\(\int \frac {(-b+a x^2) \sqrt {b x+a x^3}}{b^2 x+2 (-1+a b) x^3+a^2 x^5} \, dx\) [961]

Optimal result

Integrand size = 48, antiderivative size = 73 \[ \int \frac {\left (-b+a x^2\right ) \sqrt {b x+a x^3}}{b^2 x+2 (-1+a b) x^3+a^2 x^5} \, dx=-\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt {b x+a x^3}}{b+a x^2}\right )}{\sqrt [4]{2}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt {b x+a x^3}}{b+a x^2}\right )}{\sqrt [4]{2}} \]

[Out]

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

Rubi [C] (warning: unable to verify)

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

Time = 18.28 (sec) , antiderivative size = 2513, normalized size of antiderivative = 34.42, number of steps used = 23, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {1608, 2081, 6847, 6860, 415, 226, 418, 1231, 1721} \[ \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=-\frac {\left (1-\frac {\sqrt {a} \sqrt {b}}{\sqrt {-a b-\sqrt {1-2 a b}+1}}\right ) \left (\sqrt {a} x+\sqrt {b}\right ) \sqrt {\frac {a x^2+b}{\left (\sqrt {a} x+\sqrt {b}\right )^2}} \sqrt {a x^3+b x} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right ) \left (1-\sqrt {1-2 a b}\right )^2}{4 \sqrt [4]{a} \sqrt [4]{b} \left (-2 a b-\sqrt {1-2 a b}+1\right ) \sqrt {x} \left (a x^2+b\right )}-\frac {\left (\frac {\sqrt {a} \sqrt {b}}{\sqrt {-a b-\sqrt {1-2 a b}+1}}+1\right ) \left (\sqrt {a} x+\sqrt {b}\right ) \sqrt {\frac {a x^2+b}{\left (\sqrt {a} x+\sqrt {b}\right )^2}} \sqrt {a x^3+b x} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right ) \left (1-\sqrt {1-2 a b}\right )^2}{4 \sqrt [4]{a} \sqrt [4]{b} \left (-2 a b-\sqrt {1-2 a b}+1\right ) \sqrt {x} \left (a x^2+b\right )}-\frac {\left (2 a^2 b^2-a \left (5-3 \sqrt {1-2 a b}\right ) b-2 \sqrt {1-2 a b}+2\right ) \sqrt {a x^3+b x} \arctan \left (\frac {\sqrt {1-\sqrt {1-2 a b}} \sqrt {x}}{\sqrt [4]{-a b-\sqrt {1-2 a b}+1} \sqrt {a x^2+b}}\right ) \left (1-\sqrt {1-2 a b}\right )^{3/2}}{4 \left (-2 a b-\sqrt {1-2 a b}+1\right ) \left (-a b-\sqrt {1-2 a b}+1\right )^{7/4} \sqrt {x} \sqrt {a x^2+b}}+\frac {\left (\sqrt {a} x+\sqrt {b}\right ) \sqrt {\frac {a x^2+b}{\left (\sqrt {a} x+\sqrt {b}\right )^2}} \sqrt {a x^3+b x} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right ) \left (1-\sqrt {1-2 a b}\right )}{2 \sqrt [4]{a} \sqrt [4]{b} \sqrt {x} \left (a x^2+b\right )}-\frac {\left (\sqrt {1-2 a b}-1\right )^{3/2} \left (2 a^2 b^2-a \left (5-3 \sqrt {1-2 a b}\right ) b-2 \sqrt {1-2 a b}+2\right ) \sqrt {a x^3+b x} \arctan \left (\frac {\sqrt {\sqrt {1-2 a b}-1} \sqrt {x}}{\sqrt [4]{-a b-\sqrt {1-2 a b}+1} \sqrt {a x^2+b}}\right )}{4 \left (-2 a b-\sqrt {1-2 a b}+1\right ) \left (-a b-\sqrt {1-2 a b}+1\right )^{7/4} \sqrt {x} \sqrt {a x^2+b}}-\frac {\left (-\sqrt {1-2 a b}-1\right )^{3/2} \sqrt {a x^3+b x} \arctan \left (\frac {\sqrt {-\sqrt {1-2 a b}-1} \sqrt {x}}{\sqrt [4]{-a b+\sqrt {1-2 a b}+1} \sqrt {a x^2+b}}\right )}{4 \left (-a b+\sqrt {1-2 a b}+1\right )^{3/4} \sqrt {x} \sqrt {a x^2+b}}-\frac {\left (\sqrt {1-2 a b}+1\right )^{3/2} \sqrt {a x^3+b x} \arctan \left (\frac {\sqrt {\sqrt {1-2 a b}+1} \sqrt {x}}{\sqrt [4]{-a b+\sqrt {1-2 a b}+1} \sqrt {a