3.16.58 \(\int \frac {-b^3+a^3 x^3}{(b^3+a^3 x^3) \sqrt {b^4+a^4 x^4}} \, dx\)

Optimal. Leaf size=107 \[ -\frac {4 \tan ^{-1}\left (\frac {a b x}{\sqrt {a^4 x^4+b^4}+a^2 x^2-a b x+b^2}\right )}{3 a b}-\frac {\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} a b x}{\sqrt {a^4 x^4+b^4}+a^2 x^2+2 a b x+b^2}\right )}{3 a b} \]

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Rubi [C]  time = 2.66, antiderivative size = 662, normalized size of antiderivative = 6.19, number of steps used = 29, number of rules used = 13, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.310, Rules used = {6725, 220, 2074, 1725, 1211, 1699, 208, 1248, 725, 206, 6728, 1217, 1707} \begin {gather*} -\frac {2 \tan ^{-1}\left (\frac {a b x}{\sqrt {a^4 x^4+b^4}}\right )}{3 a b}-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} a b x}{\sqrt {a^4 x^4+b^4}}\right )}{3 \sqrt {2} a b}+\frac {\tanh ^{-1}\left (\frac {a^2 x^2+b^2}{\sqrt {2} \sqrt {a^4 x^4+b^4}}\right )}{3 \sqrt {2} a b}-\frac {\left (a-\sqrt {3} \sqrt {-a^2}\right ) \left (a^2 x^2+b^2\right ) \sqrt {\frac {a^4 x^4+b^4}{\left (a^2 x^2+b^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {a x}{b}\right )|\frac {1}{2}\right )}{6 a^2 b \sqrt {a^4 x^4+b^4}}-\frac {\left (\sqrt {3} \sqrt {-a^2}+a\right ) \left (a^2 x^2+b^2\right ) \sqrt {\frac {a^4 x^4+b^4}{\left (a^2 x^2+b^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {a x}{b}\right )|\frac {1}{2}\right )}{6 a^2 b \sqrt {a^4 x^4+b^4}}+\frac {\left (a^2 x^2+b^2\right ) \sqrt {\frac {a^4 x^4+b^4}{\left (a^2 x^2+b^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {a x}{b}\right )|\frac {1}{2}\right )}{3 a b \sqrt {a^4 x^4+b^4}}-\frac {\left (a-\sqrt {3} \sqrt {-a^2}\right ) \tanh ^{-1}\left (\frac {\sqrt {a} \left (\left (a-\sqrt {3} \sqrt {-a^2}\right )^2 x^2+4 b^2\right )}{2 \sqrt {2} \sqrt {\sqrt {3} \sqrt {-a^2}+a} \sqrt {a^4 x^4+b^4}}\right )}{3 \sqrt {2} a^{3/2} \sqrt {\sqrt {3} \sqrt {-a^2}+a} b}-\frac {\left (\sqrt {3} \sqrt {-a^2}+a\right ) \tanh ^{-1}\left (\frac {\sqrt {a} \left (\left (\sqrt {3} \sqrt {-a^2}+a\right )^2 x^2+4 b^2\right )}{2 \sqrt {2} \sqrt {a-\sqrt {3} \sqrt {-a^2}} \sqrt {a^4 x^4+b^4}}\right )}{3 \sqrt {2} a^{3/2} \sqrt {a-\sqrt {3} \sqrt {-a^2}} b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-b^3 + a^3*x^3)/((b^3 + a^3*x^3)*Sqrt[b^4 + a^4*x^4]),x]

[Out]

(-2*ArcTan[(a*b*x)/Sqrt[b^4 + a^4*x^4]])/(3*a*b) - ArcTanh[(Sqrt[2]*a*b*x)/Sqrt[b^4 + a^4*x^4]]/(3*Sqrt[2]*a*b
) + ArcTanh[(b^2 + a^2*x^2)/(Sqrt[2]*Sqrt[b^4 + a^4*x^4])]/(3*Sqrt[2]*a*b) - ((a - Sqrt[3]*Sqrt[-a^2])*ArcTanh
[(Sqrt[a]*(4*b^2 + (a - Sqrt[3]*Sqrt[-a^2])^2*x^2))/(2*Sqrt[2]*Sqrt[a + Sqrt[3]*Sqrt[-a^2]]*Sqrt[b^4 + a^4*x^4
])])/(3*Sqrt[2]*a^(3/2)*Sqrt[a + Sqrt[3]*Sqrt[-a^2]]*b) - ((a + Sqrt[3]*Sqrt[-a^2])*ArcTanh[(Sqrt[a]*(4*b^2 +
(a + Sqrt[3]*Sqrt[-a^2])^2*x^2))/(2*Sqrt[2]*Sqrt[a - Sqrt[3]*Sqrt[-a^2]]*Sqrt[b^4 + a^4*x^4])])/(3*Sqrt[2]*a^(
3/2)*Sqrt[a - Sqrt[3]*Sqrt[-a^2]]*b) + ((b^2 + a^2*x^2)*Sqrt[(b^4 + a^4*x^4)/(b^2 + a^2*x^2)^2]*EllipticF[2*Ar
cTan[(a*x)/b], 1/2])/(3*a*b*Sqrt[b^4 + a^4*x^4]) - ((a - Sqrt[3]*Sqrt[-a^2])*(b^2 + a^2*x^2)*Sqrt[(b^4 + a^4*x
^4)/(b^2 + a^2*x^2)^2]*EllipticF[2*ArcTan[(a*x)/b], 1/2])/(6*a^2*b*Sqrt[b^4 + a^4*x^4]) - ((a + Sqrt[3]*Sqrt[-
a^2])*(b^2 + a^2*x^2)*Sqrt[(b^4 + a^4*x^4)/(b^2 + a^2*x^2)^2]*EllipticF[2*ArcTan[(a*x)/b], 1/2])/(6*a^2*b*Sqrt
[b^4 + a^4*x^4])

