\(\int \frac {-b^6+a^6 x^6}{\sqrt {b^2 x+a^2 x^3} (b^6+a^6 x^6)} \, dx\) [2243]

Optimal result

Integrand size = 44, antiderivative size = 167 \[ \int \frac {-b^6+a^6 x^6}{\sqrt {b^2 x+a^2 x^3} \left (b^6+a^6 x^6\right )} \, dx=-\frac {2 \sqrt {b^2 x+a^2 x^3}}{3 \left (b^2+a^2 x^2\right )}-\frac {2 \arctan \left (\frac {\sqrt [4]{3} \sqrt {a} \sqrt {b} \sqrt {b^2 x+a^2 x^3}}{b^2+a^2 x^2}\right )}{3 \sqrt [4]{3} \sqrt {a} \sqrt {b}}-\frac {2 \text {arctanh}\left (\frac {\sqrt [4]{3} \sqrt {a} \sqrt {b} \sqrt {b^2 x+a^2 x^3}}{b^2+a^2 x^2}\right )}{3 \sqrt [4]{3} \sqrt {a} \sqrt {b}} \]

[Out]

-2*(a^2*x^3+b^2*x)^(1/2)/(3*a^2*x^2+3*b^2)-2/9*arctan(3^(1/4)*a^(1/2)*b^(1/2)*(a^2*x^3+b^2*x)^(1/2)/(a^2*x^2+b
^2))*3^(3/4)/a^(1/2)/b^(1/2)-2/9*arctanh(3^(1/4)*a^(1/2)*b^(1/2)*(a^2*x^3+b^2*x)^(1/2)/(a^2*x^2+b^2))*3^(3/4)/
a^(1/2)/b^(1/2)

Rubi [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 12.48 (sec) , antiderivative size = 2697, normalized size of antiderivative = 16.15, number of steps used = 27, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {2081, 6847, 6857, 226, 2098, 425, 537, 418, 1231, 1721} \[ \int \frac {-b^6+a^6 x^6}{\sqrt {b^2 x+a^2 x^3} \left (b^6+a^6 x^6\right )} \, dx=-\frac {4 \sqrt [4]{2} \sqrt {x} \sqrt {b^2+a^2 x^2} \arctan \left (\frac {\sqrt {3 a^2-\sqrt {3} \sqrt {-a^4}} \sqrt {b} \sqrt {x}}{\sqrt [4]{2} \sqrt [4]{a^2-\sqrt {3} \sqrt {-a^4}} \sqrt {b^2+a^2 x^2}}\right ) a^6}{\sqrt {3} \sqrt {-a^4} \left (a^2-\sqrt {3} \sqrt {-a^4}\right )^{3/4} \left (3 a^2-\sqrt {3} \sqrt {-a^4}\right )^{3/2} \sqrt {b} \sqrt {a^2 x^3+b^2 x}}+\frac {4 \sqrt [4]{2} \sqrt {x} \sqrt {b^2+a^2 x^2} \arctan \left (\frac {\sqrt {\sqrt {3} \sqrt {-a^4}-3 a^2} \sqrt {b} \sqrt {x}}{\sqrt [4]{2} \sqrt [4]{a^2-\sqrt {3} \sqrt {-a^4}} \sqrt {b^2+a^2 x^2}}\right ) a^6}{\sqrt {3} \sqrt {-a^4} \left (a^2-\sqrt {3} \sqrt {-a^4}\right )^{3/4} \left (\sqrt {3} \sqrt {-a^4}-3 a^2\right )^{3/2} \sqrt {b} \sqrt {a^2 x^3+b^2 x}}-\frac {4 \sqrt [4]{2} \sqrt {x} \sqrt {b^2+a^2 x^2} \arctan \left (\frac {\sqrt {-3 a^2-\sqrt {3} \sqrt {-a^4}} \sqrt {b} \sqrt {x}}{\sqrt [4]{2} \sqrt [4]{a^2+\sqrt {3} \sqrt {-a^4}} \sqrt {b^2+a^2 x^2}}\right ) a^6}{\sqrt {3} \sqrt {-a^4} \left (-3 a^2-\sqrt {3} \sqrt {-a^4}\right )^{3/2} \left (a^2+\sqrt {3} \sqrt {-a^4}\right )^{3/4} \sqrt {b} \sqrt {a^2 x^3+b^2 x}}+\frac {4 \sqrt [4]{2} \sqrt {x} \sqrt {b^2+a^2 x^2} \arctan \left (\frac {\sqrt {3 a^2+\sqrt {3} \sqrt {-a^4}} \sqrt {b} \sqrt {x}}{\sqrt [4]{2} \sqrt [4]{a^2+\sqrt {3} \sqrt {-a^4}} \sqrt {b^2+a^2 x^2}}\right ) a^6}{\sqrt {3} \sqrt {-a^4} \left (a^2+\sqrt {3} \sqrt {-a^4}\right )^{3/4} \left (3 a^2+\sqrt {3} \sqrt {-a^4}\right )^{3/2} \sqrt {b} \sqrt {a^2 x^3+b^2 x}}+\frac {2 \left (1-\frac {\sqrt {2} a}{\sqrt {a^2-\sqrt {3} \sqrt {-a^4}}}\right ) \sqrt {x} (b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),\frac {1}{2}\right ) a^{7/2}}{\sqrt {3} \sqrt {-a^4} \left (3 a^2+\sqrt {3} \sqrt {-a^4}\right ) \sqrt {b} \sqrt {a^2 x^3+b^2 x}}+\frac {2 \left (\frac {\sqrt {2} a}{\sqrt {a^2-\sqrt {3} \sqrt {-a^4}}}+1\right ) \sqrt {x} (b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),\frac {1}{2}\right ) a^{7/2}}{\sqrt {3} \sqrt {-a^4} \left (3 a^2+\sqrt {3} \sqrt {-a^4}\right ) \sqrt {b} \sqrt {a^2 x^3+b^2 x}}-\frac {2 \left (1-\frac {\sqrt {2} a}{\sqrt {a^2+\sqrt {3} \sqrt {-a^4}}}\right ) \sqrt {x} (b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),\frac {1}{2}\right ) a^{7/2}}{\sqrt {3} \sqrt {-a^4} \left (3 a^2-\sqrt {3} \sqrt {-a^4}\right ) \sqrt {b} \sqrt {a^2 x^3+b^2 x}}-\frac {2 \left (\frac {\sqrt {2} a}{\sqrt {a^2+\sqrt {3} \sqrt {-a^4}}}+1\right ) \sqrt {x} (b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),\frac {1}{2}\right ) a^{7/2}}{\sqrt {3} \sqrt {-a^4} \left (3 a^2-\sqrt {3} \sqrt {-a^4}\right ) \sqrt {b} \sqrt {a^2 x^3+b^2 x}}-\frac {4 x a^2}{\sqrt {3} \left (\sqrt {3} a^2-3 \sqrt {-a^4}\right ) \sqrt {a^2 x^3+b^2 x}}-\frac {4 x a^2}{\sqrt {3} \left (\sqrt {3} a^2+3 \sqrt {-a^4}\right ) \sqrt {a^2 x^3+b^2 x}}-\frac {2 \sqrt {x} (b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),\frac {1}{2}\right ) a^{3/2}}{\sqrt {3} \left (\sqrt {3} a^2-3 \sqrt {-a^4}\right ) \sqrt {b} \sqrt {a^2 x^3+b^2 x}}-\frac {2 \sqrt {x} (b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),\frac {1}{2}\right ) a^{3/2}}{\sqrt {3} \left (\sqrt {3} a^2+3 \sqrt {-a^4}\right ) \sqrt {b} \sqrt {a^2 x^3+b^2 x}}+\frac {\sqrt {x} (b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),\frac {1}{2}\right )}{\sqrt {b} \sqrt {a^2 x^3+b^2 x} \sqrt {a}}-\frac {\left (\sqrt {2} a+\sqrt {a^2-\sqrt {3} \sqrt {-a^4}}\right )^2 \sqrt {x} (b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} \operatorname {EllipticPi}\left (-\frac {\left (\sqrt {2} a-\sqrt {a^2-\sqrt {3} \sqrt {-a^4}}\right )^2}{4 \sqrt {2} a \sqrt {a^2-\sqrt {3} \sqrt {-a^4}}},2 \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),\frac {1}{2}\right )}{2 \sqrt {3} \left (\sqrt {3} a^2+3 \sqrt {-a^4}\right ) \sqrt {b} \sqrt {a^2 x^3+b^2 x} \sqrt {a}}-\frac {\left (\sqrt {2} a-\sqrt {a^2-\sqrt {3} \sqrt {-a^4}}\right )^2 \sqrt {x} (b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} \operatorname {EllipticPi}\left (\frac {\left (\sqrt {2} a+\sqrt {a^2-\sqrt {3} \sqrt {-a^4}}\right )^2}{4 \sqrt {2} a \sqrt {a^2-\sqrt {3} \sqrt {-a^4}}},2 \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),\frac {1}{2}\right )}{2 \sqrt {3} \left (\sqrt {3} a^2+3 \sqrt {-a^4}\right ) \sqrt {b} \sqrt {a^2 x^3+b^2 x} \sqrt {a}}-\frac {\left (\sqrt {2} a+\sqrt {a^2+\sqrt {3} \sqrt {-a^4}}\right )^2 \sqrt {x} (b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} \operatorname {EllipticPi}\left (-\frac {\left (\sqrt {2} a-\sqrt {a^2+\sqrt {3} \sqrt {-a^4}}\right )^2}{4 \sqrt {2} a \sqrt {a^2+\sqrt {3} \sqrt {-a^4}}},2 \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),\frac {1}{2}\right )}{2 \sqrt {3} \left (\sqrt {3} a^2-3 \sqrt {-a^4}\right ) \sqrt {b} \sqrt {a^2 x^3+b^2 x} \sqrt {a}}-\frac {\left (\sqrt {2} a-\sqrt {a^2+\sqrt {3} \sqrt {-a^4}}\right )^2 \sqrt {x} (b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} \operatorname {EllipticPi}\left (\frac {\left (\sqrt {2} a+\sqrt {a^2+\sqrt {3} \sqrt {-a^4}}\right )^2}{4 \sqrt {2} a \sqrt {a^2+\sqrt {3} \sqrt {-a^4}}},2 \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),\frac {1}{2}\right )}{2 \sqrt {3} \left (\sqrt {3} a^2-3 \sqrt {-a^4}\right ) \sqrt {b} \sqrt {a^2 x^3+b^2 x} \sqrt {a}} \]

