3.19.10 \(\int \frac {-b^3+a^3 x^3}{\sqrt {b^2 x+a^2 x^3} (b^3+a^3 x^3)} \, dx\)

Optimal. Leaf size=123 \[ -\frac {\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a^2 x^3+b^2 x}}{a^2 x^2+b^2}\right )}{3 \sqrt {a} \sqrt {b}}-\frac {4 \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {b} \sqrt {a^2 x^3+b^2 x}}{a^2 x^2+b^2}\right )}{3 \sqrt {a} \sqrt {b}} \]

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Rubi [C]  time = 3.48, antiderivative size = 728, normalized size of antiderivative = 5.92, number of steps used = 19, number of rules used = 11, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2056, 6715, 6725, 220, 2073, 1211, 1699, 205, 6728, 1217, 1707} \begin {gather*} -\frac {4 \sqrt {x} \sqrt {a^2 x^2+b^2} \tan ^{-1}\left (\frac {\sqrt {-a} \sqrt {b} \sqrt {x}}{\sqrt {a^2 x^2+b^2}}\right )}{3 \sqrt {-a} \sqrt {b} \sqrt {a^2 x^3+b^2 x}}-\frac {\sqrt {2} \sqrt {x} \sqrt {a^2 x^2+b^2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {x}}{\sqrt {a^2 x^2+b^2}}\right )}{3 \sqrt {a} \sqrt {b} \sqrt {a^2 x^3+b^2 x}}+\frac {2 \sqrt {x} (a x+b) \sqrt {\frac {a^2 x^2+b^2}{(a x+b)^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )|\frac {1}{2}\right )}{3 \sqrt {a} \sqrt {b} \sqrt {a^2 x^3+b^2 x}}-\frac {2 \left (a-\sqrt {3} \sqrt {-a^2}\right ) \sqrt {x} (a x+b) \sqrt {\frac {a^2 x^2+b^2}{(a x+b)^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )|\frac {1}{2}\right )}{3 \sqrt {a} \left (3 a-\sqrt {3} \sqrt {-a^2}\right ) \sqrt {b} \sqrt {a^2 x^3+b^2 x}}-\frac {2 \left (\sqrt {3} \sqrt {-a^2}+a\right ) \sqrt {x} (a x+b) \sqrt {\frac {a^2 x^2+b^2}{(a x+b)^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )|\frac {1}{2}\right )}{3 \sqrt {a} \left (\sqrt {3} \sqrt {-a^2}+3 a\right ) \sqrt {b} \sqrt {a^2 x^3+b^2 x}}-\frac {\sqrt {a} \sqrt {x} (a x+b) \sqrt {\frac {a^2 x^2+b^2}{(a x+b)^2}} \Pi \left (\frac {3}{4};2 \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )|\frac {1}{2}\right )}{3 \sqrt {3} \sqrt {-a^2} \sqrt {b} \sqrt {a^2 x^3+b^2 x}}-\frac {\sqrt {-a^2} \sqrt {x} (a x+b) \sqrt {\frac {a^2 x^2+b^2}{(a x+b)^2}} \Pi \left (\frac {3}{4};2 \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )|\frac {1}{2}\right )}{3 \sqrt {3} a^{3/2} \sqrt {b} \sqrt {a^2 x^3+b^2 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[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 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 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 2056

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 2073

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 6715

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

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Mathematica [C]  time = 1.22, size = 208, normalized size = 1.69 \begin {gather*} -\frac {(-1)^{5/6} x^{3/2} \sqrt {\frac {b^2}{a^2 x^2}+1} \left (\left (3-3 i \sqrt {3}\right ) F\left (\left .i \sinh ^{-1}\left (\frac {\sqrt {\frac {i b}{a}}}{\sqrt {x}}\right )\right |-1\right )+4 \left ((-1)^{2/3} \Pi \left (-i;\left .i \sinh ^{-1}\left (\frac {\sqrt {\frac {i b}{a}}}{\sqrt {x}}\right )\right |-1\right )+\left (\sqrt [3]{-1}-1\right ) \Pi \left (\sqrt [6]{-1};\left .i \sinh ^{-1}\left (\frac {\sqrt {\frac {i b}{a}}}{\sqrt {x}}\right )\right |-1\right )+(-1)^{2/3} \Pi \left ((-1)^{5/6};\left .i \sinh ^{-1}\left (\frac {\sqrt {\frac {i b}{a}}}{\sqrt {x}}\right )\right |-1\right )\right )\right )}{3 \sqrt {\frac {i b}{a}} \sqrt {x \left (a^2 x^2+b^2\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*((-1)^(5/6)*Sqrt[1 + b^2/(a^2*x^2)]*x^(3/2)*((3 - (3*I)*Sqrt[3])*EllipticF[I*ArcSinh[Sqrt[(I*b)/a]/Sqrt[x
]], -1] + 4*((-1)^(2/3)*EllipticPi[-I, I*ArcSinh[Sqrt[(I*b)/a]/Sqrt[x]], -1] + (-1 + (-1)^(1/3))*EllipticPi[(-
1)^(1/6), I*ArcSinh[Sqrt[(I*b)/a]/Sqrt[x]], -1] + (-1)^(2/3)*EllipticPi[(-1)^(5/6), I*ArcSinh[Sqrt[(I*b)/a]/Sq
rt[x]], -1])))/(Sqrt[(I*b)/a]*Sqrt[x*(b^2 + a^2*x^2)])

