3.58.3 \(\int \frac {(e^{1+x} (4+x))^{(3 x+3 x^2) \log (8)+x \log (8) \log (x)} ((15 x+18 x^2+3 x^3) \log (8)+(5 x+x^2) \log (8) \log (x)+((16+28 x+6 x^2) \log (8)+(4+x) \log (8) \log (x)) \log (e^{1+x} (4+x)))}{4+x} \, dx\)

Optimal. Leaf size=24 \[ e^{x \log (8) (3 (1+x)+\log (x)) \log \left (e^{1+x} (4+x)\right )} \]

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Rubi [F]  time = 3.38, 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 (e^{1+x} (4+x)\right )^{\left (3 x+3 x^2\right ) \log (8)+x \log (8) \log (x)} \left (\left (15 x+18 x^2+3 x^3\right ) \log (8)+\left (5 x+x^2\right ) \log (8) \log (x)+\left (\left (16+28 x+6 x^2\right ) \log (8)+(4+x) \log (8) \log (x)\right ) \log \left (e^{1+x} (4+x)\right )\right )}{4+x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((E^(1 + x)*(4 + x))^((3*x + 3*x^2)*Log[8] + x*Log[8]*Log[x])*((15*x + 18*x^2 + 3*x^3)*Log[8] + (5*x + x^2
)*Log[8]*Log[x] + ((16 + 28*x + 6*x^2)*Log[8] + (4 + x)*Log[8]*Log[x])*Log[E^(1 + x)*(4 + x)]))/(4 + x),x]

[Out]