[In]

Int[(-b^6 + a^6*x^6)/(Sqrt[b^2*x + a^2*x^3]*(b^6 + a^6*x^6)),x]

[Out]

(-4*a^2*x)/(Sqrt[3]*(Sqrt[3]*a^2 - 3*Sqrt[-a^4])*Sqrt[b^2*x + a^2*x^3]) - (4*a^2*x)/(Sqrt[3]*(Sqrt[3]*a^2 + 3*
Sqrt[-a^4])*Sqrt[b^2*x + a^2*x^3]) - (4*2^(1/4)*a^6*Sqrt[x]*Sqrt[b^2 + a^2*x^2]*ArcTan[(Sqrt[3*a^2 - Sqrt[3]*S
qrt[-a^4]]*Sqrt[b]*Sqrt[x])/(2^(1/4)*(a^2 - Sqrt[3]*Sqrt[-a^4])^(1/4)*Sqrt[b^2 + a^2*x^2])])/(Sqrt[3]*Sqrt[-a^
4]*(a^2 - Sqrt[3]*Sqrt[-a^4])^(3/4)*(3*a^2 - Sqrt[3]*Sqrt[-a^4])^(3/2)*Sqrt[b]*Sqrt[b^2*x + a^2*x^3]) + (4*2^(
1/4)*a^6*Sqrt[x]*Sqrt[b^2 + a^2*x^2]*ArcTan[(Sqrt[-3*a^2 + Sqrt[3]*Sqrt[-a^4]]*Sqrt[b]*Sqrt[x])/(2^(1/4)*(a^2
- Sqrt[3]*Sqrt[-a^4])^(1/4)*Sqrt[b^2 + a^2*x^2])])/(Sqrt[3]*Sqrt[-a^4]*(a^2 - Sqrt[3]*Sqrt[-a^4])^(3/4)*(-3*a^
2 + Sqrt[3]*Sqrt[-a^4])^(3/2)*Sqrt[b]*Sqrt[b^2*x + a^2*x^3]) - (4*2^(1/4)*a^6*Sqrt[x]*Sqrt[b^2 + a^2*x^2]*ArcT
an[(Sqrt[-3*a^2 - Sqrt[3]*Sqrt[-a^4]]*Sqrt[b]*Sqrt[x])/(2^(1/4)*(a^2 + Sqrt[3]*Sqrt[-a^4])^(1/4)*Sqrt[b^2 + a^
2*x^2])])/(Sqrt[3]*Sqrt[-a^4]*(-3*a^2 - Sqrt[3]*Sqrt[-a^4])^(3/2)*(a^2 + Sqrt[3]*Sqrt[-a^4])^(3/4)*Sqrt[b]*Sqr
t[b^2*x + a^2*x^3]) + (4*2^(1/4)*a^6*Sqrt[x]*Sqrt[b^2 + a^2*x^2]*ArcTan[(Sqrt[3*a^2 + Sqrt[3]*Sqrt[-a^4]]*Sqrt
[b]*Sqrt[x])/(2^(1/4)*(a^2 + Sqrt[3]*Sqrt[-a^4])^(1/4)*Sqrt[b^2 + a^2*x^2])])/(Sqrt[3]*Sqrt[-a^4]*(a^2 + Sqrt[
3]*Sqrt[-a^4])^(3/4)*(3*a^2 + Sqrt[3]*Sqrt[-a^4])^(3/2)*Sqrt[b]*Sqrt[b^2*x + a^2*x^3]) + (Sqrt[x]*(b + a*x)*Sq
rt[(b^2 + a^2*x^2)/(b + a*x)^2]*EllipticF[2*ArcTan[(Sqrt[a]*Sqrt[x])/Sqrt[b]], 1/2])/(Sqrt[a]*Sqrt[b]*Sqrt[b^2
*x + a^2*x^3]) - (2*a^(3/2)*Sqrt[x]*(b + a*x)*Sqrt[(b^2 + a^2*x^2)/(b + a*x)^2]*EllipticF[2*ArcTan[(Sqrt[a]*Sq
rt[x])/Sqrt[b]], 1/2])/(Sqrt[3]*(Sqrt[3]*a^2 - 3*Sqrt[-a^4])*Sqrt[b]*Sqrt[b^2*x + a^2*x^3]) - (2*a^(3/2)*Sqrt[
x]*(b + a*x)*Sqrt[(b^2 + a^2*x^2)/(b + a*x)^2]*EllipticF[2*ArcTan[(Sqrt[a]*Sqrt[x])/Sqrt[b]], 1/2])/(Sqrt[3]*(
Sqrt[3]*a^2 + 3*Sqrt[-a^4])*Sqrt[b]*Sqrt[b^2*x + a^2*x^3]) + (2*a^(7/2)*(1 - (Sqrt[2]*a)/Sqrt[a^2 - Sqrt[3]*Sq
rt[-a^4]])*Sqrt[x]*(b + a*x)*Sqrt[(b^2 + a^2*x^2)/(b + a*x)^2]*EllipticF[2*ArcTan[(Sqrt[a]*Sqrt[x])/Sqrt[b]],
1/2])/(Sqrt[3]*Sqrt[-a^4]*(3*a^2 + Sqrt[3]*Sqrt[-a^4])*Sqrt[b]*Sqrt[b^2*x + a^2*x^3]) + (2*a^(7/2)*(1 + (Sqrt[
2]*a)/Sqrt[a^2 - Sqrt[3]*Sqrt[-a^4]])*Sqrt[x]*(b + a*x)*Sqrt[(b^2 + a^2*x^2)/(b + a*x)^2]*EllipticF[2*ArcTan[(
Sqrt[a]*Sqrt[x])/Sqrt[b]], 1/2])/(Sqrt[3]*Sqrt[-a^4]*(3*a^2 + Sqrt[3]*Sqrt[-a^4])*Sqrt[b]*Sqrt[b^2*x + a^2*x^3
]) - (2*a^(7/2)*(1 - (Sqrt[2]*a)/Sqrt[a^2 + Sqrt[3]*Sqrt[-a^4]])*Sqrt[x]*(b + a*x)*Sqrt[(b^2 + a^2*x^2)/(b + a
*x)^2]*EllipticF[2*ArcTan[(Sqrt[a]*Sqrt[x])/Sqrt[b]], 1/2])/(Sqrt[3]*Sqrt[-a^4]*(3*a^2 - Sqrt[3]*Sqrt[-a^4])*S
qrt[b]*Sqrt[b^2*x + a^2*x^3]) - (2*a^(7/2)*(1 + (Sqrt[2]*a)/Sqrt[a^2 + Sqrt[3]*Sqrt[-a^4]])*Sqrt[x]*(b + a*x)*
Sqrt[(b^2 + a^2*x^2)/(b + a*x)^2]*EllipticF[2*ArcTan[(Sqrt[a]*Sqrt[x])/Sqrt[b]], 1/2])/(Sqrt[3]*Sqrt[-a^4]*(3*
a^2 - Sqrt[3]*Sqrt[-a^4])*Sqrt[b]*Sqrt[b^2*x + a^2*x^3]) - ((Sqrt[2]*a + Sqrt[a^2 - Sqrt[3]*Sqrt[-a^4]])^2*Sqr
t[x]*(b + a*x)*Sqrt[(b^2 + a^2*x^2)/(b + a*x)^2]*EllipticPi[-1/4*(Sqrt[2]*a - Sqrt[a^2 - Sqrt[3]*Sqrt[-a^4]])^
2/(Sqrt[2]*a*Sqrt[a^2 - Sqrt[3]*Sqrt[-a^4]]), 2*ArcTan[(Sqrt[a]*Sqrt[x])/Sqrt[b]], 1/2])/(2*Sqrt[3]*Sqrt[a]*(S
qrt[3]*a^2 + 3*Sqrt[-a^4])*Sqrt[b]*Sqrt[b^2*x + a^2*x^3]) - ((Sqrt[2]*a - Sqrt[a^2 - Sqrt[3]*Sqrt[-a^4]])^2*Sq
rt[x]*(b + a*x)*Sqrt[(b^2 + a^2*x^2)/(b + a*x)^2]*EllipticPi[(Sqrt[2]*a + Sqrt[a^2 - Sqrt[3]*Sqrt[-a^4]])^2/(4
*Sqrt[2]*a*Sqrt[a^2 - Sqrt[3]*Sqrt[-a^4]]), 2*ArcTan[(Sqrt[a]*Sqrt[x])/Sqrt[b]], 1/2])/(2*Sqrt[3]*Sqrt[a]*(Sqr
t[3]*a^2 + 3*Sqrt[-a^4])*Sqrt[b]*Sqrt[b^2*x + a^2*x^3]) - ((Sqrt[2]*a + Sqrt[a^2 + Sqrt[3]*Sqrt[-a^4]])^2*Sqrt
[x]*(b + a*x)*Sqrt[(b^2 + a^2*x^2)/(b + a*x)^2]*EllipticPi[-1/4*(Sqrt[2]*a - Sqrt[a^2 + Sqrt[3]*Sqrt[-a^4]])^2
/(Sqrt[2]*a*Sqrt[a^2 + Sqrt[3]*Sqrt[-a^4]]), 2*ArcTan[(Sqrt[a]*Sqrt[x])/Sqrt[b]], 1/2])/(2*Sqrt[3]*Sqrt[a]*(Sq
rt[3]*a^2 - 3*Sqrt[-a^4])*Sqrt[b]*Sqrt[b^2*x + a^2*x^3]) - ((Sqrt[2]*a - Sqrt[a^2 + Sqrt[3]*Sqrt[-a^4]])^2*Sqr
t[x]*(b + a*x)*Sqrt[(b^2 + a^2*x^2)/(b + a*x)^2]*EllipticPi[(Sqrt[2]*a + Sqrt[a^2 + Sqrt[3]*Sqrt[-a^4]])^2/(4*
Sqrt[2]*a*Sqrt[a^2 + Sqrt[3]*Sqrt[-a^4]]), 2*ArcTan[(Sqrt[a]*Sqrt[x])/Sqrt[b]], 1/2])/(2*Sqrt[3]*Sqrt[a]*(Sqrt
[3]*a^2 - 3*Sqrt[-a^4])*Sqrt[b]*Sqrt[b^2*x + a^2*x^3])

