3.71.72 \(\int \frac {1}{16} (192 x^3+80 x^4+576 x^5+224 x^6+576 x^7+216 x^8+240 x^9+88 x^{10}+36 x^{11}+13 x^{12}) \log ^2(2) \, dx\)

Optimal. Leaf size=19 \[ (3+x) \left (x+\frac {x^3}{2}\right )^4 \log ^2(2) \]

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Rubi [B]  time = 0.02, antiderivative size = 102, normalized size of antiderivative = 5.37, number of steps used = 2, number of rules used = 1, integrand size = 59, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.017, Rules used = {12} \begin {gather*} \frac {1}{16} x^{13} \log ^2(2)+\frac {3}{16} x^{12} \log ^2(2)+\frac {1}{2} x^{11} \log ^2(2)+\frac {3}{2} x^{10} \log ^2(2)+\frac {3}{2} x^9 \log ^2(2)+\frac {9}{2} x^8 \log ^2(2)+2 x^7 \log ^2(2)+6 x^6 \log ^2(2)+x^5 \log ^2(2)+3 x^4 \log ^2(2) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((192*x^3 + 80*x^4 + 576*x^5 + 224*x^6 + 576*x^7 + 216*x^8 + 240*x^9 + 88*x^10 + 36*x^11 + 13*x^12)*Log[2]
^2)/16,x]

[Out]

3*x^4*Log[2]^2 + x^5*Log[2]^2 + 6*x^6*Log[2]^2 + 2*x^7*Log[2]^2 + (9*x^8*Log[2]^2)/2 + (3*x^9*Log[2]^2)/2 + (3
*x^10*Log[2]^2)/2 + (x^11*Log[2]^2)/2 + (3*x^12*Log[2]^2)/16 + (x^13*Log[2]^2)/16

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 {1}{16} \log ^2(2) \int \left (192 x^3+80 x^4+576 x^5+224 x^6+576 x^7+216 x^8+240 x^9+88 x^{10}+36 x^{11}+13 x^{12}\right ) \, dx\\ &=3 x^4 \log ^2(2)+x^5 \log ^2(2)+6 x^6 \log ^2(2)+2 x^7 \log ^2(2)+\frac {9}{2} x^8 \log ^2(2)+\frac {3}{2} x^9 \log ^2(2)+\frac {3}{2} x^{10} \log ^2(2)+\frac {1}{2} x^{11} \log ^2(2)+\frac {3}{16} x^{12} \log ^2(2)+\frac {1}{16} x^{13} \log ^2(2)\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.01, size = 57, normalized size = 3.00 \begin {gather*} \frac {1}{16} \left (48 x^4+16 x^5+96 x^6+32 x^7+72 x^8+24 x^9+24 x^{10}+8 x^{11}+3 x^{12}+x^{13}\right ) \log ^2(2) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((192*x^3 + 80*x^4 + 576*x^5 + 224*x^6 + 576*x^7 + 216*x^8 + 240*x^9 + 88*x^10 + 36*x^11 + 13*x^12)*
Log[2]^2)/16,x]

[Out]

((48*x^4 + 16*x^5 + 96*x^6 + 32*x^7 + 72*x^8 + 24*x^9 + 24*x^10 + 8*x^11 + 3*x^12 + x^13)*Log[2]^2)/16

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fricas [B]  time = 0.68, size = 55, normalized size = 2.89 \begin {gather*} \frac {1}{16} \, {\left (x^{13} + 3 \, x^{12} + 8 \, x^{11} + 24 \, x^{10} + 24 \, x^{9} + 72 \, x^{8} + 32 \, x^{7} + 96 \, x^{6} + 16 \, x^{5} + 48 \, x^{4}\right )} \log \relax (2)^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/16*(13*x^12+36*x^11+88*x^10+240*x^9+216*x^8+576*x^7+224*x^6+576*x^5+80*x^4+192*x^3)*log(2)^2,x, al
gorithm="fricas")

[Out]

1/16*(x^13 + 3*x^12 + 8*x^11 + 24*x^10 + 24*x^9 + 72*x^8 + 32*x^7 + 96*x^6 + 16*x^5 + 48*x^4)*log(2)^2

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giac [B]  time = 0.13, size = 55, normalized size = 2.89 \begin {gather*} \frac {1}{16} \, {\left (x^{13} + 3 \, x^{12} + 8 \, x^{11} + 24 \, x^{10} + 24 \, x^{9} + 72 \, x^{8} + 32 \, x^{7} + 96 \, x^{6} + 16 \, x^{5} + 48 \, x^{4}\right )} \log \relax (2)^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/16*(13*x^12+36*x^11+88*x^10+240*x^9+216*x^8+576*x^7+224*x^6+576*x^5+80*x^4+192*x^3)*log(2)^2,x, al
gorithm="giac")

[Out]

1/16*(x^13 + 3*x^12 + 8*x^11 + 24*x^10 + 24*x^9 + 72*x^8 + 32*x^7 + 96*x^6 + 16*x^5 + 48*x^4)*log(2)^2

