\(\int \frac {4+x+e^{\frac {128+e^6 (64-8 x)+8 e^6 \log (4+x)}{e^6}} (24+8 x)}{4+x} \, dx\) [9243]

Optimal result

Integrand size = 41, antiderivative size = 25 \[ \int \frac {4+x+e^{\frac {128+e^6 (64-8 x)+8 e^6 \log (4+x)}{e^6}} (24+8 x)}{4+x} \, dx=-e^{4 \left (16+2 \left (\frac {16}{e^6}-x+\log (4+x)\right )\right )}+x \]

[Out]

x-exp(64+128/exp(3)^2-8*x+8*ln(4+x))

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96, number of steps used = 22, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.122, Rules used = {6820, 6874, 2227, 2207, 2225} \[ \int \frac {4+x+e^{\frac {128+e^6 (64-8 x)+8 e^6 \log (4+x)}{e^6}} (24+8 x)}{4+x} \, dx=x-e^{64 \left (1+\frac {2}{e^6}\right )-8 x} (x+4)^8 \]

[In]

Int[(4 + x + E^((128 + E^6*(64 - 8*x) + 8*E^6*Log[4 + x])/E^6)*(24 + 8*x))/(4 + x),x]

[Out]

x - E^(64*(1 + 2/E^6) - 8*x)*(4 + x)^8

Rule 2207

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(c + d*x)^m*
((b*F^(g*(e + f*x)))^n/(f*g*n*Log[F])), x] - Dist[d*(m/(f*g*n*Log[F])), Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x
)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2*m] &&  !TrueQ[$UseGamma]

Rule 2225

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 2227

Int[(F_)^((c_.)*(v_))*(u_), x_Symbol] :> Int[ExpandIntegrand[F^(c*ExpandToSum[v, x]), u, x], x] /; FreeQ[{F, c
}, x] && PolynomialQ[u, x] && LinearQ[v, x] &&  !TrueQ[$UseGamma]

Rule 6820

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rule 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps \begin{align*} \text {integral}& = \int \frac {4+x+8 e^{64+\frac {128}{e^6}-8 x} (3+x) (4+x)^8}{4+x} \, dx \\ & = \int \left (1+8 e^{64+\frac {128}{e^6}-8 x} (3+x) (4+x)^7\right ) \, dx \\ & = x+8 \int e^{64+\frac {128}{e^6}-8 x} (3+x) (4+x)^7 \, dx \\ & = x+8 \int \left (-e^{64 \left (1+\frac {2}{e^6}\right )-8 x} (4+x)^7+e^{64 \left (1+\frac {2}{e^6}\right )-8 x} (4+x)^8\right ) \, dx \\ & = x-8 \int e^{64 \left (1+\frac {2}{e^6}\right )-8 x} (4+x)^7 \, dx+8 \int e^{64 \left (1+\frac {2}{e^6}\right )-8 x} (4+x)^8 \, dx \\ & = x+e^{64 \left (1+\frac {2}{e^6}\right )-8 x} (4+x)^7-e^{64 \left (1+\frac {2}{e^6}\right )-8 x} (4+x)^8-7 \int e^{64 \left (1+\frac {2}{e^6}\right )-8 x} (4+x)^6 \, dx+8 \int e^{64 \left (1+\frac {2}{e^6}\right )-8 x} (4+x)^7 \, dx \\ & = x+\frac {7}{8} e^{64 \left (1+\frac {2}{e^6}\right )-8 x} (4+x)^6-e^{64 \left (1+\frac {2}{e^6}\right )-8 x} (4+x)^8-\frac {21}{4} \int e^{64 \left (1+\frac {2}{e^6}\right )-8 x} (4+x)^5 \, dx+7 \int e^{64 \left (1+\frac {2}{e^6}\right )-8 x} (4+x)^6 \, dx \\ & = x+\frac {21}{32} e^{64 \left (1+\frac {2}{e^6}\right )-8 x} (4+x)^5-e^{64 \left (1+\frac {2}{e^6}\right )-8 x} (4+x)^8-\frac {105}{32} \int e^{64 \left (1+\frac {2}{e^6}\right )-8 x} (4+x)^4 \, dx+\frac {21}{4} \int e^{64 \left (1+\frac {2}{e^6}\right )-8 x} (4+x)^5 \, dx \\ & = x+\frac {105}{256} e^{64 \left (1+\frac {2}{e^6}\right )-8 x} (4+x)^4-e^{64 \left (1+\frac {2}{e^6}\right )-8 x} (4+x)^8-\frac {105}{64} \int e^{64 \left (1+\frac {2}{e^6}\right )-8 x} (4+x)^3 \, dx+\frac {105}{32} \int e^{64 \left (1+\frac {2}{e^6}\right )-8 x} (4+x)^4 \, dx \\ & = x+\frac {105}{512} e^{64 \left (1+\frac {2}{e^6}\right )-8 x} (4+x)^3-e^{64 \left (1+\frac {2}{e^6}\right )-8 x} (4+x)^8-\frac {315}{512} \int e^{64 \left (1+\frac {2}{e^6}\right )-8 x} (4+x)^2 \, dx+\frac {105}{64} \int e^{64 \left (1+\frac {2}{e^6}\right )-8 x} (4+x)^3 \, dx \\ & = x+\frac {315 e^{64 \left (1+\frac {2}{e^6}\right )-8 x} (4+x)^2}{4096}-e^{64 \left (1+\frac {2}{e^6}\right )-8 x} (4+x)^8-\frac {315 \int e^{64 \left (1+\frac {2}{e^6}\right )-8 x} (4+x) \, dx}{2048}+\frac {315}{512} \int e^{64 \left (1+\frac {2}{e^6}\right )-8 x} (4+x)^2 \, dx \\ & = x+\frac {315 e^{64 \left (1+\frac {2}{e^6}\right )-8 x} (4+x)}{16384}-e^{64 \left (1+\frac {2}{e^6}\right )-8 x} (4+x)^8-\frac {315 \int e^{64 \left (1+\frac {2}{e^6}\right )-8 x} \, dx}{16384}+\frac {315 \int e^{64 \left (1+\frac {2}{e^6}\right )-8 x} (4+x) \, dx}{2048} \\ & = \frac {315 e^{64 \left (1+\frac {2}{e^6}\right )-8 x}}{131072}+x-e^{64 \left (1+\frac {2}{e^6}\right )-8 x} (4+x)^8+\frac {315 \int e^{64 \left (1+\frac {2}{e^6}\right )-8 x} \, dx}{16384} \\ & = x-e^{64 \left (1+\frac {2}{e^6}\right )-8 x} (4+x)^8 \\ \end{align*}

