3.2.38 \(\int \frac {x^2}{a^5+x^5} \, dx\) [138]

Optimal. Leaf size=201 \[ \frac {\sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \tan ^{-1}\left (\frac {\left (1-\sqrt {5}\right ) a-4 x}{\sqrt {2 \left (5+\sqrt {5}\right )} a}\right )}{5 a^2}-\frac {\sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \tan ^{-1}\left (\frac {\sqrt {\frac {1}{10} \left (5+\sqrt {5}\right )} \left (\left (1+\sqrt {5}\right ) a-4 x\right )}{2 a}\right )}{5 a^2}+\frac {\log (a+x)}{5 a^2}-\frac {\left (1+\sqrt {5}\right ) \log \left (a^2-\frac {1}{2} \left (1-\sqrt {5}\right ) a x+x^2\right )}{20 a^2}-\frac {\left (1-\sqrt {5}\right ) \log \left (a^2-\frac {1}{2} \left (1+\sqrt {5}\right ) a x+x^2\right )}{20 a^2} \]

[Out]

1/5*ln(a+x)/a^2-1/20*ln(a^2+x^2-1/2*a*x*(5^(1/2)+1))*(-5^(1/2)+1)/a^2-1/20*ln(a^2+x^2-1/2*a*x*(-5^(1/2)+1))*(5
^(1/2)+1)/a^2+1/10*arctan((-4*x+a*(-5^(1/2)+1))/a/(10+2*5^(1/2))^(1/2))*(10-2*5^(1/2))^(1/2)/a^2-1/10*arctan(1
/20*(-4*x+a*(5^(1/2)+1))*(50+10*5^(1/2))^(1/2)/a)*(10+2*5^(1/2))^(1/2)/a^2

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Rubi [A]
time = 0.24, antiderivative size = 201, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {299, 648, 632, 210, 642, 31} \begin {gather*} \frac {\sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \text {ArcTan}\left (\frac {\left (1-\sqrt {5}\right ) a-4 x}{\sqrt {2 \left (5+\sqrt {5}\right )} a}\right )}{5 a^2}-\frac {\sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \text {ArcTan}\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^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^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^2}+\frac {\log (a+x)}{5 a^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2/(a^5 + x^5),x]

[Out]

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

Rule 31

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

Rule 210

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

Rule 299

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)/(a*n*s^m))*Int[1/(r + s*x), x] + 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 632

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 642

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 648

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 {x^2}{a^5+x^5} \, dx &=\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^2}+\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^2}+\frac {\int \frac {1}{a+x} \, dx}{5 a^2}\\ &=\frac {\log (a+x)}{5 a^2}-\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^2}-\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^2}-\frac {\int \frac {1}{a^2-\frac {1}{2} \left (1-\sqrt {5}\right ) a x+x^2} \, dx}{2 \sqrt {5} a}+\frac {\int \frac {1}{a^2-\frac {1}{2} \left (1+\sqrt {5}\right ) a x+x^2} \, dx}{2 \sqrt {5} a}\\ &=\frac {\log (a+x)}{5 a^2}-\frac {\left (1-\sqrt {5}\right ) \log \left (2 a^2-a x-\sqrt {5} a x+2 x^2\right )}{20 a^2}-\frac {\left (1+\sqrt {5}\right ) \log \left (2 a^2-a x+\sqrt {5} a x+2 x^2\right )}{20 a^2}-\frac {\text {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}+\frac {\text {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}\\ &=\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^2}-\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^2}+\frac {\log (a+x)}{5 a^2}-\frac {\left (1-\sqrt {5}\right ) \log \left (2 a^2-a x-\sqrt {5} a x+2 x^2\right )}{20 a^2}-\frac {\left (1+\sqrt {5}\right ) \log \left (2 a^2-a x+\sqrt {5} a x+2 x^2\right )}{20 a^2}\\ \end {align*}