Rule 206

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

Rule 208

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

Rule 220

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)]*EllipticF[2*ArcTan[q*x], 1/2])/(2*q*Sqrt[a + b*x^4]), x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rule 725

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]

Rule 1211

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> Dist[1/(2*d), Int[1/Sqrt[a + c*x^4], x],
x] + Dist[1/(2*d), Int[(d - e*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] && EqQ[c*d^2 - a*e^2, 0]

Rule 1217

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 1248

Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[(d + e*x)^q
*(a + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, c, d, e, p, q}, x]

Rule 1699

Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> Dist[A, Subst[Int[1/
(d + 2*a*e*x^2), x], x, x/Sqrt[a + c*x^4]], x] /; FreeQ[{a, c, d, e, A, B}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ
[c*d^2 - a*e^2, 0] && EqQ[B*d + A*e, 0]

Rule 1707

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)]*EllipticPi[Cancel[-((B*d - A*e)^2
/(4*d*e*A*B))], 2*ArcTan[q*x], 1/2])/(4*d*e*A*q*Sqrt[a + c*x^4]), 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 1725

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

Rule 2074

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P]}, Int[ExpandIntegrand[PP^p*Q^q, x], x] /;  !SumQ[N
onfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x] && PolyQ[Q, x] && IntegerQ[p] && NeQ[P, x]

Rule 6725

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]

