3.90.51 \(\int \frac {-6640625-516 x^4+(796875+20 x^4) \log (x^2)-31875 \log ^2(x^2)+425 \log ^3(x^2)}{-390625+46875 \log (x^2)-1875 \log ^2(x^2)+25 \log ^3(x^2)} \, dx\)

Optimal. Leaf size=23 \[ 2+x+4 x \left (4+\frac {x^4}{25 \left (-25+\log \left (x^2\right )\right )^2}\right ) \]

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Rubi [A]  time = 0.31, antiderivative size = 21, normalized size of antiderivative = 0.91, number of steps used = 11, number of rules used = 6, integrand size = 62, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {6741, 12, 6742, 2306, 2310, 2178} \begin {gather*} \frac {4 x^5}{25 \left (25-\log \left (x^2\right )\right )^2}+17 x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-6640625 - 516*x^4 + (796875 + 20*x^4)*Log[x^2] - 31875*Log[x^2]^2 + 425*Log[x^2]^3)/(-390625 + 46875*Log
[x^2] - 1875*Log[x^2]^2 + 25*Log[x^2]^3),x]

[Out]

17*x + (4*x^5)/(25*(25 - Log[x^2])^2)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2178

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - (c*f)/d))*ExpIntegral
Ei[(f*g*(c + d*x)*Log[F])/d])/d, x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !$UseGamma === True

Rule 2306

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

Rule 2310

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_)*((d_.)*(x_))^(m_.), x_Symbol] :> Dist[(d*x)^(m + 1)/(d*n*(c*x^n
)^((m + 1)/n)), Subst[Int[E^(((m + 1)*x)/n)*(a + b*x)^p, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, d, m, n, p}
, x]

Rule 6741

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =!= u]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {6640625+516 x^4-\left (796875+20 x^4\right ) \log \left (x^2\right )+31875 \log ^2\left (x^2\right )-425 \log ^3\left (x^2\right )}{25 \left (25-\log \left (x^2\right )\right )^3} \, dx\\ &=\frac {1}{25} \int \frac {6640625+516 x^4-\left (796875+20 x^4\right ) \log \left (x^2\right )+31875 \log ^2\left (x^2\right )-425 \log ^3\left (x^2\right )}{\left (25-\log \left (x^2\right )\right )^3} \, dx\\ &=\frac {1}{25} \int \left (425-\frac {16 x^4}{\left (-25+\log \left (x^2\right )\right )^3}+\frac {20 x^4}{\left (-25+\log \left (x^2\right )\right )^2}\right ) \, dx\\ &=17 x-\frac {16}{25} \int \frac {x^4}{\left (-25+\log \left (x^2\right )\right )^3} \, dx+\frac {4}{5} \int \frac {x^4}{\left (-25+\log \left (x^2\right )\right )^2} \, dx\\ &=17 x+\frac {4 x^5}{25 \left (25-\log \left (x^2\right )\right )^2}+\frac {2 x^5}{5 \left (25-\log \left (x^2\right )\right )}-\frac {4}{5} \int \frac {x^4}{\left (-25+\log \left (x^2\right )\right )^2} \, dx+2 \int \frac {x^4}{-25+\log \left (x^2\right )} \, dx\\ &=17 x+\frac {4 x^5}{25 \left (25-\log \left (x^2\right )\right )^2}-2 \int \frac {x^4}{-25+\log \left (x^2\right )} \, dx+\frac {x^5 \operatorname {Subst}\left (\int \frac {e^{5 x/2}}{-25+x} \, dx,x,\log \left (x^2\right )\right )}{\left (x^2\right )^{5/2}}\\ &=17 x+\frac {e^{125/2} x^5 \text {Ei}\left (-\frac {5}{2} \left (25-\log \left (x^2\right )\right )\right )}{\left (x^2\right )^{5/2}}+\frac {4 x^5}{25 \left (25-\log \left (x^2\right )\right )^2}-\frac {x^5 \operatorname {Subst}\left (\int \frac {e^{5 x/2}}{-25+x} \, dx,x,\log \left (x^2\right )\right )}{\left (x^2\right )^{5/2}}\\ &=17 x+\frac {4 x^5}{25 \left (25-\log \left (x^2\right )\right )^2}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.15, size = 21, normalized size = 0.91 \begin {gather*} \frac {1}{25} \left (425 x+\frac {4 x^5}{\left (-25+\log \left (x^2\right )\right )^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-6640625 - 516*x^4 + (796875 + 20*x^4)*Log[x^2] - 31875*Log[x^2]^2 + 425*Log[x^2]^3)/(-390625 + 468
75*Log[x^2] - 1875*Log[x^2]^2 + 25*Log[x^2]^3),x]

