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

Optimal. Leaf size=20 \[ (2+x)^4 \left (x^2+\frac {64 e^{32} x}{\log ^2(2)}\right ) \]

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Rubi [B]  time = 0.03, antiderivative size = 82, normalized size of antiderivative = 4.10, number of steps used = 4, number of rules used = 1, integrand size = 59, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.017, Rules used = {12} \begin {gather*} 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)} \end {gather*}

Antiderivative was successfully verified.

[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 {gather*} \begin {aligned} \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 {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 23, normalized size = 1.15 \begin {gather*} \frac {x (2+x)^4 \left (64 e^{32}+x \log ^2(2)\right )}{\log ^2(2)} \end {gather*}

Antiderivative was successfully verified.

[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

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fricas [B]  time = 0.52, size = 61, normalized size = 3.05 \begin {gather*} \frac {{\left (x^{6} + 8 \, x^{5} + 24 \, x^{4} + 32 \, x^{3} + 16 \, x^{2}\right )} \log \relax (2)^{2} + 64 \, {\left (x^{5} + 8 \, x^{4} + 24 \, x^{3} + 32 \, x^{2} + 16 \, x\right )} e^{32}}{\log \relax (2)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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giac [B]  time = 0.15, size = 61, normalized size = 3.05 \begin {gather*} \frac {{\left (x^{6} + 8 \, x^{5} + 24 \, x^{4} + 32 \, x^{3} + 16 \, x^{2}\right )} \log \relax (2)^{2} + 64 \, {\left (x^{5} + 8 \, x^{4} + 24 \, x^{3} + 32 \, x^{2} + 16 \, x\right )} e^{32}}{\log \relax (2)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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maple [B]  time = 0.05, size = 65, normalized size = 3.25

method	result	size
default	\(\frac {\ln \relax (2)^{2} \left (x^{6}+8 x^{5}+24 x^{4}+32 x^{3}+16 x^{2}\right )+{\mathrm e}^{32} \left (64 x^{5}+512 x^{4}+1536 x^{3}+2048 x^{2}+1024 x \right )}{\ln \relax (2)^{2}}\)	\(65\)
risch	\(x^{6}+8 x^{5}+\frac {64 \,{\mathrm e}^{32} x^{5}}{\ln \relax (2)^{2}}+24 x^{4}+\frac {512 x^{4} {\mathrm e}^{32}}{\ln \relax (2)^{2}}+32 x^{3}+\frac {1536 \,{\mathrm e}^{32} x^{3}}{\ln \relax (2)^{2}}+16 x^{2}+\frac {2048 \,{\mathrm e}^{32} x^{2}}{\ln \relax (2)^{2}}+\frac {1024 \,{\mathrm e}^{32} x}{\ln \relax (2)^{2}}\)	\(78\)
gosper	\(\frac {x \left (x^{5} \ln \relax (2)^{2}+8 x^{4} \ln \relax (2)^{2}+64 x^{4} {\mathrm e}^{32}+24 x^{3} \ln \relax (2)^{2}+512 \,{\mathrm e}^{32} x^{3}+32 x^{2} \ln \relax (2)^{2}+1536 \,{\mathrm e}^{32} x^{2}+16 x \ln \relax (2)^{2}+2048 \,{\mathrm e}^{32} x +1024 \,{\mathrm e}^{32}\right )}{\ln \relax (2)^{2}}\)	\(90\)
norman	\(\frac {x^{6} \ln \relax (2)+\frac {8 \left (\ln \relax (2)^{2}+8 \,{\mathrm e}^{32}\right ) x^{5}}{\ln \relax (2)}+\frac {32 \left (\ln \relax (2)^{2}+48 \,{\mathrm e}^{32}\right ) x^{3}}{\ln \relax (2)}+\frac {16 \left (\ln \relax (2)^{2}+128 \,{\mathrm e}^{32}\right ) x^{2}}{\ln \relax (2)}+\frac {8 \left (3 \ln \relax (2)^{2}+64 \,{\mathrm e}^{32}\right ) x^{4}}{\ln \relax (2)}+\frac {1024 \,{\mathrm e}^{32} x}{\ln \relax (2)}}{\ln \relax (2)}\)	\(106\)

Verification of antiderivative is not currently implemented for this CAS.

[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]

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

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maxima [B]  time = 0.36, size = 61, normalized size = 3.05 \begin {gather*} \frac {{\left (x^{6} + 8 \, x^{5} + 24 \, x^{4} + 32 \, x^{3} + 16 \, x^{2}\right )} \log \relax (2)^{2} + 64 \, {\left (x^{5} + 8 \, x^{4} + 24 \, x^{3} + 32 \, x^{2} + 16 \, x\right )} e^{32}}{\log \relax (2)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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mupad [B]  time = 0.07, size = 93, normalized size = 4.65 \begin {gather*} x^6+\frac {\left (320\,{\mathrm {e}}^{32}+40\,{\ln \relax (2)}^2\right )\,x^5}{5\,{\ln \relax (2)}^2}+\frac {\left (2048\,{\mathrm {e}}^{32}+96\,{\ln \relax (2)}^2\right )\,x^4}{4\,{\ln \relax (2)}^2}+\frac {\left (4608\,{\mathrm {e}}^{32}+96\,{\ln \relax (2)}^2\right )\,x^3}{3\,{\ln \relax (2)}^2}+\frac {\left (4096\,{\mathrm {e}}^{32}+32\,{\ln \relax (2)}^2\right )\,x^2}{2\,{\ln \relax (2)}^2}+\frac {1024\,{\mathrm {e}}^{32}\,x}{{\ln \relax (2)}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [B]  time = 0.08, size = 95, normalized size = 4.75 \begin {gather*} x^{6} + \frac {x^{5} \left (8 \log {\relax (2 )}^{2} + 64 e^{32}\right )}{\log {\relax (2 )}^{2}} + \frac {x^{4} \left (24 \log {\relax (2 )}^{2} + 512 e^{32}\right )}{\log {\relax (2 )}^{2}} + \frac {x^{3} \left (32 \log {\relax (2 )}^{2} + 1536 e^{32}\right )}{\log {\relax (2 )}^{2}} + \frac {x^{2} \left (16 \log {\relax (2 )}^{2} + 2048 e^{32}\right )}{\log {\relax (2 )}^{2}} + \frac {1024 x e^{32}}{\log {\relax (2 )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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