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Mathematica [A]
time = 0.04, size = 204, normalized size = 1.01 \begin {gather*} -\frac {2 \sqrt {10-2 \sqrt {5}} \tan ^{-1}\left (\frac {\left (-1+\sqrt {5}\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 {-\left (\left (1+\sqrt {5}\right ) a\right )+4 x}{\sqrt {10-2 \sqrt {5}} a}\right )-4 \log (a+x)+\log \left (a^2+\frac {1}{2} \left (-1+\sqrt {5}\right ) a x+x^2\right )+\sqrt {5} \log \left (a^2+\frac {1}{2} \left (-1+\sqrt {5}\right ) a x+x^2\right )+\log \left (a^2-\frac {1}{2} \left (1+\sqrt {5}\right ) a x+x^2\right )-\sqrt {5} \log \left (a^2-\frac {1}{2} \left (1+\sqrt {5}\right ) a x+x^2\right )}{20 a^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^2/(a^5 + x^5),x]

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.05, size = 101, normalized size = 0.50

method	result	size
risch	\(\frac {\ln \left (a +x \right )}{5 a^{2}}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (a^{8} \textit {\_Z}^{4}+a^{6} \textit {\_Z}^{3}+a^{4} \textit {\_Z}^{2}+a^{2} \textit {\_Z} +1\right )}{\sum }\textit {\_R} \ln \left (x \,\textit {\_R}^{3} a^{5}+1\right )\right )}{5}\)	\(58\)
default	\(\frac {\ln \left (a +x \right )}{5 a^{2}}+\frac {\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}-a \,\textit {\_Z}^{3}+\textit {\_Z}^{2} a^{2}-a^{3} \textit {\_Z} +a^{4}\right )}{\sum }\frac {\left (-\textit {\_R}^{3}+2 \textit {\_R}^{2} a +2 a^{2} \textit {\_R} -a^{3}\right ) \ln \left (-\textit {\_R} +x \right )}{4 \textit {\_R}^{3}-3 \textit {\_R}^{2} a +2 a^{2} \textit {\_R} -a^{3}}}{5 a^{2}}\)	\(101\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(a^5+x^5),x,method=_RETURNVERBOSE)

[Out]

1/5/a^2*ln(a+x)+1/5/a^2*sum((-_R^3+2*_R^2*a+2*_R*a^2-a^3)/(4*_R^3-3*_R^2*a+2*_R*a^2-a^3)*ln(-_R+x),_R=RootOf(_
Z^4-_Z^3*a+_Z^2*a^2-_Z*a^3+a^4))

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Maxima [A]
time = 1.55, size = 160, normalized size = 0.80 \begin {gather*} -\frac {2 \, \sqrt {5} \arctan \left (\frac {a {\left (\sqrt {5} - 1\right )} + 4 \, x}{a \sqrt {2 \, \sqrt {5} + 10}}\right )}{5 \, 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 )}{5 \, a^{2} \sqrt {-2 \, \sqrt {5} + 10}} + \frac {\log \left (a + x\right )}{5 \, a^{2}} + \frac {\log \left (-a x {\left (\sqrt {5} + 1\right )} + 2 \, a^{2} + 2 \, x^{2}\right )}{5 \, a^{2} {\left (\sqrt {5} + 1\right )}} - \frac {\log \left (a x {\left (\sqrt {5} - 1\right )} + 2 \, a^{2} + 2 \, x^{2}\right )}{5 \, a^{2} {\left (\sqrt {5} - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(a^5+x^5),x, algorithm="maxima")

[Out]

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

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Fricas [C] Result contains complex when optimal does not.
time = 1.47, size = 12656, normalized size = 62.97 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(a^5+x^5),x, algorithm="fricas")

[Out]

