3.20.10 \(\int \frac {-32+2 x^4+384 x^5-160 x^7+2048 x^{10}+(-8 x+6 x^3+896 x^4-480 x^6+6144 x^9) \log (4) \log (5)+(-8+6 x^2+512 x^3-480 x^5+6144 x^8) \log ^2(4) \log ^2(5)+(2 x-160 x^4+2048 x^7) \log ^3(4) \log ^3(5)}{x^3+3 x^2 \log (4) \log (5)+3 x \log ^2(4) \log ^2(5)+\log ^3(4) \log ^3(5)} \, dx\)

Optimal. Leaf size=20 \[ \left (x-16 x^4-\frac {4}{x+\log (4) \log (5)}\right )^2 \]

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Rubi [B]  time = 0.24, antiderivative size = 77, normalized size of antiderivative = 3.85, number of steps used = 2, number of rules used = 1, integrand size = 141, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.007, Rules used = {2074} \begin {gather*} 256 x^8-32 x^5+128 x^3+x^2 (1-128 \log (4) \log (5))+\frac {8 \log (4) \log (5) \left (1+16 \log ^3(4) \log ^3(5)\right )}{x+\log (4) \log (5)}+128 x \log ^2(4) \log ^2(5)+\frac {16}{(x+\log (4) \log (5))^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-32 + 2*x^4 + 384*x^5 - 160*x^7 + 2048*x^10 + (-8*x + 6*x^3 + 896*x^4 - 480*x^6 + 6144*x^9)*Log[4]*Log[5]
+ (-8 + 6*x^2 + 512*x^3 - 480*x^5 + 6144*x^8)*Log[4]^2*Log[5]^2 + (2*x - 160*x^4 + 2048*x^7)*Log[4]^3*Log[5]^
3)/(x^3 + 3*x^2*Log[4]*Log[5] + 3*x*Log[4]^2*Log[5]^2 + Log[4]^3*Log[5]^3),x]

[Out]

128*x^3 - 32*x^5 + 256*x^8 + 128*x*Log[4]^2*Log[5]^2 + x^2*(1 - 128*Log[4]*Log[5]) + 16/(x + Log[4]*Log[5])^2
+ (8*Log[4]*Log[5]*(1 + 16*Log[4]^3*Log[5]^3))/(x + Log[4]*Log[5])

Rule 2074

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P]}, Int[ExpandIntegrand[PP^p*Q^q, x], x] /;  !SumQ[N
onfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x] && PolyQ[Q, x] && IntegerQ[p] && NeQ[P, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (384 x^2-160 x^4+2048 x^7+128 \log ^2(4) \log ^2(5)-\frac {32}{(x+\log (4) \log (5))^3}-2 x (-1+128 \log (4) \log (5))-\frac {8 \left (\log (4) \log (5)+16 \log ^4(4) \log ^4(5)\right )}{(x+\log (4) \log (5))^2}\right ) \, dx\\ &=128 x^3-32 x^5+256 x^8+128 x \log ^2(4) \log ^2(5)+x^2 (1-128 \log (4) \log (5))+\frac {16}{(x+\log (4) \log (5))^2}+\frac {8 \log (4) \log (5) \left (1+16 \log ^3(4) \log ^3(5)\right )}{x+\log (4) \log (5)}\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.12, size = 221, normalized size = 11.05 \begin {gather*} \frac {16-32 x^7+256 x^{10}-64 x^6 \log (4) \log (5)+512 x^9 \log (4) \log (5)+8 \log ^2(4) \log ^2(5)+256 x^8 \log ^2(4) \log ^2(5)-\log ^4(4) \log ^4(5)+512 \log ^5(4) \log ^5(5)-32 \log ^7(4) \log ^7(5)-256 \log ^{10}(4) \log ^{10}(5)+x^4 (1+128 \log (4) \log (5))-32 x^5 \left (-4+\log ^2(4) \log ^2(5)\right )-32 x^2 \log ^3(4) \log ^3(5) \left (-16+\log ^2(4) \log ^2(5)+8 \log ^5(4) \log ^5(5)\right )-2 x \log (4) \log (5) \left (-4+\log ^2(4) \log ^2(5)-512 \log ^3(4) \log ^3(5)+32 \log ^5(4) \log ^5(5)+256 \log ^8(4) \log ^8(5)\right )+x^3 \log (5) \log (16)}{(x+\log (4) \log (5))^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-32 + 2*x^4 + 384*x^5 - 160*x^7 + 2048*x^10 + (-8*x + 6*x^3 + 896*x^4 - 480*x^6 + 6144*x^9)*Log[4]*
Log[5] + (-8 + 6*x^2 + 512*x^3 - 480*x^5 + 6144*x^8)*Log[4]^2*Log[5]^2 + (2*x - 160*x^4 + 2048*x^7)*Log[4]^3*L
og[5]^3)/(x^3 + 3*x^2*Log[4]*Log[5] + 3*x*Log[4]^2*Log[5]^2 + Log[4]^3*Log[5]^3),x]

