3.10.61 \(\int \frac {(-b+a x^2) \sqrt {b x+a x^3}}{b^2 x+2 (-1+a b) x^3+a^2 x^5} \, dx\)

Optimal. Leaf size=73 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt {a x^3+b x}}{a x^2+b}\right )}{\sqrt [4]{2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt {a x^3+b x}}{a x^2+b}\right )}{\sqrt [4]{2}} \]

________________________________________________________________________________________

Rubi [C]  time = 15.30, antiderivative size = 2513, normalized size of antiderivative = 34.42, number of steps used = 23, number of rules used = 9, integrand size = 48, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {1594, 2056, 6715, 6728, 406, 220, 409, 1217, 1707}

result too large to display

Warning: Unable to verify antiderivative.

[In]

Int[((-b + a*x^2)*Sqrt[b*x + a*x^3])/(b^2*x + 2*(-1 + a*b)*x^3 + a^2*x^5),x]

[Out]

-1/4*((1 - Sqrt[1 - 2*a*b])^(3/2)*(2 + 2*a^2*b^2 - 2*Sqrt[1 - 2*a*b] - a*b*(5 - 3*Sqrt[1 - 2*a*b]))*Sqrt[b*x +
a*x^3]*ArcTan[(Sqrt[1 - Sqrt[1 - 2*a*b]]*Sqrt[x])/((1 - a*b - Sqrt[1 - 2*a*b])^(1/4)*Sqrt[b + a*x^2])])/((1 -
2*a*b - Sqrt[1 - 2*a*b])*(1 - a*b - Sqrt[1 - 2*a*b])^(7/4)*Sqrt[x]*Sqrt[b + a*x^2]) - ((-1 + Sqrt[1 - 2*a*b])
^(3/2)*(2 + 2*a^2*b^2 - 2*Sqrt[1 - 2*a*b] - a*b*(5 - 3*Sqrt[1 - 2*a*b]))*Sqrt[b*x + a*x^3]*ArcTan[(Sqrt[-1 + S
qrt[1 - 2*a*b]]*Sqrt[x])/((1 - a*b - Sqrt[1 - 2*a*b])^(1/4)*Sqrt[b + a*x^2])])/(4*(1 - 2*a*b - Sqrt[1 - 2*a*b]
)*(1 - a*b - Sqrt[1 - 2*a*b])^(7/4)*Sqrt[x]*Sqrt[b + a*x^2]) - ((-1 - Sqrt[1 - 2*a*b])^(3/2)*Sqrt[b*x + a*x^3]
*ArcTan[(Sqrt[-1 - Sqrt[1 - 2*a*b]]*Sqrt[x])/((1 - a*b + Sqrt[1 - 2*a*b])^(1/4)*Sqrt[b + a*x^2])])/(4*(1 - a*b
+ Sqrt[1 - 2*a*b])^(3/4)*Sqrt[x]*Sqrt[b + a*x^2]) - ((1 + Sqrt[1 - 2*a*b])^(3/2)*Sqrt[b*x + a*x^3]*ArcTan[(Sq
rt[1 + Sqrt[1 - 2*a*b]]*Sqrt[x])/((1 - a*b + Sqrt[1 - 2*a*b])^(1/4)*Sqrt[b + a*x^2])])/(4*(1 - a*b + Sqrt[1 -
2*a*b])^(3/4)*Sqrt[x]*Sqrt[b + a*x^2]) + ((1 - Sqrt[1 - 2*a*b])*(Sqrt[b] + Sqrt[a]*x)*Sqrt[(b + a*x^2)/(Sqrt[b
] + Sqrt[a]*x)^2]*Sqrt[b*x + a*x^3]*EllipticF[2*ArcTan[(a^(1/4)*Sqrt[x])/b^(1/4)], 1/2])/(2*a^(1/4)*b^(1/4)*Sq
rt[x]*(b + a*x^2)) + ((1 + Sqrt[1 - 2*a*b])*(Sqrt[b] + Sqrt[a]*x)*Sqrt[(b + a*x^2)/(Sqrt[b] + Sqrt[a]*x)^2]*Sq
rt[b*x + a*x^3]*EllipticF[2*ArcTan[(a^(1/4)*Sqrt[x])/b^(1/4)], 1/2])/(2*a^(1/4)*b^(1/4)*Sqrt[x]*(b + a*x^2)) -
((1 - Sqrt[1 - 2*a*b])^2*(1 - (Sqrt[a]*Sqrt[b])/Sqrt[1 - a*b - Sqrt[1 - 2*a*b]])*(Sqrt[b] + Sqrt[a]*x)*Sqrt[(
b + a*x^2)/(Sqrt[b] + Sqrt[a]*x)^2]*Sqrt[b*x + a*x^3]*EllipticF[2*ArcTan[(a^(1/4)*Sqrt[x])/b^(1/4)], 1/2])/(4*
a^(1/4)*b^(1/4)*(1 - 2*a*b - Sqrt[1 - 2*a*b])*Sqrt[x]*(b + a*x^2)) - ((1 - Sqrt[1 - 2*a*b])^2*(1 + (Sqrt[a]*Sq
rt[b])/Sqrt[1 - a*b - Sqrt[1 - 2*a*b]])*(Sqrt[b] + Sqrt[a]*x)*Sqrt[(b + a*x^2)/(Sqrt[b] + Sqrt[a]*x)^2]*Sqrt[b
*x + a*x^3]*EllipticF[2*ArcTan[(a^(1/4)*Sqrt[x])/b^(1/4)], 1/2])/(4*a^(1/4)*b^(1/4)*(1 - 2*a*b - Sqrt[1 - 2*a*
b])*Sqrt[x]*(b + a*x^2)) - ((1 + Sqrt[1 - 2*a*b])^2*(1 - (Sqrt[a]*Sqrt[b])/Sqrt[1 - a*b + Sqrt[1 - 2*a*b]])*(S
qrt[b] + Sqrt[a]*x)*Sqrt[(b + a*x^2)/(Sqrt[b] + Sqrt[a]*x)^2]*Sqrt[b*x + a*x^3]*EllipticF[2*ArcTan[(a^(1/4)*Sq
rt[x])/b^(1/4)], 1/2])/(4*a^(1/4)*b^(1/4)*(1 - 2*a*b + Sqrt[1 - 2*a*b])*Sqrt[x]*(b + a*x^2)) - ((1 + Sqrt[1 -
2*a*b])^2*(1 + (Sqrt[a]*Sqrt[b])/Sqrt[1 - a*b + Sqrt[1 - 2*a*b]])*(Sqrt[b] + Sqrt[a]*x)*Sqrt[(b + a*x^2)/(Sqrt
[b] + Sqrt[a]*x)^2]*Sqrt[b*x + a*x^3]*EllipticF[2*ArcTan[(a^(1/4)*Sqrt[x])/b^(1/4)], 1/2])/(4*a^(1/4)*b^(1/4)*
(1 - 2*a*b + Sqrt[1 - 2*a*b])*Sqrt[x]*(b + a*x^2)) + ((1 - Sqrt[1 - 2*a*b] + 2*Sqrt[a]*Sqrt[b]*Sqrt[1 - a*b -
Sqrt[1 - 2*a*b]])*(Sqrt[b] + Sqrt[a]*x)*Sqrt[(b + a*x^2)/(Sqrt[b] + Sqrt[a]*x)^2]*Sqrt[b*x + a*x^3]*EllipticPi
[-1/4*(Sqrt[a]*Sqrt[b] - Sqrt[1 - a*b - Sqrt[1 - 2*a*b]])^2/(Sqrt[a]*Sqrt[b]*Sqrt[1 - a*b - Sqrt[1 - 2*a*b]]),
2*ArcTan[(a^(1/4)*Sqrt[x])/b^(1/4)], 1/2])/(4*a^(1/4)*b^(1/4)*(1 - 2*a*b - Sqrt[1 - 2*a*b])*Sqrt[x]*(b + a*x^
2)) + ((1 - Sqrt[1 - 2*a*b] - 2*Sqrt[a]*Sqrt[b]*Sqrt[1 - a*b - Sqrt[1 - 2*a*b]])*(Sqrt[b] + Sqrt[a]*x)*Sqrt[(b
+ a*x^2)/(Sqrt[b] + Sqrt[a]*x)^2]*Sqrt[b*x + a*x^3]*EllipticPi[(Sqrt[a]*Sqrt[b] + Sqrt[1 - a*b - Sqrt[1 - 2*a
*b]])^2/(4*Sqrt[a]*Sqrt[b]*Sqrt[1 - a*b - Sqrt[1 - 2*a*b]]), 2*ArcTan[(a^(1/4)*Sqrt[x])/b^(1/4)], 1/2])/(4*a^(
1/4)*b^(1/4)*(1 - 2*a*b - Sqrt[1 - 2*a*b])*Sqrt[x]*(b + a*x^2)) + ((1 + Sqrt[1 - 2*a*b] + 2*Sqrt[a]*Sqrt[b]*Sq
rt[1 - a*b + Sqrt[1 - 2*a*b]])*(Sqrt[b] + Sqrt[a]*x)*Sqrt[(b + a*x^2)/(Sqrt[b] + Sqrt[a]*x)^2]*Sqrt[b*x + a*x^
3]*EllipticPi[-1/4*(Sqrt[a]*Sqrt[b] - Sqrt[1 - a*b + Sqrt[1 - 2*a*b]])^2/(Sqrt[a]*Sqrt[b]*Sqrt[1 - a*b + Sqrt[
1 - 2*a*b]]), 2*ArcTan[(a^(1/4)*Sqrt[x])/b^(1/4)], 1/2])/(4*a^(1/4)*b^(1/4)*(1 - 2*a*b + Sqrt[1 - 2*a*b])*Sqrt
[x]*(b + a*x^2)) + ((1 + Sqrt[1 - 2*a*b] - 2*Sqrt[a]*Sqrt[b]*Sqrt[1 - a*b + Sqrt[1 - 2*a*b]])*(Sqrt[b] + Sqrt[
a]*x)*Sqrt[(b + a*x^2)/(Sqrt[b] + Sqrt[a]*x)^2]*Sqrt[b*x + a*x^3]*EllipticPi[(Sqrt[a]*Sqrt[b] + Sqrt[1 - a*b +
Sqrt[1 - 2*a*b]])^2/(4*Sqrt[a]*Sqrt[b]*Sqrt[1 - a*b + Sqrt[1 - 2*a*b]]), 2*ArcTan[(a^(1/4)*Sqrt[x])/b^(1/4)],
1/2])/(4*a^(1/4)*b^(1/4)*(1 - 2*a*b + Sqrt[1 - 2*a*b])*Sqrt[x]*(b + a*x^2))