-1/4800*(2*(800*(1/2)^(2/3)*(-I*sqrt(3) + 1)*((sqrt(2)*a^2*sqrt((sqrt(5) - 5)/a^4) + sqrt(5) - 3)^2/a^4 - 6*(s
qrt(2)*(sqrt(5)*a^2*sqrt((sqrt(5) - 5)/a^4) - a^2*sqrt((sqrt(5) - 5)/a^4)) + 4)/a^4)/(16000*(sqrt(2)*a^2*sqrt(
(sqrt(5) - 5)/a^4) + sqrt(5) - 3)^3/a^6 - 144000*(sqrt(2)*a^2*sqrt((sqrt(5) - 5)/a^4) + sqrt(5) - 3)*(sqrt(2)*
(sqrt(5)*a^2*sqrt((sqrt(5) - 5)/a^4) - a^2*sqrt((sqrt(5) - 5)/a^4)) + 4)/a^6 + 432000*(sqrt(2)*(a^6*((sqrt(5)
- 5)/a^4)^(3/2) - sqrt(5)*a^2*sqrt((sqrt(5) - 5)/a^4) + 13*a^2*sqrt((sqrt(5) - 5)/a^4)) - 8*sqrt(5) - 8)/a^6 +
9*sqrt(-2048000000*sqrt(5)*sqrt(2)*a^10*((sqrt(5) - 5)/a^4)^(5/2) - 20480000000/3*(9*sqrt(5)*sqrt(2) + 5*sqrt
(2))*a^6*((sqrt(5) - 5)/a^4)^(3/2) - 102400000000/3*(11*sqrt(5)*sqrt(2) + 7*sqrt(2))*a^2*sqrt((sqrt(5) - 5)/a^
4) - 1024000000/3*(9*sqrt(5) - 305)*(sqrt(5) - 5)^2 + 15872000000/3*(sqrt(5) - 5)^3 - 35840000000/3*(sqrt(5) -
5)*(sqrt(5) - 51) + 778240000000/3*sqrt(5) + 2662400000000/3)/a^6)^(1/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(16000
*(sqrt(2)*a^2*sqrt((sqrt(5) - 5)/a^4) + sqrt(5) - 3)^3/a^6 - 144000*(sqrt(2)*a^2*sqrt((sqrt(5) - 5)/a^4) + sqr
t(5) - 3)*(sqrt(2)*(sqrt(5)*a^2*sqrt((sqrt(5) - 5)/a^4) - a^2*sqrt((sqrt(5) - 5)/a^4)) + 4)/a^6 + 432000*(sqrt
(2)*(a^6*((sqrt(5) - 5)/a^4)^(3/2) - sqrt(5)*a^2*sqrt((sqrt(5) - 5)/a^4) + 13*a^2*sqrt((sqrt(5) - 5)/a^4)) - 8
*sqrt(5) - 8)/a^6 + 9*sqrt(-2048000000*sqrt(5)*sqrt(2)*a^10*((sqrt(5) - 5)/a^4)^(5/2) - 20480000000/3*(9*sqrt(
5)*sqrt(2) + 5*sqrt(2))*a^6*((sqrt(5) - 5)/a^4)^(3/2) - 102400000000/3*(11*sqrt(5)*sqrt(2) + 7*sqrt(2))*a^2*sq
rt((sqrt(5) - 5)/a^4) - 1024000000/3*(9*sqrt(5) - 305)*(sqrt(5) - 5)^2 + 15872000000/3*(sqrt(5) - 5)^3 - 35840
000000/3*(sqrt(5) - 5)*(sqrt(5) - 51) + 778240000000/3*sqrt(5) + 2662400000000/3)/a^6)^(1/3) - 40*(sqrt(2)*a^2
*sqrt((sqrt(5) - 5)/a^4) + sqrt(5) - 3)/a^2)*a^2*log(1/230400*(800*(1/2)^(2/3)*(-I*sqrt(3) + 1)*((sqrt(2)*a^2*
sqrt((sqrt(5) - 5)/a^4) + sqrt(5) - 3)^2/a^4 - 6*(sqrt(2)*(sqrt(5)*a^2*sqrt((sqrt(5) - 5)/a^4) - a^2*sqrt((sqr
t(5) - 5)/a^4)) + 4)/a^4)/(16000*(sqrt(2)*a^2*sqrt((sqrt(5) - 5)/a^4) + sqrt(5) - 3)^3/a^6 - 144000*(sqrt(2)*a
^2*sqrt((sqrt(5) - 5)/a^4) + sqrt(5) - 3)*(sqrt(2)*(sqrt(5)*a^2*sqrt((sqrt(5) - 5)/a^4) - a^2*sqrt((sqrt(5) -
5)/a^4)) + 4)/a^6 + 432000*(sqrt(2)*(a^6*((sqrt(5) - 5)/a^4)^(3/2) - sqrt(5)*a^2*sqrt((sqrt(5) - 5)/a^4) + 13*
a^2*sqrt((sqrt(5) - 5)/a^4)) - 8*sqrt(5) - 8)/a^6 + 9*sqrt(-2048000000*sqrt(5)*sqrt(2)*a^10*((sqrt(5) - 5)/a^4
)^(5/2) - 20480000000/3*(9*sqrt(5)*sqrt(2) + 5*sqrt(2))*a^6*((sqrt(5) - 5)/a^4)^(3/2) - 102400000000/3*(11*sqr