[Out]

(16 - 32*x^7 + 256*x^10 - 64*x^6*Log[4]*Log[5] + 512*x^9*Log[4]*Log[5] + 8*Log[4]^2*Log[5]^2 + 256*x^8*Log[4]^
2*Log[5]^2 - Log[4]^4*Log[5]^4 + 512*Log[4]^5*Log[5]^5 - 32*Log[4]^7*Log[5]^7 - 256*Log[4]^10*Log[5]^10 + x^4*
(1 + 128*Log[4]*Log[5]) - 32*x^5*(-4 + Log[4]^2*Log[5]^2) - 32*x^2*Log[4]^3*Log[5]^3*(-16 + Log[4]^2*Log[5]^2
+ 8*Log[4]^5*Log[5]^5) - 2*x*Log[4]*Log[5]*(-4 + Log[4]^2*Log[5]^2 - 512*Log[4]^3*Log[5]^3 + 32*Log[4]^5*Log[5
]^5 + 256*Log[4]^8*Log[5]^8) + x^3*Log[5]*Log[16])/(x + Log[4]*Log[5])^2

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fricas [B]  time = 0.67, size = 131, normalized size = 6.55 \begin {gather*} \frac {256 \, x^{10} + 4096 \, \log \relax (5)^{5} \log \relax (2)^{5} + 4096 \, x \log \relax (5)^{4} \log \relax (2)^{4} + 1024 \, x^{2} \log \relax (5)^{3} \log \relax (2)^{3} - 32 \, x^{7} + 128 \, x^{5} + 4 \, {\left (256 \, x^{8} - 32 \, x^{5} + x^{2} + 8\right )} \log \relax (5)^{2} \log \relax (2)^{2} + x^{4} + 4 \, {\left (256 \, x^{9} - 32 \, x^{6} + 64 \, x^{4} + x^{3} + 4 \, x\right )} \log \relax (5) \log \relax (2) + 16}{4 \, \log \relax (5)^{2} \log \relax (2)^{2} + 4 \, x \log \relax (5) \log \relax (2) + x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*(2048*x^7-160*x^4+2*x)*log(2)^3*log(5)^3+4*(6144*x^8-480*x^5+512*x^3+6*x^2-8)*log(2)^2*log(5)^2+2
*(6144*x^9-480*x^6+896*x^4+6*x^3-8*x)*log(2)*log(5)+2048*x^10-160*x^7+384*x^5+2*x^4-32)/(8*log(2)^3*log(5)^3+1
2*x*log(2)^2*log(5)^2+6*x^2*log(2)*log(5)+x^3),x, algorithm="fricas")

[Out]

(256*x^10 + 4096*log(5)^5*log(2)^5 + 4096*x*log(5)^4*log(2)^4 + 1024*x^2*log(5)^3*log(2)^3 - 32*x^7 + 128*x^5
+ 4*(256*x^8 - 32*x^5 + x^2 + 8)*log(5)^2*log(2)^2 + x^4 + 4*(256*x^9 - 32*x^6 + 64*x^4 + x^3 + 4*x)*log(5)*lo
g(2) + 16)/(4*log(5)^2*log(2)^2 + 4*x*log(5)*log(2) + x^2)