Rule 6728

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int \frac {-b^3+a^3 x^3}{\left (b^3+a^3 x^3\right ) \sqrt {b^4+a^4 x^4}} \, dx &=\int \left (\frac {1}{\sqrt {b^4+a^4 x^4}}-\frac {2 b^3}{\left (b^3+a^3 x^3\right ) \sqrt {b^4+a^4 x^4}}\right ) \, dx\\ &=-\left (\left (2 b^3\right ) \int \frac {1}{\left (b^3+a^3 x^3\right ) \sqrt {b^4+a^4 x^4}} \, dx\right )+\int \frac {1}{\sqrt {b^4+a^4 x^4}} \, dx\\ &=\frac {\left (b^2+a^2 x^2\right ) \sqrt {\frac {b^4+a^4 x^4}{\left (b^2+a^2 x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {a x}{b}\right )|\frac {1}{2}\right )}{2 a b \sqrt {b^4+a^4 x^4}}-\left (2 b^3\right ) \int \left (\frac {1}{3 b^2 (b+a x) \sqrt {b^4+a^4 x^4}}+\frac {2 b-a x}{3 b^2 \left (b^2-a b x+a^2 x^2\right ) \sqrt {b^4+a^4 x^4}}\right ) \, dx\\ &=\frac {\left (b^2+a^2 x^2\right ) \sqrt {\frac {b^4+a^4 x^4}{\left (b^2+a^2 x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {a x}{b}\right )|\frac {1}{2}\right )}{2 a b \sqrt {b^4+a^4 x^4}}-\frac {1}{3} (2 b) \int \frac {1}{(b+a x) \sqrt {b^4+a^4 x^4}} \, dx-\frac {1}{3} (2 b) \int \frac {2 b-a x}{\left (b^2-a b x+a^2 x^2\right ) \sqrt {b^4+a^4 x^4}} \, dx\\ &=\frac {\left (b^2+a^2 x^2\right ) \sqrt {\frac {b^4+a^4 x^4}{\left (b^2+a^2 x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {a x}{b}\right )|\frac {1}{2}\right )}{2 a b \sqrt {b^4+a^4 x^4}}-\frac {1}{3} (2 b) \int \left (\frac {-a-\sqrt {3} \sqrt {-a^2}}{\left (-a b-\sqrt {3} \sqrt {-a^2} b+2 a^2 x\right ) \sqrt {b^4+a^4 x^4}}+\frac {-a+\sqrt {3} \sqrt {-a^2}}{\left (-a b+\sqrt {3} \sqrt {-a^2} b+2 a^2 x\right ) \sqrt {b^4+a^4 x^4}}\right ) \, dx+\frac {1}{3} (2 a b) \int \frac {x}{\left (b^2-a^2 x^2\right ) \sqrt {b^4+a^4 x^4}} \, dx-\frac {1}{3} \left (2 b^2\right ) \int \frac {1}{\left (b^2-a^2 x^2\right ) \sqrt {b^4+a^4 x^4}} \, dx\\ &=\frac {\left (b^2+a^2 x^2\right ) \sqrt {\frac {b^4+a^4 x^4}{\left (b^2+a^2 x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {a x}{b}\right )|\frac {1}{2}\right )}{2 a b \sqrt {b^4+a^4 x^4}}-\frac {1}{3} \int \frac {1}{\sqrt {b^4+a^4 x^4}} \, dx-\frac {1}{3} \int \frac {b^2+a^2 x^2}{\left (b^2-a^2 x^2\right ) \sqrt {b^4+a^4 x^4}} \, dx+\frac {1}{3} (a b) \operatorname {Subst}\left (\int \frac {1}{\left (b^2-a^2 x\right ) \sqrt {b^4+a^4 x^2}} \, dx,x,x^2\right )+\frac {1}{3} \left (2 \left (a-\sqrt {3} \sqrt {-a^2}\right ) b\right ) \int \frac {1}{\left (-a b+\sqrt {3} \sqrt {-a^2} b+2 a^2 x\right ) \sqrt {b^4+a^4 x^4}} \, dx+\frac {1}{3} \left (2 \left (a+\sqrt {3} \sqrt {-a^2}\right ) b\right ) \int \frac {1}{\left (-a b-\sqrt {3} \sqrt {-a^2} b+2 a^2 x\right ) \sqrt {b^4+a^4 x^4}} \, dx\\ &=\frac {\left (b^2+a^2 x^2\right ) \sqrt {\frac {b^4+a^4 x^4}{\left (b^2+a^2 x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {a x}{b}\right )|\frac {1}{2}\right )}{3 a b \sqrt {b^4+a^4 x^4}}-\frac {1}{3} (a b) \operatorname {Subst}\left (\int \frac {1}{2 a^4 b^4-x^2} \, dx,x,\frac {-a^2 b^4-a^4 b^2 x^2}{\sqrt {b^4+a^4 x^4}}\right )-\frac {1}{3} \left (4 a^2 \left (a-\sqrt {3} \sqrt {-a^2}\right ) b\right ) \int \frac {x}{\left (\left (-a b+\sqrt {3} \sqrt {-a^2} b\right )^2-4 a^4 x^2\right ) \sqrt {b^4+a^4 x^4}} \, dx-\frac {1}{3} \left (4 a^2 \left (a+\sqrt {3} \sqrt {-a^2}\right ) b\right ) \int \frac {x}{\left (\left (-a b-\sqrt {3} \sqrt {-a^2} b\right )^2-4 a^4 x^2\right ) \sqrt {b^4+a^4 x^4}} \, dx-\frac {1}{3} b^2 \operatorname {Subst}\left (\int \frac {1}{b^2-2 a^2 b^4 x^2} \, dx,x,\frac {x}{\sqrt {b^4+a^4 x^4}}\right )-\frac {1}{3} \left (2 \left (a-\sqrt {3} \sqrt {-a^2}\right )^2 b^2\right ) \int \frac {1}{\left (\left (-a b+\sqrt {3} \sqrt {-a^2} b\right )^2-4 a^4 x^2\right ) \sqrt {b^4+a^4 x^4}} \, dx-\frac {1}{3} \left (2 \left (a+\sqrt {3} \sqrt {-a^2}\right )^2 b^2\right ) \int \frac {1}{\left (\left (-a b-\sqrt {3} \sqrt {-a^2} b\right )^2-4 a^4 x^2\right ) \sqrt {b^4+a^4 x^4}} \, dx\\ &=-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} a b x}{\sqrt {b^4+a^4 x^4}}\right )}{3 \sqrt {2} a b}+\frac {\tanh ^{-1}\left (\frac {b^2+a^2 x^2}{\sqrt {2} \sqrt {b^4+a^4 x^4}}\right )}{3 \sqrt {2} a b}+\frac {\left (b^2+a^2 x^2\right ) \sqrt {\frac {b^4+a^4 x^4}{\left (b^2+a^2 x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {a x}{b}\right )|\frac {1}{2}\right )}{3 a b \sqrt {b^4+a^4 x^4}}-\frac {\left (a-\sqrt {3} \sqrt {-a^2}\right ) \int \frac {1}{\sqrt {b^4+a^4 x^4}} \, dx}{3 a}-\frac {\left (a+\sqrt {3} \sqrt {-a^2}\right ) \int \frac {1}{\sqrt {b^4+a^4 x^4}} \, dx}{3 a}-\frac {1}{3} \left (2 a^2 \left (a-\sqrt {3} \sqrt {-a^2}\right ) b\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\left (-a b+\sqrt {3} \sqrt {-a^2} b\right )^2-4 a^4 x\right ) \sqrt {b^4+a^4 x^2}} \, dx,x,x^2\right )-\frac {1}{3} \left (2 a^2 \left (a+\sqrt {3} \sqrt {-a^2}\right ) b\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\left (-a b-\sqrt {3} \sqrt {-a^2} b\right )^2-4 a^4 x\right ) \sqrt {b^4+a^4 x^2}} \, dx,x,x^2\right )-\frac {1}{3} \left (4 a \left (a-\sqrt {3} \sqrt {-a^2}\right ) b^2\right ) \int \frac {1+\frac {a^2 x^2}{b^2}}{\left (\left (-a b+\sqrt {3} \sqrt {-a^2} b\right )^2-4 a^4 x^2\right ) \sqrt {b^4+a^4 x^4}} \, dx-\frac {1}{3} \left (4 a \left (a+\sqrt {3} \sqrt {-a^2}\right ) b^2\right ) \int \frac {1+\frac {a^2 x^2}{b^2}}{\left (\left (-a b-\sqrt {3} \sqrt {-a^2} b\right )^2-4 a^4 x^2\right ) \sqrt {b^4+a^4 x^4}} \, dx\\ &=-\frac {2 \tan ^{-1}\left (\frac {a b x}{\sqrt {b^4+a^4 x^4}}\right )}{3 a b}-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} a b x}{\sqrt {b^4+a^4 x^4}}\right )}{3 \sqrt {2} a b}+\frac {\tanh ^{-1}\left (\frac {b^2+a^2 x^2}{\sqrt {2} \sqrt {b^4+a^4 x^4}}\right )}{3 \sqrt {2} a b}+\frac {\left (b^2+a^2 x^2\right ) \sqrt {\frac {b^4+a^4 x^4}{\left (b^2+a^2 x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {a x}{b}\right )|\frac {1}{2}\right )}{3 a b \sqrt {b^4+a^4 x^4}}-\frac {\left (a-\sqrt {3} \sqrt {-a^2}\right ) \left (b^2+a^2 x^2\right ) \sqrt {\frac {b^4+a^4 x^4}{\left (b^2+a^2 x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {a x}{b}\right )|\frac {1}{2}\right )}{6 a^2 b \sqrt {b^4+a^4 x^4}}-\frac {\left (a+\sqrt {3} \sqrt {-a^2}\right ) \left (b^2+a^2 x^2\right ) \sqrt {\frac {b^4+a^4 x^4}{\left (b^2+a^2 x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {a x}{b}\right )|\frac {1}{2}\right )}{6 a^2 b \sqrt {b^4+a^4 x^4}}+\frac {1}{3} \left (2 a^2 \left (a-\sqrt {3} \sqrt {-a^2}\right ) b\right ) \operatorname {Subst}\left (\int \frac {1}{16 a^8 b^4+a^4 \left (-a b+\sqrt {3} \sqrt {-a^2} b\right )^4-x^2} \, dx,x,\frac {-4 a^4 b^4-a^4 \left (-a b+\sqrt {3} \sqrt {-a^2} b\right )^2 x^2}{\sqrt {b^4+a^4 x^4}}\right )+\frac {1}{3} \left (2 a^2 \left (a+\sqrt {3} \sqrt {-a^2}\right ) b\right ) \operatorname {Subst}\left (\int \frac {1}{16 a^8 b^4+a^4 \left (-a b-\sqrt {3} \sqrt {-a^2} b\right )^4-x^2} \, dx,x,\frac {-4 a^4 b^4-a^4 \left (-a b-\sqrt {3} \sqrt {-a^2} b\right )^2 x^2}{\sqrt {b^4+a^4 x^4}}\right )\\ &=-\frac {2 \tan ^{-1}\left (\frac {a b x}{\sqrt {b^4+a^4 x^4}}\right )}{3 a b}-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} a b x}{\sqrt {b^4+a^4 x^4}}\right )}{3 \sqrt {2} a b}+\frac {\tanh ^{-1}\left (\frac {b^2+a^2 x^2}{\sqrt {2} \sqrt {b^4+a^4 x^4}}\right )}{3 \sqrt {2} a b}-\frac {\left (a-\sqrt {3} \sqrt {-a^2}\right ) \tanh ^{-1}\left (\frac {\sqrt {a} \left (4 b^2+\left (a-\sqrt {3} \sqrt {-a^2}\right )^2 x^2\right )}{2 \sqrt {2} \sqrt {a+\sqrt {3} \sqrt {-a^2}} \sqrt {b^4+a^4 x^4}}\right )}{3 \sqrt {2} a^{3/2} \sqrt {a+\sqrt {3} \sqrt {-a^2}} b}-\frac {\left (a+\sqrt {3} \sqrt {-a^2}\right ) \tanh ^{-1}\left (\frac {\sqrt {a} \left (4 b^2+\left (a+\sqrt {3} \sqrt {-a^2}\right )^2 x^2\right )}{2 \sqrt {2} \sqrt {a-\sqrt {3} \sqrt {-a^2}} \sqrt {b^4+a^4 x^4}}\right )}{3 \sqrt {2} a^{3/2} \sqrt {a-\sqrt {3} \sqrt {-a^2}} b}+\frac {\left (b^2+a^2 x^2\right ) \sqrt {\frac {b^4+a^4 x^4}{\left (b^2+a^2 x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {a x}{b}\right )|\frac {1}{2}\right )}{3 a b \sqrt {b^4+a^4 x^4}}-\frac {\left (a-\sqrt {3} \sqrt {-a^2}\right ) \left (b^2+a^2 x^2\right ) \sqrt {\frac {b^4+a^4 x^4}{\left (b^2+a^2 x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {a x}{b}\right )|\frac {1}{2}\right )}{6 a^2 b \sqrt {b^4+a^4 x^4}}-\frac {\left (a+\sqrt {3} \sqrt {-a^2}\right ) \left (b^2+a^2 x^2\right ) \sqrt {\frac {b^4+a^4 x^4}{\left (b^2+a^2 x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {a x}{b}\right )|\frac {1}{2}\right )}{6 a^2 b \sqrt {b^4+a^4 x^4}}\\ \end {align*}