-9*Log[8]*Defer[Int][(E^(1 + x)*(4 + x))^(x*Log[8]*(3 + 3*x + Log[x])), x] + 6*Log[8]*Defer[Int][x*(E^(1 + x)*
(4 + x))^(x*Log[8]*(3 + 3*x + Log[x])), x] + 3*Log[8]*Defer[Int][x^2*(E^(1 + x)*(4 + x))^(x*Log[8]*(3 + 3*x +
Log[x])), x] + 36*Log[8]*Defer[Int][(E^(1 + x)*(4 + x))^(x*Log[8]*(3 + 3*x + Log[x]))/(4 + x), x] + Log[8]*Def
er[Int][(E^(1 + x)*(4 + x))^(x*Log[8]*(3 + 3*x + Log[x]))*Log[x], x] + Log[8]*Defer[Int][x*(E^(1 + x)*(4 + x))
^(x*Log[8]*(3 + 3*x + Log[x]))*Log[x], x] - 4*Log[8]*Defer[Int][((E^(1 + x)*(4 + x))^(x*Log[8]*(3 + 3*x + Log[
x]))*Log[x])/(4 + x), x] + 4*Log[8]*Defer[Int][(E^(1 + x)*(4 + x))^(x*Log[8]*(3 + 3*x + Log[x]))*Log[E^(1 + x)
*(4 + x)], x] + 6*Log[8]*Defer[Int][x*(E^(1 + x)*(4 + x))^(x*Log[8]*(3 + 3*x + Log[x]))*Log[E^(1 + x)*(4 + x)]
, x] + Log[8]*Defer[Int][(E^(1 + x)*(4 + x))^(x*Log[8]*(3 + 3*x + Log[x]))*Log[x]*Log[E^(1 + x)*(4 + x)], x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\left (e^{1+x} (4+x)\right )^{x \log (8) (3+3 x+\log (x))} \left (\left (15 x+18 x^2+3 x^3\right ) \log (8)+\left (5 x+x^2\right ) \log (8) \log (x)+\left (\left (16+28 x+6 x^2\right ) \log (8)+(4+x) \log (8) \log (x)\right ) \log \left (e^{1+x} (4+x)\right )\right )}{4+x} \, dx\\ &=\int \left (\frac {x \left (e^{1+x} (4+x)\right )^{x \log (8) (3+3 x+\log (x))} (5+x) \log (8) (3+3 x+\log (x))}{4+x}+\left (e^{1+x} (4+x)\right )^{x \log (8) (3+3 x+\log (x))} \log (8) (4+6 x+\log (x)) \log \left (e^{1+x} (4+x)\right )\right ) \, dx\\ &=\log (8) \int \frac {x \left (e^{1+x} (4+x)\right )^{x \log (8) (3+3 x+\log (x))} (5+x) (3+3 x+\log (x))}{4+x} \, dx+\log (8) \int \left (e^{1+x} (4+x)\right )^{x \log (8) (3+3 x+\log (x))} (4+6 x+\log (x)) \log \left (e^{1+x} (4+x)\right ) \, dx\\ &=\log (8) \int \left (\frac {3 x \left (e^{1+x} (4+x)\right )^{x \log (8) (3+3 x+\log (x))} \left (5+6 x+x^2\right )}{4+x}+\frac {x \left (e^{1+x} (4+x)\right )^{x \log (8) (3+3 x+\log (x))} (5+x) \log (x)}{4+x}\right ) \, dx+\log (8) \int \left (4 \left (e^{1+x} (4+x)\right )^{x \log (8) (3+3 x+\log (x))} \log \left (e^{1+x} (4+x)\right )+6 x \left (e^{1+x} (4+x)\right )^{x \log (8) (3+3 x+\log (x))} \log \left (e^{1+x} (4+x)\right )+\left (e^{1+x} (4+x)\right )^{x \log (8) (3+3 x+\log (x))} \log (x) \log \left (e^{1+x} (4+x)\right )\right ) \, dx\\ &=\log (8) \int \frac {x \left (e^{1+x} (4+x)\right )^{x \log (8) (3+3 x+\log (x))} (5+x) \log (x)}{4+x} \, dx+\log (8) \int \left (e^{1+x} (4+x)\right )^{x \log (8) (3+3 x+\log (x))} \log (x) \log \left (e^{1+x} (4+x)\right ) \, dx+(3 \log (8)) \int \frac {x \left (e^{1+x} (4+x)\right )^{x \log (8) (3+3 x+\log (x))} \left (5+6 x+x^2\right )}{4+x} \, dx+(4 \log (8)) \int \left (e^{1+x} (4+x)\right )^{x \log (8) (3+3 x+\log (x))} \log \left (e^{1+x} (4+x)\right ) \, dx+(6 \log (8)) \int x \left (e^{1+x} (4+x)\right )^{x \log (8) (3+3 x+\log (x))} \log \left (e^{1+x} (4+x)\right ) \, dx\\ &=\log (8) \int \left (\left (e^{1+x} (4+x)\right )^{x \log (8) (3+3 x+\log (x))} \log (x)+x \left (e^{1+x} (4+x)\right )^{x \log (8) (3+3 x+\log (x))} \log (x)-\frac {4 \left (e^{1+x} (4+x)\right )^{x \log (8) (3+3 x+\log (x))} \log (x)}{4+x}\right ) \, dx+\log (8) \int \left (e^{1+x} (4+x)\right )^{x \log (8) (3+3 x+\log (x))} \log (x) \log \left (e^{1+x} (4+x)\right ) \, dx+(3 \log (8)) \int \left (-3 \left (e^{1+x} (4+x)\right )^{x \log (8) (3+3 x+\log (x))}+2 x \left (e^{1+x} (4+x)\right )^{x \log (8) (3+3 x+\log (x))}+x^2 \left (e^{1+x} (4+x)\right )^{x \log (8) (3+3 x+\log (x))}+\frac {12 \left (e^{1+x} (4+x)\right )^{x \log (8) (3+3 x+\log (x))}}{4+x}\right ) \, dx+(4 \log (8)) \int \left (e^{1+x} (4+x)\right )^{x \log (8) (3+3 x+\log (x))} \log \left (e^{1+x} (4+x)\right ) \, dx+(6 \log (8)) \int x \left (e^{1+x} (4+x)\right )^{x \log (8) (3+3 x+\log (x))} \log \left (e^{1+x} (4+x)\right ) \, dx\\ &=\log (8) \int \left (e^{1+x} (4+x)\right )^{x \log (8) (3+3 x+\log (x))} \log (x) \, dx+\log (8) \int x \left (e^{1+x} (4+x)\right )^{x \log (8) (3+3 x+\log (x))} \log (x) \, dx+\log (8) \int \left (e^{1+x} (4+x)\right )^{x \log (8) (3+3 x+\log (x))} \log (x) \log \left (e^{1+x} (4+x)\right ) \, dx+(3 \log (8)) \int x^2 \left (e^{1+x} (4+x)\right )^{x \log (8) (3+3 x+\log (x))} \, dx-(4 \log (8)) \int \frac {\left (e^{1+x} (4+x)\right )^{x \log (8) (3+3 x+\log (x))} \log (x)}{4+x} \, dx+(4 \log (8)) \int \left (e^{1+x} (4+x)\right )^{x \log (8) (3+3 x+\log (x))} \log \left (e^{1+x} (4+x)\right ) \, dx+(6 \log (8)) \int x \left (e^{1+x} (4+x)\right )^{x \log (8) (3+3 x+\log (x))} \, dx+(6 \log (8)) \int x \left (e^{1+x} (4+x)\right )^{x \log (8) (3+3 x+\log (x))} \log \left (e^{1+x} (4+x)\right ) \, dx-(9 \log (8)) \int \left (e^{1+x} (4+x)\right )^{x \log (8) (3+3 x+\log (x))} \, dx+(36 \log (8)) \int \frac {\left (e^{1+x} (4+x)\right )^{x \log (8) (3+3 x+\log (x))}}{4+x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.22, size = 21, normalized size = 0.88 \begin {gather*} \left (e^{1+x} (4+x)\right )^{x \log (8) (3+3 x+\log (x))} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((E^(1 + x)*(4 + x))^((3*x + 3*x^2)*Log[8] + x*Log[8]*Log[x])*((15*x + 18*x^2 + 3*x^3)*Log[8] + (5*x
+ x^2)*Log[8]*Log[x] + ((16 + 28*x + 6*x^2)*Log[8] + (4 + x)*Log[8]*Log[x])*Log[E^(1 + x)*(4 + x)]))/(4 + x),
x]