t(5)*sqrt(2) + 7*sqrt(2))*a^2*sqrt((sqrt(5) - 5)/a^4) - 1024000000/3*(9*sqrt(5) - 305)*(sqrt(5) - 5)^2 + 15872
000000/3*(sqrt(5) - 5)^3 - 35840000000/3*(sqrt(5) - 5)*(sqrt(5) - 51) + 778240000000/3*sqrt(5) + 2662400000000
/3)/a^6)^(1/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(16000*(sqrt(2)*a^2*sqrt((sqrt(5) - 5)/a^4) + sqrt(5) - 3)^3/a^6
- 144000*(sqrt(2)*a^2*sqrt((sqrt(5) - 5)/a^4) + sqrt(5) - 3)*(sqrt(2)*(sqrt(5)*a^2*sqrt((sqrt(5) - 5)/a^4) - a
^2*sqrt((sqrt(5) - 5)/a^4)) + 4)/a^6 + 432000*(sqrt(2)*(a^6*((sqrt(5) - 5)/a^4)^(3/2) - sqrt(5)*a^2*sqrt((sqrt
(5) - 5)/a^4) + 13*a^2*sqrt((sqrt(5) - 5)/a^4)) - 8*sqrt(5) - 8)/a^6 + 9*sqrt(-2048000000*sqrt(5)*sqrt(2)*a^10
*((sqrt(5) - 5)/a^4)^(5/2) - 20480000000/3*(9*sqrt(5)*sqrt(2) + 5*sqrt(2))*a^6*((sqrt(5) - 5)/a^4)^(3/2) - 102
400000000/3*(11*sqrt(5)*sqrt(2) + 7*sqrt(2))*a^2*sqrt((sqrt(5) - 5)/a^4) - 1024000000/3*(9*sqrt(5) - 305)*(sqr
t(5) - 5)^2 + 15872000000/3*(sqrt(5) - 5)^3 - 35840000000/3*(sqrt(5) - 5)*(sqrt(5) - 51) + 778240000000/3*sqrt
(5) + 2662400000000/3)/a^6)^(1/3) - 40*(sqrt(2)*a^2*sqrt((sqrt(5) - 5)/a^4) + sqrt(5) - 3)/a^2)^2*a^5 + x) + 2
40*a^2*(2*sqrt(1/2)*sqrt(sqrt(5)/a^4 - 5/a^4) + sqrt(5)/a^2 + 1/a^2)*log(1/16*a^5*(2*sqrt(1/2)*sqrt(sqrt(5)/a^
4 - 5/a^4) + sqrt(5)/a^2 + 1/a^2)^2 + x) - ((800*(1/2)^(2/3)*(-I*sqrt(3) + 1)*((sqrt(2)*a^2*sqrt((sqrt(5) - 5)
/a^4) + sqrt(5) - 3)^2/a^4 - 6*(sqrt(2)*(sqrt(5)*a^2*sqrt((sqrt(5) - 5)/a^4) - a^2*sqrt((sqrt(5) - 5)/a^4)) +
4)/a^4)/(16000*(sqrt(2)*a^2*sqrt((sqrt(5) - 5)/a^4) + sqrt(5) - 3)^3/a^6 - 144000*(sqrt(2)*a^2*sqrt((sqrt(5) -
5)/a^4) + sqrt(5) - 3)*(sqrt(2)*(sqrt(5)*a^2*sqrt((sqrt(5) - 5)/a^4) - a^2*sqrt((sqrt(5) - 5)/a^4)) + 4)/a^6
+ 432000*(sqrt(2)*(a^6*((sqrt(5) - 5)/a^4)^(3/2) - sqrt(5)*a^2*sqrt((sqrt(5) - 5)/a^4) + 13*a^2*sqrt((sqrt(5)
- 5)/a^4)) - 8*sqrt(5) - 8)/a^6 + 9*sqrt(-2048000000*sqrt(5)*sqrt(2)*a^10*((sqrt(5) - 5)/a^4)^(5/2) - 20480000
000/3*(9*sqrt(5)*sqrt(2) + 5*sqrt(2))*a^6*((sqrt(5) - 5)/a^4)^(3/2) - 102400000000/3*(11*sqrt(5)*sqrt(2) + 7*s
qrt(2))*a^2*sqrt((sqrt(5) - 5)/a^4) - 1024000000/3*(9*sqrt(5) - 305)*(sqrt(5) - 5)^2 + 15872000000/3*(sqrt(5)
- 5)^3 - 35840000000/3*(sqrt(5) - 5)*(sqrt(5) - 51) + 778240000000/3*sqrt(5) + 2662400000000/3)/a^6)^(1/3) + (
1/2)^(1/3)*(I*sqrt(3) + 1)*(16000*(sqrt(2)*a^2*sqrt((sqrt(5) - 5)/a^4) + sqrt(5) - 3)^3/a^6 - 144000*(sqrt(2)*
a^2*sqrt((sqrt(5) - 5)/a^4) + sqrt(5) - 3)*(sqrt(2)*(sqrt(5)*a^2*sqrt((sqrt(5) - 5)/a^4) - a^2*sqrt((sqrt(5) -
5)/a^4)) + 4)/a^6 + 432000*(sqrt(2)*(a^6*((sqr...