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 406

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

Rule 409

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 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 1594

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^
(q - p) + c*x^(r - p))^n, x] /; FreeQ[{a, b, c, p, q, r}, x] && IntegerQ[n] && PosQ[q - p] && PosQ[r - p]

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 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 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 {\left (-b+a x^2\right ) \sqrt {b x+a x^3}}{b^2 x+2 (-1+a b) x^3+a^2 x^5} \, dx &=\int \frac {\left (-b+a x^2\right ) \sqrt {b x+a x^3}}{x \left (b^2+2 (-1+a b) x^2+a^2 x^4\right )} \, dx\\ &=\frac {\sqrt {b x+a x^3} \int \frac {\left (-b+a x^2\right ) \sqrt {b+a x^2}}{\sqrt {x} \left (b^2+2 (-1+a b) x^2+a^2 x^4\right )} \, dx}{\sqrt {x} \sqrt {b+a x^2}}\\ &=\frac {\left (2 \sqrt {b x+a x^3}\right ) \operatorname {Subst}\left (\int \frac {\left (-b+a x^4\right ) \sqrt {b+a x^4}}{b^2+2 (-1+a b) x^4+a^2 x^8} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {b+a x^2}}\\ &=\frac {\left (2 \sqrt {b x+a x^3}\right ) \operatorname {Subst}\left (\int \left (\frac {\left (a+a \sqrt {1-2 a b}\right ) \sqrt {b+a x^4}}{-2 \sqrt {1-2 a b}+2 (-1+a b)+2 a^2 x^4}+\frac {\left (a-a \sqrt {1-2 a b}\right ) \sqrt {b+a x^4}}{2 \sqrt {1-2 a b}+2 (-1+a b)+2 a^2 x^4}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {b+a x^2}}\\ &=\frac {\left (2 a \left (1-\sqrt {1-2 a b}\right ) \sqrt {b x+a x^3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {b+a x^4}}{2 \sqrt {1-2 a b}+2 (-1+a b)+2 a^2 x^4} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {b+a x^2}}+\frac {\left (2 a \left (1+\sqrt {1-2 a b}\right ) \sqrt {b x+a x^3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {b+a x^4}}{-2 \sqrt {1-2 a b}+2 (-1+a b)+2 a^2 x^4} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {b+a x^2}}\\ &=\frac {\left (\left (1-\sqrt {1-2 a b}\right ) \sqrt {b x+a x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b+a x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {b+a x^2}}+\frac {\left (2 \left (1-\sqrt {1-2 a b}\right )^2 \sqrt {b x+a x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b+a x^4} \left (2 \sqrt {1-2 a b}+2 (-1+a b)+2 a^2 x^4\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {b+a x^2}}+\frac {\left (\left (1+\sqrt {1-2 a b}\right ) \sqrt {b x+a x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b+a x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {b+a x^2}}+\frac {\left (2 \left (1+\sqrt {1-2 a b}\right )^2 \sqrt {b x+a x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b+a x^4} \left (-2 \sqrt {1-2 a b}+2 (-1+a b)+2 a^2 x^4\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {b+a x^2}}\\ &=\frac {\left (1-\sqrt {1-2 a b}\right ) \left (\sqrt {b}+\sqrt {a} x\right ) \sqrt {\frac {b+a x^2}{\left (\sqrt {b}+\sqrt {a} x\right )^2}} \sqrt {b x+a x^3} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt [4]{b} \sqrt {x} \left (b+a x^2\right )}+\frac {\left (1+\sqrt {1-2 a b}\right ) \left (\sqrt {b}+\sqrt {a} x\right ) \sqrt {\frac {b+a x^2}{\left (\sqrt {b}+\sqrt {a} x\right )^2}} \sqrt {b x+a x^3} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt [4]{b} \sqrt {x} \left (b+a x^2\right )}-\frac {\left (\left (1-\sqrt {1-2 a b}\right )^2 \sqrt {b x+a x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-\frac {a x^2}{\sqrt {1-a b-\sqrt {1-2 a b}}}\right ) \sqrt {b+a x^4}} \, dx,x,\sqrt {x}\right )}{2 \left (1-a b-\sqrt {1-2 a b}\right ) \sqrt {x} \sqrt {b+a x^2}}-\frac {\left (\left (1-\sqrt {1-2 a b}\right )^2 \sqrt {b x+a x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+\frac {a x^2}{\sqrt {1-a b-\sqrt {1-2 a b}}}\right ) \sqrt {b+a x^4}} \, dx,x,\sqrt {x}\right )}{2 \left (1-a b-\sqrt {1-2 a b}\right ) \sqrt {x} \sqrt {b+a x^2}}-\frac {\left (\left (1+\sqrt {1-2 a b}\right )^2 \sqrt {b x+a x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-\frac {a x^2}{\sqrt {1-a b+\sqrt {1-2 a b}}}\right ) \sqrt {b+a x^4}} \, dx,x,\sqrt {x}\right )}{2 \left (1-a b+\sqrt {1-2 a b}\right ) \sqrt {x} \sqrt {b+a x^2}}-\frac {\left (\left (1+\sqrt {1-2 a b}\right )^2 \sqrt {b x+a x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+\frac {a x^2}{\sqrt {1-a b+\sqrt {1-2 a b}}}\right ) \sqrt {b+a x^4}} \, dx,x,\sqrt {x}\right )}{2 \left (1-a b+\sqrt {1-2 a b}\right ) \sqrt {x} \sqrt {b+a x^2}}\\ &=\frac {\left (1-\sqrt {1-2 a b}\right ) \left (\sqrt {b}+\sqrt {a} x\right ) \sqrt {\frac {b+a x^2}{\left (\sqrt {b}+\sqrt {a} x\right )^2}} \sqrt {b x+a x^3} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt [4]{b} \sqrt {x} \left (b+a x^2\right )}+\frac {\left (1+\sqrt {1-2 a b}\right ) \left (\sqrt {b}+\sqrt {a} x\right ) \sqrt {\frac {b+a x^2}{\left (\sqrt {b}+\sqrt {a} x\right )^2}} \sqrt {b x+a x^3} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt [4]{b} \sqrt {x} \left (b+a x^2\right )}-\frac {\left (\left (1-\sqrt {1-2 a b}\right )^2 \left (1-\frac {\sqrt {a} \sqrt {b}}{\sqrt {1-a b-\sqrt {1-2 a b}}}\right ) \sqrt {b x+a x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b+a x^4}} \, dx,x,\sqrt {x}\right )}{2 \left (1-a b-\sqrt {1-2 a b}\right ) \left (1-\frac {a b}{1-a b-\sqrt {1-2 a b}}\right ) \sqrt {x} \sqrt {b+a x^2}}-\frac {\left (\left (1-\sqrt {1-2 a b}\right )^2 \left (1+\frac {\sqrt {a} \sqrt {b}}{\sqrt {1-a b-\sqrt {1-2 a b}}}\right ) \sqrt {b x+a x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b+a x^4}} \, dx,x,\sqrt {x}\right )}{2 \left (1-a b-\sqrt {1-2 a b}\right ) \left (1-\frac {a b}{1-a b-\sqrt {1-2 a b}}\right ) \sqrt {x} \sqrt {b+a x^2}}+\frac {\left (\sqrt {a} \sqrt {b} \left (1-\sqrt {1-2 a b}\right )^2 \left (\sqrt {a} \sqrt {b}-\sqrt {1-a b-\sqrt {1-2 a b}}\right ) \sqrt {b x+a x^3}\right ) \operatorname {Subst}\left (\int \frac {1+\frac {\sqrt {a} x^2}{\sqrt {b}}}{\left (1-\frac {a x^2}{\sqrt {1-a b-\sqrt {1-2 a b}}}\right ) \sqrt {b+a x^4}} \, dx,x,\sqrt {x}\right )}{2 \left (1-2 a b-\sqrt {1-2 a b}\right ) \left (1-a b-\sqrt {1-2 a b}\right ) \sqrt {x} \sqrt {b+a x^2}}+\frac {\left (\sqrt {a} \sqrt {b} \left (1-\sqrt {1-2 a b}\right )^2 \left (\sqrt {a} \sqrt {b}+\sqrt {1-a b-\sqrt {1-2 a b}}\right ) \sqrt {b x+a x^3}\right ) \operatorname {Subst}\left (\int \frac {1+\frac {\sqrt {a} x^2}{\sqrt {b}}}{\left (1+\frac {a x^2}{\sqrt {1-a b-\sqrt {1-2 a b}}}\right ) \sqrt {b+a x^4}} \, dx,x,\sqrt {x}\right )}{2 \left (1-2 a b-\sqrt {1-2 a b}\right ) \left (1-a b-\sqrt {1-2 a b}\right ) \sqrt {x} \sqrt {b+a x^2}}-\frac {\left (\left (1+\sqrt {1-2 a b}\right )^2 \left (1-\frac {\sqrt {a} \sqrt {b}}{\sqrt {1-a b+\sqrt {1-2 a b}}}\right ) \sqrt {b x+a x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b+a x^4}} \, dx,x,\sqrt {x}\right )}{2 \left (1-a b+\sqrt {1-2 a b}\right ) \left (1-\frac {a b}{1-a b+\sqrt {1-2 a b}}\right ) \sqrt {x} \sqrt {b+a x^2}}-\frac {\left (\left (1+\sqrt {1-2 a b}\right )^2 \left (1+\frac {\sqrt {a} \sqrt {b}}{\sqrt {1-a b+\sqrt {1-2 a b}}}\right ) \sqrt {b x+a x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b+a x^4}} \, dx,x,\sqrt {x}\right )}{2 \left (1-a b+\sqrt {1-2 a b}\right ) \left (1-\frac {a b}{1-a b+\sqrt {1-2 a b}}\right ) \sqrt {x} \sqrt {b+a x^2}}+\frac {\left (\sqrt {a} \sqrt {b} \left (1+\sqrt {1-2 a b}\right )^2 \left (\sqrt {a} \sqrt {b}-\sqrt {1-a b+\sqrt {1-2 a b}}\right ) \sqrt {b x+a x^3}\right ) \operatorname {Subst}\left (\int \frac {1+\frac {\sqrt {a} x^2}{\sqrt {b}}}{\left (1-\frac {a x^2}{\sqrt {1-a b+\sqrt {1-2 a b}}}\right ) \sqrt {b+a x^4}} \, dx,x,\sqrt {x}\right )}{2 \left (1-2 a b+\sqrt {1-2 a b}\right ) \left (1-a b+\sqrt {1-2 a b}\right ) \sqrt {x} \sqrt {b+a x^2}}+\frac {\left (\sqrt {a} \sqrt {b} \left (1+\sqrt {1-2 a b}\right )^2 \left (\sqrt {a} \sqrt {b}+\sqrt {1-a b+\sqrt {1-2 a b}}\right ) \sqrt {b x+a x^3}\right ) \operatorname {Subst}\left (\int \frac {1+\frac {\sqrt {a} x^2}{\sqrt {b}}}{\left (1+\frac {a x^2}{\sqrt {1-a b+\sqrt {1-2 a b}}}\right ) \sqrt {b+a x^4}} \, dx,x,\sqrt {x}\right )}{2 \left (1-2 a b+\sqrt {1-2 a b}\right ) \left (1-a b+\sqrt {1-2 a b}\right ) \sqrt {x} \sqrt {b+a x^2}}\\ &=\frac {\left (1-\sqrt {1-2 a b}\right )^{3/2} \left (\sqrt {a} \sqrt {b}-\sqrt {1-a b-\sqrt {1-2 a b}}\right ) \left (\sqrt {a} \sqrt {b}+\sqrt {1-a b-\sqrt {1-2 a b}}\right ) \sqrt {b x+a x^3} \tan ^{-1}\left (\frac {\sqrt {1-\sqrt {1-2 a b}} \sqrt {x}}{\sqrt [4]{1-a b-\sqrt {1-2 a b}} \sqrt {b+a x^2}}\right )}{4 \left (1-2 a b-\sqrt {1-2 a b}\right ) \left (1-a b-\sqrt {1-2 a b}\right )^{3/4} \sqrt {x} \sqrt {b+a x^2}}+\frac {\left (-1+\sqrt {1-2 a b}\right )^{3/2} \left (\sqrt {a} \sqrt {b}-\sqrt {1-a b-\sqrt {1-2 a b}}\right ) \left (\sqrt {a} \sqrt {b}+\sqrt {1-a b-\sqrt {1-2 a b}}\right ) \sqrt {b x+a x^3} \tan ^{-1}\left (\frac {\sqrt {-1+\sqrt {1-2 a b}} \sqrt {x}}{\sqrt [4]{1-a b-\sqrt {1-2 a b}} \sqrt {b+a x^2}}\right )}{4 \left (1-2 a b-\sqrt {1-2 a b}\right ) \left (1-a b-\sqrt {1-2 a b}\right )^{3/4} \sqrt {x} \sqrt {b+a x^2}}-\frac {\left (2 a^2 b^2+2 \left (1+\sqrt {1-2 a b}\right )-a b \left (5+3 \sqrt {1-2 a b}\right )\right ) \sqrt {b x+a x^3} \tan ^{-1}\left (\frac {\sqrt {-1-\sqrt {1-2 a b}} \sqrt {x}}{\sqrt [4]{1-a b+\sqrt {1-2 a b}} \sqrt {b+a x^2}}\right )}{2 \sqrt {-1-\sqrt {1-2 a b}} \left (1-2 a b+\sqrt {1-2 a b}\right ) \left (1-a b+\sqrt {1-2 a b}\right )^{3/4} \sqrt {x} \sqrt {b+a x^2}}+\frac {\left (1+\sqrt {1-2 a b}\right )^{3/2} \left (\sqrt {a} \sqrt {b}-\sqrt {1-a b+\sqrt {1-2 a b}}\right ) \left (\sqrt {a} \sqrt {b}+\sqrt {1-a b+\sqrt {1-2 a b}}\right ) \sqrt {b x+a x^3} \tan ^{-1}\left (\frac {\sqrt {1+\sqrt {1-2 a b}} \sqrt {x}}{\sqrt [4]{1-a b+\sqrt {1-2 a b}} \sqrt {b+a x^2}}\right )}{4 \left (1-2 a b+\sqrt {1-2 a b}\right ) \left (1-a b+\sqrt {1-2 a b}\right )^{3/4} \sqrt {x} \sqrt {b+a x^2}}+\frac {\left (1-\sqrt {1-2 a b}\right ) \left (\sqrt {b}+\sqrt {a} x\right ) \sqrt {\frac {b+a x^2}{\left (\sqrt {b}+\sqrt {a} x\right )^2}} \sqrt {b x+a x^3} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt [4]{b} \sqrt {x} \left (b+a x^2\right )}+\frac {\left (1+\sqrt {1-2 a b}\right ) \left (\sqrt {b}+\sqrt {a} x\right ) \sqrt {\frac {b+a x^2}{\left (\sqrt {b}+\sqrt {a} x\right )^2}} \sqrt {b x+a x^3} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt [4]{b} \sqrt {x} \left (b+a x^2\right )}+\frac {\left (1-\sqrt {1-2 a b}\right )^2 \left (\sqrt {a} \sqrt {b}-\sqrt {1-a b-\sqrt {1-2 a b}}\right ) \left (\sqrt {b}+\sqrt {a} x\right ) \sqrt {\frac {b+a x^2}{\left (\sqrt {b}+\sqrt {a} x\right )^2}} \sqrt {b x+a x^3} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{b} \left (1-2 a b-\sqrt {1-2 a b}\right ) \sqrt {1-a b-\sqrt {1-2 a b}} \sqrt {x} \left (b+a x^2\right )}-\frac {\left (1-\sqrt {1-2 a b}\right )^2 \left (\sqrt {a} \sqrt {b}+\sqrt {1-a b-\sqrt {1-2 a b}}\right ) \left (\sqrt {b}+\sqrt {a} x\right ) \sqrt {\frac {b+a x^2}{\left (\sqrt {b}+\sqrt {a} x\right )^2}} \sqrt {b x+a x^3} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{b} \left (1-2 a b-\sqrt {1-2 a b}\right ) \sqrt {1-a b-\sqrt {1-2 a b}} \sqrt {x} \left (b+a x^2\right )}+\frac {\sqrt {1-a b+\sqrt {1-2 a b}} \left (\sqrt {a} \sqrt {b}-\sqrt {1-a b+\sqrt {1-2 a b}}\right ) \left (\sqrt {b}+\sqrt {a} x\right ) \sqrt {\frac {b+a x^2}{\left (\sqrt {b}+\sqrt {a} x\right )^2}} \sqrt {b x+a