3.2.39 \(\int \frac {x^3}{a^5+x^5} \, dx\) [139]

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

[Out]

-1/5*ln(a+x)/a+1/20*ln(a^2+x^2-1/2*a*x*(-5^(1/2)+1))*(-5^(1/2)+1)/a+1/20*ln(a^2+x^2-1/2*a*x*(5^(1/2)+1))*(5^(1
/2)+1)/a-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-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

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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 {\left (1-\sqrt {5}\right ) \log \left (a^2-\frac {1}{2} \left (1-\sqrt {5}\right ) a x+x^2\right )}{20 a}+\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}-\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}-\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}-\frac {\log (a+x)}{5 a} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3/(a^5 + x^5),x]

[Out]

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

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

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Mathematica [A]
time = 0.02, size = 204, normalized size = 1.01 \begin {gather*} \frac {2 \sqrt {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 )+2 \sqrt {10-2 \sqrt {5}} \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} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^3/(a^5 + x^5),x]

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.05, size = 97, normalized size = 0.48

method	result	size
risch	\(-\frac {\ln \left (a +x \right )}{5 a}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (a^{4} \textit {\_Z}^{4}-a^{3} \textit {\_Z}^{3}+\textit {\_Z}^{2} a^{2}-a \textit {\_Z} +1\right )}{\sum }\textit {\_R} \ln \left (\textit {\_R}^{3} a^{4}-\textit {\_R}^{2} a^{3}+a^{2} \textit {\_R} -a +x \right )\right )}{5}\)	\(73\)
default	\(-\frac {\ln \left (a +x \right )}{5 a}+\frac {\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}-a \,\textit {\_Z}^{3}+\textit {\_Z}^{2} a^{2}-a^{3} \textit {\_Z} +a^{4}\right )}{\sum }\frac {\left (\textit {\_R}^{3}+3 \textit {\_R}^{2} a -2 a^{2} \textit {\_R} +a^{3}\right ) \ln \left (-\textit {\_R} +x \right )}{4 \textit {\_R}^{3}-3 \textit {\_R}^{2} a +2 a^{2} \textit {\_R} -a^{3}}}{5 a}\)	\(97\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/5*ln(a+x)/a+1/5/a*sum((_R^3+3*_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 = 2.40, size = 180, normalized size = 0.90 \begin {gather*} \frac {\sqrt {5} {\left (\sqrt {5} + 1\right )} \arctan \left (\frac {a {\left (\sqrt {5} - 1\right )} + 4 \, x}{a \sqrt {2 \, \sqrt {5} + 10}}\right )}{5 \, a \sqrt {2 \, \sqrt {5} + 10}} + \frac {\sqrt {5} {\left (\sqrt {5} - 1\right )} \arctan \left (-\frac {a {\left (\sqrt {5} + 1\right )} - 4 \, x}{a \sqrt {-2 \, \sqrt {5} + 10}}\right )}{5 \, a \sqrt {-2 \, \sqrt {5} + 10}} + \frac {{\left (\sqrt {5} + 3\right )} \log \left (-a x {\left (\sqrt {5} + 1\right )} + 2 \, a^{2} + 2 \, x^{2}\right )}{10 \, a {\left (\sqrt {5} + 1\right )}} + \frac {{\left (\sqrt {5} - 3\right )} \log \left (a x {\left (\sqrt {5} - 1\right )} + 2 \, a^{2} + 2 \, x^{2}\right )}{10 \, a {\left (\sqrt {5} - 1\right )}} - \frac {\log \left (a + x\right )}{5 \, a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [C] Result contains complex when optimal does not.
time = 2.34, size = 17865, normalized size = 88.88 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6000*(2*(3125*(1/25)^(2/3)*(-I*sqrt(3) + 1)*((2*sqrt(5)*sqrt(1/2)*a*sqrt(-sqrt(5)*(sqrt(5) + 1)/a^2) + 3*sq
rt(5) + 5)^2/a^2 - 12*sqrt(5)*(sqrt(5)*sqrt(1/2)*a*sqrt(-sqrt(5)*(sqrt(5) + 1)/a^2) + 5*sqrt(1/2)*a*sqrt(-sqrt
(5)*(sqrt(5) + 1)/a^2) + 2*sqrt(5))/a^2)/(15625*sqrt(5)*(2*sqrt(5)*sqrt(1/2)*a*sqrt(-sqrt(5)*(sqrt(5) + 1)/a^2
) + 3*sqrt(5) + 5)^3/a^3 - 1406250*(2*sqrt(5)*sqrt(1/2)*a*sqrt(-sqrt(5)*(sqrt(5) + 1)/a^2) + 3*sqrt(5) + 5)*(s
qrt(5)*sqrt(1/2)*a*sqrt(-sqrt(5)*(sqrt(5) + 1)/a^2) + 5*sqrt(1/2)*a*sqrt(-sqrt(5)*(sqrt(5) + 1)/a^2) + 2*sqrt(
5))/a^3 + 4218750*sqrt(5)*(sqrt(5)*sqrt(1/2)*a^3*(-sqrt(5)*(sqrt(5) + 1)/a^2)^(3/2) + 13*sqrt(5)*sqrt(1/2)*a*s
qrt(-sqrt(5)*(sqrt(5) + 1)/a^2) + 5*sqrt(1/2)*a*sqrt(-sqrt(5)*(sqrt(5) + 1)/a^2) + 4*sqrt(5) - 20)/a^3 + 18*sq
rt(-610351562500*sqrt(5)*sqrt(1/2)*a^5*(-sqrt(5)*(sqrt(5) + 1)/a^2)^(5/2) - 6103515625000/3*sqrt(1/2)*a^3*(9*s
qrt(5) - 5)*(-sqrt(5)*(sqrt(5) + 1)/a^2)^(3/2) - 23651123046875/6*sqrt(5)*(sqrt(5) + 1)^3 + 762939453125/3*(9*
sqrt(5) + 305)*(sqrt(5) + 1)^2 - 30517578125000/3*sqrt(1/2)*a*(11*sqrt(5) - 7)*sqrt(-sqrt(5)*(sqrt(5) + 1)/a^2
) - 5340576171875/3*sqrt(5)*(sqrt(5) + 51)*(sqrt(5) + 1) - 115966796875000/3*sqrt(5) + 396728515625000/3)/a^3)
^(1/3) + (1/25)^(1/3)*(I*sqrt(3) + 1)*(15625*sqrt(5)*(2*sqrt(5)*sqrt(1/2)*a*sqrt(-sqrt(5)*(sqrt(5) + 1)/a^2) +
3*sqrt(5) + 5)^3/a^3 - 1406250*(2*sqrt(5)*sqrt(1/2)*a*sqrt(-sqrt(5)*(sqrt(5) + 1)/a^2) + 3*sqrt(5) + 5)*(sqrt
(5)*sqrt(1/2)*a*sqrt(-sqrt(5)*(sqrt(5) + 1)/a^2) + 5*sqrt(1/2)*a*sqrt(-sqrt(5)*(sqrt(5) + 1)/a^2) + 2*sqrt(5))
/a^3 + 4218750*sqrt(5)*(sqrt(5)*sqrt(1/2)*a^3*(-sqrt(5)*(sqrt(5) + 1)/a^2)^(3/2) + 13*sqrt(5)*sqrt(1/2)*a*sqrt
(-sqrt(5)*(sqrt(5) + 1)/a^2) + 5*sqrt(1/2)*a*sqrt(-sqrt(5)*(sqrt(5) + 1)/a^2) + 4*sqrt(5) - 20)/a^3 + 18*sqrt(
-610351562500*sqrt(5)*sqrt(1/2)*a^5*(-sqrt(5)*(sqrt(5) + 1)/a^2)^(5/2) - 6103515625000/3*sqrt(1/2)*a^3*(9*sqrt
(5) - 5)*(-sqrt(5)*(sqrt(5) + 1)/a^2)^(3/2) - 23651123046875/6*sqrt(5)*(sqrt(5) + 1)^3 + 762939453125/3*(9*sqr
t(5) + 305)*(sqrt(5) + 1)^2 - 30517578125000/3*sqrt(1/2)*a*(11*sqrt(5) - 7)*sqrt(-sqrt(5)*(sqrt(5) + 1)/a^2) -
5340576171875/3*sqrt(5)*(sqrt(5) + 51)*(sqrt(5) + 1) - 115966796875000/3*sqrt(5) + 396728515625000/3)/a^3)^(1
