3.81.95 \(\int \frac {3 x^2+8 x^3-1280 x^6-32 x^4 \log (5)+3 x^2 \log ^2(5)}{1+24 x+192 x^2+512 x^3+768 x^4+12288 x^5+49152 x^6+196608 x^8+1572864 x^9+16777216 x^{12}+(96 x^2+1536 x^3+6144 x^4+49152 x^6+393216 x^7+6291456 x^{10}) \log (5)+(3+48 x+192 x^2+4608 x^4+36864 x^5+983040 x^8) \log ^2(5)+(192 x^2+1536 x^3+81920 x^6) \log ^3(5)+(3+24 x+3840 x^4) \log ^4(5)+96 x^2 \log ^5(5)+\log ^6(5)} \, dx\)

Optimal. Leaf size=25 \[ \frac {x}{\left (4+\frac {1+4 x+\left (16 x^2+\log (5)\right )^2}{x}\right )^2} \]

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Rubi [F]  time = 0.89, 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 {3 x^2+8 x^3-1280 x^6-32 x^4 \log (5)+3 x^2 \log ^2(5)}{1+24 x+192 x^2+512 x^3+768 x^4+12288 x^5+49152 x^6+196608 x^8+1572864 x^9+16777216 x^{12}+\left (96 x^2+1536 x^3+6144 x^4+49152 x^6+393216 x^7+6291456 x^{10}\right ) \log (5)+\left (3+48 x+192 x^2+4608 x^4+36864 x^5+983040 x^8\right ) \log ^2(5)+\left (192 x^2+1536 x^3+81920 x^6\right ) \log ^3(5)+\left (3+24 x+3840 x^4\right ) \log ^4(5)+96 x^2 \log ^5(5)+\log ^6(5)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(3*x^2 + 8*x^3 - 1280*x^6 - 32*x^4*Log[5] + 3*x^2*Log[5]^2)/(1 + 24*x + 192*x^2 + 512*x^3 + 768*x^4 + 1228
8*x^5 + 49152*x^6 + 196608*x^8 + 1572864*x^9 + 16777216*x^12 + (96*x^2 + 1536*x^3 + 6144*x^4 + 49152*x^6 + 393
216*x^7 + 6291456*x^10)*Log[5] + (3 + 48*x + 192*x^2 + 4608*x^4 + 36864*x^5 + 983040*x^8)*Log[5]^2 + (192*x^2
+ 1536*x^3 + 81920*x^6)*Log[5]^3 + (3 + 24*x + 3840*x^4)*Log[5]^4 + 96*x^2*Log[5]^5 + Log[5]^6),x]

[Out]