x^3} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt [4]{b} \left (1-2 a b+\sqrt {1-2 a b}\right ) \sqrt {x} \left (b+a x^2\right )}-\frac {\sqrt {1-a b+\sqrt {1-2 a b}} \left (\sqrt {a} \sqrt {b}+\sqrt {1-a b+\sqrt {1-2 a b}}\right ) \left (\sqrt {b}+\sqrt {a} x\right ) \sqrt {\frac {b+a x^2}{\left (\sqrt {b}+\sqrt {a} x\right )^2}} \sqrt {b x+a x^3} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt [4]{b} \left (1-2 a b+\sqrt {1-2 a b}\right ) \sqrt {x} \left (b+a x^2\right )}+\frac {\left (1-\sqrt {1-2 a b}+2 \sqrt {a} \sqrt {b} \sqrt {1-a b-\sqrt {1-2 a b}}\right ) \left (\sqrt {b}+\sqrt {a} x\right ) \sqrt {\frac {b+a x^2}{\left (\sqrt {b}+\sqrt {a} x\right )^2}} \sqrt {b x+a x^3} \Pi \left (-\frac {\left (\sqrt {a} \sqrt {b}-\sqrt {1-a b-\sqrt {1-2 a b}}\right )^2}{4 \sqrt {a} \sqrt {b} \sqrt {1-a b-\sqrt {1-2 a b}}};2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{b} \left (1-2 a b-\sqrt {1-2 a b}\right ) \sqrt {x} \left (b+a x^2\right )}+\frac {\left (1-\sqrt {1-2 a b}-2 \sqrt {a} \sqrt {b} \sqrt {1-a b-\sqrt {1-2 a b}}\right ) \left (\sqrt {b}+\sqrt {a} x\right ) \sqrt {\frac {b+a x^2}{\left (\sqrt {b}+\sqrt {a} x\right )^2}} \sqrt {b x+a x^3} \Pi \left (\frac {\left (\sqrt {a} \sqrt {b}+\sqrt {1-a b-\sqrt {1-2 a b}}\right )^2}{4 \sqrt {a} \sqrt {b} \sqrt {1-a b-\sqrt {1-2 a b}}};2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{b} \left (1-2 a b-\sqrt {1-2 a b}\right ) \sqrt {x} \left (b+a x^2\right )}+\frac {\left (1+\sqrt {1-2 a b}+2 \sqrt {a} \sqrt {b} \sqrt {1-a b+\sqrt {1-2 a b}}\right ) \left (\sqrt {b}+\sqrt {a} x\right ) \sqrt {\frac {b+a x^2}{\left (\sqrt {b}+\sqrt {a} x\right )^2}} \sqrt {b x+a x^3} \Pi \left (-\frac {\left (\sqrt {a} \sqrt {b}-\sqrt {1-a b+\sqrt {1-2 a b}}\right )^2}{4 \sqrt {a} \sqrt {b} \sqrt {1-a b+\sqrt {1-2 a b}}};2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{b} \left (1-2 a b+\sqrt {1-2 a b}\right ) \sqrt {x} \left (b+a x^2\right )}+\frac {\left (1+\sqrt {1-2 a b}-2 \sqrt {a} \sqrt {b} \sqrt {1-a b+\sqrt {1-2 a b}}\right ) \left (\sqrt {b}+\sqrt {a} x\right ) \sqrt {\frac {b+a x^2}{\left (\sqrt {b}+\sqrt {a} x\right )^2}} \sqrt {b x+a x^3} \Pi \left (\frac {\left (\sqrt {a} \sqrt {b}+\sqrt {1-a b+\sqrt {1-2 a b}}\right )^2}{4 \sqrt {a} \sqrt {b} \sqrt {1-a b+\sqrt {1-2 a b}}};2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{b} \left (1-2 a b+\sqrt {1-2 a b}\right ) \sqrt {x} \left (b+a x^2\right )}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 1.42, size = 372, normalized size = 5.10 \begin {gather*} -\frac {i x^{3/2} \sqrt {\frac {b}{a x^2}+1} \left (-\Pi \left (-\frac {i \sqrt {a}}{\sqrt {b} \sqrt {\frac {-a b+\sqrt {1-2 a b}+1}{b^2}}};\left .i \sinh ^{-1}\left (\frac {\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}}{\sqrt {x}}\right )\right |-1\right )-\Pi \left (\frac {i \sqrt {a}}{\sqrt {b} \sqrt {\frac {-a b+\sqrt {1-2 a b}+1}{b^2}}};\left .i \sinh ^{-1}\left (\frac {\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}}{\sqrt {x}}\right )\right |-1\right )-\Pi \left (-\frac {i \sqrt {a}}{\sqrt {b} \sqrt {-\frac {a b+\sqrt {1-2 a b}-1}{b^2}}};\left .i \sinh ^{-1}\left (\frac {\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}}{\sqrt {x}}\right )\right |-1\right )-\Pi \left (\frac {i \sqrt {a}}{\sqrt {b} \sqrt {-\frac {a b+\sqrt {1-2 a b}-1}{b^2}}};\left .i \sinh ^{-1}\left (\frac {\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}}{\sqrt {x}}\right )\right |-1\right )+2 F\left (\left .i \sinh ^{-1}\left (\frac {\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}}{\sqrt {x}}\right )\right |-1\right )\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}} \sqrt {x \left (a x^2+b\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-b + a*x^2)*Sqrt[b*x + a*x^3])/(b^2*x + 2*(-1 + a*b)*x^3 + a^2*x^5),x]

