3.9.74 \(\int \frac {2 x^3+3 x^8 \log (5)+(8 x-36 x^6 \log (5)) \log (16)+144 x^4 \log (5) \log ^2(16)-192 x^2 \log (5) \log ^3(16)}{-x^6 \log (5)+12 x^4 \log (5) \log (16)-48 x^2 \log (5) \log ^2(16)+64 \log (5) \log ^3(16)} \, dx\)

Optimal. Leaf size=27 \[ x^2 \left (-x+\frac {1}{x^2 \log (5) \left (x-\frac {4 \log (16)}{x}\right )^2}\right ) \]

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Rubi [A]  time = 0.16, antiderivative size = 39, normalized size of antiderivative = 1.44, number of steps used = 4, number of rules used = 2, integrand size = 88, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.023, Rules used = {2073, 261} \begin {gather*} -x^3+\frac {1}{\log (5) \left (x^2-4 \log (16)\right )}+\frac {4 \log (16)}{\log (5) \left (x^2-4 \log (16)\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2*x^3 + 3*x^8*Log[5] + (8*x - 36*x^6*Log[5])*Log[16] + 144*x^4*Log[5]*Log[16]^2 - 192*x^2*Log[5]*Log[16]^
3)/(-(x^6*Log[5]) + 12*x^4*Log[5]*Log[16] - 48*x^2*Log[5]*Log[16]^2 + 64*Log[5]*Log[16]^3),x]

[Out]

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

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rule 2073

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P /. x -> Sqrt[x]]}, Int[ExpandIntegrand[(PP /. x ->
x^2)^p*Q^q, x], x] /;  !SumQ[NonfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x^2] && PolyQ[Q, x] && ILtQ[p,
0]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-3 x^2-\frac {2 x}{\log (5) \left (x^2-4 \log (16)\right )^2}-\frac {16 x \log (16)}{\log (5) \left (x^2-4 \log (16)\right )^3}\right ) \, dx\\ &=-x^3-\frac {2 \int \frac {x}{\left (x^2-4 \log (16)\right )^2} \, dx}{\log (5)}-\frac {(16 \log (16)) \int \frac {x}{\left (x^2-4 \log (16)\right )^3} \, dx}{\log (5)}\\ &=-x^3+\frac {1}{\log (5) \left (x^2-4 \log (16)\right )}+\frac {4 \log (16)}{\log (5) \left (x^2-4 \log (16)\right )^2}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 35, normalized size = 1.30 \begin {gather*} -\frac {x^2 \left (-1+x \log (5) \left (x^2-4 \log (16)\right )^2\right )}{\log (5) \left (x^2-4 \log (16)\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2*x^3 + 3*x^8*Log[5] + (8*x - 36*x^6*Log[5])*Log[16] + 144*x^4*Log[5]*Log[16]^2 - 192*x^2*Log[5]*Lo
g[16]^3)/(-(x^6*Log[5]) + 12*x^4*Log[5]*Log[16] - 48*x^2*Log[5]*Log[16]^2 + 64*Log[5]*Log[16]^3),x]

[Out]

-((x^2*(-1 + x*Log[5]*(x^2 - 4*Log[16])^2))/(Log[5]*(x^2 - 4*Log[16])^2))

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fricas [A]  time = 0.50, size = 52, normalized size = 1.93 \begin {gather*} \frac {x^{2} - {\left (x^{7} - 32 \, x^{5} \log \relax (2) + 256 \, x^{3} \log \relax (2)^{2}\right )} \log \relax (5)}{{\left (x^{4} - 32 \, x^{2} \log \relax (2) + 256 \, \log \relax (2)^{2}\right )} \log \relax (5)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-12288*x^2*log(5)*log(2)^3+2304*x^4*log(5)*log(2)^2+4*(-36*x^6*log(5)+8*x)*log(2)+3*x^8*log(5)+2*x^
3)/(4096*log(5)*log(2)^3-768*x^2*log(5)*log(2)^2+48*x^4*log(5)*log(2)-x^6*log(5)),x, algorithm="fricas")

[Out]

(x^2 - (x^7 - 32*x^5*log(2) + 256*x^3*log(2)^2)*log(5))/((x^4 - 32*x^2*log(2) + 256*log(2)^2)*log(5))