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maple [B]  time = 0.03, size = 53, normalized size = 2.79

method	result	size
gosper	\(\frac {x^{4} \left (x^{9}+3 x^{8}+8 x^{7}+24 x^{6}+24 x^{5}+72 x^{4}+32 x^{3}+96 x^{2}+16 x +48\right ) \ln \relax (2)^{2}}{16}\)	\(53\)
default	\(\frac {\ln \relax (2)^{2} \left (x^{13}+3 x^{12}+8 x^{11}+24 x^{10}+24 x^{9}+72 x^{8}+32 x^{7}+96 x^{6}+16 x^{5}+48 x^{4}\right )}{16}\)	\(56\)
norman	\(x^{5} \ln \relax (2)^{2}+3 x^{4} \ln \relax (2)^{2}+6 x^{6} \ln \relax (2)^{2}+2 x^{7} \ln \relax (2)^{2}+\frac {9 \ln \relax (2)^{2} x^{8}}{2}+\frac {3 \ln \relax (2)^{2} x^{9}}{2}+\frac {3 \ln \relax (2)^{2} x^{10}}{2}+\frac {\ln \relax (2)^{2} x^{11}}{2}+\frac {3 \ln \relax (2)^{2} x^{12}}{16}+\frac {\ln \relax (2)^{2} x^{13}}{16}\)	\(91\)
risch	\(x^{5} \ln \relax (2)^{2}+3 x^{4} \ln \relax (2)^{2}+6 x^{6} \ln \relax (2)^{2}+2 x^{7} \ln \relax (2)^{2}+\frac {9 \ln \relax (2)^{2} x^{8}}{2}+\frac {3 \ln \relax (2)^{2} x^{9}}{2}+\frac {3 \ln \relax (2)^{2} x^{10}}{2}+\frac {\ln \relax (2)^{2} x^{11}}{2}+\frac {3 \ln \relax (2)^{2} x^{12}}{16}+\frac {\ln \relax (2)^{2} x^{13}}{16}\)	\(91\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/16*(13*x^12+36*x^11+88*x^10+240*x^9+216*x^8+576*x^7+224*x^6+576*x^5+80*x^4+192*x^3)*ln(2)^2,x,method=_RE
TURNVERBOSE)

[Out]

1/16*x^4*(x^9+3*x^8+8*x^7+24*x^6+24*x^5+72*x^4+32*x^3+96*x^2+16*x+48)*ln(2)^2

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maxima [B]  time = 0.40, size = 55, normalized size = 2.89 \begin {gather*} \frac {1}{16} \, {\left (x^{13} + 3 \, x^{12} + 8 \, x^{11} + 24 \, x^{10} + 24 \, x^{9} + 72 \, x^{8} + 32 \, x^{7} + 96 \, x^{6} + 16 \, x^{5} + 48 \, x^{4}\right )} \log \relax (2)^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/16*(13*x^12+36*x^11+88*x^10+240*x^9+216*x^8+576*x^7+224*x^6+576*x^5+80*x^4+192*x^3)*log(2)^2,x, al
gorithm="maxima")

[Out]

1/16*(x^13 + 3*x^12 + 8*x^11 + 24*x^10 + 24*x^9 + 72*x^8 + 32*x^7 + 96*x^6 + 16*x^5 + 48*x^4)*log(2)^2

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mupad [B]  time = 0.07, size = 90, normalized size = 4.74 \begin {gather*} \frac {{\ln \relax (2)}^2\,x^{13}}{16}+\frac {3\,{\ln \relax (2)}^2\,x^{12}}{16}+\frac {{\ln \relax (2)}^2\,x^{11}}{2}+\frac {3\,{\ln \relax (2)}^2\,x^{10}}{2}+\frac {3\,{\ln \relax (2)}^2\,x^9}{2}+\frac {9\,{\ln \relax (2)}^2\,x^8}{2}+2\,{\ln \relax (2)}^2\,x^7+6\,{\ln \relax (2)}^2\,x^6+{\ln \relax (2)}^2\,x^5+3\,{\ln \relax (2)}^2\,x^4 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(2)^2*(192*x^3 + 80*x^4 + 576*x^5 + 224*x^6 + 576*x^7 + 216*x^8 + 240*x^9 + 88*x^10 + 36*x^11 + 13*x^1
2))/16,x)

[Out]

3*x^4*log(2)^2 + x^5*log(2)^2 + 6*x^6*log(2)^2 + 2*x^7*log(2)^2 + (9*x^8*log(2)^2)/2 + (3*x^9*log(2)^2)/2 + (3
*x^10*log(2)^2)/2 + (x^11*log(2)^2)/2 + (3*x^12*log(2)^2)/16 + (x^13*log(2)^2)/16

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sympy [B]  time = 0.07, size = 105, normalized size = 5.53 \begin {gather*} \frac {x^{13} \log {\relax (2 )}^{2}}{16} + \frac {3 x^{12} \log {\relax (2 )}^{2}}{16} + \frac {x^{11} \log {\relax (2 )}^{2}}{2} + \frac {3 x^{10} \log {\relax (2 )}^{2}}{2} + \frac {3 x^{9} \log {\relax (2 )}^{2}}{2} + \frac {9 x^{8} \log {\relax (2 )}^{2}}{2} + 2 x^{7} \log {\relax (2 )}^{2} + 6 x^{6} \log {\relax (2 )}^{2} + x^{5} \log {\relax (2 )}^{2} + 3 x^{4} \log {\relax (2 )}^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/16*(13*x**12+36*x**11+88*x**10+240*x**9+216*x**8+576*x**7+224*x**6+576*x**5+80*x**4+192*x**3)*ln(2
)**2,x)

[Out]

x**13*log(2)**2/16 + 3*x**12*log(2)**2/16 + x**11*log(2)**2/2 + 3*x**10*log(2)**2/2 + 3*x**9*log(2)**2/2 + 9*x
**8*log(2)**2/2 + 2*x**7*log(2)**2 + 6*x**6*log(2)**2 + x**5*log(2)**2 + 3*x**4*log(2)**2

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