[Out]

((-I)*Sqrt[1 + b/(a*x^2)]*x^(3/2)*(2*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[b])/Sqrt[a]]/Sqrt[x]], -1] - EllipticPi[
((-I)*Sqrt[a])/(Sqrt[b]*Sqrt[(1 - a*b + Sqrt[1 - 2*a*b])/b^2]), I*ArcSinh[Sqrt[(I*Sqrt[b])/Sqrt[a]]/Sqrt[x]],
-1] - EllipticPi[(I*Sqrt[a])/(Sqrt[b]*Sqrt[(1 - a*b + Sqrt[1 - 2*a*b])/b^2]), I*ArcSinh[Sqrt[(I*Sqrt[b])/Sqrt[
a]]/Sqrt[x]], -1] - EllipticPi[((-I)*Sqrt[a])/(Sqrt[b]*Sqrt[-((-1 + a*b + Sqrt[1 - 2*a*b])/b^2)]), I*ArcSinh[S
qrt[(I*Sqrt[b])/Sqrt[a]]/Sqrt[x]], -1] - EllipticPi[(I*Sqrt[a])/(Sqrt[b]*Sqrt[-((-1 + a*b + Sqrt[1 - 2*a*b])/b
^2)]), I*ArcSinh[Sqrt[(I*Sqrt[b])/Sqrt[a]]/Sqrt[x]], -1]))/(Sqrt[(I*Sqrt[b])/Sqrt[a]]*Sqrt[x*(b + a*x^2)])