/3) - 10*sqrt(5)*(2*sqrt(5)*sqrt(1/2)*a*sqrt(-sqrt(5)*(sqrt(5) + 1)/a^2) + 3*sqrt(5) + 5)/a)*a*log(1/1440000*(
3125*(1/25)^(2/3)*(-I*sqrt(3) + 1)*((2*sqrt(5)*sqrt(1/2)*a*sqrt(-sqrt(5)*(sqrt(5) + 1)/a^2) + 3*sqrt(5) + 5)^2
/a^2 - 12*sqrt(5)*(sqrt(5)*sqrt(1/2)*a*sqrt(-sqrt(5)*(sqrt(5) + 1)/a^2) + 5*sqrt(1/2)*a*sqrt(-sqrt(5)*(sqrt(5)
+ 1)/a^2) + 2*sqrt(5))/a^2)/(15625*sqrt(5)*(2*sqrt(5)*sqrt(1/2)*a*sqrt(-sqrt(5)*(sqrt(5) + 1)/a^2) + 3*sqrt(5
) + 5)^3/a^3 - 1406250*(2*sqrt(5)*sqrt(1/2)*a*sqrt(-sqrt(5)*(sqrt(5) + 1)/a^2) + 3*sqrt(5) + 5)*(sqrt(5)*sqrt(
1/2)*a*sqrt(-sqrt(5)*(sqrt(5) + 1)/a^2) + 5*sqrt(1/2)*a*sqrt(-sqrt(5)*(sqrt(5) + 1)/a^2) + 2*sqrt(5))/a^3 + 42
18750*sqrt(5)*(sqrt(5)*sqrt(1/2)*a^3*(-sqrt(5)*(sqrt(5) + 1)/a^2)^(3/2) + 13*sqrt(5)*sqrt(1/2)*a*sqrt(-sqrt(5)
*(sqrt(5) + 1)/a^2) + 5*sqrt(1/2)*a*sqrt(-sqrt(5)*(sqrt(5) + 1)/a^2) + 4*sqrt(5) - 20)/a^3 + 18*sqrt(-61035156
2500*sqrt(5)*sqrt(1/2)*a^5*(-sqrt(5)*(sqrt(5) + 1)/a^2)^(5/2) - 6103515625000/3*sqrt(1/2)*a^3*(9*sqrt(5) - 5)*
(-sqrt(5)*(sqrt(5) + 1)/a^2)^(3/2) - 23651123046875/6*sqrt(5)*(sqrt(5) + 1)^3 + 762939453125/3*(9*sqrt(5) + 30
5)*(sqrt(5) + 1)^2 - 30517578125000/3*sqrt(1/2)*a*(11*sqrt(5) - 7)*sqrt(-sqrt(5)*(sqrt(5) + 1)/a^2) - 53405761
71875/3*sqrt(5)*(sqrt(5) + 51)*(sqrt(5) + 1) - 115966796875000/3*sqrt(5) + 396728515625000/3)/a^3)^(1/3) + (1/
25)^(1/3)*(I*sqrt(3) + 1)*(15625*sqrt(5)*(2*sqrt(5)*sqrt(1/2)*a*sqrt(-sqrt(5)*(sqrt(5) + 1)/a^2) + 3*sqrt(5) +
5)^3/a^3 - 1406250*(2*sqrt(5)*sqrt(1/2)*a*sqrt(-sqrt(5)*(sqrt(5) + 1)/a^2) + 3*sqrt(5) + 5)*(sqrt(5)*sqrt(1/2
)*a*sqrt(-sqrt(5)*(sqrt(5) + 1)/a^2) + 5*sqrt(1/2)*a*sqrt(-sqrt(5)*(sqrt(5) + 1)/a^2) + 2*sqrt(5))/a^3 + 42187
50*sqrt(5)*(sqrt(5)*sqrt(1/2)*a^3*(-sqrt(5)*(sqrt(5) + 1)/a^2)^(3/2) + 13*sqrt(5)*sqrt(1/2)*a*sqrt(-sqrt(5)*(s
qrt(5) + 1)/a^2) + 5*sqrt(1/2)*a*sqrt(-sqrt(5)*(sqrt(5) + 1)/a^2) + 4*sqrt(5) - 20)/a^3 + 18*sqrt(-61035156250
0*sqrt(5)*sqrt(1/2)*a^5*(-sqrt(5)*(sqrt(5) + 1)/a^2)^(5/2) - 6103515625000/3*sqrt(1/2)*a^3*(9*sqrt(5) - 5)*(-s
qrt(5)*(sqrt(5) + 1)/a^2)^(3/2) - 23651123046875/6*sqrt(5)*(sqrt(5) + 1)^3 + 762939453125/3*(9*sqrt(5) + 305)*
(sqrt(5) + 1)^2 - 30517578125000/3*sqrt(1/2)*a*(11*sqrt(5) - 7)*sqrt(-sqrt(5)*(sqrt(5) + 1)/a^2) - 53405761718
75/3*sqrt(5)*(sqrt(5) + 51)*(sqrt(5) + 1) - 115966796875000/3*sqrt(5) + 396728515625000/3)/a^3)^(1/3) - 10*sqr
t(5)*(2*sqrt(5)*sqrt(1/2)*a*sqrt(-sqrt(5)*(sqrt(5) + 1)/a^2) + 3*sqrt(5) + 5)/a)^2*a^4*(10*sqrt(-1/50*sqrt(5)/
a^2 - 1/10/a^2) + sqrt(5)/a - 1/a) + 1/64*a^4*(10*sqrt(-1/50*sqrt(5)/a^2 - 1/10/a^2) + sqrt(5)/a - 1/a)^3 + 1/
16*a^3*(10*sqrt(-1/50*sqrt(5)/a^2 - 1/10/a^2) + sqrt(5)/a - 1/a)^2 + 1/4*a^2*(10*sqrt(-1/50*sqrt(5)/a^2 - 1/10
/a^2) + sqrt(5)/a - 1/a) + 1/9600*(a^4*(10*sqrt(-1/50*sqrt(5)/a^2 - 1/10/a^2) + sqrt(5)/a - 1/a)^2 + 4*a^3*(10
*sqrt(-1/50*sqrt(5)/a^2 - 1/10/a^2) + sqrt(5)/a...

