\right)}^{(-1+x)x}}{2+2x}dx-\int \frac{\int {\left(\frac{6+2x}{x(1+x)}\right)}^{(-1+x)x}dx}{-3-x}dx-\int \frac{\int {\left(\frac{6+2x}{x(1+x)}\right)}^{(-1+x)x}dx}{x}dx-\int \frac{\int {\left(\frac{6+2x}{x(1+x)}\right)}^{(-1+x)x}dx}{1+x}dx\end{array}\end{array}$$

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Mathematica [A]  time = 0.25, size = 19, normalized size = 0.86 $$\begin{array}{c}{\left(\frac{6+2x}{x+x^2}\right)}^{(-1+x)x}\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(((6 + 2*x)/(x + x^2))^(-x + x^2)*(3 + 3*x - 5*x^2 - x^3 + (-3 + 2*x + 7*x^2 + 2*x^3)*Log[(6 + 2*x)/
(x + x^2)]))/(3 + 4*x + x^2),x]

[Out]

((6 + 2*x)/(x + x^2))^((-1 + x)*x)

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fricas [A]  time = 0.61, size = 20, normalized size = 0.91 $$\begin{array}{c}{\left(\frac{2(x+3)}{x^2+x}\right)}^{x^2-x}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^3+7*x^2+2*x-3)*log((2*x+6)/(x^2+x))-x^3-5*x^2+3*x+3)*exp((x^2-x)*log((2*x+6)/(x^2+x)))/(x^2+4*
x+3),x, algorithm="fricas")

[Out]

(2*(x + 3)/(x^2 + x))^(x^2 - x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 $$\begin{array}{c}\int -\frac{(x^3+5x^2-(2x^3+7x^2+2x-3)\log \left(\frac{2(x+3)}{x^2+x}\right)-3x-3){\left(\frac{2(x+3)}{x^2+x}\right)}^{x^2-x}}{x^2+4x+3}dx\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^3+7*x^2+2*x-3)*log((2*x+6)/(x^2+x))-x^3-5*x^2+3*x+3)*exp((x^2-x)*log((2*x+6)/(x^2+x)))/(x^2+4*
x+3),x, algorithm="giac")

[Out]

integrate(-(x^3 + 5*x^2 - (2*x^3 + 7*x^2 + 2*x - 3)*log(2*(x + 3)/(x^2 + x)) - 3*x - 3)*(2*(x + 3)/(x^2 + x))^
(x^2 - x)/(x^2 + 4*x + 3), x)

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maple [A]  time = 0.21, size = 20, normalized size = 0.91

method	result	size
risch	${\left(\frac{2x+6}{x^2+x}\right)}^{x(x-1)}$	$20$
norman	${\mathrm{e}}^{(x^2-x)\ln \left(\frac{2x+6}{x^2+x}\right)}$	$24$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x^3+7*x^2+2*x-3)*ln((2*x+6)/(x^2+x))-x^3-5*x^2+3*x+3)*exp((x^2-x)*ln((2*x+6)/(x^2+x)))/(x^2+4*x+3),x,m
ethod=_RETURNVERBOSE)

[Out]

((2*x+6)/(x^2+x))^(x*(x-1))

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maxima [B]  time = 0.57, size = 54, normalized size = 2.45 $$\begin{array}{c}{e}^{(x^2\log (2)+x^2\log (x+3)-x^2\log (x+1)-x^2\log (x)-x\log (2)-x\log (x+3)+x\log (x+1)+x\log (x))}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^3+7*x^2+2*x-3)*log((2*x+6)/(x^2+x))-x^3-5*x^2+3*x+3)*exp((x^2-x)*log((2*x+6)/(x^2+x)))/(x^2+4*
x+3),x, algorithm="maxima")

[Out]

e^(x^2*log(2) + x^2*log(x + 3) - x^2*log(x + 1) - x^2*log(x) - x*log(2) - x*log(x + 3) + x*log(x + 1) + x*log(
x))

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mupad [B]  time = 3.75, size = 21, normalized size = 0.95 $$\begin{array}{c}{\left(\frac{2x+6}{x^2+x}\right)}^{x^2-x}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-log((2*x + 6)/(x + x^2))*(x - x^2))*(3*x + log((2*x + 6)/(x + x^2))*(2*x + 7*x^2 + 2*x^3 - 3) - 5*x^
2 - x^3 + 3))/(4*x + x^2 + 3),x)

[Out]

((2*x + 6)/(x + x^2))^(x^2 - x)

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sympy [A]  time = 0.60, size = 17, normalized size = 0.77 $$\begin{array}{c}{e}^{(x^2-x)\log \left(\frac{2x+6}{x^2+x}\right)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x**3+7*x**2+2*x-3)*ln((2*x+6)/(x**2+x))-x**3-5*x**2+3*x+3)*exp((x**2-x)*ln((2*x+6)/(x**2+x)))/(x
**2+4*x+3),x)

[Out]

exp((x**2 - x)*log((2*x + 6)/(x**2 + x)))

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