-3/(128*(1 + 8*x + 256*x^4 + 32*x^2*Log[5] + Log[5]^2)^2) - ((3 + 4*Log[5]^3 + Log[625])*Defer[Int][(1 + 8*x +
256*x^4 + 32*x^2*Log[5] + Log[5]^2)^(-3), x])/8 - 7*Log[5]*Defer[Int][x/(1 + 8*x + 256*x^4 + 32*x^2*Log[5] +
Log[5]^2)^3, x] + 8*(1 - Log[5]^2)*Defer[Int][x^2/(1 + 8*x + 256*x^4 + 32*x^2*Log[5] + Log[5]^2)^3, x] + (Log[
5]*Defer[Int][(1 + 8*x + 256*x^4 + 32*x^2*Log[5] + Log[5]^2)^(-2), x])/2 - 5*Defer[Int][x^2/(1 + 8*x + 256*x^4
+ 32*x^2*Log[5] + Log[5]^2)^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {8 x^3-1280 x^6-32 x^4 \log (5)+x^2 \left (3+3 \log ^2(5)\right )}{1+24 x+192 x^2+512 x^3+768 x^4+12288 x^5+49152 x^6+196608 x^8+1572864 x^9+16777216 x^{12}+\left (96 x^2+1536 x^3+6144 x^4+49152 x^6+393216 x^7+6291456 x^{10}\right ) \log (5)+\left (3+48 x+192 x^2+4608 x^4+36864 x^5+983040 x^8\right ) \log ^2(5)+\left (192 x^2+1536 x^3+81920 x^6\right ) \log ^3(5)+\left (3+24 x+3840 x^4\right ) \log ^4(5)+96 x^2 \log ^5(5)+\log ^6(5)} \, dx\\ &=\int \frac {8 x^3-1280 x^6-32 x^4 \log (5)+x^2 \left (3+3 \log ^2(5)\right )}{1+24 x+512 x^3+768 x^4+12288 x^5+49152 x^6+196608 x^8+1572864 x^9+16777216 x^{12}+\left (96 x^2+1536 x^3+6144 x^4+49152 x^6+393216 x^7+6291456 x^{10}\right ) \log (5)+\left (3+48 x+192 x^2+4608 x^4+36864 x^5+983040 x^8\right ) \log ^2(5)+\left (192 x^2+1536 x^3+81920 x^6\right ) \log ^3(5)+\left (3+24 x+3840 x^4\right ) \log ^4(5)+\log ^6(5)+x^2 \left (192+96 \log ^5(5)\right )} \, dx\\ &=\int \left (\frac {-10 x^2+\log (5)}{2 \left (1+8 x+256 x^4+32 x^2 \log (5)+\log ^2(5)\right )^2}+\frac {96 x^3-8 x \log (5)+16 x^2 \left (1-\log ^2(5)\right )-\log (5) \left (1+\log ^2(5)\right )}{2 \left (1+8 x+256 x^4+32 x^2 \log (5)+\log ^2(5)\right )^3}\right ) \, dx\\ &=\frac {1}{2} \int \frac {-10 x^2+\log (5)}{\left (1+8 x+256 x^4+32 x^2 \log (5)+\log ^2(5)\right )^2} \, dx+\frac {1}{2} \int \frac {96 x^3-8 x \log (5)+16 x^2 \left (1-\log ^2(5)\right )-\log (5) \left (1+\log ^2(5)\right )}{\left (1+8 x+256 x^4+32 x^2 \log (5)+\log ^2(5)\right )^3} \, dx\\ &=-\frac {3}{128 \left (1+8 x+256 x^4+32 x^2 \log (5)+\log ^2(5)\right )^2}+\frac {\int \frac {-14336 x \log (5)+16384 x^2 \left (1-\log ^2(5)\right )-256 \left (3+4 \log ^3(5)+\log (625)\right )}{\left (1+8 x+256 x^4+32 x^2 \log (5)+\log ^2(5)\right )^3} \, dx}{2048}+\frac {1}{2} \int \left (-\frac {10 x^2}{\left (1+8 x+256 x^4+32 x^2 \log (5)+\log ^2(5)\right )^2}+\frac {\log (5)}{\left (1+8 x+256 x^4+32 x^2 \log (5)+\log ^2(5)\right )^2}\right ) \, dx\\ &=-\frac {3}{128 \left (1+8 x+256 x^4+32 x^2 \log (5)+\log ^2(5)\right )^2}+\frac {\int \left (-\frac {14336 x \log (5)}{\left (1+8 x+256 x^4+32 x^2 \log (5)+\log ^2(5)\right )^3}-\frac {16384 x^2 \left (-1+\log ^2(5)\right )}{\left (1+8 x+256 x^4+32 x^2 \log (5)+\log ^2(5)\right )^3}-\frac {256 \left (3+4 \log ^3(5)+\log (625)\right )}{\left (1+8 x+256 x^4+32 x^2 \log (5)+\log ^2(5)\right )^3}\right ) \, dx}{2048}-5 \int \frac {x^2}{\left (1+8 x+256 x^4+32 x^2 \log (5)+\log ^2(5)\right )^2} \, dx+\frac {1}{2} \log (5) \int \frac {1}{\left (1+8 x+256 x^4+32 x^2 \log (5)+\log ^2(5)\right )^2} \, dx\\ &=-\frac {3}{128 \left (1+8 x+256 x^4+32 x^2 \log (5)+\log ^2(5)\right )^2}-5 \int \frac {x^2}{\left (1+8 x+256 x^4+32 x^2 \log (5)+\log ^2(5)\right )^2} \, dx+\frac {1}{2} \log (5) \int \frac {1}{\left (1+8 x+256 x^4+32 x^2 \log (5)+\log ^2(5)\right )^2} \, dx-(7 \log (5)) \int \frac {x}{\left (1+8 x+256 x^4+32 x^2 \log (5)+\log ^2(5)\right )^3} \, dx+\left (8 \left (1-\log ^2(5)\right )\right ) \int \frac {x^2}{\left (1+8 x+256 x^4+32 x^2 \log (5)+\log ^2(5)\right )^3} \, dx+\frac {1}{8} \left (-3-4 \log ^3(5)-\log (625)\right ) \int \frac {1}{\left (1+8 x+256 x^4+32 x^2 \log (5)+\log ^2(5)\right )^3} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.03, size = 27, normalized size = 1.08 \begin {gather*} \frac {x^3}{\left (1+8 x+256 x^4+32 x^2 \log (5)+\log ^2(5)\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(3*x^2 + 8*x^3 - 1280*x^6 - 32*x^4*Log[5] + 3*x^2*Log[5]^2)/(1 + 24*x + 192*x^2 + 512*x^3 + 768*x^4
+ 12288*x^5 + 49152*x^6 + 196608*x^8 + 1572864*x^9 + 16777216*x^12 + (96*x^2 + 1536*x^3 + 6144*x^4 + 49152*x^6
+ 393216*x^7 + 6291456*x^10)*Log[5] + (3 + 48*x + 192*x^2 + 4608*x^4 + 36864*x^5 + 983040*x^8)*Log[5]^2 + (19
2*x^2 + 1536*x^3 + 81920*x^6)*Log[5]^3 + (3 + 24*x + 3840*x^4)*Log[5]^4 + 96*x^2*Log[5]^5 + Log[5]^6),x]