Mathematica [A] (verified)

Time = 8.14 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.32 \[ \int \frac {4+x+e^{\frac {128+e^6 (64-8 x)+8 e^6 \log (4+x)}{e^6}} (24+8 x)}{4+x} \, dx=-e^{-8 x} (4+x) \left (-e^{8 x}+e^{64+\frac {128}{e^6}} (4+x)^7\right ) \]

[In]

Integrate[(4 + x + E^((128 + E^6*(64 - 8*x) + 8*E^6*Log[4 + x])/E^6)*(24 + 8*x))/(4 + x),x]

[Out]

-(((4 + x)*(-E^(8*x) + E^(64 + 128/E^6)*(4 + x)^7))/E^(8*x))

Maple [A] (verified)

Time = 0.72 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.32

method	result	size
default	\(x -{\mathrm e}^{\left (8 \,{\mathrm e}^{6} \ln \left (4+x \right )+\left (-8 x +64\right ) {\mathrm e}^{6}+128\right ) {\mathrm e}^{-6}}\)	\(33\)
norman	\(x -{\mathrm e}^{\left (8 \,{\mathrm e}^{6} \ln \left (4+x \right )+\left (-8 x +64\right ) {\mathrm e}^{6}+128\right ) {\mathrm e}^{-6}}\)	\(33\)
parts	\(x -{\mathrm e}^{\left (8 \,{\mathrm e}^{6} \ln \left (4+x \right )+\left (-8 x +64\right ) {\mathrm e}^{6}+128\right ) {\mathrm e}^{-6}}\)	\(33\)
parallelrisch	\(x -{\mathrm e}^{\left (8 \,{\mathrm e}^{6} \ln \left (4+x \right )+\left (-8 x +64\right ) {\mathrm e}^{6}+128\right ) {\mathrm e}^{-6}}-8\)	\(34\)
risch	\(x +\left (-x^{8}-32 x^{7}-448 x^{6}-3584 x^{5}-17920 x^{4}-57344 x^{3}-114688 x^{2}-131072 x -65536\right ) {\mathrm e}^{-8 x +64+128 \,{\mathrm e}^{-6}}\)	\(54\)

[In]

int(((8*x+24)*exp((8*exp(3)^2*ln(4+x)+(-8*x+64)*exp(3)^2+128)/exp(3)^2)+4+x)/(4+x),x,method=_RETURNVERBOSE)