Rule 226

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^4)/(a*(
1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^4]))*EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rule 418

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

Rule 425

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-b)*x*(a + b*x^n)^(p + 1)*
((c + d*x^n)^(q + 1)/(a*n*(p + 1)*(b*c - a*d))), x] + Dist[1/(a*n*(p + 1)*(b*c - a*d)), Int[(a + b*x^n)^(p + 1
)*(c + d*x^n)^q*Simp[b*c + n*(p + 1)*(b*c - a*d) + d*b*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c,
d, n, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &&  !( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomi
alQ[a, b, c, d, n, p, q, x]

Rule 537

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*Sqrt[(c_) + (d_.)*(x_)^(n_)]), x_Symbol] :> Dist[f/b, I
nt[1/Sqrt[c + d*x^n], x], x] + Dist[(b*e - a*f)/b, Int[1/((a + b*x^n)*Sqrt[c + d*x^n]), x], x] /; FreeQ[{a, b,
c, d, e, f, n}, x]

Rule 1231

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[c/a, 2]}, Dist[(c*d + a*e*q
)/(c*d^2 - a*e^2), Int[1/Sqrt[a + c*x^4], x], x] - Dist[(a*e*(e + d*q))/(c*d^2 - a*e^2), Int[(1 + q*x^2)/((d +
e*x^2)*Sqrt[a + c*x^4]), x], x]] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0]
&& PosQ[c/a]

Rule 1721

Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[B/A, 2]
}, Simp[(-(B*d - A*e))*(ArcTan[Rt[c*(d/e) + a*(e/d), 2]*(x/Sqrt[a + c*x^4])]/(2*d*e*Rt[c*(d/e) + a*(e/d), 2]))
, x] + Simp[(B*d + A*e)*(A + B*x^2)*(Sqrt[A^2*((a + c*x^4)/(a*(A + B*x^2)^2))]/(4*d*e*A*q*Sqrt[a + c*x^4]))*El
lipticPi[Cancel[-(B*d - A*e)^2/(4*d*e*A*B)], 2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, c, d, e, A, B}, x] && NeQ[c
*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a] && EqQ[c*A^2 - a*B^2, 0]

Rule 2081

Int[(u_.)*(P_)^(p_.), x_Symbol] :> With[{m = MinimumMonomialExponent[P, x]}, Dist[P^FracPart[p]/(x^(m*FracPart
[p])*Distrib[1/x^m, P]^FracPart[p]), Int[u*x^(m*p)*Distrib[1/x^m, P]^p, x], x]] /; FreeQ[p, x] &&  !IntegerQ[p
] && SumQ[P] && EveryQ[BinomialQ[#1, x] & , P] &&  !PolyQ[P, x, 2]

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]

Rule 6847

Int[(u_)*(x_)^(m_.), x_Symbol] :> Dist[1/(m + 1), Subst[Int[SubstFor[x^(m + 1), u, x], x], x, x^(m + 1)], x] /
; FreeQ[m, x] && NeQ[m, -1] && FunctionOfQ[x^(m + 1), u, x]