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IntegrateAlgebraic [A]  time = 0.51, size = 123, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {b^2 x+a^2 x^3}}{b^2+a^2 x^2}\right )}{3 \sqrt {a} \sqrt {b}}-\frac {4 \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {b} \sqrt {b^2 x+a^2 x^3}}{b^2+a^2 x^2}\right )}{3 \sqrt {a} \sqrt {b}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.54, size = 421, normalized size = 3.42 \begin {gather*} \left [-\frac {\sqrt {2} a b \sqrt {\frac {1}{a b}} \arctan \left (\frac {2 \, \sqrt {2} \sqrt {a^{2} x^{3} + b^{2} x} a b \sqrt {\frac {1}{a b}}}{a^{2} x^{2} - 2 \, a b x + b^{2}}\right ) - 2 \, \sqrt {a b} \log \left (\frac {a^{4} x^{4} + 6 \, a^{3} b x^{3} + 3 \, a^{2} b^{2} x^{2} + 6 \, a b^{3} x + b^{4} - 4 \, \sqrt {a^{2} x^{3} + b^{2} x} {\left (a^{2} x^{2} + a b x + b^{2}\right )} \sqrt {a b}}{a^{4} x^{4} - 2 \, a^{3} b x^{3} + 3 \, a^{2} b^{2} x^{2} - 2 \, a b^{3} x + b^{4}}\right )}{6 \, a b}, \frac {\sqrt {2} a b \sqrt {-\frac {1}{a b}} \log \left (\frac {a^{4} x^{4} - 12 \, a^{3} b x^{3} + 6 \, a^{2} b^{2} x^{2} - 12 \, a b^{3} x + b^{4} + 4 \, \sqrt {2} {\left (a^{3} b x^{2} - 2 \, a^{2} b^{2} x + a b^{3}\right )} \sqrt {a^{2} x^{3} + b^{2} x} \sqrt {-\frac {1}{a b}}}{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 \, \sqrt {-a b} \arctan \left (\frac {\sqrt {a^{2} x^{3} + b^{2} x} {\left (a^{2} x^{2} + a b x + b^{2}\right )} \sqrt {-a b}}{2 \, {\left (a^{3} b x^{3} + a b^{3} x\right )}}\right )}{12 \, a b}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/6*(sqrt(2)*a*b*sqrt(1/(a*b))*arctan(2*sqrt(2)*sqrt(a^2*x^3 + b^2*x)*a*b*sqrt(1/(a*b))/(a^2*x^2 - 2*a*b*x +
b^2)) - 2*sqrt(a*b)*log((a^4*x^4 + 6*a^3*b*x^3 + 3*a^2*b^2*x^2 + 6*a*b^3*x + b^4 - 4*sqrt(a^2*x^3 + b^2*x)*(a
^2*x^2 + a*b*x + b^2)*sqrt(a*b))/(a^4*x^4 - 2*a^3*b*x^3 + 3*a^2*b^2*x^2 - 2*a*b^3*x + b^4)))/(a*b), 1/12*(sqrt
(2)*a*b*sqrt(-1/(a*b))*log((a^4*x^4 - 12*a^3*b*x^3 + 6*a^2*b^2*x^2 - 12*a*b^3*x + b^4 + 4*sqrt(2)*(a^3*b*x^2 -
2*a^2*b^2*x + a*b^3)*sqrt(a^2*x^3 + b^2*x)*sqrt(-1/(a*b)))/(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*sqrt(-a*b)*arctan(1/2*sqrt(a^2*x^3 + b^2*x)*(a^2*x^2 + a*b*x + b^2)*sqrt(-a*b)/(a^3*b*x^3 + a*b^3
*x)))/(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}}{{\left (a^{3} x^{3} + b^{3}\right )} \sqrt {a^{2} x^{3} + b^{2} x}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.21, size = 391, normalized size = 3.18