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giac [A]  time = 0.37, size = 24, normalized size = 0.89 \begin {gather*} -x^{3} + \frac {x^{2}}{{\left (x^{2} - 16 \, \log \relax (2)\right )}^{2} \log \relax (5)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-12288*x^2*log(5)*log(2)^3+2304*x^4*log(5)*log(2)^2+4*(-36*x^6*log(5)+8*x)*log(2)+3*x^8*log(5)+2*x^
3)/(4096*log(5)*log(2)^3-768*x^2*log(5)*log(2)^2+48*x^4*log(5)*log(2)-x^6*log(5)),x, algorithm="giac")

[Out]

-x^3 + x^2/((x^2 - 16*log(2))^2*log(5))

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maple [A]  time = 0.10, size = 34, normalized size = 1.26

method	result	size
risch	\(-x^{3}+\frac {x^{2}}{\ln \relax (5) \left (x^{4}-32 x^{2} \ln \relax (2)+256 \ln \relax (2)^{2}\right )}\)	\(34\)
default	\(\frac {-x^{3} \ln \relax (5)+\frac {1}{-16 \ln \relax (2)+x^{2}}+\frac {16 \ln \relax (2)}{\left (-16 \ln \relax (2)+x^{2}\right )^{2}}}{\ln \relax (5)}\)	\(38\)
norman	\(\frac {\frac {x^{2}}{\ln \relax (5)}-x^{7}-256 x^{3} \ln \relax (2)^{2}+32 x^{5} \ln \relax (2)}{\left (16 \ln \relax (2)-x^{2}\right )^{2}}\)	\(44\)
gosper	\(-\frac {x^{2} \left (x^{5} \ln \relax (5)-32 x^{3} \ln \relax (5) \ln \relax (2)+256 x \ln \relax (2)^{2} \ln \relax (5)-1\right )}{\ln \relax (5) \left (x^{4}-32 x^{2} \ln \relax (2)+256 \ln \relax (2)^{2}\right )}\)	\(55\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-12288*x^2*ln(5)*ln(2)^3+2304*x^4*ln(5)*ln(2)^2+4*(-36*x^6*ln(5)+8*x)*ln(2)+3*x^8*ln(5)+2*x^3)/(4096*ln(5
)*ln(2)^3-768*x^2*ln(5)*ln(2)^2+48*x^4*ln(5)*ln(2)-x^6*ln(5)),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.41, size = 36, normalized size = 1.33 \begin {gather*} -x^{3} + \frac {x^{2}}{x^{4} \log \relax (5) - 32 \, x^{2} \log \relax (5) \log \relax (2) + 256 \, \log \relax (5) \log \relax (2)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-12288*x^2*log(5)*log(2)^3+2304*x^4*log(5)*log(2)^2+4*(-36*x^6*log(5)+8*x)*log(2)+3*x^8*log(5)+2*x^
3)/(4096*log(5)*log(2)^3-768*x^2*log(5)*log(2)^2+48*x^4*log(5)*log(2)-x^6*log(5)),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.25, size = 26, normalized size = 0.96 \begin {gather*} \frac {x^2}{\ln \relax (5)\,{\left (16\,\ln \relax (2)-x^2\right )}^2}-x^3 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x^8*log(5) + 4*log(2)*(8*x - 36*x^6*log(5)) + 2*x^3 - 12288*x^2*log(2)^3*log(5) + 2304*x^4*log(2)^2*log
(5))/(4096*log(2)^3*log(5) - x^6*log(5) + 48*x^4*log(2)*log(5) - 768*x^2*log(2)^2*log(5)),x)

[Out]

x^2/(log(5)*(16*log(2) - x^2)^2) - x^3

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sympy [A]  time = 0.29, size = 34, normalized size = 1.26 \begin {gather*} - x^{3} + \frac {x^{2}}{x^{4} \log {\relax (5 )} - 32 x^{2} \log {\relax (2 )} \log {\relax (5 )} + 256 \log {\relax (2 )}^{2} \log {\relax (5 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-12288*x**2*ln(5)*ln(2)**3+2304*x**4*ln(5)*ln(2)**2+4*(-36*x**6*ln(5)+8*x)*ln(2)+3*x**8*ln(5)+2*x**
3)/(4096*ln(5)*ln(2)**3-768*x**2*ln(5)*ln(2)**2+48*x**4*ln(5)*ln(2)-x**6*ln(5)),x)

[Out]

-x**3 + x**2/(x**4*log(5) - 32*x**2*log(2)*log(5) + 256*log(2)**2*log(5))

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