Rule 6857

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \int \frac {-b^6+a^6 x^6}{\sqrt {x} \sqrt {b^2+a^2 x^2} \left (b^6+a^6 x^6\right )} \, dx}{\sqrt {b^2 x+a^2 x^3}} \\ & = \frac {\left (2 \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {-b^6+a^6 x^{12}}{\sqrt {b^2+a^2 x^4} \left (b^6+a^6 x^{12}\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {b^2 x+a^2 x^3}} \\ & = \frac {\left (2 \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \left (\frac {1}{\sqrt {b^2+a^2 x^4}}-\frac {2 b^6}{\sqrt {b^2+a^2 x^4} \left (b^6+a^6 x^{12}\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {b^2 x+a^2 x^3}} \\ & = \frac {\left (2 \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {b^2 x+a^2 x^3}}-\frac {\left (4 b^6 \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b^2+a^2 x^4} \left (b^6+a^6 x^{12}\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {b^2 x+a^2 x^3}} \\ & = \frac {\sqrt {x} (b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),\frac {1}{2}\right )}{\sqrt {a} \sqrt {b} \sqrt {b^2 x+a^2 x^3}}-\frac {\left (4 b^6 \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \left (-\frac {2 a^4}{\sqrt {3} \sqrt {-a^4} b^2 \left (b^2+a^2 x^4\right )^{3/2} \left (a^2 b^2+\sqrt {3} \sqrt {-a^4} b^2-2 a^4 x^4\right )}-\frac {2 a^4}{\sqrt {3} \sqrt {-a^4} b^2 \left (b^2+a^2 x^4\right )^{3/2} \left (-a^2 b^2+\sqrt {3} \sqrt {-a^4} b^2+2 a^4 x^4\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {b^2 x+a^2 x^3}} \\ & = \frac {\sqrt {x} (b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),\frac {1}{2}\right )}{\sqrt {a} \sqrt {b} \sqrt {b^2 x+a^2 x^3}}-\frac {\left (8 \sqrt {-a^4} b^4 \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (b^2+a^2 x^4\right )^{3/2} \left (a^2 b^2+\sqrt {3} \sqrt {-a^4} b^2-2 a^4 x^4\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {3} \sqrt {b^2 x+a^2 x^3}}-\frac {\left (8 \sqrt {-a^4} b^4 \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (b^2+a^2 x^4\right )^{3/2} \left (-a^2 b^2+\sqrt {3} \sqrt {-a^4} b^2+2 a^4 x^4\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {3} \sqrt {b^2 x+a^2 x^3}} \\ & = \frac {4 \sqrt {-a^4} x}{\sqrt {3} \left (3 a^2-\sqrt {3} \sqrt {-a^4}\right ) \sqrt {b^2 x+a^2 x^3}}-\frac {4 \sqrt {-a^4} x}{\sqrt {3} \left (3 a^2+\sqrt {3} \sqrt {-a^4}\right ) \sqrt {b^2 x+a^2 x^3}}+\frac {\sqrt {x} (b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),\frac {1}{2}\right )}{\sqrt {a} \sqrt {b} \sqrt {b^2 x+a^2 x^3}}-\frac {\left (4 \sqrt {-a^4} \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {a^2 \left (5 a^2-\sqrt {3} \sqrt {-a^4}\right ) b^2-2 a^6 x^4}{\sqrt {b^2+a^2 x^4} \left (-a^2 b^2+\sqrt {3} \sqrt {-a^4} b^2+2 a^4 x^4\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {3} a^2 \left (3 a^2-\sqrt {3} \sqrt {-a^4}\right ) \sqrt {b^2 x+a^2 x^3}}+\frac {\left (4 \sqrt {-a^4} \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {-a^2 \left (5 a^2+\sqrt {3} \sqrt {-a^4}\right ) b^2+2 a^6 x^4}{\sqrt {b^2+a^2 x^4} \left (a^2 b^2+\sqrt {3} \sqrt {-a^4} b^2-2 a^4 x^4\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {3} a^2 \left (3 a^2+\sqrt {3} \sqrt {-a^4}\right ) \sqrt {b^2 x+a^2 x^3}} \\ & = \frac {4 \sqrt {-a^4} x}{\sqrt {3} \left (3 a^2-\sqrt {3} \sqrt {-a^4}\right ) \sqrt {b^2 x+a^2 x^3}}-\frac {4 \sqrt {-a^4} x}{\sqrt {3} \left (3 a^2+\sqrt {3} \sqrt {-a^4}\right ) \sqrt {b^2 x+a^2 x^3}}+\frac {\sqrt {x} (b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),\frac {1}{2}\right )}{\sqrt {a} \sqrt {b} \sqrt {b^2 x+a^2 x^3}}+\frac {\left (4 \sqrt {-a^4} \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {3} \left (3 a^2-\sqrt {3} \sqrt {-a^4}\right ) \sqrt {b^2 x+a^2 x^3}}-\frac {\left (4 \sqrt {-a^4} \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {3} \left (3 a^2+\sqrt {3} \sqrt {-a^4}\right ) \sqrt {b^2 x+a^2 x^3}}-\frac {\left (16 a^2 \sqrt {-a^4} b^2 \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b^2+a^2 x^4} \left (-a^2 b^2+\sqrt {3} \sqrt {-a^4} b^2+2 a^4 x^4\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {3} \left (3 a^2-\sqrt {3} \sqrt {-a^4}\right ) \sqrt {b^2 x+a^2 x^3}}-\frac {\left (16 a^2 \sqrt {-a^4} b^2 \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b^2+a^2 x^4} \left (a^2 b^2+\sqrt {3} \sqrt {-a^4} b^2-2 a^4 x^4\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {3} \left (3 a^2+\sqrt {3} \sqrt {-a^4}\right ) \sqrt {b^2 x+a^2 x^3}} \\ & = \frac {4 \sqrt {-a^4} x}{\sqrt {3} \left (3 a^2-\sqrt {3} \sqrt {-a^4}\right ) \sqrt {b^2 x+a^2 x^3}}-\frac {4 \sqrt {-a^4} x}{\sqrt {3} \left (3 a^2+\sqrt {3} \sqrt {-a^4}\right ) \sqrt {b^2 x+a^2 x^3}}+\frac {\sqrt {x} (b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),\frac {1}{2}\right )}{\sqrt {a} \sqrt {b} \sqrt {b^2 x+a^2 x^3}}+\frac {2 \sqrt {-a^4} \sqrt {x} (b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),\frac {1}{2}\right )}{\sqrt {3} \sqrt {a} \left (3 a^2-\sqrt {3} \sqrt {-a^4}\right ) \sqrt {b} \sqrt {b^2 x+a^2 x^3}}-\frac {2 \sqrt {-a^4} \sqrt {x} (b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),\frac {1}{2}\right )}{\sqrt {3} \sqrt {a} \left (3 a^2+\sqrt {3} \sqrt {-a^4}\right ) \sqrt {b} \sqrt {b^2 x+a^2 x^3}}+\frac {\left (8 a^2 \sqrt {-a^4} \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (1-\frac {\sqrt {2} a^2 x^2}{\sqrt {a^2-\sqrt {3} \sqrt {-a^4}} b}\right ) \sqrt {b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {3} \left (a^2-\sqrt {3} \sqrt {-a^4}\right ) \left (3 a^2-\sqrt {3} \sqrt {-a^4}\right ) \sqrt {b^2 x+a^2 x^3}}+\frac {\left (8 a^2 \sqrt {-a^4} \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (1+\frac {\sqrt {2} a^2 x^2}{\sqrt {a^2-\sqrt {3} \sqrt {-a^4}} b}\right ) \sqrt {b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {3} \left (a^2-\sqrt {3} \sqrt {-a^4}\right ) \left (3 a^2-\sqrt {3} \sqrt {-a^4}\right ) \sqrt {b^2 x+a^2 x^3}}-\frac {\left (8 a^2 \sqrt {-a^4} \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (1-\frac {\sqrt {2} a^2 x^2}{\sqrt {a^2+\sqrt {3} \sqrt {-a^4}} b}\right ) \sqrt {b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {3} \left (a^2+\sqrt {3} \sqrt {-a^4}\right ) \left (3 a^2+\sqrt {3} \sqrt {-a^4}\right ) \sqrt {b^2 x+a^2 x^3}}-\frac {\left (8 a^2 \sqrt {-a^4} \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (1+\frac {\sqrt {2} a^2 x^2}{\sqrt {a^2+\sqrt {3} \sqrt {-a^4}} b}\right ) \sqrt {b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {3} \left (a^2+\sqrt {3} \sqrt {-a^4}\right ) \left (3 a^2+\sqrt {3} \sqrt {-a^4}\right ) \sqrt {b^2 x+a^2 x^3}} \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.85 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.02 \[ \int \frac {-b^6+a^6 x^6}{\sqrt {b^2 x+a^2 x^3} \left (b^6+a^6 x^6\right )} \, dx=-\frac {2 \sqrt {x} \left (3 \sqrt {a} \sqrt {b} \sqrt {x}+3^{3/4} \sqrt {b^2+a^2 x^2} \arctan \left (\frac {\sqrt [4]{3} \sqrt {a} \sqrt {b} \sqrt {x}}{\sqrt {b^2+a^2 x^2}}\right )+3^{3/4} \sqrt {b^2+a^2 x^2} \text {arctanh}\left (\frac {\sqrt [4]{3} \sqrt {a} \sqrt {b} \sqrt {x}}{\sqrt {b^2+a^2 x^2}}\right )\right )}{9 \sqrt {a} \sqrt {b} \sqrt {x \left (b^2+a^2 x^2\right )}} \]

