3.143 \(\int \frac {1}{x^3 (a^5+x^5)} \, dx\)

Optimal. Leaf size=211 \[ -\frac {\log (a+x)}{5 a^7}-\frac {\sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \tan ^{-1}\left (\frac {\left (1-\sqrt {5}\right ) a-4 x}{\sqrt {2 \left (5+\sqrt {5}\right )} a}\right )}{5 a^7}+\frac {\sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \tan ^{-1}\left (\frac {\sqrt {\frac {1}{10} \left (5+\sqrt {5}\right )} \left (\left (1+\sqrt {5}\right ) a-4 x\right )}{2 a}\right )}{5 a^7}-\frac {1}{2 a^5 x^2}+\frac {\left (1+\sqrt {5}\right ) \log \left (a^2-\frac {1}{2} \left (1-\sqrt {5}\right ) a x+x^2\right )}{20 a^7}+\frac {\left (1-\sqrt {5}\right ) \log \left (a^2-\frac {1}{2} \left (1+\sqrt {5}\right ) a x+x^2\right )}{20 a^7} \]

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Rubi [A]  time = 0.33, antiderivative size = 211, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {325, 293, 634, 618, 204, 628, 31} \[ -\frac {1}{2 a^5 x^2}+\frac {\left (1+\sqrt {5}\right ) \log \left (a^2-\frac {1}{2} \left (1-\sqrt {5}\right ) a x+x^2\right )}{20 a^7}+\frac {\left (1-\sqrt {5}\right ) \log \left (a^2-\frac {1}{2} \left (1+\sqrt {5}\right ) a x+x^2\right )}{20 a^7}-\frac {\log (a+x)}{5 a^7}-\frac {\sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \tan ^{-1}\left (\frac {\left (1-\sqrt {5}\right ) a-4 x}{\sqrt {2 \left (5+\sqrt {5}\right )} a}\right )}{5 a^7}+\frac {\sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \tan ^{-1}\left (\frac {\sqrt {\frac {1}{10} \left (5+\sqrt {5}\right )} \left (\left (1+\sqrt {5}\right ) a-4 x\right )}{2 a}\right )}{5 a^7} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^3*(a^5 + x^5)),x]

[Out]

-1/(2*a^5*x^2) - (Sqrt[(5 - Sqrt[5])/2]*ArcTan[((1 - Sqrt[5])*a - 4*x)/(Sqrt[2*(5 + Sqrt[5])]*a)])/(5*a^7) + (
Sqrt[(5 + Sqrt[5])/2]*ArcTan[(Sqrt[(5 + Sqrt[5])/10]*((1 + Sqrt[5])*a - 4*x))/(2*a)])/(5*a^7) - Log[a + x]/(5*
a^7) + ((1 + Sqrt[5])*Log[a^2 - ((1 - Sqrt[5])*a*x)/2 + x^2])/(20*a^7) + ((1 - Sqrt[5])*Log[a^2 - ((1 + Sqrt[5
])*a*x)/2 + x^2])/(20*a^7)

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 293

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Module[{r = Numerator[Rt[a/b, n]], s = Denominator[Rt[a/
b, n]], k, u}, Simp[u = Int[(r*Cos[((2*k - 1)*m*Pi)/n] - s*Cos[((2*k - 1)*(m + 1)*Pi)/n]*x)/(r^2 - 2*r*s*Cos[(
(2*k - 1)*Pi)/n]*x + s^2*x^2), x]; -(((-r)^(m + 1)*Int[1/(r + s*x), x])/(a*n*s^m)) + Dist[(2*r^(m + 1))/(a*n*s
^m), Sum[u, {k, 1, (n - 1)/2}], x], x]] /; FreeQ[{a, b}, x] && IGtQ[(n - 1)/2, 0] && IGtQ[m, 0] && LtQ[m, n -
1] && PosQ[a/b]

Rule 325

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*
c*(m + 1)), x] - Dist[(b*(m + n*(p + 1) + 1))/(a*c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rubi steps