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Mathematica [C]  time = 3.40, size = 562, normalized size = 5.25 \begin {gather*} \frac {a^3 \left (18 i a \sqrt [4]{a^4} b \sqrt {\frac {a^4 x^4}{b^4}+1} F\left (\left .i \sinh ^{-1}\left (\sqrt {\frac {i a^2}{b^2}} x\right )\right |-1\right )-\frac {3}{2} \left (8 i a \sqrt [4]{a^4} b \sqrt {\frac {a^4 x^4}{b^4}+1} \Pi \left (-\frac {i}{2}-\frac {\sqrt {3}}{2};\left .i \sinh ^{-1}\left (\sqrt {\frac {i a^2}{b^2}} x\right )\right |-1\right )+8 i a \sqrt [4]{a^4} b \sqrt {\frac {a^4 x^4}{b^4}+1} \Pi \left (\frac {1}{2} \left (-i+\sqrt {3}\right );\left .i \sinh ^{-1}\left (\sqrt {\frac {i a^2}{b^2}} x\right )\right |-1\right )+\sqrt {2} \sqrt {\frac {i a^2}{b^2}} \left ((4+4 i) a b \sqrt [4]{b^4} \sqrt {\frac {a^4 x^4}{b^4}+1} \Pi \left (\frac {i \sqrt {a^4} \sqrt {b^4}}{a^2 b^2};\left .i \sinh ^{-1}\left (\frac {(1+i) \sqrt [4]{a^4} x}{\sqrt {2} \sqrt [4]{b^4}}\right )\right |-1\right )+\sqrt [4]{a^4} \sqrt {a^4 x^4+b^4} \left (\sqrt {1+i \sqrt {3}} \left (\sqrt {3}-i\right ) \tan ^{-1}\left (\frac {\left (\sqrt {3}+i\right ) b^2-2 i a^2 x^2}{\sqrt {2-2 i \sqrt {3}} \sqrt {a^4 x^4+b^4}}\right )+\sqrt {1-i \sqrt {3}} \left (\sqrt {3}+i\right ) \tan ^{-1}\left (\frac {2 i a^2 x^2+\left (\sqrt {3}-i\right ) b^2}{\sqrt {2+2 i \sqrt {3}} \sqrt {a^4 x^4+b^4}}\right )+2 \tanh ^{-1}\left (\frac {a^2 x^2+b^2}{\sqrt {2} \sqrt {a^4 x^4+b^4}}\right )\right )\right )\right )\right )}{18 \sqrt [4]{a^4} b^5 \left (\frac {i a^2}{b^2}\right )^{5/2} \sqrt {a^4 x^4+b^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-b^3 + a^3*x^3)/((b^3 + a^3*x^3)*Sqrt[b^4 + a^4*x^4]),x]