[Out]

x-exp((8*exp(3)^2*ln(4+x)+(-8*x+64)*exp(3)^2+128)/exp(3)^2)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {4+x+e^{\frac {128+e^6 (64-8 x)+8 e^6 \log (4+x)}{e^6}} (24+8 x)}{4+x} \, dx=x - e^{\left (-8 \, {\left ({\left (x - 8\right )} e^{6} - e^{6} \log \left (x + 4\right ) - 16\right )} e^{\left (-6\right )}\right )} \]

[In]

integrate(((8*x+24)*exp((8*exp(3)^2*log(4+x)+(-8*x+64)*exp(3)^2+128)/exp(3)^2)+4+x)/(4+x),x, algorithm="fricas
")

[Out]

x - e^(-8*((x - 8)*e^6 - e^6*log(x + 4) - 16)*e^(-6))

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 590 vs. \(2 (17) = 34\).

Time = 4.42 (sec) , antiderivative size = 590, normalized size of antiderivative = 23.60 \[ \int \frac {4+x+e^{\frac {128+e^6 (64-8 x)+8 e^6 \log (4+x)}{e^6}} (24+8 x)}{4+x} \, dx=x + 819200 \left (- \frac {x e^{- 8 x}}{8} - \frac {e^{- 8 x}}{64}\right ) e^{64} e^{\frac {128}{e^{6}}} + 745472 \left (- \frac {x^{2} e^{- 8 x}}{8} - \frac {x e^{- 8 x}}{32} - \frac {e^{- 8 x}}{256}\right ) e^{64} e^{\frac {128}{e^{6}}} + 387072 \left (- \frac {x^{3} e^{- 8 x}}{8} - \frac {3 x^{2} e^{- 8 x}}{64} - \frac {3 x e^{- 8 x}}{256} - \frac {3 e^{- 8 x}}{2048}\right ) e^{64} e^{\frac {128}{e^{6}}} + 125440 \left (- \frac {x^{4} e^{- 8 x}}{8} - \frac {x^{3} e^{- 8 x}}{16} - \frac {3 x^{2} e^{- 8 x}}{128} - \frac {3 x e^{- 8 x}}{512} - \frac {3 e^{- 8 x}}{4096}\right ) e^{64} e^{\frac {128}{e^{6}}} + 25984 \left (- \frac {x^{5} e^{- 8 x}}{8} - \frac {5 x^{4} e^{- 8 x}}{64} - \frac {5 x^{3} e^{- 8 x}}{128} - \frac {15 x^{2} e^{- 8 x}}{1024} - \frac {15 x e^{- 8 x}}{4096} - \frac {15 e^{- 8 x}}{32768}\right ) e^{64} e^{\frac {128}{e^{6}}} + 3360 \left (- \frac {x^{6} e^{- 8 x}}{8} - \frac {3 x^{5} e^{- 8 x}}{32} - \frac {15 x^{4} e^{- 8 x}}{256} - \frac {15 x^{3} e^{- 8 x}}{512} - \frac {45 x^{2} e^{- 8 x}}{4096} - \frac {45 x e^{- 8 x}}{16384} - \frac {45 e^{- 8 x}}{131072}\right ) e^{64} e^{\frac {128}{e^{6}}} + 248 \left (- \frac {x^{7} e^{- 8 x}}{8} - \frac {7 x^{6} e^{- 8 x}}{64} - \frac {21 x^{5} e^{- 8 x}}{256} - \frac {105 x^{4} e^{- 8 x}}{2048} - \frac {105 x^{3} e^{- 8 x}}{4096} - \frac {315 x^{2} e^{- 8 x}}{32768} - \frac {315 x e^{- 8 x}}{131072} - \frac {315 e^{- 8 x}}{1048576}\right ) e^{64} e^{\frac {128}{e^{6}}} + 8 \left (- \frac {x^{8} e^{- 8 x}}{8} - \frac {x^{7} e^{- 8 x}}{8} - \frac {7 x^{6} e^{- 8 x}}{64} - \frac {21 x^{5} e^{- 8 x}}{256} - \frac {105 x^{4} e^{- 8 x}}{2048} - \frac {105 x^{3} e^{- 8 x}}{4096} - \frac {315 x^{2} e^{- 8 x}}{32768} - \frac {315 x e^{- 8 x}}{131072} - \frac {315 e^{- 8 x}}{1048576}\right ) e^{64} e^{\frac {128}{e^{6}}} - 49152 e^{64} e^{- 8 x} e^{\frac {128}{e^{6}}} \]