\begin {align*} \int \frac {1}{x^3 \left (a^5+x^5\right )} \, dx &=-\frac {1}{2 a^5 x^2}-\frac {\int \frac {x^2}{a^5+x^5} \, dx}{a^5}\\ &=-\frac {1}{2 a^5 x^2}-\frac {\int \frac {1}{a+x} \, dx}{5 a^7}-\frac {2 \int \frac {\frac {1}{4} \left (-1-\sqrt {5}\right ) a-\frac {1}{4} \left (1+\sqrt {5}\right ) x}{a^2-\frac {1}{2} \left (1-\sqrt {5}\right ) a x+x^2} \, dx}{5 a^7}-\frac {2 \int \frac {\frac {1}{4} \left (-1+\sqrt {5}\right ) a-\frac {1}{4} \left (1-\sqrt {5}\right ) x}{a^2-\frac {1}{2} \left (1+\sqrt {5}\right ) a x+x^2} \, dx}{5 a^7}\\ &=-\frac {1}{2 a^5 x^2}-\frac {\log (a+x)}{5 a^7}+\frac {\left (1-\sqrt {5}\right ) \int \frac {-\frac {1}{2} \left (1+\sqrt {5}\right ) a+2 x}{a^2-\frac {1}{2} \left (1+\sqrt {5}\right ) a x+x^2} \, dx}{20 a^7}+\frac {\left (1+\sqrt {5}\right ) \int \frac {-\frac {1}{2} \left (1-\sqrt {5}\right ) a+2 x}{a^2-\frac {1}{2} \left (1-\sqrt {5}\right ) a x+x^2} \, dx}{20 a^7}+\frac {\int \frac {1}{a^2-\frac {1}{2} \left (1-\sqrt {5}\right ) a x+x^2} \, dx}{2 \sqrt {5} a^6}-\frac {\int \frac {1}{a^2-\frac {1}{2} \left (1+\sqrt {5}\right ) a x+x^2} \, dx}{2 \sqrt {5} a^6}\\ &=-\frac {1}{2 a^5 x^2}-\frac {\log (a+x)}{5 a^7}+\frac {\left (1-\sqrt {5}\right ) \log \left (2 a^2-a x-\sqrt {5} a x+2 x^2\right )}{20 a^7}+\frac {\left (1+\sqrt {5}\right ) \log \left (2 a^2-a x+\sqrt {5} a x+2 x^2\right )}{20 a^7}+\frac {\operatorname {Subst}\left (\int \frac {1}{-\frac {1}{2} \left (5-\sqrt {5}\right ) a^2-x^2} \, dx,x,-\frac {1}{2} \left (1+\sqrt {5}\right ) a+2 x\right )}{\sqrt {5} a^6}-\frac {\operatorname {Subst}\left (\int \frac {1}{-\frac {1}{2} \left (5+\sqrt {5}\right ) a^2-x^2} \, dx,x,-\frac {1}{2} \left (1-\sqrt {5}\right ) a+2 x\right )}{\sqrt {5} a^6}\\ &=-\frac {1}{2 a^5 x^2}-\frac {\sqrt {\frac {2}{5 \left (5+\sqrt {5}\right )}} \tan ^{-1}\left (\frac {\left (1-\sqrt {5}\right ) a-4 x}{\sqrt {2 \left (5+\sqrt {5}\right )} a}\right )}{a^7}+\frac {\sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \tan ^{-1}\left (\frac {\sqrt {\frac {1}{10} \left (5+\sqrt {5}\right )} \left (\left (1+\sqrt {5}\right ) a-4 x\right )}{2 a}\right )}{5 a^7}-\frac {\log (a+x)}{5 a^7}+\frac {\left (1-\sqrt {5}\right ) \log \left (2 a^2-a x-\sqrt {5} a x+2 x^2\right )}{20 a^7}+\frac {\left (1+\sqrt {5}\right ) \log \left (2 a^2-a x+\sqrt {5} a x+2 x^2\right )}{20 a^7}\\ \end {align*}

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Mathematica [A]  time = 0.17, size = 174, normalized size = 0.82 \[ -\frac {\frac {10 a^2}{x^2}-\left (1+\sqrt {5}\right ) \log \left (a^2+\frac {1}{2} \left (\sqrt {5}-1\right ) a x+x^2\right )+\left (\sqrt {5}-1\right ) \log \left (a^2-\frac {1}{2} \left (1+\sqrt {5}\right ) a x+x^2\right )+4 \log (a+x)-2 \sqrt {10-2 \sqrt {5}} \tan ^{-1}\left (\frac {\left (\sqrt {5}-1\right ) a+4 x}{\sqrt {2 \left (5+\sqrt {5}\right )} a}\right )+2 \sqrt {2 \left (5+\sqrt {5}\right )} \tan ^{-1}\left (\frac {4 x-\left (1+\sqrt {5}\right ) a}{\sqrt {10-2 \sqrt {5}} a}\right )}{20 a^7} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/(x^3*(a^5 + x^5)),x]