method	result	size
default	\(\frac {i b \sqrt {-\frac {i \left (x +\frac {i b}{a}\right ) a}{b}}\, \sqrt {2}\, \sqrt {\frac {i \left (x -\frac {i b}{a}\right ) a}{b}}\, \sqrt {\frac {i x a}{b}}\, \EllipticF \left (\sqrt {-\frac {i \left (x +\frac {i b}{a}\right ) a}{b}}, \frac {\sqrt {2}}{2}\right )}{a \sqrt {a^{2} x^{3}+b^{2} x}}+\frac {2 i \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{2} a^{2}-\textit {\_Z} a b +b^{2}\right )}{\sum }\frac {\left (-\underline {\hspace {1.25 ex}}\alpha a +2 b \right ) \left (i \underline {\hspace {1.25 ex}}\alpha a -i b +b \right ) \sqrt {-\frac {i \left (x +\frac {i b}{a}\right ) a}{b}}\, \sqrt {\frac {i \left (x -\frac {i b}{a}\right ) a}{b}}\, \sqrt {\frac {i x a}{b}}\, \EllipticPi \left (\sqrt {-\frac {i \left (x +\frac {i b}{a}\right ) a}{b}}, -\frac {\underline {\hspace {1.25 ex}}\alpha a -i b -b}{b}, \frac {\sqrt {2}}{2}\right )}{\left (2 \underline {\hspace {1.25 ex}}\alpha a -b \right ) \sqrt {x \left (a^{2} x^{2}+b^{2}\right )}}\right )}{3 a}-\frac {2 i b^{2} \sqrt {-\frac {i \left (x +\frac {i b}{a}\right ) a}{b}}\, \sqrt {2}\, \sqrt {\frac {i \left (x -\frac {i b}{a}\right ) a}{b}}\, \sqrt {\frac {i x a}{b}}\, \EllipticPi \left (\sqrt {-\frac {i \left (x +\frac {i b}{a}\right ) a}{b}}, -\frac {i b}{a \left (-\frac {i b}{a}+\frac {b}{a}\right )}, \frac {\sqrt {2}}{2}\right )}{3 a^{2} \sqrt {a^{2} x^{3}+b^{2} x}\, \left (-\frac {i b}{a}+\frac {b}{a}\right )}\)	\(391\)
elliptic	\(\frac {i b \sqrt {-\frac {i x a}{b}+1}\, \sqrt {2}\, \sqrt {\frac {i x a}{b}+1}\, \sqrt {\frac {i x a}{b}}\, \EllipticF \left (\sqrt {-\frac {i \left (x +\frac {i b}{a}\right ) a}{b}}, \frac {\sqrt {2}}{2}\right )}{a \sqrt {a^{2} x^{3}+b^{2} x}}-\frac {2 a \sqrt {2}\, \sqrt {-\frac {i \left (a x +i b \right )}{b}}\, \sqrt {-\frac {i \left (-a x +i b \right )}{b}}\, \sqrt {\frac {i x a}{b}}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{2} a^{2}-\textit {\_Z} a b +b^{2}\right )}{\sum }\frac {\EllipticPi \left (\sqrt {-\frac {i \left (a x +i b \right )}{b}}, \frac {-\underline {\hspace {1.25 ex}}\alpha a +i b +b}{b}, \frac {\sqrt {2}}{2}\right ) \underline {\hspace {1.25 ex}}\alpha ^{2}}{2 \underline {\hspace {1.25 ex}}\alpha a -b}\right )}{3 \sqrt {x \left (a^{2} x^{2}+b^{2}\right )}}+\frac {2 \sqrt {2}\, \sqrt {-\frac {i \left (a x +i b \right )}{b}}\, \sqrt {-\frac {i \left (-a x +i b \right )}{b}}\, \sqrt {\frac {i x a}{b}}\, b \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{2} a^{2}-\textit {\_Z} a b +b^{2}\right )}{\sum }\frac {\EllipticPi \left (\sqrt {-\frac {i \left (a x +i b \right )}{b}}, \frac {-\underline {\hspace {1.25 ex}}\alpha a +i b +b}{b}, \frac {\sqrt {2}}{2}\right ) \underline {\hspace {1.25 ex}}\alpha }{2 \underline {\hspace {1.25 ex}}\alpha a -b}\right )}{\sqrt {x \left (a^{2} x^{2}+b^{2}\right )}}-\frac {4 \sqrt {2}\, \sqrt {-\frac {i \left (a x +i b \right )}{b}}\, \sqrt {-\frac {i \left (-a x +i b \right )}{b}}\, \sqrt {\frac {i x a}{b}}\, b^{2} \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{2} a^{2}-\textit {\_Z} a b +b^{2}\right )}{\sum }\frac {\EllipticPi \left (\sqrt {-\frac {i \left (a x +i b \right )}{b}}, \frac {-\underline {\hspace {1.25 ex}}\alpha a +i b +b}{b}, \frac {\sqrt {2}}{2}\right )}{2 \underline {\hspace {1.25 ex}}\alpha a -b}\right )}{3 a \sqrt {x \left (a^{2} x^{2}+b^{2}\right )}}+\frac {2 i \sqrt {2}\, \sqrt {-\frac {i \left (a x +i b \right )}{b}}\, \sqrt {-\frac {i \left (-a x +i b \right )}{b}}\, \sqrt {\frac {i x a}{b}}\, b \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{2} a^{2}-\textit {\_Z} a b +b^{2}\right )}{\sum }\frac {\EllipticPi \left (\sqrt {-\frac {i \left (a x +i b \right )}{b}}, \frac {-\underline {\hspace {1.25 ex}}\alpha a +i b +b}{b}, \frac {\sqrt {2}}{2}\right ) \underline {\hspace {1.25 ex}}\alpha }{2 \underline {\hspace {1.25 ex}}\alpha a -b}\right )}{3 \sqrt {x \left (a^{2} x^{2}+b^{2}\right )}}-\frac {4 i \sqrt {2}\, \sqrt {-\frac {i \left (a x +i b \right )}{b}}\, \sqrt {-\frac {i \left (-a x +i b \right )}{b}}\, \sqrt {\frac {i x a}{b}}\, b^{2} \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{2} a^{2}-\textit {\_Z} a b +b^{2}\right )}{\sum }\frac {\EllipticPi \left (\sqrt {-\frac {i \left (a x +i b \right )}{b}}, \frac {-\underline {\hspace {1.25 ex}}\alpha a +i b +b}{b}, \frac {\sqrt {2}}{2}\right )}{2 \underline {\hspace {1.25 ex}}\alpha a -b}\right )}{3 a \sqrt {x \left (a^{2} x^{2}+b^{2}\right )}}-\frac {2 i b^{2} \sqrt {-\frac {i x a}{b}+1}\, \sqrt {2}\, \sqrt {\frac {i x a}{b}+1}\, \sqrt {\frac {i x a}{b}}\, \EllipticPi \left (\sqrt {-\frac {i \left (x +\frac {i b}{a}\right ) a}{b}}, -\frac {i b}{a \left (-\frac {i b}{a}+b \right )}, \frac {\sqrt {2}}{2}\right )}{3 a \sqrt {a^{2} x^{3}+b^{2} x}\, \left (-\frac {i b}{a}+b \right )}\)	\(871\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