[In]

Integrate[(-b^6 + a^6*x^6)/(Sqrt[b^2*x + a^2*x^3]*(b^6 + a^6*x^6)),x]

[Out]

(-2*Sqrt[x]*(3*Sqrt[a]*Sqrt[b]*Sqrt[x] + 3^(3/4)*Sqrt[b^2 + a^2*x^2]*ArcTan[(3^(1/4)*Sqrt[a]*Sqrt[b]*Sqrt[x])/
Sqrt[b^2 + a^2*x^2]] + 3^(3/4)*Sqrt[b^2 + a^2*x^2]*ArcTanh[(3^(1/4)*Sqrt[a]*Sqrt[b]*Sqrt[x])/Sqrt[b^2 + a^2*x^
2]]))/(9*Sqrt[a]*Sqrt[b]*Sqrt[x*(b^2 + a^2*x^2)])

Maple [A] (verified)

Time = 1.34 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.07

method	result	size
default	\(\frac {\left (2 \arctan \left (\frac {\sqrt {x \left (a^{2} x^{2}+b^{2}\right )}\, 3^{\frac {3}{4}}}{3 x \left (a^{2} b^{2}\right )^{\frac {1}{4}}}\right ) \sqrt {x \left (a^{2} x^{2}+b^{2}\right )}-\ln \left (\frac {x 3^{\frac {1}{4}} \left (a^{2} b^{2}\right )^{\frac {1}{4}}+\sqrt {x \left (a^{2} x^{2}+b^{2}\right )}}{-x 3^{\frac {1}{4}} \left (a^{2} b^{2}\right )^{\frac {1}{4}}+\sqrt {x \left (a^{2} x^{2}+b^{2}\right )}}\right ) \sqrt {x \left (a^{2} x^{2}+b^{2}\right )}-2 x 3^{\frac {1}{4}} \left (a^{2} b^{2}\right )^{\frac {1}{4}}\right ) 3^{\frac {3}{4}}}{9 \sqrt {x \left (a^{2} x^{2}+b^{2}\right )}\, \left (a^{2} b^{2}\right )^{\frac {1}{4}}}\)	\(178\)
pseudoelliptic	\(\frac {\left (2 \arctan \left (\frac {\sqrt {x \left (a^{2} x^{2}+b^{2}\right )}\, 3^{\frac {3}{4}}}{3 x \left (a^{2} b^{2}\right )^{\frac {1}{4}}}\right ) \sqrt {x \left (a^{2} x^{2}+b^{2}\right )}-\ln \left (\frac {x 3^{\frac {1}{4}} \left (a^{2} b^{2}\right )^{\frac {1}{4}}+\sqrt {x \left (a^{2} x^{2}+b^{2}\right )}}{-x 3^{\frac {1}{4}} \left (a^{2} b^{2}\right )^{\frac {1}{4}}+\sqrt {x \left (a^{2} x^{2}+b^{2}\right )}}\right ) \sqrt {x \left (a^{2} x^{2}+b^{2}\right )}-2 x 3^{\frac {1}{4}} \left (a^{2} b^{2}\right )^{\frac {1}{4}}\right ) 3^{\frac {3}{4}}}{9 \sqrt {x \left (a^{2} x^{2}+b^{2}\right )}\, \left (a^{2} b^{2}\right )^{\frac {1}{4}}}\)	\(178\)
elliptic	\(-\frac {2 x}{3 \sqrt {\left (x^{2}+\frac {b^{2}}{a^{2}}\right ) a^{2} x}}+\frac {2 i b \sqrt {-\frac {i \left (x +\frac {i b}{a}\right ) a}{b}}\, \sqrt {2}\, \sqrt {\frac {i \left (x -\frac {i b}{a}\right ) a}{b}}\, \sqrt {\frac {i x a}{b}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {i \left (x +\frac {i b}{a}\right ) a}{b}}, \frac {\sqrt {2}}{2}\right )}{3 a \sqrt {a^{2} x^{3}+b^{2} x}}+\frac {i \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{4} a^{4}-\textit {\_Z}^{2} a^{2} b^{2}+b^{4}\right )}{\sum }\frac {\left (\underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}-2 b^{2}\right ) \left (a^{3} \underline {\hspace {1.25 ex}}\alpha ^{3}-i \underline {\hspace {1.25 ex}}\alpha ^{2} a^{2} b -2 \underline {\hspace {1.25 ex}}\alpha a \,b^{2}+2 i b^{3}\right ) \sqrt {-\frac {i \left (x +\frac {i b}{a}\right ) a}{b}}\, \sqrt {\frac {i \left (x -\frac {i b}{a}\right ) a}{b}}\, \sqrt {\frac {i x a}{b}}\, \operatorname {EllipticPi}\left (\sqrt {-\frac {i \left (x +\frac {i b}{a}\right ) a}{b}}, -\frac {i \underline {\hspace {1.25 ex}}\alpha ^{3} a^{3}+b \,\underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}-2 i \underline {\hspace {1.25 ex}}\alpha a \,b^{2}-2 b^{3}}{3 b^{3}}, \frac {\sqrt {2}}{2}\right )}{\underline {\hspace {1.25 ex}}\alpha \left (2 \underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}-b^{2}\right ) \sqrt {x \left (a^{2} x^{2}+b^{2}\right )}}\right )}{9 b \,a^{2}}\)	\(341\)

[In]

int((a^6*x^6-b^6)/(a^2*x^3+b^2*x)^(1/2)/(a^6*x^6+b^6),x,method=_RETURNVERBOSE)

[Out]

1/9*(2*arctan(1/3*(x*(a^2*x^2+b^2))^(1/2)/x*3^(3/4)/(a^2*b^2)^(1/4))*(x*(a^2*x^2+b^2))^(1/2)-ln((x*3^(1/4)*(a^
2*b^2)^(1/4)+(x*(a^2*x^2+b^2))^(1/2))/(-x*3^(1/4)*(a^2*b^2)^(1/4)+(x*(a^2*x^2+b^2))^(1/2)))*(x*(a^2*x^2+b^2))^
(1/2)-2*x*3^(1/4)*(a^2*b^2)^(1/4))/(x*(a^2*x^2+b^2))^(1/2)*3^(3/4)/(a^2*b^2)^(1/4)