[Out]

-1/20*((10*a^2)/x^2 - 2*Sqrt[10 - 2*Sqrt[5]]*ArcTan[((-1 + Sqrt[5])*a + 4*x)/(Sqrt[2*(5 + Sqrt[5])]*a)] + 2*Sq
rt[2*(5 + Sqrt[5])]*ArcTan[(-((1 + Sqrt[5])*a) + 4*x)/(Sqrt[10 - 2*Sqrt[5]]*a)] + 4*Log[a + x] - (1 + Sqrt[5])
*Log[a^2 + ((-1 + Sqrt[5])*a*x)/2 + x^2] + (-1 + Sqrt[5])*Log[a^2 - ((1 + Sqrt[5])*a*x)/2 + x^2])/a^7

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IntegrateAlgebraic [A]  time = 0.23, size = 270, normalized size = 1.28 \[ -\frac {\log (a+x)}{5 a^7}+\frac {\sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \tan ^{-1}\left (\frac {\sqrt {2-\frac {2}{\sqrt {5}}} x}{a}-\frac {1}{2} \sqrt {\frac {1}{10} \left (5-\sqrt {5}\right )}+\frac {1}{2} \sqrt {\frac {1}{10} \left (25-5 \sqrt {5}\right )}\right )}{5 a^7}+\frac {\sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \tan ^{-1}\left (-\frac {\sqrt {2+\frac {2}{\sqrt {5}}} x}{a}+\frac {1}{2} \sqrt {\frac {1}{10} \left (25+5 \sqrt {5}\right )}+\frac {1}{2} \sqrt {\frac {1}{10} \left (5+\sqrt {5}\right )}\right )}{5 a^7}-\frac {1}{2 a^5 x^2}+\frac {\left (1-\sqrt {5}\right ) \log \left (2 a^2-\sqrt {5} a x-a x+2 x^2\right )}{20 a^7}+\frac {\left (1+\sqrt {5}\right ) \log \left (2 a^2+\sqrt {5} a x-a x+2 x^2\right )}{20 a^7} \]

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/(x^3*(a^5 + x^5)),x]

[Out]

-1/2*1/(a^5*x^2) + (Sqrt[(5 - Sqrt[5])/2]*ArcTan[Sqrt[(25 - 5*Sqrt[5])/10]/2 - Sqrt[(5 - Sqrt[5])/10]/2 + (Sqr
t[2 - 2/Sqrt[5]]*x)/a])/(5*a^7) + (Sqrt[(5 + Sqrt[5])/2]*ArcTan[Sqrt[(5 + Sqrt[5])/10]/2 + Sqrt[(25 + 5*Sqrt[5
])/10]/2 - (Sqrt[2 + 2/Sqrt[5]]*x)/a])/(5*a^7) - Log[a + x]/(5*a^7) + ((1 - Sqrt[5])*Log[2*a^2 - a*x - Sqrt[5]
*a*x + 2*x^2])/(20*a^7) + ((1 + Sqrt[5])*Log[2*a^2 - a*x + Sqrt[5]*a*x + 2*x^2])/(20*a^7)

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(a^5+x^5),x, algorithm="fricas")

[Out]

Timed out

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giac [A]  time = 0.99, size = 185, normalized size = 0.88 \[ \frac {\sqrt {-2 \, \sqrt {5} + 10} \arctan \left (\frac {a {\left (\sqrt {5} - 1\right )} + 4 \, x}{a \sqrt {2 \, \sqrt {5} + 10}}\right )}{10 \, a^{7}} - \frac {\sqrt {2 \, \sqrt {5} + 10} \arctan \left (-\frac {a {\left (\sqrt {5} + 1\right )} - 4 \, x}{a \sqrt {-2 \, \sqrt {5} + 10}}\right )}{10 \, a^{7}} - \frac {\sqrt {5} \log \left (a^{2} - \frac {1}{2} \, {\left (\sqrt {5} a + a\right )} x + x^{2}\right )}{20 \, a^{7}} + \frac {\sqrt {5} \log \left (a^{2} + \frac {1}{2} \, {\left (\sqrt {5} a - a\right )} x + x^{2}\right )}{20 \, a^{7}} + \frac {\log \left ({\left | a^{4} - a^{3} x + a^{2} x^{2} - a x^{3} + x^{4} \right |}\right )}{20 \, a^{7}} - \frac {\log \left ({\left | a + x \right |}\right )}{5 \, a^{7}} - \frac {1}{2 \, a^{5} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(a^5+x^5),x, algorithm="giac")

