\(\int \frac {e^{32} (1024+4096 x+4608 x^2+2048 x^3+320 x^4)+(32 x+96 x^2+96 x^3+40 x^4+6 x^5) \log ^2(2)}{\log ^2(2)} \, dx\) [5226]

Optimal result

Integrand size = 59, antiderivative size = 20 \[ \int \frac {e^{32} \left (1024+4096 x+4608 x^2+2048 x^3+320 x^4\right )+\left (32 x+96 x^2+96 x^3+40 x^4+6 x^5\right ) \log ^2(2)}{\log ^2(2)} \, dx=(2+x)^4 \left (x^2+\frac {64 e^{32} x}{\log ^2(2)}\right ) \]

[Out]

(x^2+64/ln(2)^2*exp(16)^2*x)*(2+x)^4

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(82\) vs. \(2(20)=40\).

Time = 0.02 (sec) , antiderivative size = 82, normalized size of antiderivative = 4.10, number of steps used = 4, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.017, Rules used = {12} \[ \int \frac {e^{32} \left (1024+4096 x+4608 x^2+2048 x^3+320 x^4\right )+\left (32 x+96 x^2+96 x^3+40 x^4+6 x^5\right ) \log ^2(2)}{\log ^2(2)} \, dx=x^6+8 x^5+\frac {64 e^{32} x^5}{\log ^2(2)}+24 x^4+\frac {512 e^{32} x^4}{\log ^2(2)}+32 x^3+\frac {1536 e^{32} x^3}{\log ^2(2)}+16 x^2+\frac {2048 e^{32} x^2}{\log ^2(2)}+\frac {1024 e^{32} x}{\log ^2(2)} \]

[In]

Int[(E^32*(1024 + 4096*x + 4608*x^2 + 2048*x^3 + 320*x^4) + (32*x + 96*x^2 + 96*x^3 + 40*x^4 + 6*x^5)*Log[2]^2
)/Log[2]^2,x]

[Out]

16*x^2 + 32*x^3 + 24*x^4 + 8*x^5 + x^6 + (1024*E^32*x)/Log[2]^2 + (2048*E^32*x^2)/Log[2]^2 + (1536*E^32*x^3)/L
og[2]^2 + (512*E^32*x^4)/Log[2]^2 + (64*E^32*x^5)/Log[2]^2

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rubi steps \begin{align*} \text {integral}& = \frac {\int \left (e^{32} \left (1024+4096 x+4608 x^2+2048 x^3+320 x^4\right )+\left (32 x+96 x^2+96 x^3+40 x^4+6 x^5\right ) \log ^2(2)\right ) \, dx}{\log ^2(2)} \\ & = \frac {e^{32} \int \left (1024+4096 x+4608 x^2+2048 x^3+320 x^4\right ) \, dx}{\log ^2(2)}+\int \left (32 x+96 x^2+96 x^3+40 x^4+6 x^5\right ) \, dx \\ & = 16 x^2+32 x^3+24 x^4+8 x^5+x^6+\frac {1024 e^{32} x}{\log ^2(2)}+\frac {2048 e^{32} x^2}{\log ^2(2)}+\frac {1536 e^{32} x^3}{\log ^2(2)}+\frac {512 e^{32} x^4}{\log ^2(2)}+\frac {64 e^{32} x^5}{\log ^2(2)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.15 \[ \int \frac {e^{32} \left (1024+4096 x+4608 x^2+2048 x^3+320 x^4\right )+\left (32 x+96 x^2+96 x^3+40 x^4+6 x^5\right ) \log ^2(2)}{\log ^2(2)} \, dx=\frac {x (2+x)^4 \left (64 e^{32}+x \log ^2(2)\right )}{\log ^2(2)} \]

[In]

Integrate[(E^32*(1024 + 4096*x + 4608*x^2 + 2048*x^3 + 320*x^4) + (32*x + 96*x^2 + 96*x^3 + 40*x^4 + 6*x^5)*Lo
g[2]^2)/Log[2]^2,x]

[Out]

(x*(2 + x)^4*(64*E^32 + x*Log[2]^2))/Log[2]^2

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(77\) vs. \(2(21)=42\).

