3.68.89 \(\int e^{(4+x)^{x^2} (-10 x+10 x^3)} (4+x)^{-1+x^2} (-40-10 x+120 x^2+20 x^3+10 x^5+(-80 x^2-20 x^3+80 x^4+20 x^5) \log (4+x)) \, dx\)

Optimal. Leaf size=21 \[ e^{5 (4+x)^{x^2} (-2+2 x) \left (x+x^2\right )} \]

________________________________________________________________________________________

Rubi [F]  time = 6.21, 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 e^{(4+x)^{x^2} \left (-10 x+10 x^3\right )} (4+x)^{-1+x^2} \left (-40-10 x+120 x^2+20 x^3+10 x^5+\left (-80 x^2-20 x^3+80 x^4+20 x^5\right ) \log (4+x)\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[E^((4 + x)^x^2*(-10*x + 10*x^3))*(4 + x)^(-1 + x^2)*(-40 - 10*x + 120*x^2 + 20*x^3 + 10*x^5 + (-80*x^2 - 2
0*x^3 + 80*x^4 + 20*x^5)*Log[4 + x]),x]

[Out]

-20*Log[4 + x]*Defer[Int][E^(10*x*(4 + x)^x^2*(-1 + x^2))*x^2*(4 + x)^x^2, x] + 20*Log[4 + x]*Defer[Int][E^(10
*x*(4 + x)^x^2*(-1 + x^2))*x^4*(4 + x)^x^2, x] - 40*Defer[Int][E^(x*(4 + x)^x^2*(-10 + 10*x^2))*(4 + x)^(-1 +
x^2), x] - 10*Defer[Int][E^(x*(4 + x)^x^2*(-10 + 10*x^2))*x*(4 + x)^(-1 + x^2), x] + 120*Defer[Int][E^(x*(4 +
x)^x^2*(-10 + 10*x^2))*x^2*(4 + x)^(-1 + x^2), x] + 20*Defer[Int][E^(x*(4 + x)^x^2*(-10 + 10*x^2))*x^3*(4 + x)
^(-1 + x^2), x] + 10*Defer[Int][E^(x*(4 + x)^x^2*(-10 + 10*x^2))*x^5*(4 + x)^(-1 + x^2), x] + 20*Defer[Int][De
fer[Int][E^(10*x*(4 + x)^x^2*(-1 + x^2))*x^2*(4 + x)^x^2, x]/(4 + x), x] - 20*Defer[Int][Defer[Int][E^(10*x*(4
+ x)^x^2*(-1 + x^2))*x^4*(4 + x)^x^2, x]/(4 + x), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int e^{x (4+x)^{x^2} \left (-10+10 x^2\right )} (4+x)^{-1+x^2} \left (-40-10 x+120 x^2+20 x^3+10 x^5+\left (-80 x^2-20 x^3+80 x^4+20 x^5\right ) \log (4+x)\right ) \, dx\\ &=\int \left (-40 e^{x (4+x)^{x^2} \left (-10+10 x^2\right )} (4+x)^{-1+x^2}-10 e^{x (4+x)^{x^2} \left (-10+10 x^2\right )} x (4+x)^{-1+x^2}+120 e^{x (4+x)^{x^2} \left (-10+10 x^2\right )} x^2 (4+x)^{-1+x^2}+20 e^{x (4+x)^{x^2} \left (-10+10 x^2\right )} x^3 (4+x)^{-1+x^2}+10 e^{x (4+x)^{x^2} \left (-10+10 x^2\right )} x^5 (4+x)^{-1+x^2}+20 e^{x (4+x)^{x^2} \left (-10+10 x^2\right )} x^2 (4+x)^{-1+x^2} \left (-4-x+4 x^2+x^3\right ) \log (4+x)\right ) \, dx\\ &=-\left (10 \int e^{x (4+x)^{x^2} \left (-10+10 x^2\right )} x (4+x)^{-1+x^2} \, dx\right )+10 \int e^{x (4+x)^{x^2} \left (-10+10 x^2\right )} x^5 (4+x)^{-1+x^2} \, dx+20 \int e^{x (4+x)^{x^2} \left (-10+10 x^2\right )} x^3 (4+x)^{-1+x^2} \, dx+20 \int e^{x (4+x)^{x^2} \left (-10+10 x^2\right )} x^2 (4+x)^{-1+x^2} \left (-4-x+4 x^2+x^3\right ) \log (4+x) \, dx-40 \int e^{x (4+x)^{x^2} \left (-10+10 x^2\right )} (4+x)^{-1+x^2} \, dx+120 \int e^{x (4+x)^{x^2} \left (-10+10 x^2\right )} x^2 (4+x)^{-1+x^2} \, dx\\ &=-\left (10 \int e^{x (4+x)^{x^2} \left (-10+10 x^2\right )} x (4+x)^{-1+x^2} \, dx\right )+10 \int e^{x (4+x)^{x^2} \left (-10+10 x^2\right )} x^5 (4+x)^{-1+x^2} \, dx+20 \int e^{x (4+x)^{x^2} \left (-10+10 x^2\right )} x^3 (4+x)^{-1+x^2} \, dx-20 \int \frac {-\int e^{10 x (4+x)^{x^2} \left (-1+x^2\right )} x^2 (4+x)^{x^2} \, dx+\int e^{10 x (4+x)^{x^2} \left (-1+x^2\right )} x^4 (4+x)^{x^2} \, dx}{4+x} \, dx-40 \int e^{x (4+x)^{x^2} \left (-10+10 x^2\right )} (4+x)^{-1+x^2} \, dx+120 \int e^{x (4+x)^{x^2} \left (-10+10 x^2\right )} x^2 (4+x)^{-1+x^2} \, dx-(20 \log (4+x)) \int e^{10 x (4+x)^{x^2} \left (-1+x^2\right )} x^2 (4+x)^{x^2} \, dx+(20 \log (4+x)) \int e^{10 x (4+x)^{x^2} \left (-1+x^2\right )} x^4 (4+x)^{x^2} \, dx\\ &=-\left (10 \int e^{x (4+x)^{x^2} \left (-10+10 x^2\right )} x (4+x)^{-1+x^2} \, dx\right )+10 \int e^{x (4+x)^{x^2} \left (-10+10 x^2\right )} x^5 (4+x)^{-1+x^2} \, dx+20 \int e^{x (4+x)^{x^2} \left (-10+10 x^2\right )} x^3 (4+x)^{-1+x^2} \, dx-20 \int \left (-\frac {\int e^{10 x (4+x)^{x^2} \left (-1+x^2\right )} x^2 (4+x)^{x^2} \, dx}{4+x}+\frac {\int e^{10 x (4+x)^{x^2} \left (-1+x^2\right )} x^4 (4+x)^{x^2} \, dx}{4+x}\right ) \, dx-40 \int e^{x (4+x)^{x^2} \left (-10+10 x^2\right )} (4+x)^{-1+x^2} \, dx+120 \int e^{x (4+x)^{x^2} \left (-10+10 x^2\right )} x^2 (4+x)^{-1+x^2} \, dx-(20 \log (4+x)) \int e^{10 x (4+x)^{x^2} \left (-1+x^2\right )} x^2 (4+x)^{x^2} \, dx+(20 \log (4+x)) \int e^{10 x (4+x)^{x^2} \left (-1+x^2\right )} x^4 (4+x)^{x^2} \, dx\\ &=-\left (10 \int e^{x (4+x)^{x^2} \left (-10+10 x^2\right )} x (4+x)^{-1+x^2} \, dx\right )+10 \int e^{x (4+x)^{x^2} \left (-10+10 x^2\right )} x^5 (4+x)^{-1+x^2} \, dx+20 \int e^{x (4+x)^{x^2} \left (-10+10 x^2\right )} x^3 (4+x)^{-1+x^2} \, dx+20 \int \frac {\int e^{10 x (4+x)^{x^2} \left (-1+x^2\right )} x^2 (4+x)^{x^2} \, dx}{4+x} \, dx-20 \int \frac {\int e^{10 x (4+x)^{x^2} \left (-1+x^2\right )} x^4 (4+x)^{x^2} \, dx}{4+x} \, dx-40 \int e^{x (4+x)^{x^2} \left (-10+10 x^2\right )} (4+x)^{-1+x^2} \, dx+120 \int e^{x (4+x)^{x^2} \left (-10+10 x^2\right )} x^2 (4+x)^{-1+x^2} \, dx-(20 \log (4+x)) \int e^{10 x (4+x)^{x^2} \left (-1+x^2\right )} x^2 (4+x)^{x^2} \, dx+(20 \log (4+x)) \int e^{10 x (4+x)^{x^2} \left (-1+x^2\right )} x^4 (4+x)^{x^2} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 2.43, size = 17, normalized size = 0.81 \begin {gather*} e^{10 x (4+x)^{x^2} \left (-1+x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^((4 + x)^x^2*(-10*x + 10*x^3))*(4 + x)^(-1 + x^2)*(-40 - 10*x + 120*x^2 + 20*x^3 + 10*x^5 + (-80*x
^2 - 20*x^3 + 80*x^4 + 20*x^5)*Log[4 + x]),x]