[Out]

(a^3*((18*I)*a*(a^4)^(1/4)*b*Sqrt[1 + (a^4*x^4)/b^4]*EllipticF[I*ArcSinh[Sqrt[(I*a^2)/b^2]*x], -1] - (3*((8*I)
*a*(a^4)^(1/4)*b*Sqrt[1 + (a^4*x^4)/b^4]*EllipticPi[-1/2*I - Sqrt[3]/2, I*ArcSinh[Sqrt[(I*a^2)/b^2]*x], -1] +
(8*I)*a*(a^4)^(1/4)*b*Sqrt[1 + (a^4*x^4)/b^4]*EllipticPi[(-I + Sqrt[3])/2, I*ArcSinh[Sqrt[(I*a^2)/b^2]*x], -1]
+ Sqrt[2]*Sqrt[(I*a^2)/b^2]*((a^4)^(1/4)*Sqrt[b^4 + a^4*x^4]*(Sqrt[1 + I*Sqrt[3]]*(-I + Sqrt[3])*ArcTan[((I +
Sqrt[3])*b^2 - (2*I)*a^2*x^2)/(Sqrt[2 - (2*I)*Sqrt[3]]*Sqrt[b^4 + a^4*x^4])] + Sqrt[1 - I*Sqrt[3]]*(I + Sqrt[
3])*ArcTan[((-I + Sqrt[3])*b^2 + (2*I)*a^2*x^2)/(Sqrt[2 + (2*I)*Sqrt[3]]*Sqrt[b^4 + a^4*x^4])] + 2*ArcTanh[(b^
2 + a^2*x^2)/(Sqrt[2]*Sqrt[b^4 + a^4*x^4])]) + (4 + 4*I)*a*b*(b^4)^(1/4)*Sqrt[1 + (a^4*x^4)/b^4]*EllipticPi[(I
*Sqrt[a^4]*Sqrt[b^4])/(a^2*b^2), I*ArcSinh[((1 + I)*(a^4)^(1/4)*x)/(Sqrt[2]*(b^4)^(1/4))], -1])))/2))/(18*(a^4
)^(1/4)*((I*a^2)/b^2)^(5/2)*b^5*Sqrt[b^4 + a^4*x^4])