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.48, size = 73, normalized size = 1.00 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt {b x+a x^3}}{b+a x^2}\right )}{\sqrt [4]{2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt {b x+a x^3}}{b+a x^2}\right )}{\sqrt [4]{2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((-b + a*x^2)*Sqrt[b*x + a*x^3])/(b^2*x + 2*(-1 + a*b)*x^3 + a^2*x^5),x]

[Out]

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

________________________________________________________________________________________

fricas [B]  time = 0.73, size = 221, normalized size = 3.03 \begin {gather*} -\frac {1}{2} \cdot 2^{\frac {3}{4}} \arctan \left (\frac {2^{\frac {1}{4}} \sqrt {a x^{3} + b x}}{a x^{2} + b}\right ) - \frac {1}{8} \cdot 2^{\frac {3}{4}} \log \left (\frac {a^{2} x^{4} + 2 \, {\left (a b + 1\right )} x^{2} + b^{2} + 2 \, \sqrt {2} {\left (a x^{3} + b x\right )} + 2 \, \sqrt {a x^{3} + b x} {\left (2^{\frac {3}{4}} x + 2^{\frac {1}{4}} {\left (a x^{2} + b\right )}\right )}}{a^{2} x^{4} + 2 \, {\left (a b - 1\right )} x^{2} + b^{2}}\right ) + \frac {1}{8} \cdot 2^{\frac {3}{4}} \log \left (\frac {a^{2} x^{4} + 2 \, {\left (a b + 1\right )} x^{2} + b^{2} + 2 \, \sqrt {2} {\left (a x^{3} + b x\right )} - 2 \, \sqrt {a x^{3} + b x} {\left (2^{\frac {3}{4}} x + 2^{\frac {1}{4}} {\left (a x^{2} + b\right )}\right )}}{a^{2} x^{4} + 2 \, {\left (a b - 1\right )} x^{2} + b^{2}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x^2-b)*(a*x^3+b*x)^(1/2)/(b^2*x+2*(a*b-1)*x^3+a^2*x^5),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a x^{3} + b x} {\left (a x^{2} - b\right )}}{a^{2} x^{5} + 2 \, {\left (a b - 1\right )} x^{3} + b^{2} x}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x^2-b)*(a*x^3+b*x)^(1/2)/(b^2*x+2*(a*b-1)*x^3+a^2*x^5),x, algorithm="giac")

[Out]

integrate(sqrt(a*x^3 + b*x)*(a*x^2 - b)/(a^2*x^5 + 2*(a*b - 1)*x^3 + b^2*x), x)