[Out]

(425*x + (4*x^5)/(-25 + Log[x^2])^2)/25

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fricas [B]  time = 0.49, size = 43, normalized size = 1.87 \begin {gather*} \frac {4 \, x^{5} + 425 \, x \log \left (x^{2}\right )^{2} - 21250 \, x \log \left (x^{2}\right ) + 265625 \, x}{25 \, {\left (\log \left (x^{2}\right )^{2} - 50 \, \log \left (x^{2}\right ) + 625\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((425*log(x^2)^3-31875*log(x^2)^2+(20*x^4+796875)*log(x^2)-516*x^4-6640625)/(25*log(x^2)^3-1875*log(x
^2)^2+46875*log(x^2)-390625),x, algorithm="fricas")

[Out]

1/25*(4*x^5 + 425*x*log(x^2)^2 - 21250*x*log(x^2) + 265625*x)/(log(x^2)^2 - 50*log(x^2) + 625)

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giac [A]  time = 0.22, size = 25, normalized size = 1.09 \begin {gather*} \frac {4 \, x^{5}}{25 \, {\left (\log \left (x^{2}\right )^{2} - 50 \, \log \left (x^{2}\right ) + 625\right )}} + 17 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((425*log(x^2)^3-31875*log(x^2)^2+(20*x^4+796875)*log(x^2)-516*x^4-6640625)/(25*log(x^2)^3-1875*log(x
^2)^2+46875*log(x^2)-390625),x, algorithm="giac")

[Out]

4/25*x^5/(log(x^2)^2 - 50*log(x^2) + 625) + 17*x

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maple [A]  time = 0.04, size = 18, normalized size = 0.78

method	result	size
risch	\(17 x +\frac {4 x^{5}}{25 \left (\ln \left (x^{2}\right )-25\right )^{2}}\)	\(18\)
norman	\(\frac {10625 x +\frac {4 x^{5}}{25}-850 x \ln \left (x^{2}\right )+17 x \ln \left (x^{2}\right )^{2}}{\left (\ln \left (x^{2}\right )-25\right )^{2}}\)	\(35\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((425*ln(x^2)^3-31875*ln(x^2)^2+(20*x^4+796875)*ln(x^2)-516*x^4-6640625)/(25*ln(x^2)^3-1875*ln(x^2)^2+46875
*ln(x^2)-390625),x,method=_RETURNVERBOSE)

[Out]

17*x+4/25*x^5/(ln(x^2)-25)^2

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maxima [A]  time = 0.36, size = 37, normalized size = 1.61 \begin {gather*} \frac {4 \, x^{5} + 1700 \, x \log \relax (x)^{2} - 42500 \, x \log \relax (x) + 265625 \, x}{25 \, {\left (4 \, \log \relax (x)^{2} - 100 \, \log \relax (x) + 625\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((425*log(x^2)^3-31875*log(x^2)^2+(20*x^4+796875)*log(x^2)-516*x^4-6640625)/(25*log(x^2)^3-1875*log(x
^2)^2+46875*log(x^2)-390625),x, algorithm="maxima")

[Out]

1/25*(4*x^5 + 1700*x*log(x)^2 - 42500*x*log(x) + 265625*x)/(4*log(x)^2 - 100*log(x) + 625)

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mupad [B]  time = 7.74, size = 28, normalized size = 1.22 \begin {gather*} 17\,x-\frac {10625\,x-\frac {x\,\left (4\,x^4+265625\right )}{25}}{{\left (\ln \left (x^2\right )-25\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(31875*log(x^2)^2 - log(x^2)*(20*x^4 + 796875) - 425*log(x^2)^3 + 516*x^4 + 6640625)/(46875*log(x^2) - 18
75*log(x^2)^2 + 25*log(x^2)^3 - 390625),x)

[Out]

17*x - (10625*x - (x*(4*x^4 + 265625))/25)/(log(x^2) - 25)^2

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sympy [A]  time = 0.13, size = 24, normalized size = 1.04 \begin {gather*} \frac {4 x^{5}}{25 \log {\left (x^{2} \right )}^{2} - 1250 \log {\left (x^{2} \right )} + 15625} + 17 x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((425*ln(x**2)**3-31875*ln(x**2)**2+(20*x**4+796875)*ln(x**2)-516*x**4-6640625)/(25*ln(x**2)**3-1875*
ln(x**2)**2+46875*ln(x**2)-390625),x)

[Out]

4*x**5/(25*log(x**2)**2 - 1250*log(x**2) + 15625) + 17*x

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