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IntegrateAlgebraic [A]  time = 2.31, size = 107, normalized size = 1.00 \begin {gather*} -\frac {4 \tan ^{-1}\left (\frac {a b x}{b^2-a b x+a^2 x^2+\sqrt {b^4+a^4 x^4}}\right )}{3 a b}-\frac {\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} a b x}{b^2+2 a b x+a^2 x^2+\sqrt {b^4+a^4 x^4}}\right )}{3 a b} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(-b^3 + a^3*x^3)/((b^3 + a^3*x^3)*Sqrt[b^4 + a^4*x^4]),x]

[Out]

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

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fricas [A]  time = 0.77, size = 165, normalized size = 1.54 \begin {gather*} \frac {\sqrt {2} \log \left (-\frac {3 \, a^{4} x^{4} + 4 \, a^{3} b x^{3} + 6 \, a^{2} b^{2} x^{2} + 4 \, a b^{3} x + 3 \, b^{4} + 2 \, \sqrt {2} \sqrt {a^{4} x^{4} + b^{4}} {\left (a^{2} x^{2} + a b x + b^{2}\right )}}{a^{4} x^{4} + 4 \, a^{3} b x^{3} + 6 \, a^{2} b^{2} x^{2} + 4 \, a b^{3} x + b^{4}}\right ) - 8 \, \arctan \left (\frac {\sqrt {a^{4} x^{4} + b^{4}}}{a^{2} x^{2} - 2 \, a b x + b^{2}}\right )}{12 \, a b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a^3*x^3-b^3)/(a^3*x^3+b^3)/(a^4*x^4+b^4)^(1/2),x, algorithm="fricas")

[Out]

1/12*(sqrt(2)*log(-(3*a^4*x^4 + 4*a^3*b*x^3 + 6*a^2*b^2*x^2 + 4*a*b^3*x + 3*b^4 + 2*sqrt(2)*sqrt(a^4*x^4 + b^4
)*(a^2*x^2 + a*b*x + b^2))/(a^4*x^4 + 4*a^3*b*x^3 + 6*a^2*b^2*x^2 + 4*a*b^3*x + b^4)) - 8*arctan(sqrt(a^4*x^4
+ b^4)/(a^2*x^2 - 2*a*b*x + b^2)))/(a*b)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a^{3} x^{3} - b^{3}}{\sqrt {a^{4} x^{4} + b^{4}} {\left (a^{3} x^{3} + b^{3}\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a^3*x^3-b^3)/(a^3*x^3+b^3)/(a^4*x^4+b^4)^(1/2),x, algorithm="giac")

[Out]

integrate((a^3*x^3 - b^3)/(sqrt(a^4*x^4 + b^4)*(a^3*x^3 + b^3)), x)

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maple [C]  time = 0.22, size = 457, normalized size = 4.27