Time = 0.16 (sec) , antiderivative size = 78, normalized size of antiderivative = 3.90

method	result	size
risch	\(x^{6}+8 x^{5}+\frac {64 x^{5} {\mathrm e}^{32}}{\ln \left (2\right )^{2}}+24 x^{4}+\frac {512 x^{4} {\mathrm e}^{32}}{\ln \left (2\right )^{2}}+32 x^{3}+\frac {1536 x^{3} {\mathrm e}^{32}}{\ln \left (2\right )^{2}}+16 x^{2}+\frac {2048 x^{2} {\mathrm e}^{32}}{\ln \left (2\right )^{2}}+\frac {1024 \,{\mathrm e}^{32} x}{\ln \left (2\right )^{2}}\)	\(78\)
gosper	\(\frac {x \left (x^{5} \ln \left (2\right )^{2}+8 x^{4} \ln \left (2\right )^{2}+64 x^{4} {\mathrm e}^{32}+24 x^{3} \ln \left (2\right )^{2}+512 x^{3} {\mathrm e}^{32}+32 x^{2} \ln \left (2\right )^{2}+1536 \,{\mathrm e}^{32} x^{2}+16 x \ln \left (2\right )^{2}+2048 x \,{\mathrm e}^{32}+1024 \,{\mathrm e}^{32}\right )}{\ln \left (2\right )^{2}}\)	\(90\)
default	\(\frac {x^{6} \ln \left (2\right )^{2}+\frac {2 \left (160 \,{\mathrm e}^{32}+20 \ln \left (2\right )^{2}\right ) x^{5}}{5}+\frac {\left (1024 \,{\mathrm e}^{32}+48 \ln \left (2\right )^{2}\right ) x^{4}}{2}+\frac {2 \left (2304 \,{\mathrm e}^{32}+48 \ln \left (2\right )^{2}\right ) x^{3}}{3}+\left (2048 \,{\mathrm e}^{32}+16 \ln \left (2\right )^{2}\right ) x^{2}+1024 x \,{\mathrm e}^{32}}{\ln \left (2\right )^{2}}\)	\(93\)
parallelrisch	\(\frac {x^{6} \ln \left (2\right )^{2}+64 x^{5} {\mathrm e}^{32}+8 x^{5} \ln \left (2\right )^{2}+512 x^{4} {\mathrm e}^{32}+24 x^{4} \ln \left (2\right )^{2}+1536 x^{3} {\mathrm e}^{32}+32 x^{3} \ln \left (2\right )^{2}+2048 \,{\mathrm e}^{32} x^{2}+16 x^{2} \ln \left (2\right )^{2}+1024 x \,{\mathrm e}^{32}}{\ln \left (2\right )^{2}}\)	\(94\)
norman	\(\frac {x^{6} \ln \left (2\right )+\frac {8 \left (8 \,{\mathrm e}^{32}+\ln \left (2\right )^{2}\right ) x^{5}}{\ln \left (2\right )}+\frac {32 \left (48 \,{\mathrm e}^{32}+\ln \left (2\right )^{2}\right ) x^{3}}{\ln \left (2\right )}+\frac {8 \left (64 \,{\mathrm e}^{32}+3 \ln \left (2\right )^{2}\right ) x^{4}}{\ln \left (2\right )}+\frac {16 \left (128 \,{\mathrm e}^{32}+\ln \left (2\right )^{2}\right ) x^{2}}{\ln \left (2\right )}+\frac {1024 \,{\mathrm e}^{32} x}{\ln \left (2\right )}}{\ln \left (2\right )}\)	\(106\)

[In]

int(((6*x^5+40*x^4+96*x^3+96*x^2+32*x)*ln(2)^2+(320*x^4+2048*x^3+4608*x^2+4096*x+1024)*exp(16)^2)/ln(2)^2,x,me
thod=_RETURNVERBOSE)

[Out]

x^6+8*x^5+64/ln(2)^2*x^5*exp(32)+24*x^4+512/ln(2)^2*x^4*exp(32)+32*x^3+1536/ln(2)^2*x^3*exp(32)+16*x^2+2048/ln
(2)^2*x^2*exp(32)+1024/ln(2)^2*exp(32)*x

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (19) = 38\).