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Sympy [A]
time = 0.05, size = 39, normalized size = 0.19 \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 (625 t^{4} a + x \right )} \right )\right )}}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3/(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(625*_t**4*a + x))))/a

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Giac [A]
time = 0.58, 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} + \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} + \frac {\sqrt {5} \log \left (a^{2} - \frac {1}{2} \, {\left (\sqrt {5} a + a\right )} x + x^{2}\right )}{20 \, a} - \frac {\sqrt {5} \log \left (a^{2} + \frac {1}{2} \, {\left (\sqrt {5} a - a\right )} x + x^{2}\right )}{20 \, a} + \frac {\log \left ({\left | a^{4} - a^{3} x + a^{2} x^{2} - a x^{3} + x^{4} \right |}\right )}{20 \, a} - \frac {\log \left ({\left | a + x \right |}\right )}{5 \, a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Mupad [B]
time = 0.41, size = 202, normalized size = 1.00 \begin {gather*} \frac {\ln \left (5\,a^{10}-\frac {5\,a^9\,x\,\left (\sqrt {5}+\sqrt {2\,\sqrt {5}-10}+1\right )}{4}\right )\,\left (\sqrt {5}+\sqrt {2\,\sqrt {5}-10}+1\right )}{20\,a}-\frac {\ln \left (5\,a^{10}+\frac {5\,x\,\left (\sqrt {5}+\sqrt {-2\,\sqrt {5}-10}-1\right )\,a^9}{4}\right )\,\left (\sqrt {5}+\sqrt {-2\,\sqrt {5}-10}-1\right )}{20\,a}-\frac {\ln \left (a+x\right )}{5\,a}+\frac {\ln \left (5\,a^{10}-\frac {5\,a^9\,x\,\left (\sqrt {5}-\sqrt {2\,\sqrt {5}-10}+1\right )}{4}\right )\,\left (\sqrt {5}-\sqrt {2\,\sqrt {5}-10}+1\right )}{20\,a}+\frac {\ln \left (5\,a^{10}-\frac {5\,a^9\,x\,\left (\sqrt {-2\,\sqrt {5}-10}-\sqrt {5}+1\right )}{4}\right )\,\left (\sqrt {-2\,\sqrt {5}-10}-\sqrt {5}+1\right )}{20\,a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(a^5 + x^5),x)

[Out]

(log(5*a^10 - (5*a^9*x*(5^(1/2) + (2*5^(1/2) - 10)^(1/2) + 1))/4)*(5^(1/2) + (2*5^(1/2) - 10)^(1/2) + 1))/(20*
a) - (log(5*a^10 + (5*a^9*x*(5^(1/2) + (- 2*5^(1/2) - 10)^(1/2) - 1))/4)*(5^(1/2) + (- 2*5^(1/2) - 10)^(1/2) -
1))/(20*a) - log(a + x)/(5*a) + (log(5*a^10 - (5*a^9*x*(5^(1/2) - (2*5^(1/2) - 10)^(1/2) + 1))/4)*(5^(1/2) -
(2*5^(1/2) - 10)^(1/2) + 1))/(20*a) + (log(5*a^10 - (5*a^9*x*((- 2*5^(1/2) - 10)^(1/2) - 5^(1/2) + 1))/4)*((-
2*5^(1/2) - 10)^(1/2) - 5^(1/2) + 1))/(20*a)

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