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giac [B]  time = 0.29, size = 90, normalized size = 4.50 \begin {gather*} 256 \, x^{8} - 32 \, x^{5} + 512 \, x \log \relax (5)^{2} \log \relax (2)^{2} - 256 \, x^{2} \log \relax (5) \log \relax (2) + 128 \, x^{3} + x^{2} + \frac {16 \, {\left (256 \, \log \relax (5)^{5} \log \relax (2)^{5} + 128 \, x \log \relax (5)^{4} \log \relax (2)^{4} + 2 \, \log \relax (5)^{2} \log \relax (2)^{2} + x \log \relax (5) \log \relax (2) + 1\right )}}{{\left (2 \, \log \relax (5) \log \relax (2) + x\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*(2048*x^7-160*x^4+2*x)*log(2)^3*log(5)^3+4*(6144*x^8-480*x^5+512*x^3+6*x^2-8)*log(2)^2*log(5)^2+2
*(6144*x^9-480*x^6+896*x^4+6*x^3-8*x)*log(2)*log(5)+2048*x^10-160*x^7+384*x^5+2*x^4-32)/(8*log(2)^3*log(5)^3+1
2*x*log(2)^2*log(5)^2+6*x^2*log(2)*log(5)+x^3),x, algorithm="giac")

[Out]

256*x^8 - 32*x^5 + 512*x*log(5)^2*log(2)^2 - 256*x^2*log(5)*log(2) + 128*x^3 + x^2 + 16*(256*log(5)^5*log(2)^5
+ 128*x*log(5)^4*log(2)^4 + 2*log(5)^2*log(2)^2 + x*log(5)*log(2) + 1)/(2*log(5)*log(2) + x)^2

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maple [B]  time = 0.14, size = 80, normalized size = 4.00

method	result	size
default	\(256 x^{8}+512 x \ln \relax (2)^{2} \ln \relax (5)^{2}-32 x^{5}-256 x^{2} \ln \relax (2) \ln \relax (5)+128 x^{3}+x^{2}+\frac {16}{\left (2 \ln \relax (2) \ln \relax (5)+x \right )^{2}}+\frac {16 \ln \relax (2) \ln \relax (5) \left (128 \ln \relax (2)^{3} \ln \relax (5)^{3}+1\right )}{2 \ln \relax (2) \ln \relax (5)+x}\)	\(80\)
risch	\(256 x^{8}+512 x \ln \relax (2)^{2} \ln \relax (5)^{2}-32 x^{5}-256 x^{2} \ln \relax (2) \ln \relax (5)+128 x^{3}+x^{2}+\frac {\left (512 \ln \relax (5)^{4} \ln \relax (2)^{4}+4 \ln \relax (2) \ln \relax (5)\right ) x +1024 \ln \relax (2)^{5} \ln \relax (5)^{5}+8 \ln \relax (2)^{2} \ln \relax (5)^{2}+4}{\ln \relax (2)^{2} \ln \relax (5)^{2}+x \ln \relax (2) \ln \relax (5)+\frac {x^{2}}{4}}\)	\(105\)
norman	\(\frac {\left (256 \ln \relax (2) \ln \relax (5)+1\right ) x^{4}+\left (-128 \ln \relax (2)^{2} \ln \relax (5)^{2}+128\right ) x^{5}+\left (-16 \ln \relax (2)^{3} \ln \relax (5)^{3}+16 \ln \relax (2) \ln \relax (5)\right ) x -32 x^{7}+256 x^{10}+4 \ln \relax (2) \ln \relax (5) x^{3}-128 \ln \relax (5) \ln \relax (2) x^{6}+1024 \ln \relax (5) \ln \relax (2) x^{9}+1024 \ln \relax (5)^{2} \ln \relax (2)^{2} x^{8}+16-16 \ln \relax (5)^{4} \ln \relax (2)^{4}+32 \ln \relax (2)^{2} \ln \relax (5)^{2}}{\left (2 \ln \relax (2) \ln \relax (5)+x \right )^{2}}\)	\(131\)
gosper	\(-\frac {-1024 \ln \relax (5)^{2} \ln \relax (2)^{2} x^{8}-1024 \ln \relax (5) \ln \relax (2) x^{9}-256 x^{10}+128 \ln \relax (5)^{2} \ln \relax (2)^{2} x^{5}+16 \ln \relax (5)^{4} \ln \relax (2)^{4}+128 \ln \relax (5) \ln \relax (2) x^{6}+16 \ln \relax (5)^{3} \ln \relax (2)^{3} x +32 x^{7}-256 x^{4} \ln \relax (5) \ln \relax (2)-4 \ln \relax (2) \ln \relax (5) x^{3}-128 x^{5}-32 \ln \relax (2)^{2} \ln \relax (5)^{2}-x^{4}-16 x \ln \relax (2) \ln \relax (5)-16}{4 \ln \relax (2)^{2} \ln \relax (5)^{2}+4 x \ln \relax (2) \ln \relax (5)+x^{2}}\)	\(148\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((8*(2048*x^7-160*x^4+2*x)*ln(2)^3*ln(5)^3+4*(6144*x^8-480*x^5+512*x^3+6*x^2-8)*ln(2)^2*ln(5)^2+2*(6144*x^9
-480*x^6+896*x^4+6*x^3-8*x)*ln(2)*ln(5)+2048*x^10-160*x^7+384*x^5+2*x^4-32)/(8*ln(2)^3*ln(5)^3+12*x*ln(2)^2*ln
(5)^2+6*x^2*ln(2)*ln(5)+x^3),x,method=_RETURNVERBOSE)

