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

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

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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 {gather*} \int \frac {\left (\frac {6+2 x}{x+x^2}\right )^{-x+x^2} \left (3+3 x-5 x^2-x^3+\left (-3+2 x+7 x^2+2 x^3\right ) \log \left (\frac {6+2 x}{x+x^2}\right )\right )}{3+4 x+x^2} \, dx \end {gather*}

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

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

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 {gather*} \left (\frac {2 \, {\left (x + 3\right )}}{x^{2} + x}\right )^{x^{2} - x} \end {gather*}

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 {gather*} \int -\frac {{\left (x^{3} + 5 \, x^{2} - {\left (2 \, x^{3} + 7 \, x^{2} + 2 \, x - 3\right )} \log \left (\frac {2 \, {\left (x + 3\right )}}{x^{2} + x}\right ) - 3 \, x - 3\right )} \left (\frac {2 \, {\left (x + 3\right )}}{x^{2} + x}\right )^{x^{2} - x}}{x^{2} + 4 \, x + 3}\,{d x} \end {gather*}

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

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 {gather*} e^{\left (x^{2} \log \relax (2) + x^{2} \log \left (x + 3\right ) - x^{2} \log \left (x + 1\right ) - x^{2} \log \relax (x) - x \log \relax (2) - x \log \left (x + 3\right ) + x \log \left (x + 1\right ) + x \log \relax (x)\right )} \end {gather*}

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 {gather*} {\left (\frac {2\,x+6}{x^2+x}\right )}^{x^2-x} \end {gather*}

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 {gather*} e^{\left (x^{2} - x\right ) \log {\left (\frac {2 x + 6}{x^{2} + x} \right )}} \end {gather*}

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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