[Out]

x^3/(1 + 8*x + 256*x^4 + 32*x^2*Log[5] + Log[5]^2)^2

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fricas [B]  time = 1.02, size = 78, normalized size = 3.12 \begin {gather*} \frac {x^{3}}{65536 \, x^{8} + 4096 \, x^{5} + 64 \, x^{2} \log \relax (5)^{3} + 512 \, x^{4} + \log \relax (5)^{4} + 2 \, {\left (768 \, x^{4} + 8 \, x + 1\right )} \log \relax (5)^{2} + 64 \, x^{2} + 64 \, {\left (256 \, x^{6} + 8 \, x^{3} + x^{2}\right )} \log \relax (5) + 16 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^2*log(5)^2-32*x^4*log(5)-1280*x^6+8*x^3+3*x^2)/(log(5)^6+96*x^2*log(5)^5+(3840*x^4+24*x+3)*log(
5)^4+(81920*x^6+1536*x^3+192*x^2)*log(5)^3+(983040*x^8+36864*x^5+4608*x^4+192*x^2+48*x+3)*log(5)^2+(6291456*x^
10+393216*x^7+49152*x^6+6144*x^4+1536*x^3+96*x^2)*log(5)[phone]*x^12+1572864*x^9+196608*x^8+49152*x^6+12288*
x^5+768*x^4+512*x^3+192*x^2+24*x+1),x, algorithm="fricas")

[Out]

x^3/(65536*x^8 + 4096*x^5 + 64*x^2*log(5)^3 + 512*x^4 + log(5)^4 + 2*(768*x^4 + 8*x + 1)*log(5)^2 + 64*x^2 + 6
4*(256*x^6 + 8*x^3 + x^2)*log(5) + 16*x + 1)