Time = 0.25 (sec) , antiderivative size = 61, normalized size of antiderivative = 3.05 \[ \int \frac {e^{32} \left (1024+4096 x+4608 x^2+2048 x^3+320 x^4\right )+\left (32 x+96 x^2+96 x^3+40 x^4+6 x^5\right ) \log ^2(2)}{\log ^2(2)} \, dx=\frac {{\left (x^{6} + 8 \, x^{5} + 24 \, x^{4} + 32 \, x^{3} + 16 \, x^{2}\right )} \log \left (2\right )^{2} + 64 \, {\left (x^{5} + 8 \, x^{4} + 24 \, x^{3} + 32 \, x^{2} + 16 \, x\right )} e^{32}}{\log \left (2\right )^{2}} \]

[In]

integrate(((6*x^5+40*x^4+96*x^3+96*x^2+32*x)*log(2)^2+(320*x^4+2048*x^3+4608*x^2+4096*x+1024)*exp(16)^2)/log(2
)^2,x, algorithm="fricas")

[Out]

((x^6 + 8*x^5 + 24*x^4 + 32*x^3 + 16*x^2)*log(2)^2 + 64*(x^5 + 8*x^4 + 24*x^3 + 32*x^2 + 16*x)*e^32)/log(2)^2

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 95 vs. \(2 (19) = 38\).

Time = 0.03 (sec) , antiderivative size = 95, normalized size of antiderivative = 4.75 \[ \int \frac {e^{32} \left (1024+4096 x+4608 x^2+2048 x^3+320 x^4\right )+\left (32 x+96 x^2+96 x^3+40 x^4+6 x^5\right ) \log ^2(2)}{\log ^2(2)} \, dx=x^{6} + \frac {x^{5} \cdot \left (8 \log {\left (2 \right )}^{2} + 64 e^{32}\right )}{\log {\left (2 \right )}^{2}} + \frac {x^{4} \cdot \left (24 \log {\left (2 \right )}^{2} + 512 e^{32}\right )}{\log {\left (2 \right )}^{2}} + \frac {x^{3} \cdot \left (32 \log {\left (2 \right )}^{2} + 1536 e^{32}\right )}{\log {\left (2 \right )}^{2}} + \frac {x^{2} \cdot \left (16 \log {\left (2 \right )}^{2} + 2048 e^{32}\right )}{\log {\left (2 \right )}^{2}} + \frac {1024 x e^{32}}{\log {\left (2 \right )}^{2}} \]

[In]

integrate(((6*x**5+40*x**4+96*x**3+96*x**2+32*x)*ln(2)**2+(320*x**4+2048*x**3+4608*x**2+4096*x+1024)*exp(16)**
2)/ln(2)**2,x)

[Out]

x**6 + x**5*(8*log(2)**2 + 64*exp(32))/log(2)**2 + x**4*(24*log(2)**2 + 512*exp(32))/log(2)**2 + x**3*(32*log(
2)**2 + 1536*exp(32))/log(2)**2 + x**2*(16*log(2)**2 + 2048*exp(32))/log(2)**2 + 1024*x*exp(32)/log(2)**2

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (19) = 38\).