[Out]

(E^(1 + x)*(4 + x))^(x*Log[8]*(3 + 3*x + Log[x]))

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fricas [A]  time = 0.84, size = 26, normalized size = 1.08 \begin {gather*} \left ({\left (x + 4\right )} e^{\left (x + 1\right )}\right )^{3 \, x \log \relax (2) \log \relax (x) + 9 \, {\left (x^{2} + x\right )} \log \relax (2)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*(4+x)*log(2)*log(x)+3*(6*x^2+28*x+16)*log(2))*log((4+x)*exp(x+1))+3*(x^2+5*x)*log(2)*log(x)+3*(3
*x^3+18*x^2+15*x)*log(2))*exp((3*x*log(2)*log(x)+3*(3*x^2+3*x)*log(2))*log((4+x)*exp(x+1)))/(4+x),x, algorithm
="fricas")

[Out]

((x + 4)*e^(x + 1))^(3*x*log(2)*log(x) + 9*(x^2 + x)*log(2))

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giac [A]  time = 0.70, size = 34, normalized size = 1.42 \begin {gather*} {\left (x e^{\left (x + 1\right )} + 4 \, e^{\left (x + 1\right )}\right )}^{9 \, x^{2} \log \relax (2) + 3 \, x \log \relax (2) \log \relax (x) + 9 \, x \log \relax (2)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*(4+x)*log(2)*log(x)+3*(6*x^2+28*x+16)*log(2))*log((4+x)*exp(x+1))+3*(x^2+5*x)*log(2)*log(x)+3*(3
*x^3+18*x^2+15*x)*log(2))*exp((3*x*log(2)*log(x)+3*(3*x^2+3*x)*log(2))*log((4+x)*exp(x+1)))/(4+x),x, algorithm
="giac")

[Out]

(x*e^(x + 1) + 4*e^(x + 1))^(9*x^2*log(2) + 3*x*log(2)*log(x) + 9*x*log(2))