x^2+b}}\right )}{4 \left (-a b+\sqrt {1-2 a b}+1\right )^{3/4} \sqrt {x} \sqrt {a x^2+b}}+\frac {\left (\sqrt {1-2 a b}+1\right ) \left (\sqrt {a} x+\sqrt {b}\right ) \sqrt {\frac {a x^2+b}{\left (\sqrt {a} x+\sqrt {b}\right )^2}} \sqrt {a x^3+b x} \operatorname {EllipticF}\left (2 \arctan \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 (a x^2+b\right )}-\frac {\left (\sqrt {1-2 a b}+1\right )^2 \left (1-\frac {\sqrt {a} \sqrt {b}}{\sqrt {-a b+\sqrt {1-2 a b}+1}}\right ) \left (\sqrt {a} x+\sqrt {b}\right ) \sqrt {\frac {a x^2+b}{\left (\sqrt {a} x+\sqrt {b}\right )^2}} \sqrt {a x^3+b x} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{b} \left (-2 a b+\sqrt {1-2 a b}+1\right ) \sqrt {x} \left (a x^2+b\right )}-\frac {\left (\sqrt {1-2 a b}+1\right )^2 \left (\frac {\sqrt {a} \sqrt {b}}{\sqrt {-a b+\sqrt {1-2 a b}+1}}+1\right ) \left (\sqrt {a} x+\sqrt {b}\right ) \sqrt {\frac {a x^2+b}{\left (\sqrt {a} x+\sqrt {b}\right )^2}} \sqrt {a x^3+b x} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{b} \left (-2 a b+\sqrt {1-2 a b}+1\right ) \sqrt {x} \left (a x^2+b\right )}+\frac {\left (-\sqrt {1-2 a b}+2 \sqrt {a} \sqrt {b} \sqrt {-a b-\sqrt {1-2 a b}+1}+1\right ) \left (\sqrt {a} x+\sqrt {b}\right ) \sqrt {\frac {a x^2+b}{\left (\sqrt {a} x+\sqrt {b}\right )^2}} \sqrt {a x^3+b x} \operatorname {EllipticPi}\left (-\frac {\left (\sqrt {a} \sqrt {b}-\sqrt {-a b-\sqrt {1-2 a b}+1}\right )^2}{4 \sqrt {a} \sqrt {b} \sqrt {-a b-\sqrt {1-2 a b}+1}},2 \arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{b} \left (-2 a b-\sqrt {1-2 a b}+1\right ) \sqrt {x} \left (a x^2+b\right )}+\frac {\left (-\sqrt {1-2 a b}-2 \sqrt {a} \sqrt {b} \sqrt {-a b-\sqrt {1-2 a b}+1}+1\right ) \left (\sqrt {a} x+\sqrt {b}\right ) \sqrt {\frac {a x^2+b}{\left (\sqrt {a} x+\sqrt {b}\right )^2}} \sqrt {a x^3+b x} \operatorname {EllipticPi}\left (\frac {\left (\sqrt {a} \sqrt {b}+\sqrt {-a b-\sqrt {1-2 a b}+1}\right )^2}{4 \sqrt {a} \sqrt {b} \sqrt {-a b-\sqrt {1-2 a b}+1}},2 \arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{b} \left (-2 a b-\sqrt {1-2 a b}+1\right ) \sqrt {x} \left (a x^2+b\right )}+\frac {\left (\sqrt {1-2 a b}+2 \sqrt {a} \sqrt {b} \sqrt {-a b+\sqrt {1-2 a b}+1}+1\right ) \left (\sqrt {a} x+\sqrt {b}\right ) \sqrt {\frac {a x^2+b}{\left (\sqrt {a} x+\sqrt {b}\right )^2}} \sqrt {a x^3+b x} \operatorname {EllipticPi}\left (-\frac {\left (\sqrt {a} \sqrt {b}-\sqrt {-a b+\sqrt {1-2 a b}+1}\right )^2}{4 \sqrt {a} \sqrt {b} \sqrt {-a b+\sqrt {1-2 a b}+1}},2 \arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{b} \left (-2 a b+\sqrt {1-2 a b}+1\right ) \sqrt {x} \left (a x^2+b\right )}+\frac {\left (\sqrt {1-2 a b}-2 \sqrt {a} \sqrt {b} \sqrt {-a b+\sqrt {1-2 a b}+1}+1\right ) \left (\sqrt {a} x+\sqrt {b}\right ) \sqrt {\frac {a x^2+b}{\left (\sqrt {a} x+\sqrt {b}\right )^2}} \sqrt {a x^3+b x} \operatorname {EllipticPi}\left (\frac {\left (\sqrt {a} \sqrt {b}+\sqrt {-a b+\sqrt {1-2 a b}+1}\right )^2}{4 \sqrt {a} \sqrt {b} \sqrt {-a b+\sqrt {1-2 a b}+1}},2 \arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{b} \left (-2 a b+\sqrt {1-2 a b}+1\right ) \sqrt {x} \left (a x^2+b\right )} \]