[In]

integrate(((8*x+24)*exp((8*exp(3)**2*ln(4+x)+(-8*x+64)*exp(3)**2+128)/exp(3)**2)+4+x)/(4+x),x)

[Out]

x + 819200*(-x*exp(-8*x)/8 - exp(-8*x)/64)*exp(64)*exp(128*exp(-6)) + 745472*(-x**2*exp(-8*x)/8 - x*exp(-8*x)/
32 - exp(-8*x)/256)*exp(64)*exp(128*exp(-6)) + 387072*(-x**3*exp(-8*x)/8 - 3*x**2*exp(-8*x)/64 - 3*x*exp(-8*x)
/256 - 3*exp(-8*x)/2048)*exp(64)*exp(128*exp(-6)) + 125440*(-x**4*exp(-8*x)/8 - x**3*exp(-8*x)/16 - 3*x**2*exp
(-8*x)/128 - 3*x*exp(-8*x)/512 - 3*exp(-8*x)/4096)*exp(64)*exp(128*exp(-6)) + 25984*(-x**5*exp(-8*x)/8 - 5*x**
4*exp(-8*x)/64 - 5*x**3*exp(-8*x)/128 - 15*x**2*exp(-8*x)/1024 - 15*x*exp(-8*x)/4096 - 15*exp(-8*x)/32768)*exp
(64)*exp(128*exp(-6)) + 3360*(-x**6*exp(-8*x)/8 - 3*x**5*exp(-8*x)/32 - 15*x**4*exp(-8*x)/256 - 15*x**3*exp(-8
*x)/512 - 45*x**2*exp(-8*x)/4096 - 45*x*exp(-8*x)/16384 - 45*exp(-8*x)/131072)*exp(64)*exp(128*exp(-6)) + 248*
(-x**7*exp(-8*x)/8 - 7*x**6*exp(-8*x)/64 - 21*x**5*exp(-8*x)/256 - 105*x**4*exp(-8*x)/2048 - 105*x**3*exp(-8*x
)/4096 - 315*x**2*exp(-8*x)/32768 - 315*x*exp(-8*x)/131072 - 315*exp(-8*x)/1048576)*exp(64)*exp(128*exp(-6)) +
8*(-x**8*exp(-8*x)/8 - x**7*exp(-8*x)/8 - 7*x**6*exp(-8*x)/64 - 21*x**5*exp(-8*x)/256 - 105*x**4*exp(-8*x)/20
48 - 105*x**3*exp(-8*x)/4096 - 315*x**2*exp(-8*x)/32768 - 315*x*exp(-8*x)/131072 - 315*exp(-8*x)/1048576)*exp(
64)*exp(128*exp(-6)) - 49152*exp(64)*exp(-8*x)*exp(128*exp(-6))

Maxima [F]

\[ \int \frac {4+x+e^{\frac {128+e^6 (64-8 x)+8 e^6 \log (4+x)}{e^6}} (24+8 x)}{4+x} \, dx=\int { \frac {8 \, {\left (x + 3\right )} e^{\left (-8 \, {\left ({\left (x - 8\right )} e^{6} - e^{6} \log \left (x + 4\right ) - 16\right )} e^{\left (-6\right )}\right )} + x + 4}{x + 4} \,d x } \]

[In]

integrate(((8*x+24)*exp((8*exp(3)^2*log(4+x)+(-8*x+64)*exp(3)^2+128)/exp(3)^2)+4+x)/(4+x),x, algorithm="maxima
")

[Out]

-1572864*e^(64*(e^6 + 2)*e^(-6) + 32)*exp_integral_e(1, 8*x + 32) + x - (x^9*e^(128*e^(-6) + 64) + 36*x^8*e^(1
28*e^(-6) + 64) + 576*x^7*e^(128*e^(-6) + 64) + 5376*x^6*e^(128*e^(-6) + 64) + 32256*x^5*e^(128*e^(-6) + 64) +
129024*x^4*e^(128*e^(-6) + 64) + 344064*x^3*e^(128*e^(-6) + 64) + 589824*x^2*e^(128*e^(-6) + 64) + 589824*x*e
^(128*e^(-6) + 64))*e^(-8*x)/(x + 4) + integrate(262144*(2*x*e^(128*e^(-6) + 64) + 9*e^(128*e^(-6) + 64))*e^(-
8*x)/(x^2 + 8*x + 16), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 159 vs. \(2 (20) = 40\).