[Out]

1/10*sqrt(-2*sqrt(5) + 10)*arctan((a*(sqrt(5) - 1) + 4*x)/(a*sqrt(2*sqrt(5) + 10)))/a^7 - 1/10*sqrt(2*sqrt(5)
+ 10)*arctan(-(a*(sqrt(5) + 1) - 4*x)/(a*sqrt(-2*sqrt(5) + 10)))/a^7 - 1/20*sqrt(5)*log(a^2 - 1/2*(sqrt(5)*a +
a)*x + x^2)/a^7 + 1/20*sqrt(5)*log(a^2 + 1/2*(sqrt(5)*a - a)*x + x^2)/a^7 + 1/20*log(abs(a^4 - a^3*x + a^2*x^
2 - a*x^3 + x^4))/a^7 - 1/5*log(abs(a + x))/a^7 - 1/2/(a^5*x^2)

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maple [C]  time = 0.35, size = 78, normalized size = 0.37

method	result	size
risch	\(-\frac {1}{2 a^{5} x^{2}}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (a^{28} \textit {\_Z}^{4}-a^{21} \textit {\_Z}^{3}+a^{14} \textit {\_Z}^{2}-a^{7} \textit {\_Z} +1\right )}{\sum }\textit {\_R} \ln \left (\left (-6 \textit {\_R}^{5} a^{35}-5\right ) x +a^{15} \textit {\_R}^{2}\right )\right )}{5}-\frac {\ln \left (a +x \right )}{5 a^{7}}\)	\(78\)
default	\(-\frac {\ln \left (a +x \right )}{5 a^{7}}-\frac {1}{2 a^{5} x^{2}}+\frac {\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}-a \,\textit {\_Z}^{3}+\textit {\_Z}^{2} a^{2}-a^{3} \textit {\_Z} +a^{4}\right )}{\sum }\frac {\left (\textit {\_R}^{3}-2 \textit {\_R}^{2} a -2 \textit {\_R} \,a^{2}+a^{3}\right ) \ln \left (x -\textit {\_R} \right )}{4 \textit {\_R}^{3}-3 \textit {\_R}^{2} a +2 \textit {\_R} \,a^{2}-a^{3}}}{5 a^{7}}\)	\(105\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^3/(a^5+x^5),x,method=_RETURNVERBOSE)

[Out]

-1/2/a^5/x^2+1/5*sum(_R*ln((-6*_R^5*a^35-5)*x+a^15*_R^2),_R=RootOf(_Z^4*a^28-_Z^3*a^21+_Z^2*a^14-_Z*a^7+1))-1/
5*ln(a+x)/a^7

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maxima [A]  time = 0.98, size = 173, normalized size = 0.82 \[ \frac {\frac {2 \, \sqrt {5} \arctan \left (\frac {a {\left (\sqrt {5} - 1\right )} + 4 \, x}{a \sqrt {2 \, \sqrt {5} + 10}}\right )}{a^{2} \sqrt {2 \, \sqrt {5} + 10}} - \frac {2 \, \sqrt {5} \arctan \left (-\frac {a {\left (\sqrt {5} + 1\right )} - 4 \, x}{a \sqrt {-2 \, \sqrt {5} + 10}}\right )}{a^{2} \sqrt {-2 \, \sqrt {5} + 10}} - \frac {\log \left (a + x\right )}{a^{2}} - \frac {\log \left (-a x {\left (\sqrt {5} + 1\right )} + 2 \, a^{2} + 2 \, x^{2}\right )}{a^{2} {\left (\sqrt {5} + 1\right )}} + \frac {\log \left (a x {\left (\sqrt {5} - 1\right )} + 2 \, a^{2} + 2 \, x^{2}\right )}{a^{2} {\left (\sqrt {5} - 1\right )}}}{5 \, a^{5}} - \frac {1}{2 \, a^{5} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(a^5+x^5),x, algorithm="maxima")