[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 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 415

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 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 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 1608

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 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 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 6860

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*} \text {integral}& = \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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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} \operatorname {EllipticF}\left (2 \arctan \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} \operatorname {EllipticF}\left (2 \arctan \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 ) \text {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 ) \text {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 ) \text {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 ) \text {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}} \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.02 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.14 \[ \int \frac {\left (-b+a x^2\right ) \sqrt {b x+a x^3}}{b^2 x+2 (-1+a b) x^3+a^2 x^5} \, dx=-\frac {\sqrt {x} \sqrt {b+a x^2} \left (\arctan \left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt {b+a x^2}}\right )+\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt {b+a x^2}}\right )\right )}{\sqrt [4]{2} \sqrt {x \left (b+a x^2\right )}} \]

[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]

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

Maple [A] (verified)

Time = 3.09 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.99

method	result	size
default	\(-\frac {2^{\frac {3}{4}} \left (-2 \arctan \left (\frac {\sqrt {x \left (a \,x^{2}+b \right )}\, 2^{\frac {3}{4}}}{2 x}\right )+\ln \left (\frac {-2^{\frac {1}{4}} x -\sqrt {x \left (a \,x^{2}+b \right )}}{2^{\frac {1}{4}} x -\sqrt {x \left (a \,x^{2}+b \right )}}\right )\right )}{4}\)	\(72\)
pseudoelliptic	\(-\frac {2^{\frac {3}{4}} \left (-2 \arctan \left (\frac {\sqrt {x \left (a \,x^{2}+b \right )}\, 2^{\frac {3}{4}}}{2 x}\right )+\ln \left (\frac {-2^{\frac {1}{4}} x -\sqrt {x \left (a \,x^{2}+b \right )}}{2^{\frac {1}{4}} x -\sqrt {x \left (a \,x^{2}+b \right )}}\right )\right )}{4}\)	\(72\)
elliptic	\(\frac {\sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {x a}{\sqrt {-a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{a \sqrt {a \,x^{3}+b x}}-\frac {\sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (a^{2} \textit {\_Z}^{4}+\left (2 a b -2\right ) \textit {\_Z}^{2}+b^{2}\right )}{\sum }\frac {\left (\underline {\hspace {1.25 ex}}\alpha ^{2} a b -\underline {\hspace {1.25 ex}}\alpha ^{2}+b^{2}\right ) \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {\left (x -\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {x a}{\sqrt {-a b}}}\, \left (a \left (a^{2} \underline {\hspace {1.25 ex}}\alpha ^{3}+a b \underline {\hspace {1.25 ex}}\alpha -2 \underline {\hspace {1.25 ex}}\alpha \right )-a^{2} \sqrt {-a b}\, \underline {\hspace {1.25 ex}}\alpha ^{2}-\sqrt {-a b}\, a b +2 \sqrt {-a b}\right ) \operatorname {EllipticPi}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, -\frac {\sqrt {-a b}\, \underline {\hspace {1.25 ex}}\alpha ^{3} a^{2}+\underline {\hspace {1.25 ex}}\alpha ^{2} a^{2} b +\sqrt {-a b}\, \underline {\hspace {1.25 ex}}\alpha a b +a \,b^{2}-2 \sqrt {-a b}\, \underline {\hspace {1.25 ex}}\alpha -2 b}{2 b}, \frac {\sqrt {2}}{2}\right )}{\underline {\hspace {1.25 ex}}\alpha \left (\underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}+a b -1\right ) \sqrt {x \left (a \,x^{2}+b \right )}}\right )}{4 a b}\)	\(385\)

[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]