Time = 0.28 (sec) , antiderivative size = 159, normalized size of antiderivative = 6.36 \[ \int \frac {4+x+e^{\frac {128+e^6 (64-8 x)+8 e^6 \log (4+x)}{e^6}} (24+8 x)}{4+x} \, dx=-x^{8} e^{\left (64 \, {\left (e^{6} + 2\right )} e^{\left (-6\right )} - 8 \, x\right )} - 32 \, x^{7} e^{\left (64 \, {\left (e^{6} + 2\right )} e^{\left (-6\right )} - 8 \, x\right )} - 448 \, x^{6} e^{\left (64 \, {\left (e^{6} + 2\right )} e^{\left (-6\right )} - 8 \, x\right )} - 3584 \, x^{5} e^{\left (64 \, {\left (e^{6} + 2\right )} e^{\left (-6\right )} - 8 \, x\right )} - 17920 \, x^{4} e^{\left (64 \, {\left (e^{6} + 2\right )} e^{\left (-6\right )} - 8 \, x\right )} - 57344 \, x^{3} e^{\left (64 \, {\left (e^{6} + 2\right )} e^{\left (-6\right )} - 8 \, x\right )} - 114688 \, x^{2} e^{\left (64 \, {\left (e^{6} + 2\right )} e^{\left (-6\right )} - 8 \, x\right )} - 131072 \, x e^{\left (64 \, {\left (e^{6} + 2\right )} e^{\left (-6\right )} - 8 \, x\right )} + x - 65536 \, e^{\left (64 \, {\left (e^{6} + 2\right )} e^{\left (-6\right )} - 8 \, x\right )} \]

[In]

integrate(((8*x+24)*exp((8*exp(3)^2*log(4+x)+(-8*x+64)*exp(3)^2+128)/exp(3)^2)+4+x)/(4+x),x, algorithm="giac")

[Out]

-x^8*e^(64*(e^6 + 2)*e^(-6) - 8*x) - 32*x^7*e^(64*(e^6 + 2)*e^(-6) - 8*x) - 448*x^6*e^(64*(e^6 + 2)*e^(-6) - 8
*x) - 3584*x^5*e^(64*(e^6 + 2)*e^(-6) - 8*x) - 17920*x^4*e^(64*(e^6 + 2)*e^(-6) - 8*x) - 57344*x^3*e^(64*(e^6
+ 2)*e^(-6) - 8*x) - 114688*x^2*e^(64*(e^6 + 2)*e^(-6) - 8*x) - 131072*x*e^(64*(e^6 + 2)*e^(-6) - 8*x) + x - 6
5536*e^(64*(e^6 + 2)*e^(-6) - 8*x)

Mupad [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 132, normalized size of antiderivative = 5.28 \[ \int \frac {4+x+e^{\frac {128+e^6 (64-8 x)+8 e^6 \log (4+x)}{e^6}} (24+8 x)}{4+x} \, dx=x-65536\,{\mathrm {e}}^{128\,{\mathrm {e}}^{-6}-8\,x+64}-131072\,x\,{\mathrm {e}}^{128\,{\mathrm {e}}^{-6}-8\,x+64}-114688\,x^2\,{\mathrm {e}}^{128\,{\mathrm {e}}^{-6}-8\,x+64}-57344\,x^3\,{\mathrm {e}}^{128\,{\mathrm {e}}^{-6}-8\,x+64}-17920\,x^4\,{\mathrm {e}}^{128\,{\mathrm {e}}^{-6}-8\,x+64}-3584\,x^5\,{\mathrm {e}}^{128\,{\mathrm {e}}^{-6}-8\,x+64}-448\,x^6\,{\mathrm {e}}^{128\,{\mathrm {e}}^{-6}-8\,x+64}-32\,x^7\,{\mathrm {e}}^{128\,{\mathrm {e}}^{-6}-8\,x+64}-x^8\,{\mathrm {e}}^{128\,{\mathrm {e}}^{-6}-8\,x+64} \]

[In]

int((x + exp(exp(-6)*(8*log(x + 4)*exp(6) - exp(6)*(8*x - 64) + 128))*(8*x + 24) + 4)/(x + 4),x)

[Out]

x - 65536*exp(128*exp(-6) - 8*x + 64) - 131072*x*exp(128*exp(-6) - 8*x + 64) - 114688*x^2*exp(128*exp(-6) - 8*
x + 64) - 57344*x^3*exp(128*exp(-6) - 8*x + 64) - 17920*x^4*exp(128*exp(-6) - 8*x + 64) - 3584*x^5*exp(128*exp
(-6) - 8*x + 64) - 448*x^6*exp(128*exp(-6) - 8*x + 64) - 32*x^7*exp(128*exp(-6) - 8*x + 64) - x^8*exp(128*exp(
-6) - 8*x + 64)