Time = 0.19 (sec) , antiderivative size = 61, normalized size of antiderivative = 3.05 \[ \int \frac {e^{32} \left (1024+4096 x+4608 x^2+2048 x^3+320 x^4\right )+\left (32 x+96 x^2+96 x^3+40 x^4+6 x^5\right ) \log ^2(2)}{\log ^2(2)} \, dx=\frac {{\left (x^{6} + 8 \, x^{5} + 24 \, x^{4} + 32 \, x^{3} + 16 \, x^{2}\right )} \log \left (2\right )^{2} + 64 \, {\left (x^{5} + 8 \, x^{4} + 24 \, x^{3} + 32 \, x^{2} + 16 \, x\right )} e^{32}}{\log \left (2\right )^{2}} \]

[In]

integrate(((6*x^5+40*x^4+96*x^3+96*x^2+32*x)*log(2)^2+(320*x^4+2048*x^3+4608*x^2+4096*x+1024)*exp(16)^2)/log(2
)^2,x, algorithm="maxima")

[Out]

((x^6 + 8*x^5 + 24*x^4 + 32*x^3 + 16*x^2)*log(2)^2 + 64*(x^5 + 8*x^4 + 24*x^3 + 32*x^2 + 16*x)*e^32)/log(2)^2

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (19) = 38\).

Time = 0.27 (sec) , antiderivative size = 61, normalized size of antiderivative = 3.05 \[ \int \frac {e^{32} \left (1024+4096 x+4608 x^2+2048 x^3+320 x^4\right )+\left (32 x+96 x^2+96 x^3+40 x^4+6 x^5\right ) \log ^2(2)}{\log ^2(2)} \, dx=\frac {{\left (x^{6} + 8 \, x^{5} + 24 \, x^{4} + 32 \, x^{3} + 16 \, x^{2}\right )} \log \left (2\right )^{2} + 64 \, {\left (x^{5} + 8 \, x^{4} + 24 \, x^{3} + 32 \, x^{2} + 16 \, x\right )} e^{32}}{\log \left (2\right )^{2}} \]

[In]

integrate(((6*x^5+40*x^4+96*x^3+96*x^2+32*x)*log(2)^2+(320*x^4+2048*x^3+4608*x^2+4096*x+1024)*exp(16)^2)/log(2
)^2,x, algorithm="giac")

[Out]

((x^6 + 8*x^5 + 24*x^4 + 32*x^3 + 16*x^2)*log(2)^2 + 64*(x^5 + 8*x^4 + 24*x^3 + 32*x^2 + 16*x)*e^32)/log(2)^2

Mupad [B] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 93, normalized size of antiderivative = 4.65 \[ \int \frac {e^{32} \left (1024+4096 x+4608 x^2+2048 x^3+320 x^4\right )+\left (32 x+96 x^2+96 x^3+40 x^4+6 x^5\right ) \log ^2(2)}{\log ^2(2)} \, dx=x^6+\frac {\left (320\,{\mathrm {e}}^{32}+40\,{\ln \left (2\right )}^2\right )\,x^5}{5\,{\ln \left (2\right )}^2}+\frac {\left (2048\,{\mathrm {e}}^{32}+96\,{\ln \left (2\right )}^2\right )\,x^4}{4\,{\ln \left (2\right )}^2}+\frac {\left (4608\,{\mathrm {e}}^{32}+96\,{\ln \left (2\right )}^2\right )\,x^3}{3\,{\ln \left (2\right )}^2}+\frac {\left (4096\,{\mathrm {e}}^{32}+32\,{\ln \left (2\right )}^2\right )\,x^2}{2\,{\ln \left (2\right )}^2}+\frac {1024\,{\mathrm {e}}^{32}\,x}{{\ln \left (2\right )}^2} \]

[In]

int((log(2)^2*(32*x + 96*x^2 + 96*x^3 + 40*x^4 + 6*x^5) + exp(32)*(4096*x + 4608*x^2 + 2048*x^3 + 320*x^4 + 10
24))/log(2)^2,x)

[Out]

x^6 + (1024*x*exp(32))/log(2)^2 + (x^5*(320*exp(32) + 40*log(2)^2))/(5*log(2)^2) + (x^4*(2048*exp(32) + 96*log
(2)^2))/(4*log(2)^2) + (x^2*(4096*exp(32) + 32*log(2)^2))/(2*log(2)^2) + (x^3*(4608*exp(32) + 96*log(2)^2))/(3
*log(2)^2)