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

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.33 (sec) , antiderivative size = 389, normalized size of antiderivative = 5.33 \[ \int \frac {\left (-b+a x^2\right ) \sqrt {b x+a x^3}}{b^2 x+2 (-1+a b) x^3+a^2 x^5} \, dx=-\frac {1}{8} \cdot 2^{\frac {3}{4}} \log \left (\frac {a^{2} x^{4} + 2 \, {\left (a b + 1\right )} x^{2} + b^{2} + 2 \, \sqrt {2} {\left (a x^{3} + b x\right )} + 2 \, \sqrt {a x^{3} + b x} {\left (2^{\frac {3}{4}} x + 2^{\frac {1}{4}} {\left (a x^{2} + b\right )}\right )}}{a^{2} x^{4} + 2 \, {\left (a b - 1\right )} x^{2} + b^{2}}\right ) + \frac {1}{8} \cdot 2^{\frac {3}{4}} \log \left (\frac {a^{2} x^{4} + 2 \, {\left (a b + 1\right )} x^{2} + b^{2} + 2 \, \sqrt {2} {\left (a x^{3} + b x\right )} - 2 \, \sqrt {a x^{3} + b x} {\left (2^{\frac {3}{4}} x + 2^{\frac {1}{4}} {\left (a x^{2} + b\right )}\right )}}{a^{2} x^{4} + 2 \, {\left (a b - 1\right )} x^{2} + b^{2}}\right ) + \frac {1}{8} i \cdot 2^{\frac {3}{4}} \log \left (\frac {a^{2} x^{4} + 2 \, {\left (a b + 1\right )} x^{2} + b^{2} - 2 \, \sqrt {2} {\left (a x^{3} + b x\right )} - 2 \, \sqrt {a x^{3} + b x} {\left (i \cdot 2^{\frac {3}{4}} x + 2^{\frac {1}{4}} {\left (-i \, a x^{2} - i \, b\right )}\right )}}{a^{2} x^{4} + 2 \, {\left (a b - 1\right )} x^{2} + b^{2}}\right ) - \frac {1}{8} i \cdot 2^{\frac {3}{4}} \log \left (\frac {a^{2} x^{4} + 2 \, {\left (a b + 1\right )} x^{2} + b^{2} - 2 \, \sqrt {2} {\left (a x^{3} + b x\right )} - 2 \, \sqrt {a x^{3} + b x} {\left (-i \cdot 2^{\frac {3}{4}} x + 2^{\frac {1}{4}} {\left (i \, a x^{2} + i \, b\right )}\right )}}{a^{2} x^{4} + 2 \, {\left (a b - 1\right )} x^{2} + b^{2}}\right ) \]

[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]

-1/8*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)) + 1/8*I*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)*(I
*2^(3/4)*x + 2^(1/4)*(-I*a*x^2 - I*b)))/(a^2*x^4 + 2*(a*b - 1)*x^2 + b^2)) - 1/8*I*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)*(-I*2^(3/4)*x + 2^(1/4)*(I*a*x^2 + I*b)))/(a
^2*x^4 + 2*(a*b - 1)*x^2 + b^2))

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (-b+a x^2\right ) \sqrt {b x+a x^3}}{b^2 x+2 (-1+a b) x^3+a^2 x^5} \, dx=\text {Timed out} \]

[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]

Timed out

Maxima [F]

\[ \int \frac {\left (-b+a x^2\right ) \sqrt {b x+a x^3}}{b^2 x+2 (-1+a b) x^3+a^2 x^5} \, dx=\int { \frac {\sqrt {a x^{3} + b x} {\left (a x^{2} - b\right )}}{a^{2} x^{5} + 2 \, {\left (a b - 1\right )} x^{3} + b^{2} x} \,d x } \]

[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)

Giac [F]

\[ \int \frac {\left (-b+a x^2\right ) \sqrt {b x+a x^3}}{b^2 x+2 (-1+a b) x^3+a^2 x^5} \, dx=\int { \frac {\sqrt {a x^{3} + b x} {\left (a x^{2} - b\right )}}{a^{2} x^{5} + 2 \, {\left (a b - 1\right )} x^{3} + b^{2} x} \,d x } \]

[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)

Mupad [B] (verification not implemented)

Time = 10.26 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.63 \[ \int \frac {\left (-b+a x^2\right ) \sqrt {b x+a x^3}}{b^2 x+2 (-1+a b) x^3+a^2 x^5} \, dx=\frac {2^{3/4}\,\ln \left (\frac {2^{3/4}\,b+2\,2^{1/4}\,x-4\,\sqrt {x\,\left (a\,x^2+b\right )}+2^{3/4}\,a\,x^2}{4\,a\,x^2-4\,\sqrt {2}\,x+4\,b}\right )}{4}+\frac {2^{3/4}\,\ln \left (\frac {2^{3/4}\,b\,1{}\mathrm {i}-2^{1/4}\,x\,2{}\mathrm {i}-4\,\sqrt {x\,\left (a\,x^2+b\right )}+2^{3/4}\,a\,x^2\,1{}\mathrm {i}}{a\,x^2+\sqrt {2}\,x+b}\right )\,1{}\mathrm {i}}{4} \]

[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