[Out]

256*x^8+512*x*ln(2)^2*ln(5)^2-32*x^5-256*x^2*ln(2)*ln(5)+128*x^3+x^2+16/(2*ln(2)*ln(5)+x)^2+16*ln(2)*ln(5)*(12
8*ln(2)^3*ln(5)^3+1)/(2*ln(2)*ln(5)+x)

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maxima [B]  time = 0.55, size = 105, normalized size = 5.25 \begin {gather*} 256 \, x^{8} - 32 \, x^{5} + 512 \, x \log \relax (5)^{2} \log \relax (2)^{2} - {\left (256 \, \log \relax (5) \log \relax (2) - 1\right )} x^{2} + 128 \, x^{3} + \frac {16 \, {\left (256 \, \log \relax (5)^{5} \log \relax (2)^{5} + 2 \, \log \relax (5)^{2} \log \relax (2)^{2} + {\left (128 \, \log \relax (5)^{4} \log \relax (2)^{4} + \log \relax (5) \log \relax (2)\right )} x + 1\right )}}{4 \, \log \relax (5)^{2} \log \relax (2)^{2} + 4 \, x \log \relax (5) \log \relax (2) + x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*(2048*x^7-160*x^4+2*x)*log(2)^3*log(5)^3+4*(6144*x^8-480*x^5+512*x^3+6*x^2-8)*log(2)^2*log(5)^2+2
*(6144*x^9-480*x^6+896*x^4+6*x^3-8*x)*log(2)*log(5)+2048*x^10-160*x^7+384*x^5+2*x^4-32)/(8*log(2)^3*log(5)^3+1
2*x*log(2)^2*log(5)^2+6*x^2*log(2)*log(5)+x^3),x, algorithm="maxima")

[Out]

256*x^8 - 32*x^5 + 512*x*log(5)^2*log(2)^2 - (256*log(5)*log(2) - 1)*x^2 + 128*x^3 + 16*(256*log(5)^5*log(2)^5
+ 2*log(5)^2*log(2)^2 + (128*log(5)^4*log(2)^4 + log(5)*log(2))*x + 1)/(4*log(5)^2*log(2)^2 + 4*x*log(5)*log(
2) + x^2)