method	result	size
default	\(\frac {\sqrt {1-\frac {i a^{2} x^{2}}{b^{2}}}\, \sqrt {1+\frac {i a^{2} x^{2}}{b^{2}}}\, \EllipticF \left (x \sqrt {\frac {i a^{2}}{b^{2}}}, i\right )}{\sqrt {\frac {i a^{2}}{b^{2}}}\, \sqrt {a^{4} x^{4}+b^{4}}}-\frac {b \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (a^{2} \textit {\_Z}^{2}-a b \textit {\_Z} +b^{2}\right )}{\sum }\frac {\left (-\underline {\hspace {1.25 ex}}\alpha a +2 b \right ) \left (-\frac {\arctanh \left (\frac {\left (\underline {\hspace {1.25 ex}}\alpha a -b \right ) a b \left (a \,x^{2}-b \underline {\hspace {1.25 ex}}\alpha \right )}{\sqrt {-b^{3} \left (\underline {\hspace {1.25 ex}}\alpha a -b \right )}\, \sqrt {a^{4} x^{4}+b^{4}}}\right )}{\sqrt {-b^{3} \left (\underline {\hspace {1.25 ex}}\alpha a -b \right )}}+\frac {2 a \left (\underline {\hspace {1.25 ex}}\alpha a -b \right ) \sqrt {1-\frac {i a^{2} x^{2}}{b^{2}}}\, \sqrt {1+\frac {i a^{2} x^{2}}{b^{2}}}\, \EllipticPi \left (x \sqrt {\frac {i a^{2}}{b^{2}}}, \frac {i \underline {\hspace {1.25 ex}}\alpha a}{b}, \frac {\sqrt {-\frac {i a^{2}}{b^{2}}}}{\sqrt {\frac {i a^{2}}{b^{2}}}}\right )}{\sqrt {\frac {i a^{2}}{b^{2}}}\, b^{2} \sqrt {a^{4} x^{4}+b^{4}}}\right )}{2 \underline {\hspace {1.25 ex}}\alpha a -b}\right )}{3 a}-\frac {2 b \left (-\frac {\sqrt {2}\, \arctanh \left (\frac {\left (2 a^{2} b^{2} x^{2}+2 b^{4}\right ) \sqrt {2}}{4 \sqrt {b^{4}}\, \sqrt {a^{4} x^{4}+b^{4}}}\right )}{4 \sqrt {b^{4}}}+\frac {a \sqrt {1-\frac {i a^{2} x^{2}}{b^{2}}}\, \sqrt {1+\frac {i a^{2} x^{2}}{b^{2}}}\, \EllipticPi \left (x \sqrt {\frac {i a^{2}}{b^{2}}}, -i, \frac {\sqrt {-\frac {i a^{2}}{b^{2}}}}{\sqrt {\frac {i a^{2}}{b^{2}}}}\right )}{\sqrt {\frac {i a^{2}}{b^{2}}}\, b \sqrt {a^{4} x^{4}+b^{4}}}\right )}{3 a}\)	\(457\)
elliptic	\(-\frac {a^{3} b^{5} \sqrt {2}\, \ln \left (\frac {\frac {b^{2} \left (a^{2} b^{2}-\sqrt {-3 a^{4} b^{4}}\right )}{a^{2}}+\left (-a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}\right ) \left (x^{2}-\frac {-a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}}{2 a^{4}}\right )+\frac {\sqrt {2}\, \sqrt {\frac {b^{2} \left (a^{2} b^{2}-\sqrt {-3 a^{4} b^{4}}\right )}{a^{2}}}\, \sqrt {4 \left (x^{2}-\frac {-a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}}{2 a^{4}}\right )^{2} a^{4}+4 \left (-a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}\right ) \left (x^{2}-\frac {-a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}}{2 a^{4}}\right )+\frac {2 b^{2} \left (a^{2} b^{2}-\sqrt {-3 a^{4} b^{4}}\right )}{a^{2}}}}{2}}{x^{2}-\frac {-a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}}{2 a^{4}}}\right )}{\left (-3 a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}\right ) \sqrt {-3 a^{4} b^{4}}\, \sqrt {\frac {b^{2} \left (a^{2} b^{2}-\sqrt {-3 a^{4} b^{4}}\right )}{a^{2}}}}+\frac {a \,b^{3} \sqrt {2}\, \ln \left (\frac {\frac {b^{2} \left (a^{2} b^{2}-\sqrt {-3 a^{4} b^{4}}\right )}{a^{2}}+\left (-a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}\right ) \left (x^{2}-\frac {-a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}}{2 a^{4}}\right )+\frac {\sqrt {2}\, \sqrt {\frac {b^{2} \left (a^{2} b^{2}-\sqrt {-3 a^{4} b^{4}}\right )}{a^{2}}}\, \sqrt {4 \left (x^{2}-\frac {-a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}}{2 a^{4}}\right )^{2} a^{4}+4 \left (-a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}\right ) \left (x^{2}-\frac {-a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}}{2 a^{4}}\right )+\frac {2 b^{2} \left (a^{2} b^{2}-\sqrt {-3 a^{4} b^{4}}\right )}{a^{2}}}}{2}}{x^{2}-\frac {-a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}}{2 a^{4}}}\right )}{\left (-3 a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}\right ) \sqrt {\frac {b^{2} \left (a^{2} b^{2}-\sqrt {-3 a^{4} b^{4}}\right )}{a^{2}}}}-\frac {2 a^{3} b^{5} \sqrt {2}\, \ln \left (\frac {4 b^{4}+2 a^{2} b^{2} \left (x^{2}-\frac {b^{2}}{a^{2}}\right )+2 \sqrt {2}\, \sqrt {b^{4}}\, \sqrt {\left (x^{2}-\frac {b^{2}}{a^{2}}\right )^{2} a^{4}+2 a^{2} b^{2} \left (x^{2}-\frac {b^{2}}{a^{2}}\right )+2 b^{4}}}{x^{2}-\frac {b^{2}}{a^{2}}}\right )}{\left (-3 a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}\right ) \left (3 a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}\right ) \sqrt {b^{4}}}-\frac {a^{3} b^{5} \sqrt {2}\, \ln \left (\frac {\frac {b^{2} \left (a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}\right )}{a^{2}}+\left (-a^{2} b^{2}-\sqrt {-3 a^{4} b^{4}}\right ) \left (x^{2}+\frac {a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}}{2 a^{4}}\right )+\frac {\sqrt {2}\, \sqrt {\frac {b^{2} \left (a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}\right )}{a^{2}}}\, \sqrt {4 \left (x^{2}+\frac {a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}}{2 a^{4}}\right )^{2} a^{4}+4 \left (-a^{2} b^{2}-\sqrt {-3 a^{4} b^{4}}\right ) \left (x^{2}+\frac {a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}}{2 a^{4}}\right )+\frac {2 b^{2} \left (a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}\right )}{a^{2}}}}{2}}{x^{2}+\frac {a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}}{2 a^{4}}}\right )}{\left (3 a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}\right ) \sqrt {-3 a^{4} b^{4}}\, \sqrt {\frac {b^{2} \left (a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}\right )}{a^{2}}}}-\frac {a \,b^{3} \sqrt {2}\, \ln \left (\frac {\frac {b^{2} \left (a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}\right )}{a^{2}}+\left (-a^{2} b^{2}-\sqrt {-3 a^{4} b^{4}}\right ) \left (x^{2}+\frac {a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}}{2 a^{4}}\right )+\frac {\sqrt {2}\, \sqrt {\frac {b^{2} \left (a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}\right )}{a^{2}}}\, \sqrt {4 \left (x^{2}+\frac {a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}}{2 a^{4}}\right )^{2} a^{4}+4 \left (-a^{2} b^{2}-\sqrt {-3 a^{4} b^{4}}\right ) \left (x^{2}+\frac {a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}}{2 a^{4}}\right )+\frac {2 b^{2} \left (a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}\right )}{a^{2}}}}{2}}{x^{2}+\frac {a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}}{2 a^{4}}}\right )}{\left (3 a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}\right ) \sqrt {\frac {b^{2} \left (a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}\right )}{a^{2}}}}+\frac {\left (-\frac {\ln \left (a b +\frac {\sqrt {a^{4} x^{4}+b^{4}}\, \sqrt {2}}{2 x}\right )}{6 a b}+\frac {2 \sqrt {2}\, \arctan \left (\frac {\sqrt {a^{4} x^{4}+b^{4}}}{a b x}\right )}{3 a b}+\frac {\ln \left (-a b +\frac {\sqrt {a^{4} x^{4}+b^{4}}\, \sqrt {2}}{2 x}\right )}{6 a b}\right ) \sqrt {2}}{2}\)	\(1539\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a^3*x^3-b^3)/(a^3*x^3+b^3)/(a^4*x^4+b^4)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/(I*a^2/b^2)^(1/2)*(1-I*a^2/b^2*x^2)^(1/2)*(1+I*a^2/b^2*x^2)^(1/2)/(a^4*x^4+b^4)^(1/2)*EllipticF(x*(I*a^2/b^2
)^(1/2),I)-1/3*b/a*sum((-_alpha*a+2*b)/(2*_alpha*a-b)*(-1/(-b^3*(_alpha*a-b))^(1/2)*arctanh((_alpha*a-b)*a*b*(
a*x^2-_alpha*b)/(-b^3*(_alpha*a-b))^(1/2)/(a^4*x^4+b^4)^(1/2))+2/(I*a^2/b^2)^(1/2)*a*(_alpha*a-b)/b^2*(1-I*a^2
/b^2*x^2)^(1/2)*(1+I*a^2/b^2*x^2)^(1/2)/(a^4*x^4+b^4)^(1/2)*EllipticPi(x*(I*a^2/b^2)^(1/2),I*_alpha*a/b,(-I*a^
2/b^2)^(1/2)/(I*a^2/b^2)^(1/2))),_alpha=RootOf(_Z^2*a^2-_Z*a*b+b^2))-2/3*b/a*(-1/4*2^(1/2)/(b^4)^(1/2)*arctanh
(1/4*(2*a^2*b^2*x^2+2*b^4)*2^(1/2)/(b^4)^(1/2)/(a^4*x^4+b^4)^(1/2))+1/(I*a^2/b^2)^(1/2)/b*a*(1-I*a^2/b^2*x^2)^
(1/2)*(1+I*a^2/b^2*x^2)^(1/2)/(a^4*x^4+b^4)^(1/2)*EllipticPi(x*(I*a^2/b^2)^(1/2),-I,(-I*a^2/b^2)^(1/2)/(I*a^2/
b^2)^(1/2)))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a^{3} x^{3} - b^{3}}{\sqrt {a^{4} x^{4} + b^{4}} {\left (a^{3} x^{3} + b^{3}\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a^3*x^3-b^3)/(a^3*x^3+b^3)/(a^4*x^4+b^4)^(1/2),x, algorithm="maxima")

[Out]

integrate((a^3*x^3 - b^3)/(sqrt(a^4*x^4 + b^4)*(a^3*x^3 + b^3)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int -\frac {b^3-a^3\,x^3}{\left (a^3\,x^3+b^3\right )\,\sqrt {a^4\,x^4+b^4}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(b^3 - a^3*x^3)/((b^3 + a^3*x^3)*(b^4 + a^4*x^4)^(1/2)),x)

[Out]

int(-(b^3 - a^3*x^3)/((b^3 + a^3*x^3)*(b^4 + a^4*x^4)^(1/2)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a x - b\right ) \left (a^{2} x^{2} + a b x + b^{2}\right )}{\left (a x + b\right ) \sqrt {a^{4} x^{4} + b^{4}} \left (a^{2} x^{2} - a b x + b^{2}\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a**3*x**3-b**3)/(a**3*x**3+b**3)/(a**4*x**4+b**4)**(1/2),x)

[Out]

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

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