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.35 (sec) , antiderivative size = 742, normalized size of antiderivative = 4.44 \[ \int \frac {-b^6+a^6 x^6}{\sqrt {b^2 x+a^2 x^3} \left (b^6+a^6 x^6\right )} \, dx=-\frac {\left (\frac {1}{3}\right )^{\frac {1}{4}} {\left (a^{2} x^{2} + b^{2}\right )} \left (\frac {1}{a^{2} b^{2}}\right )^{\frac {1}{4}} \log \left (\frac {a^{4} x^{4} + 5 \, a^{2} b^{2} x^{2} + b^{4} + 6 \, \sqrt {\frac {1}{3}} {\left (a^{4} b^{2} x^{3} + a^{2} b^{4} x\right )} \sqrt {\frac {1}{a^{2} b^{2}}} + 6 \, {\left (\left (\frac {1}{3}\right )^{\frac {1}{4}} a^{2} b^{2} x \left (\frac {1}{a^{2} b^{2}}\right )^{\frac {1}{4}} + \left (\frac {1}{3}\right )^{\frac {3}{4}} {\left (a^{4} b^{2} x^{2} + a^{2} b^{4}\right )} \left (\frac {1}{a^{2} b^{2}}\right )^{\frac {3}{4}}\right )} \sqrt {a^{2} x^{3} + b^{2} x}}{a^{4} x^{4} - a^{2} b^{2} x^{2} + b^{4}}\right ) - \left (\frac {1}{3}\right )^{\frac {1}{4}} {\left (a^{2} x^{2} + b^{2}\right )} \left (\frac {1}{a^{2} b^{2}}\right )^{\frac {1}{4}} \log \left (\frac {a^{4} x^{4} + 5 \, a^{2} b^{2} x^{2} + b^{4} + 6 \, \sqrt {\frac {1}{3}} {\left (a^{4} b^{2} x^{3} + a^{2} b^{4} x\right )} \sqrt {\frac {1}{a^{2} b^{2}}} - 6 \, {\left (\left (\frac {1}{3}\right )^{\frac {1}{4}} a^{2} b^{2} x \left (\frac {1}{a^{2} b^{2}}\right )^{\frac {1}{4}} + \left (\frac {1}{3}\right )^{\frac {3}{4}} {\left (a^{4} b^{2} x^{2} + a^{2} b^{4}\right )} \left (\frac {1}{a^{2} b^{2}}\right )^{\frac {3}{4}}\right )} \sqrt {a^{2} x^{3} + b^{2} x}}{a^{4} x^{4} - a^{2} b^{2} x^{2} + b^{4}}\right ) + \left (\frac {1}{3}\right )^{\frac {1}{4}} {\left (-i \, a^{2} x^{2} - i \, b^{2}\right )} \left (\frac {1}{a^{2} b^{2}}\right )^{\frac {1}{4}} \log \left (\frac {a^{4} x^{4} + 5 \, a^{2} b^{2} x^{2} + b^{4} - 6 \, \sqrt {\frac {1}{3}} {\left (a^{4} b^{2} x^{3} + a^{2} b^{4} x\right )} \sqrt {\frac {1}{a^{2} b^{2}}} - 6 \, {\left (i \, \left (\frac {1}{3}\right )^{\frac {1}{4}} a^{2} b^{2} x \left (\frac {1}{a^{2} b^{2}}\right )^{\frac {1}{4}} + \left (\frac {1}{3}\right )^{\frac {3}{4}} {\left (-i \, a^{4} b^{2} x^{2} - i \, a^{2} b^{4}\right )} \left (\frac {1}{a^{2} b^{2}}\right )^{\frac {3}{4}}\right )} \sqrt {a^{2} x^{3} + b^{2} x}}{a^{4} x^{4} - a^{2} b^{2} x^{2} + b^{4}}\right ) + \left (\frac {1}{3}\right )^{\frac {1}{4}} {\left (i \, a^{2} x^{2} + i \, b^{2}\right )} \left (\frac {1}{a^{2} b^{2}}\right )^{\frac {1}{4}} \log \left (\frac {a^{4} x^{4} + 5 \, a^{2} b^{2} x^{2} + b^{4} - 6 \, \sqrt {\frac {1}{3}} {\left (a^{4} b^{2} x^{3} + a^{2} b^{4} x\right )} \sqrt {\frac {1}{a^{2} b^{2}}} - 6 \, {\left (-i \, \left (\frac {1}{3}\right )^{\frac {1}{4}} a^{2} b^{2} x \left (\frac {1}{a^{2} b^{2}}\right )^{\frac {1}{4}} + \left (\frac {1}{3}\right )^{\frac {3}{4}} {\left (i \, a^{4} b^{2} x^{2} + i \, a^{2} b^{4}\right )} \left (\frac {1}{a^{2} b^{2}}\right )^{\frac {3}{4}}\right )} \sqrt {a^{2} x^{3} + b^{2} x}}{a^{4} x^{4} - a^{2} b^{2} x^{2} + b^{4}}\right ) + 4 \, \sqrt {a^{2} x^{3} + b^{2} x}}{6 \, {\left (a^{2} x^{2} + b^{2}\right )}} \]

[In]

integrate((a^6*x^6-b^6)/(a^2*x^3+b^2*x)^(1/2)/(a^6*x^6+b^6),x, algorithm="fricas")

[Out]