I*b/a*(-I*(x+I*b/a)/b*a)^(1/2)*2^(1/2)*(I*(x-I*b/a)/b*a)^(1/2)*(I*x/b*a)^(1/2)/(a^2*x^3+b^2*x)^(1/2)*EllipticF
((-I*(x+I*b/a)/b*a)^(1/2),1/2*2^(1/2))+2/3*I/a*2^(1/2)*sum((-_alpha*a+2*b)/(2*_alpha*a-b)*(I*_alpha*a-I*b+b)*(
-I*(x+I*b/a)/b*a)^(1/2)*(I*(x-I*b/a)/b*a)^(1/2)*(I*x/b*a)^(1/2)/(x*(a^2*x^2+b^2))^(1/2)*EllipticPi((-I*(x+I*b/
a)/b*a)^(1/2),-(_alpha*a-b-I*b)/b,1/2*2^(1/2)),_alpha=RootOf(_Z^2*a^2-_Z*a*b+b^2))-2/3*I*b^2/a^2*(-I*(x+I*b/a)
/b*a)^(1/2)*2^(1/2)*(I*(x-I*b/a)/b*a)^(1/2)*(I*x/b*a)^(1/2)/(a^2*x^3+b^2*x)^(1/2)/(-I*b/a+b/a)*EllipticPi((-I*
(x+I*b/a)/b*a)^(1/2),-I*b/a/(-I*b/a+b/a),1/2*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}}{{\left (a^{3} x^{3} + b^{3}\right )} \sqrt {a^{2} x^{3} + b^{2} x}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F(-1)]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \text {Hanged} \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^2*x + a^2*x^3)^(1/2)),x)

[Out]

\text{Hanged}

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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 )}{\sqrt {x \left (a^{2} x^{2} + b^{2}\right )} \left (a x + b\right ) \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**2*x**3+b**2*x)**(1/2)/(a**3*x**3+b**3),x)

[Out]

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

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