[Out]

E^(10*x*(4 + x)^x^2*(-1 + x^2))

________________________________________________________________________________________

fricas [A]  time = 0.54, size = 17, normalized size = 0.81 \begin {gather*} e^{\left (10 \, {\left (x^{3} - x\right )} {\left (x + 4\right )}^{\left (x^{2}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((20*x^5+80*x^4-20*x^3-80*x^2)*log(4+x)+10*x^5+20*x^3+120*x^2-10*x-40)*exp(x^2*log(4+x))*exp((10*x^3
-10*x)*exp(x^2*log(4+x)))/(4+x),x, algorithm="fricas")

[Out]

e^(10*(x^3 - x)*(x + 4)^(x^2))

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((20*x^5+80*x^4-20*x^3-80*x^2)*log(4+x)+10*x^5+20*x^3+120*x^2-10*x-40)*exp(x^2*log(4+x))*exp((10*x^3
-10*x)*exp(x^2*log(4+x)))/(4+x),x, algorithm="giac")

[Out]

Timed out

________________________________________________________________________________________

maple [A]  time = 0.07, size = 18, normalized size = 0.86

method	result	size
risch	\({\mathrm e}^{10 \left (x -1\right ) \left (x +1\right ) x \left (4+x \right )^{x^{2}}}\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((20*x^5+80*x^4-20*x^3-80*x^2)*ln(4+x)+10*x^5+20*x^3+120*x^2-10*x-40)*exp(x^2*ln(4+x))*exp((10*x^3-10*x)*e
xp(x^2*ln(4+x)))/(4+x),x,method=_RETURNVERBOSE)

[Out]

exp(10*(x-1)*(x+1)*x*(4+x)^(x^2))

________________________________________________________________________________________

maxima [A]  time = 0.63, size = 24, normalized size = 1.14 \begin {gather*} e^{\left (10 \, {\left (x + 4\right )}^{\left (x^{2}\right )} x^{3} - 10 \, {\left (x + 4\right )}^{\left (x^{2}\right )} x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((20*x^5+80*x^4-20*x^3-80*x^2)*log(4+x)+10*x^5+20*x^3+120*x^2-10*x-40)*exp(x^2*log(4+x))*exp((10*x^3
-10*x)*exp(x^2*log(4+x)))/(4+x),x, algorithm="maxima")

[Out]

e^(10*(x + 4)^(x^2)*x^3 - 10*(x + 4)^(x^2)*x)

________________________________________________________________________________________

mupad [B]  time = 4.27, size = 25, normalized size = 1.19 \begin {gather*} {\mathrm {e}}^{-10\,x\,{\left (x+4\right )}^{x^2}}\,{\mathrm {e}}^{10\,x^3\,{\left (x+4\right )}^{x^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(-exp(x^2*log(x + 4))*(10*x - 10*x^3))*exp(x^2*log(x + 4))*(10*x - 120*x^2 - 20*x^3 - 10*x^5 + log(x
+ 4)*(80*x^2 + 20*x^3 - 80*x^4 - 20*x^5) + 40))/(x + 4),x)

[Out]

exp(-10*x*(x + 4)^(x^2))*exp(10*x^3*(x + 4)^(x^2))

________________________________________________________________________________________

sympy [A]  time = 1.58, size = 19, normalized size = 0.90 \begin {gather*} e^{\left (10 x^{3} - 10 x\right ) e^{x^{2} \log {\left (x + 4 \right )}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((20*x**5+80*x**4-20*x**3-80*x**2)*ln(4+x)+10*x**5+20*x**3+120*x**2-10*x-40)*exp(x**2*ln(4+x))*exp((
10*x**3-10*x)*exp(x**2*ln(4+x)))/(4+x),x)

[Out]

exp((10*x**3 - 10*x)*exp(x**2*log(x + 4)))

________________________________________________________________________________________