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giac [A]  time = 0.21, size = 27, normalized size = 1.08 \begin {gather*} \frac {x^{3}}{{\left (256 \, x^{4} + 32 \, x^{2} \log \relax (5) + \log \relax (5)^{2} + 8 \, x + 1\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^2*log(5)^2-32*x^4*log(5)-1280*x^6+8*x^3+3*x^2)/(log(5)^6+96*x^2*log(5)^5+(3840*x^4+24*x+3)*log(
5)^4+(81920*x^6+1536*x^3+192*x^2)*log(5)^3+(983040*x^8+36864*x^5+4608*x^4+192*x^2+48*x+3)*log(5)^2+(6291456*x^
10+393216*x^7+49152*x^6+6144*x^4+1536*x^3+96*x^2)*log(5)[phone]*x^12+1572864*x^9+196608*x^8+49152*x^6+12288*
x^5+768*x^4+512*x^3+192*x^2+24*x+1),x, algorithm="giac")

[Out]

x^3/(256*x^4 + 32*x^2*log(5) + log(5)^2 + 8*x + 1)^2

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maple [A]  time = 0.29, size = 28, normalized size = 1.12

method	result	size
default	\(\frac {x^{3}}{\left (256 x^{4}+32 x^{2} \ln \relax (5)+\ln \relax (5)^{2}+8 x +1\right )^{2}}\)	\(28\)
norman	\(\frac {x^{3}}{\left (256 x^{4}+32 x^{2} \ln \relax (5)+\ln \relax (5)^{2}+8 x +1\right )^{2}}\)	\(28\)
gosper	\(\frac {x^{3}}{65536 x^{8}+16384 x^{6} \ln \relax (5)+1536 x^{4} \ln \relax (5)^{2}+64 x^{2} \ln \relax (5)^{3}+4096 x^{5}+\ln \relax (5)^{4}+512 x^{3} \ln \relax (5)+512 x^{4}+16 x \ln \relax (5)^{2}+64 x^{2} \ln \relax (5)+2 \ln \relax (5)^{2}+64 x^{2}+16 x +1}\)	\(88\)
risch	\(\frac {x^{3}}{65536 x^{8}+16384 x^{6} \ln \relax (5)+1536 x^{4} \ln \relax (5)^{2}+64 x^{2} \ln \relax (5)^{3}+4096 x^{5}+\ln \relax (5)^{4}+512 x^{3} \ln \relax (5)+512 x^{4}+16 x \ln \relax (5)^{2}+64 x^{2} \ln \relax (5)+2 \ln \relax (5)^{2}+64 x^{2}+16 x +1}\)	\(88\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x^2*ln(5)^2-32*x^4*ln(5)-1280*x^6+8*x^3+3*x^2)/(ln(5)^6+96*x^2*ln(5)^5+(3840*x^4+24*x+3)*ln(5)^4+(81920
*x^6+1536*x^3+192*x^2)*ln(5)^3+(983040*x^8+36864*x^5+4608*x^4+192*x^2+48*x+3)*ln(5)^2+(6291456*x^10+393216*x^7
+49152*x^6+6144*x^4+1536*x^3+96*x^2)*ln(5)[phone]*x^12+1572864*x^9+196608*x^8+49152*x^6+12288*x^5+768*x^4+51
2*x^3+192*x^2+24*x+1),x,method=_RETURNVERBOSE)

[Out]