[Out]

1/5*(2*sqrt(5)*arctan((a*(sqrt(5) - 1) + 4*x)/(a*sqrt(2*sqrt(5) + 10)))/(a^2*sqrt(2*sqrt(5) + 10)) - 2*sqrt(5)
*arctan(-(a*(sqrt(5) + 1) - 4*x)/(a*sqrt(-2*sqrt(5) + 10)))/(a^2*sqrt(-2*sqrt(5) + 10)) - log(a + x)/a^2 - log
(-a*x*(sqrt(5) + 1) + 2*a^2 + 2*x^2)/(a^2*(sqrt(5) + 1)) + log(a*x*(sqrt(5) - 1) + 2*a^2 + 2*x^2)/(a^2*(sqrt(5
) - 1)))/a^5 - 1/2/(a^5*x^2)

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mupad [B]  time = 0.78, size = 210, normalized size = 1.00 \[ \frac {\ln \left (a^{20}-\frac {a^{19}\,x\,{\left (\sqrt {5}+\sqrt {2\,\sqrt {5}-10}+1\right )}^3}{64}\right )\,\left (\sqrt {5}+\sqrt {2\,\sqrt {5}-10}+1\right )}{20\,a^7}-\frac {1}{2\,a^5\,x^2}-\frac {\ln \left (a^{20}+\frac {x\,{\left (\sqrt {5}+\sqrt {-2\,\sqrt {5}-10}-1\right )}^3\,a^{19}}{64}\right )\,\left (\sqrt {5}+\sqrt {-2\,\sqrt {5}-10}-1\right )}{20\,a^7}-\frac {\ln \left (a+x\right )}{5\,a^7}+\frac {\ln \left (a^{20}-\frac {a^{19}\,x\,{\left (\sqrt {5}-\sqrt {2\,\sqrt {5}-10}+1\right )}^3}{64}\right )\,\left (\sqrt {5}-\sqrt {2\,\sqrt {5}-10}+1\right )}{20\,a^7}+\frac {\ln \left (a^{20}-\frac {a^{19}\,x\,{\left (\sqrt {-2\,\sqrt {5}-10}-\sqrt {5}+1\right )}^3}{64}\right )\,\left (\sqrt {-2\,\sqrt {5}-10}-\sqrt {5}+1\right )}{20\,a^7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^3*(a^5 + x^5)),x)

[Out]

(log(a^20 - (a^19*x*(5^(1/2) + (2*5^(1/2) - 10)^(1/2) + 1)^3)/64)*(5^(1/2) + (2*5^(1/2) - 10)^(1/2) + 1))/(20*
a^7) - 1/(2*a^5*x^2) - (log(a^20 + (a^19*x*(5^(1/2) + (- 2*5^(1/2) - 10)^(1/2) - 1)^3)/64)*(5^(1/2) + (- 2*5^(
1/2) - 10)^(1/2) - 1))/(20*a^7) - log(a + x)/(5*a^7) + (log(a^20 - (a^19*x*(5^(1/2) - (2*5^(1/2) - 10)^(1/2) +
1)^3)/64)*(5^(1/2) - (2*5^(1/2) - 10)^(1/2) + 1))/(20*a^7) + (log(a^20 - (a^19*x*((- 2*5^(1/2) - 10)^(1/2) -
5^(1/2) + 1)^3)/64)*((- 2*5^(1/2) - 10)^(1/2) - 5^(1/2) + 1))/(20*a^7)

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sympy [A]  time = 0.22, size = 51, normalized size = 0.24 \[ - \frac {1}{2 a^{5} x^{2}} + \frac {- \frac {\log {\left (a + x \right )}}{5} + \operatorname {RootSum} {\left (625 t^{4} - 125 t^{3} + 25 t^{2} - 5 t + 1, \left (t \mapsto t \log {\left (25 t^{2} a + x \right )} \right )\right )}}{a^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**3/(a**5+x**5),x)

[Out]

-1/(2*a**5*x**2) + (-log(a + x)/5 + RootSum(625*_t**4 - 125*_t**3 + 25*_t**2 - 5*_t + 1, Lambda(_t, _t*log(25*
_t**2*a + x))))/a**7

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