3.2.87 \(\int \frac {9-15 x-6 x^2+e^{x-(3+x)^x} (1-2 x) (6+14 x+4 x^2+(3+x)^x (2 x-4 x^2+(6-10 x-4 x^2) \log (3+x)))}{-3+5 x+2 x^2} \, dx\)

Optimal. Leaf size=22 \[ 2 e^{x-(3+x)^x} (1-2 x)-3 x \]

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Rubi [F]  time = 2.45, 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 {9-15 x-6 x^2+e^{x-(3+x)^x} (1-2 x) \left (6+14 x+4 x^2+(3+x)^x \left (2 x-4 x^2+\left (6-10 x-4 x^2\right ) \log (3+x)\right )\right )}{-3+5 x+2 x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(9 - 15*x - 6*x^2 + E^(x - (3 + x)^x)*(1 - 2*x)*(6 + 14*x + 4*x^2 + (3 + x)^x*(2*x - 4*x^2 + (6 - 10*x - 4
*x^2)*Log[3 + x])))/(-3 + 5*x + 2*x^2),x]

[Out]

-3*x - 2*Defer[Int][E^(x - (3 + x)^x), x] - 4*Defer[Int][E^(x - (3 + x)^x)*x, x] - 2*Defer[Int][E^(x - (3 + x)
^x)*x*(3 + x)^(-1 + x), x] + 4*Defer[Int][E^(x - (3 + x)^x)*x^2*(3 + x)^(-1 + x), x] - 2*Log[3 + x]*Defer[Int]
[E^(x - (3 + x)^x)*(3 + x)^x, x] + 4*Log[3 + x]*Defer[Int][E^(x - (3 + x)^x)*x*(3 + x)^x, x] + 2*Defer[Int][De
fer[Int][E^(x - (3 + x)^x)*(3 + x)^x, x]/(3 + x), x] - 4*Defer[Int][Defer[Int][E^(x - (3 + x)^x)*x*(3 + x)^x,
x]/(3 + x), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {6 e^{x-(3+x)^x}}{3+x}-\frac {14 e^{x-(3+x)^x} x}{3+x}-\frac {4 e^{x-(3+x)^x} x^2}{3+x}+\frac {9}{(3+x) (-1+2 x)}-\frac {15 x}{-3+5 x+2 x^2}-\frac {6 x^2}{-3+5 x+2 x^2}+2 e^{x-(3+x)^x} (3+x)^{-1+x} (-1+2 x) (x+3 \log (3+x)+x \log (3+x))\right ) \, dx\\ &=2 \int e^{x-(3+x)^x} (3+x)^{-1+x} (-1+2 x) (x+3 \log (3+x)+x \log (3+x)) \, dx-4 \int \frac {e^{x-(3+x)^x} x^2}{3+x} \, dx-6 \int \frac {e^{x-(3+x)^x}}{3+x} \, dx-6 \int \frac {x^2}{-3+5 x+2 x^2} \, dx+9 \int \frac {1}{(3+x) (-1+2 x)} \, dx-14 \int \frac {e^{x-(3+x)^x} x}{3+x} \, dx-15 \int \frac {x}{-3+5 x+2 x^2} \, dx\\ &=-3 x-\frac {9}{7} \int \frac {1}{3+x} \, dx+2 \int \left (e^{x-(3+x)^x} x (3+x)^{-1+x} (-1+2 x)+e^{x-(3+x)^x} (3+x)^{-1+x} \left (-3+5 x+2 x^2\right ) \log (3+x)\right ) \, dx-\frac {15}{7} \int \frac {1}{-1+2 x} \, dx+\frac {18}{7} \int \frac {1}{-1+2 x} \, dx-3 \int \frac {3-5 x}{-3+5 x+2 x^2} \, dx-4 \int \left (-3 e^{x-(3+x)^x}+e^{x-(3+x)^x} x+\frac {9 e^{x-(3+x)^x}}{3+x}\right ) \, dx-6 \int \frac {e^{x-(3+x)^x}}{3+x} \, dx-\frac {90}{7} \int \frac {1}{6+2 x} \, dx-14 \int \left (e^{x-(3+x)^x}-\frac {3 e^{x-(3+x)^x}}{3+x}\right ) \, dx\\ &=-3 x+\frac {3}{14} \log (1-2 x)-\frac {54}{7} \log (3+x)-\frac {3}{7} \int \frac {1}{-1+2 x} \, dx+2 \int e^{x-(3+x)^x} x (3+x)^{-1+x} (-1+2 x) \, dx+2 \int e^{x-(3+x)^x} (3+x)^{-1+x} \left (-3+5 x+2 x^2\right ) \log (3+x) \, dx-4 \int e^{x-(3+x)^x} x \, dx-6 \int \frac {e^{x-(3+x)^x}}{3+x} \, dx+12 \int e^{x-(3+x)^x} \, dx-14 \int e^{x-(3+x)^x} \, dx+\frac {108}{7} \int \frac {1}{6+2 x} \, dx-36 \int \frac {e^{x-(3+x)^x}}{3+x} \, dx+42 \int \frac {e^{x-(3+x)^x}}{3+x} \, dx\\ &=-3 x+2 \int \left (-e^{x-(3+x)^x} x (3+x)^{-1+x}+2 e^{x-(3+x)^x} x^2 (3+x)^{-1+x}\right ) \, dx-2 \int \frac {-\int e^{x-(3+x)^x} (3+x)^x \, dx+2 \int e^{x-(3+x)^x} x (3+x)^x \, dx}{3+x} \, dx-4 \int e^{x-(3+x)^x} x \, dx-6 \int \frac {e^{x-(3+x)^x}}{3+x} \, dx+12 \int e^{x-(3+x)^x} \, dx-14 \int e^{x-(3+x)^x} \, dx-36 \int \frac {e^{x-(3+x)^x}}{3+x} \, dx+42 \int \frac {e^{x-(3+x)^x}}{3+x} \, dx-(2 \log (3+x)) \int e^{x-(3+x)^x} (3+x)^x \, dx+(4 \log (3+x)) \int e^{x-(3+x)^x} x (3+x)^x \, dx\\ &=-3 x-2 \int e^{x-(3+x)^x} x (3+x)^{-1+x} \, dx-2 \int \left (-\frac {\int e^{x-(3+x)^x} (3+x)^x \, dx}{3+x}+\frac {2 \int e^{x-(3+x)^x} x (3+x)^x \, dx}{3+x}\right ) \, dx-4 \int e^{x-(3+x)^x} x \, dx+4 \int e^{x-(3+x)^x} x^2 (3+x)^{-1+x} \, dx-6 \int \frac {e^{x-(3+x)^x}}{3+x} \, dx+12 \int e^{x-(3+x)^x} \, dx-14 \int e^{x-(3+x)^x} \, dx-36 \int \frac {e^{x-(3+x)^x}}{3+x} \, dx+42 \int \frac {e^{x-(3+x)^x}}{3+x} \, dx-(2 \log (3+x)) \int e^{x-(3+x)^x} (3+x)^x \, dx+(4 \log (3+x)) \int e^{x-(3+x)^x} x (3+x)^x \, dx\\ &=-3 x-2 \int e^{x-(3+x)^x} x (3+x)^{-1+x} \, dx+2 \int \frac {\int e^{x-(3+x)^x} (3+x)^x \, dx}{3+x} \, dx-4 \int e^{x-(3+x)^x} x \, dx+4 \int e^{x-(3+x)^x} x^2 (3+x)^{-1+x} \, dx-4 \int \frac {\int e^{x-(3+x)^x} x (3+x)^x \, dx}{3+x} \, dx-6 \int \frac {e^{x-(3+x)^x}}{3+x} \, dx+12 \int e^{x-(3+x)^x} \, dx-14 \int e^{x-(3+x)^x} \, dx-36 \int \frac {e^{x-(3+x)^x}}{3+x} \, dx+42 \int \frac {e^{x-(3+x)^x}}{3+x} \, dx-(2 \log (3+x)) \int e^{x-(3+x)^x} (3+x)^x \, dx+(4 \log (3+x)) \int e^{x-(3+x)^x} x (3+x)^x \, dx\\ \end {aligned} \end {gather*}

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Mathematica [F]  time = 180.04, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(9 - 15*x - 6*x^2 + E^(x - (3 + x)^x)*(1 - 2*x)*(6 + 14*x + 4*x^2 + (3 + x)^x*(2*x - 4*x^2 + (6 - 10
*x - 4*x^2)*Log[3 + x])))/(-3 + 5*x + 2*x^2),x]

[Out]