x^3/(256*x^4+32*x^2*ln(5)+ln(5)^2+8*x+1)^2

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maxima [B]  time = 0.43, size = 77, normalized size = 3.08 \begin {gather*} \frac {x^{3}}{65536 \, x^{8} + 16384 \, x^{6} \log \relax (5) + 512 \, {\left (3 \, \log \relax (5)^{2} + 1\right )} x^{4} + 4096 \, x^{5} + 512 \, x^{3} \log \relax (5) + \log \relax (5)^{4} + 64 \, {\left (\log \relax (5)^{3} + \log \relax (5) + 1\right )} x^{2} + 16 \, {\left (\log \relax (5)^{2} + 1\right )} x + 2 \, \log \relax (5)^{2} + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^2*log(5)^2-32*x^4*log(5)-1280*x^6+8*x^3+3*x^2)/(log(5)^6+96*x^2*log(5)^5+(3840*x^4+24*x+3)*log(
5)^4+(81920*x^6+1536*x^3+192*x^2)*log(5)^3+(983040*x^8+36864*x^5+4608*x^4+192*x^2+48*x+3)*log(5)^2+(6291456*x^
10+393216*x^7+49152*x^6+6144*x^4+1536*x^3+96*x^2)*log(5)[phone]*x^12+1572864*x^9+196608*x^8+49152*x^6+12288*
x^5+768*x^4+512*x^3+192*x^2+24*x+1),x, algorithm="maxima")

[Out]

x^3/(65536*x^8 + 16384*x^6*log(5) + 512*(3*log(5)^2 + 1)*x^4 + 4096*x^5 + 512*x^3*log(5) + log(5)^4 + 64*(log(
5)^3 + log(5) + 1)*x^2 + 16*(log(5)^2 + 1)*x + 2*log(5)^2 + 1)

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mupad [B]  time = 5.55, size = 27, normalized size = 1.08 \begin {gather*} \frac {x^3}{{\left (256\,x^4+32\,\ln \relax (5)\,x^2+8\,x+{\ln \relax (5)}^2+1\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x^2*log(5)^2 - 32*x^4*log(5) + 3*x^2 + 8*x^3 - 1280*x^6)/(24*x + 96*x^2*log(5)^5 + log(5)^3*(192*x^2 +
1536*x^3 + 81920*x^6) + log(5)^2*(48*x + 192*x^2 + 4608*x^4 + 36864*x^5 + 983040*x^8 + 3) + log(5)*(96*x^2 + 1
536*x^3 + 6144*x^4 + 49152*x^6 + 393216*x^7 + 6291456*x^10) + log(5)^4*(24*x + 3840*x^4 + 3) + log(5)^6 + 192*
x^2 + 512*x^3 + 768*x^4 + 12288*x^5 + 49152*x^6 + 196608*x^8 + 1572864*x^9 + 16777216*x^12 + 1),x)

[Out]

x^3/(8*x + 32*x^2*log(5) + log(5)^2 + 256*x^4 + 1)^2

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sympy [B]  time = 8.91, size = 82, normalized size = 3.28 \begin {gather*} \frac {x^{3}}{65536 x^{8} + 16384 x^{6} \log {\relax (5 )} + 4096 x^{5} + x^{4} \left (512 + 1536 \log {\relax (5 )}^{2}\right ) + 512 x^{3} \log {\relax (5 )} + x^{2} \left (64 + 64 \log {\relax (5 )} + 64 \log {\relax (5 )}^{3}\right ) + x \left (16 + 16 \log {\relax (5 )}^{2}\right ) + 1 + 2 \log {\relax (5 )}^{2} + \log {\relax (5 )}^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x**2*ln(5)**2-32*x**4*ln(5)-1280*x**6+8*x**3+3*x**2)/(ln(5)**6+96*x**2*ln(5)**5+(3840*x**4+24*x+3
)*ln(5)**4+(81920*x**6+1536*x**3+192*x**2)*ln(5)**3+(983040*x**8+36864*x**5+4608*x**4+192*x**2+48*x+3)*ln(5)**
2+(6291456*x**10+393216*x**7+49152*x**6+6144*x**4+1536*x**3+96*x**2)*ln(5)[phone]*x**12+1572864*x**9+196608*
x**8+49152*x**6+12288*x**5+768*x**4+512*x**3+192*x**2+24*x+1),x)

[Out]

x**3/(65536*x**8 + 16384*x**6*log(5) + 4096*x**5 + x**4*(512 + 1536*log(5)**2) + 512*x**3*log(5) + x**2*(64 +
64*log(5) + 64*log(5)**3) + x*(16 + 16*log(5)**2) + 1 + 2*log(5)**2 + log(5)**4)

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