________________________________________________________________________________________

maple [C]  time = 0.17, size = 385, normalized size = 5.27

method	result	size
elliptic	\(\frac {\sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {x a}{\sqrt {-a b}}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{a \sqrt {a \,x^{3}+b x}}-\frac {\sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (a^{2} \textit {\_Z}^{4}+\left (2 a b -2\right ) \textit {\_Z}^{2}+b^{2}\right )}{\sum }\frac {\left (\underline {\hspace {1.25 ex}}\alpha ^{2} a b -\underline {\hspace {1.25 ex}}\alpha ^{2}+b^{2}\right ) \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {\left (x -\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {x a}{\sqrt {-a b}}}\, \left (a \left (a^{2} \underline {\hspace {1.25 ex}}\alpha ^{3}+a b \underline {\hspace {1.25 ex}}\alpha -2 \underline {\hspace {1.25 ex}}\alpha \right )-a^{2} \sqrt {-a b}\, \underline {\hspace {1.25 ex}}\alpha ^{2}-\sqrt {-a b}\, a b +2 \sqrt {-a b}\right ) \EllipticPi \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, -\frac {\sqrt {-a b}\, \underline {\hspace {1.25 ex}}\alpha ^{3} a^{2}+\underline {\hspace {1.25 ex}}\alpha ^{2} a^{2} b +\sqrt {-a b}\, \underline {\hspace {1.25 ex}}\alpha a b +a \,b^{2}-2 \sqrt {-a b}\, \underline {\hspace {1.25 ex}}\alpha -2 b}{2 b}, \frac {\sqrt {2}}{2}\right )}{\underline {\hspace {1.25 ex}}\alpha \left (\underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}+a b -1\right ) \sqrt {x \left (a \,x^{2}+b \right )}}\right )}{4 a b}\)	\(385\)
default	\(\frac {\frac {2 \sqrt {a \,x^{3}+b x}}{3}+\frac {5 b \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {x a}{\sqrt {-a b}}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{3 a \sqrt {a \,x^{3}+b x}}-\frac {\sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (a^{2} \textit {\_Z}^{4}+\left (2 a b -2\right ) \textit {\_Z}^{2}+b^{2}\right )}{\sum }\frac {\left (\underline {\hspace {1.25 ex}}\alpha ^{2} a b -\underline {\hspace {1.25 ex}}\alpha ^{2}+b^{2}\right ) \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {\left (x -\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {x a}{\sqrt {-a b}}}\, \left (a \left (a^{2} \underline {\hspace {1.25 ex}}\alpha ^{3}+a b \underline {\hspace {1.25 ex}}\alpha -2 \underline {\hspace {1.25 ex}}\alpha \right )-a^{2} \sqrt {-a b}\, \underline {\hspace {1.25 ex}}\alpha ^{2}-\sqrt {-a b}\, a b +2 \sqrt {-a b}\right ) \EllipticPi \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, -\frac {\sqrt {-a b}\, \underline {\hspace {1.25 ex}}\alpha ^{3} a^{2}+\underline {\hspace {1.25 ex}}\alpha ^{2} a^{2} b +\sqrt {-a b}\, \underline {\hspace {1.25 ex}}\alpha a b +a \,b^{2}-2 \sqrt {-a b}\, \underline {\hspace {1.25 ex}}\alpha -2 b}{2 b}, \frac {\sqrt {2}}{2}\right )}{\underline {\hspace {1.25 ex}}\alpha \left (\underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}+a b -1\right ) \sqrt {x \left (a \,x^{2}+b \right )}}\right )}{4 a}}{b}-\frac {\frac {2 \sqrt {a \,x^{3}+b x}}{3}+\frac {2 b \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {x a}{\sqrt {-a b}}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{3 a \sqrt {a \,x^{3}+b x}}}{b}\)	\(530\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x^2-b)*(a*x^3+b*x)^(1/2)/(b^2*x+2*(a*b-1)*x^3+a^2*x^5),x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a x^{3} + b x} {\left (a x^{2} - b\right )}}{a^{2} x^{5} + 2 \, {\left (a b - 1\right )} x^{3} + b^{2} x}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x^2-b)*(a*x^3+b*x)^(1/2)/(b^2*x+2*(a*b-1)*x^3+a^2*x^5),x, algorithm="maxima")

[Out]

integrate(sqrt(a*x^3 + b*x)*(a*x^2 - b)/(a^2*x^5 + 2*(a*b - 1)*x^3 + b^2*x), x)

________________________________________________________________________________________

mupad [B]  time = 4.36, size = 119, normalized size = 1.63 \begin {gather*} \frac {2^{3/4}\,\ln \left (\frac {2^{3/4}\,b+2\,2^{1/4}\,x-4\,\sqrt {x\,\left (a\,x^2+b\right )}+2^{3/4}\,a\,x^2}{4\,a\,x^2-4\,\sqrt {2}\,x+4\,b}\right )}{4}+\frac {2^{3/4}\,\ln \left (\frac {2^{3/4}\,b\,1{}\mathrm {i}-2^{1/4}\,x\,2{}\mathrm {i}-4\,\sqrt {x\,\left (a\,x^2+b\right )}+2^{3/4}\,a\,x^2\,1{}\mathrm {i}}{a\,x^2+\sqrt {2}\,x+b}\right )\,1{}\mathrm {i}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((b*x + a*x^3)^(1/2)*(b - a*x^2))/(b^2*x + 2*x^3*(a*b - 1) + a^2*x^5),x)

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x**2-b)*(a*x**3+b*x)**(1/2)/(b**2*x+2*(a*b-1)*x**3+a**2*x**5),x)

[Out]

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

________________________________________________________________________________________