3.9.74 \(\int \frac {2 x^3+3 x^8 \log (5)+(8 x-36 x^6 \log (5)) \log (16)+144 x^4 \log (5) \log ^2(16)-192 x^2 \log (5) \log ^3(16)}{-x^6 \log (5)+12 x^4 \log (5) \log (16)-48 x^2 \log (5) \log ^2(16)+64 \log (5) \log ^3(16)} \, dx\) [874]

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

[Out]

x^2*(1/x^2/(x-16*ln(2)/x)^2/ln(5)-x)

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Rubi [A]
time = 0.09, 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 = {2098, 267} \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 267

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 2098

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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Maple [A]
time = 0.15, size = 38, normalized size = 1.41

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

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]

1/ln(5)*(-x^3*ln(5)+1/(-16*ln(2)+x^2)+16*ln(2)/(-16*ln(2)+x^2)^2)

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Maxima [A]
time = 0.27, size = 36, normalized size = 1.33 \begin {gather*} -x^{3} + \frac {x^{2}}{x^{4} \log \left (5\right ) - 32 \, x^{2} \log \left (5\right ) \log \left (2\right ) + 256 \, \log \left (5\right ) \log \left (2\right )^{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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Fricas [A]
time = 0.34, size = 52, normalized size = 1.93 \begin {gather*} \frac {x^{2} - {\left (x^{7} - 32 \, x^{5} \log \left (2\right ) + 256 \, x^{3} \log \left (2\right )^{2}\right )} \log \left (5\right )}{{\left (x^{4} - 32 \, x^{2} \log \left (2\right ) + 256 \, \log \left (2\right )^{2}\right )} \log \left (5\right )} \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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Sympy [A]
time = 0.15, size = 34, normalized size = 1.26 \begin {gather*} - x^{3} + \frac {x^{2}}{x^{4} \log {\left (5 \right )} - 32 x^{2} \log {\left (2 \right )} \log {\left (5 \right )} + 256 \log {\left (2 \right )}^{2} \log {\left (5 \right )}} \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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Giac [A]
time = 0.41, size = 24, normalized size = 0.89 \begin {gather*} -x^{3} + \frac {x^{2}}{{\left (x^{2} - 16 \, \log \left (2\right )\right )}^{2} \log \left (5\right )} \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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Mupad [B]
time = 0.25, size = 26, normalized size = 0.96 \begin {gather*} \frac {x^2}{\ln \left (5\right )\,{\left (16\,\ln \left (2\right )-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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