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Sympy [A]
time = 0.05, size = 41, normalized size = 0.20 \begin {gather*} \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^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2/(a**5+x**5),x)

[Out]

(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**2

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Giac [A]
time = 0.60, size = 177, normalized size = 0.88 \begin {gather*} -\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^{2}} + \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^{2}} + \frac {\sqrt {5} \log \left (a^{2} - \frac {1}{2} \, {\left (\sqrt {5} a + a\right )} x + x^{2}\right )}{20 \, a^{2}} - \frac {\sqrt {5} \log \left (a^{2} + \frac {1}{2} \, {\left (\sqrt {5} a - a\right )} x + x^{2}\right )}{20 \, a^{2}} - \frac {\log \left ({\left | a^{4} - a^{3} x + a^{2} x^{2} - a x^{3} + x^{4} \right |}\right )}{20 \, a^{2}} + \frac {\log \left ({\left | a + x \right |}\right )}{5 \, a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Mupad [B]
time = 0.80, size = 202, normalized size = 1.00 \begin {gather*} \frac {\ln \left (a+x\right )}{5\,a^2}+\frac {\ln \left (a^5+\frac {x\,{\left (\sqrt {5}+\sqrt {-2\,\sqrt {5}-10}-1\right )}^3\,a^4}{64}\right )\,\left (\sqrt {5}+\sqrt {-2\,\sqrt {5}-10}-1\right )}{20\,a^2}-\frac {\ln \left (a^5-\frac {a^4\,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^2}-\frac {\ln \left (a^5-\frac {a^4\,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^2}-\frac {\ln \left (a^5-\frac {a^4\,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^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(a^5 + x^5),x)

[Out]

log(a + x)/(5*a^2) + (log(a^5 + (a^4*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^2) - (log(a^5 - (a^4*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^2) - (log(a^5 - (a^4*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^2) - (log(a^5 - (a^4*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^2)

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