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maple [C]  time = 0.33, size = 95, normalized size = 3.96

method	result	size
risch	\(2^{-\frac {3 x \left (3 x +3+\ln \relax (x )\right ) \left (i \mathrm {csgn}\left (i {\mathrm e}^{x +1} \left (4+x \right )\right ) \pi -i \pi \,\mathrm {csgn}\left (i {\mathrm e}^{x +1}\right )-i \pi \,\mathrm {csgn}\left (i \left (4+x \right )\right )+i \pi \,\mathrm {csgn}\left (i {\mathrm e}^{x +1} \left (4+x \right )\right ) \mathrm {csgn}\left (i {\mathrm e}^{x +1}\right ) \mathrm {csgn}\left (i \left (4+x \right )\right )-2 \ln \left ({\mathrm e}^{x +1}\right )-2 \ln \left (4+x \right )\right )}{2}}\)	\(95\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((3*(4+x)*ln(2)*ln(x)+3*(6*x^2+28*x+16)*ln(2))*ln((4+x)*exp(x+1))+3*(x^2+5*x)*ln(2)*ln(x)+3*(3*x^3+18*x^2+
15*x)*ln(2))*exp((3*x*ln(2)*ln(x)+3*(3*x^2+3*x)*ln(2))*ln((4+x)*exp(x+1)))/(4+x),x,method=_RETURNVERBOSE)

[Out]

2^(-3/2*x*(3*x+3+ln(x))*(I*csgn(I*exp(x+1)*(4+x))*Pi-I*Pi*csgn(I*exp(x+1))-I*Pi*csgn(I*(4+x))+I*Pi*csgn(I*exp(
x+1)*(4+x))*csgn(I*exp(x+1))*csgn(I*(4+x))-2*ln(exp(x+1))-2*ln(4+x)))

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maxima [B]  time = 0.59, size = 68, normalized size = 2.83 \begin {gather*} e^{\left (9 \, x^{3} \log \relax (2) + 9 \, x^{2} \log \relax (2) \log \left (x + 4\right ) + 3 \, x^{2} \log \relax (2) \log \relax (x) + 3 \, x \log \relax (2) \log \left (x + 4\right ) \log \relax (x) + 18 \, x^{2} \log \relax (2) + 9 \, x \log \relax (2) \log \left (x + 4\right ) + 3 \, x \log \relax (2) \log \relax (x) + 9 \, x \log \relax (2)\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*(4+x)*log(2)*log(x)+3*(6*x^2+28*x+16)*log(2))*log((4+x)*exp(x+1))+3*(x^2+5*x)*log(2)*log(x)+3*(3
*x^3+18*x^2+15*x)*log(2))*exp((3*x*log(2)*log(x)+3*(3*x^2+3*x)*log(2))*log((4+x)*exp(x+1)))/(4+x),x, algorithm
="maxima")

[Out]

e^(9*x^3*log(2) + 9*x^2*log(2)*log(x + 4) + 3*x^2*log(2)*log(x) + 3*x*log(2)*log(x + 4)*log(x) + 18*x^2*log(2)
+ 9*x*log(2)*log(x + 4) + 3*x*log(2)*log(x) + 9*x*log(2))

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mupad [B]  time = 3.97, size = 49, normalized size = 2.04 \begin {gather*} x^{3\,x\,\ln \relax (2)\,\ln \left (4\,\mathrm {e}\,{\mathrm {e}}^x+x\,\mathrm {e}\,{\mathrm {e}}^x\right )}\,{\left (4\,{\mathrm {e}}^{x+1}+x\,{\mathrm {e}}^{x+1}\right )}^{9\,\ln \relax (2)\,x^2+9\,\ln \relax (2)\,x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(log(exp(x + 1)*(x + 4))*(3*log(2)*(3*x + 3*x^2) + 3*x*log(2)*log(x)))*(log(exp(x + 1)*(x + 4))*(3*log
(2)*(28*x + 6*x^2 + 16) + 3*log(2)*log(x)*(x + 4)) + 3*log(2)*(15*x + 18*x^2 + 3*x^3) + 3*log(2)*log(x)*(5*x +
x^2)))/(x + 4),x)

[Out]

x^(3*x*log(2)*log(4*exp(1)*exp(x) + x*exp(1)*exp(x)))*(4*exp(x + 1) + x*exp(x + 1))^(9*x*log(2) + 9*x^2*log(2)
)

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sympy [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(((3*(4+x)*ln(2)*ln(x)+3*(6*x**2+28*x+16)*ln(2))*ln((4+x)*exp(x+1))+3*(x**2+5*x)*ln(2)*ln(x)+3*(3*x**
3+18*x**2+15*x)*ln(2))*exp((3*x*ln(2)*ln(x)+3*(3*x**2+3*x)*ln(2))*ln((4+x)*exp(x+1)))/(4+x),x)

[Out]

Timed out

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