$Aborted

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fricas [A]  time = 0.94, size = 22, normalized size = 1.00 \begin {gather*} -3 \, x + 2 \, e^{\left (-{\left (x + 3\right )}^{x} + x + \log \left (-2 \, x + 1\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((-4*x^2-10*x+6)*log(3+x)-4*x^2+2*x)*exp(x*log(3+x))+4*x^2+14*x+6)*exp(-exp(x*log(3+x))+log(1-2*x)
+x)-6*x^2-15*x+9)/(2*x^2+5*x-3),x, algorithm="fricas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {6 \, x^{2} + 2 \, {\left ({\left (2 \, x^{2} + {\left (2 \, x^{2} + 5 \, x - 3\right )} \log \left (x + 3\right ) - x\right )} {\left (x + 3\right )}^{x} - 2 \, x^{2} - 7 \, x - 3\right )} e^{\left (-{\left (x + 3\right )}^{x} + x + \log \left (-2 \, x + 1\right )\right )} + 15 \, x - 9}{2 \, x^{2} + 5 \, x - 3}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((-4*x^2-10*x+6)*log(3+x)-4*x^2+2*x)*exp(x*log(3+x))+4*x^2+14*x+6)*exp(-exp(x*log(3+x))+log(1-2*x)
+x)-6*x^2-15*x+9)/(2*x^2+5*x-3),x, algorithm="giac")

[Out]

integrate(-(6*x^2 + 2*((2*x^2 + (2*x^2 + 5*x - 3)*log(x + 3) - x)*(x + 3)^x - 2*x^2 - 7*x - 3)*e^(-(x + 3)^x +
x + log(-2*x + 1)) + 15*x - 9)/(2*x^2 + 5*x - 3), x)

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maple [A]  time = 0.14, size = 22, normalized size = 1.00

method	result	size
risch	\(2 \left (1-2 x \right ) {\mathrm e}^{-\left (3+x \right )^{x}+x}-3 x\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((((-4*x^2-10*x+6)*ln(3+x)-4*x^2+2*x)*exp(x*ln(3+x))+4*x^2+14*x+6)*exp(-exp(x*ln(3+x))+ln(1-2*x)+x)-6*x^2-
15*x+9)/(2*x^2+5*x-3),x,method=_RETURNVERBOSE)

[Out]

2*(1-2*x)*exp(-(3+x)^x+x)-3*x

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maxima [A]  time = 0.76, size = 21, normalized size = 0.95 \begin {gather*} -2 \, {\left (2 \, x - 1\right )} e^{\left (-{\left (x + 3\right )}^{x} + x\right )} - 3 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((-4*x^2-10*x+6)*log(3+x)-4*x^2+2*x)*exp(x*log(3+x))+4*x^2+14*x+6)*exp(-exp(x*log(3+x))+log(1-2*x)
+x)-6*x^2-15*x+9)/(2*x^2+5*x-3),x, algorithm="maxima")

[Out]

-2*(2*x - 1)*e^(-(x + 3)^x + x) - 3*x

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mupad [B]  time = 0.27, size = 29, normalized size = 1.32 \begin {gather*} 2\,{\mathrm {e}}^{-{\left (x+3\right )}^x}\,{\mathrm {e}}^x-3\,x-4\,x\,{\mathrm {e}}^{-{\left (x+3\right )}^x}\,{\mathrm {e}}^x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(15*x + 6*x^2 - exp(x - exp(x*log(x + 3)) + log(1 - 2*x))*(14*x - exp(x*log(x + 3))*(log(x + 3)*(10*x + 4
*x^2 - 6) - 2*x + 4*x^2) + 4*x^2 + 6) - 9)/(5*x + 2*x^2 - 3),x)

[Out]

2*exp(-(x + 3)^x)*exp(x) - 3*x - 4*x*exp(-(x + 3)^x)*exp(x)

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sympy [A]  time = 7.94, size = 19, normalized size = 0.86 \begin {gather*} - 3 x + \left (2 - 4 x\right ) e^{x - e^{x \log {\left (x + 3 \right )}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((-4*x**2-10*x+6)*ln(3+x)-4*x**2+2*x)*exp(x*ln(3+x))+4*x**2+14*x+6)*exp(-exp(x*ln(3+x))+ln(1-2*x)+
x)-6*x**2-15*x+9)/(2*x**2+5*x-3),x)

[Out]

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

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