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mupad [B]  time = 1.22, size = 135, normalized size = 6.75 \begin {gather*} \frac {x\,\left (16\,\ln \relax (2)\,\ln \relax (5)+2048\,{\ln \relax (2)}^4\,{\ln \relax (5)}^4\right )+32\,{\ln \relax (2)}^2\,{\ln \relax (5)}^2+4096\,{\ln \relax (2)}^5\,{\ln \relax (5)}^5+16}{x^2+4\,\ln \relax (2)\,\ln \relax (5)\,x+4\,{\ln \relax (2)}^2\,{\ln \relax (5)}^2}+x\,\left (6\,\ln \relax (2)\,\ln \relax (5)\,\left (512\,\ln \relax (2)\,\ln \relax (5)-2\right )-4608\,{\ln \relax (2)}^2\,{\ln \relax (5)}^2+4\,\ln \relax (2)\,\ln \relax (5)\,\left (512\,\ln \relax (2)\,\ln \relax (5)+3\right )\right )-x^2\,\left (256\,\ln \relax (2)\,\ln \relax (5)-1\right )+128\,x^3-32\,x^5+256\,x^8 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x^4 + 384*x^5 - 160*x^7 + 2048*x^10 + 4*log(2)^2*log(5)^2*(6*x^2 + 512*x^3 - 480*x^5 + 6144*x^8 - 8) +
2*log(2)*log(5)*(6*x^3 - 8*x + 896*x^4 - 480*x^6 + 6144*x^9) + 8*log(2)^3*log(5)^3*(2*x - 160*x^4 + 2048*x^7)
- 32)/(x^3 + 8*log(2)^3*log(5)^3 + 6*x^2*log(2)*log(5) + 12*x*log(2)^2*log(5)^2),x)

[Out]

(x*(16*log(2)*log(5) + 2048*log(2)^4*log(5)^4) + 32*log(2)^2*log(5)^2 + 4096*log(2)^5*log(5)^5 + 16)/(x^2 + 4*
log(2)^2*log(5)^2 + 4*x*log(2)*log(5)) + x*(6*log(2)*log(5)*(512*log(2)*log(5) - 2) - 4608*log(2)^2*log(5)^2 +
4*log(2)*log(5)*(512*log(2)*log(5) + 3)) - x^2*(256*log(2)*log(5) - 1) + 128*x^3 - 32*x^5 + 256*x^8

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sympy [B]  time = 0.47, size = 114, normalized size = 5.70 \begin {gather*} 256 x^{8} - 32 x^{5} + 128 x^{3} + x^{2} \left (- 256 \log {\relax (2 )} \log {\relax (5 )} + 1\right ) + 512 x \log {\relax (2 )}^{2} \log {\relax (5 )}^{2} + \frac {x \left (16 \log {\relax (2 )} \log {\relax (5 )} + 2048 \log {\relax (2 )}^{4} \log {\relax (5 )}^{4}\right ) + 16 + 32 \log {\relax (2 )}^{2} \log {\relax (5 )}^{2} + 4096 \log {\relax (2 )}^{5} \log {\relax (5 )}^{5}}{x^{2} + 4 x \log {\relax (2 )} \log {\relax (5 )} + 4 \log {\relax (2 )}^{2} \log {\relax (5 )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*(2048*x**7-160*x**4+2*x)*ln(2)**3*ln(5)**3+4*(6144*x**8-480*x**5+512*x**3+6*x**2-8)*ln(2)**2*ln(5
)**2+2*(6144*x**9-480*x**6+896*x**4+6*x**3-8*x)*ln(2)*ln(5)+2048*x**10-160*x**7+384*x**5+2*x**4-32)/(8*ln(2)**
3*ln(5)**3+12*x*ln(2)**2*ln(5)**2+6*x**2*ln(2)*ln(5)+x**3),x)

[Out]

256*x**8 - 32*x**5 + 128*x**3 + x**2*(-256*log(2)*log(5) + 1) + 512*x*log(2)**2*log(5)**2 + (x*(16*log(2)*log(
5) + 2048*log(2)**4*log(5)**4) + 16 + 32*log(2)**2*log(5)**2 + 4096*log(2)**5*log(5)**5)/(x**2 + 4*x*log(2)*lo
g(5) + 4*log(2)**2*log(5)**2)

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