-1/6*((1/3)^(1/4)*(a^2*x^2 + b^2)*(1/(a^2*b^2))^(1/4)*log((a^4*x^4 + 5*a^2*b^2*x^2 + b^4 + 6*sqrt(1/3)*(a^4*b^
2*x^3 + a^2*b^4*x)*sqrt(1/(a^2*b^2)) + 6*((1/3)^(1/4)*a^2*b^2*x*(1/(a^2*b^2))^(1/4) + (1/3)^(3/4)*(a^4*b^2*x^2
+ a^2*b^4)*(1/(a^2*b^2))^(3/4))*sqrt(a^2*x^3 + b^2*x))/(a^4*x^4 - a^2*b^2*x^2 + b^4)) - (1/3)^(1/4)*(a^2*x^2
+ b^2)*(1/(a^2*b^2))^(1/4)*log((a^4*x^4 + 5*a^2*b^2*x^2 + b^4 + 6*sqrt(1/3)*(a^4*b^2*x^3 + a^2*b^4*x)*sqrt(1/(
a^2*b^2)) - 6*((1/3)^(1/4)*a^2*b^2*x*(1/(a^2*b^2))^(1/4) + (1/3)^(3/4)*(a^4*b^2*x^2 + a^2*b^4)*(1/(a^2*b^2))^(
3/4))*sqrt(a^2*x^3 + b^2*x))/(a^4*x^4 - a^2*b^2*x^2 + b^4)) + (1/3)^(1/4)*(-I*a^2*x^2 - I*b^2)*(1/(a^2*b^2))^(
1/4)*log((a^4*x^4 + 5*a^2*b^2*x^2 + b^4 - 6*sqrt(1/3)*(a^4*b^2*x^3 + a^2*b^4*x)*sqrt(1/(a^2*b^2)) - 6*(I*(1/3)
^(1/4)*a^2*b^2*x*(1/(a^2*b^2))^(1/4) + (1/3)^(3/4)*(-I*a^4*b^2*x^2 - I*a^2*b^4)*(1/(a^2*b^2))^(3/4))*sqrt(a^2*
x^3 + b^2*x))/(a^4*x^4 - a^2*b^2*x^2 + b^4)) + (1/3)^(1/4)*(I*a^2*x^2 + I*b^2)*(1/(a^2*b^2))^(1/4)*log((a^4*x^
4 + 5*a^2*b^2*x^2 + b^4 - 6*sqrt(1/3)*(a^4*b^2*x^3 + a^2*b^4*x)*sqrt(1/(a^2*b^2)) - 6*(-I*(1/3)^(1/4)*a^2*b^2*
x*(1/(a^2*b^2))^(1/4) + (1/3)^(3/4)*(I*a^4*b^2*x^2 + I*a^2*b^4)*(1/(a^2*b^2))^(3/4))*sqrt(a^2*x^3 + b^2*x))/(a
^4*x^4 - a^2*b^2*x^2 + b^4)) + 4*sqrt(a^2*x^3 + b^2*x))/(a^2*x^2 + b^2)

Sympy [F]

\[ \int \frac {-b^6+a^6 x^6}{\sqrt {b^2 x+a^2 x^3} \left (b^6+a^6 x^6\right )} \, dx=\int \frac {\left (a x - b\right ) \left (a x + b\right ) \left (a^{2} x^{2} - a b x + b^{2}\right ) \left (a^{2} x^{2} + a b x + b^{2}\right )}{\sqrt {x \left (a^{2} x^{2} + b^{2}\right )} \left (a^{2} x^{2} + b^{2}\right ) \left (a^{4} x^{4} - a^{2} b^{2} x^{2} + b^{4}\right )}\, dx \]

[In]

integrate((a**6*x**6-b**6)/(a**2*x**3+b**2*x)**(1/2)/(a**6*x**6+b**6),x)

[Out]

Integral((a*x - b)*(a*x + b)*(a**2*x**2 - a*b*x + b**2)*(a**2*x**2 + a*b*x + b**2)/(sqrt(x*(a**2*x**2 + b**2))
*(a**2*x**2 + b**2)*(a**4*x**4 - a**2*b**2*x**2 + b**4)), x)

Maxima [F]

\[ \int \frac {-b^6+a^6 x^6}{\sqrt {b^2 x+a^2 x^3} \left (b^6+a^6 x^6\right )} \, dx=\int { \frac {a^{6} x^{6} - b^{6}}{{\left (a^{6} x^{6} + b^{6}\right )} \sqrt {a^{2} x^{3} + b^{2} x}} \,d x } \]

[In]

integrate((a^6*x^6-b^6)/(a^2*x^3+b^2*x)^(1/2)/(a^6*x^6+b^6),x, algorithm="maxima")

[Out]

integrate((a^6*x^6 - b^6)/((a^6*x^6 + b^6)*sqrt(a^2*x^3 + b^2*x)), x)

Giac [F]

\[ \int \frac {-b^6+a^6 x^6}{\sqrt {b^2 x+a^2 x^3} \left (b^6+a^6 x^6\right )} \, dx=\int { \frac {a^{6} x^{6} - b^{6}}{{\left (a^{6} x^{6} + b^{6}\right )} \sqrt {a^{2} x^{3} + b^{2} x}} \,d x } \]

[In]

integrate((a^6*x^6-b^6)/(a^2*x^3+b^2*x)^(1/2)/(a^6*x^6+b^6),x, algorithm="giac")

[Out]

integrate((a^6*x^6 - b^6)/((a^6*x^6 + b^6)*sqrt(a^2*x^3 + b^2*x)), x)

Mupad [B] (verification not implemented)

Time = 10.42 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.20 \[ \int \frac {-b^6+a^6 x^6}{\sqrt {b^2 x+a^2 x^3} \left (b^6+a^6 x^6\right )} \, dx=\frac {3^{3/4}\,\ln \left (\frac {3^{3/4}\,b^2-6\,\sqrt {a}\,\sqrt {b}\,\sqrt {a^2\,x^3+b^2\,x}+3^{3/4}\,a^2\,x^2+3\,3^{1/4}\,a\,b\,x}{a^2\,x^2-\sqrt {3}\,a\,b\,x+b^2}\right )}{9\,\sqrt {a}\,\sqrt {b}}-\frac {2\,\sqrt {a^2\,x^3+b^2\,x}}{3\,\left (a^2\,x^2+b^2\right )}+\frac {3^{3/4}\,\ln \left (\frac {3^{3/4}\,b^2+3^{3/4}\,a^2\,x^2-3\,3^{1/4}\,a\,b\,x+\sqrt {a}\,\sqrt {b}\,\sqrt {a^2\,x^3+b^2\,x}\,6{}\mathrm {i}}{a^2\,x^2+\sqrt {3}\,a\,b\,x+b^2}\right )\,1{}\mathrm {i}}{9\,\sqrt {a}\,\sqrt {b}} \]

[In]

int(-(b^6 - a^6*x^6)/((b^6 + a^6*x^6)*(b^2*x + a^2*x^3)^(1/2)),x)

[Out]

(3^(3/4)*log((3^(3/4)*b^2 - 6*a^(1/2)*b^(1/2)*(b^2*x + a^2*x^3)^(1/2) + 3^(3/4)*a^2*x^2 + 3*3^(1/4)*a*b*x)/(b^
2 + a^2*x^2 - 3^(1/2)*a*b*x)))/(9*a^(1/2)*b^(1/2)) - (2*(b^2*x + a^2*x^3)^(1/2))/(3*(b^2 + a^2*x^2)) + (3^(3/4
)*log((3^(3/4)*b^2 + a^(1/2)*b^(1/2)*(b^2*x + a^2*x^3)^(1/2)*6i + 3^(3/4)*a^2*x^2 - 3*3^(1/4)*a*b*x)/(b^2 + a^
2*x^2 + 3^(1/2)*a*b*x))*1i)/(9*a^(1/2)*b^(1/2))