\(\int \frac {b^2+c x^2+a^2 x^4}{\sqrt {b x+a x^3} (-b^2+a^2 x^4)} \, dx\) [2240]

Optimal result

Integrand size = 45, antiderivative size = 167 \[ \int \frac {b^2+c x^2+a^2 x^4}{\sqrt {b x+a x^3} \left (-b^2+a^2 x^4\right )} \, dx=\frac {(-2 a b+c) \sqrt {b x+a x^3}}{2 a b \left (b+a x^2\right )}-\frac {(2 a b+c) \arctan \left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {b x+a x^3}}{b+a x^2}\right )}{4 \sqrt {2} a^{5/4} b^{5/4}}-\frac {(2 a b+c) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {b x+a x^3}}{b+a x^2}\right )}{4 \sqrt {2} a^{5/4} b^{5/4}} \]

[Out]

1/2*(-2*a*b+c)*(a*x^3+b*x)^(1/2)/a/b/(a*x^2+b)-1/8*(2*a*b+c)*arctan(2^(1/2)*a^(1/4)*b^(1/4)*(a*x^3+b*x)^(1/2)/
(a*x^2+b))*2^(1/2)/a^(5/4)/b^(5/4)-1/8*(2*a*b+c)*arctanh(2^(1/2)*a^(1/4)*b^(1/4)*(a*x^3+b*x)^(1/2)/(a*x^2+b))*
2^(1/2)/a^(5/4)/b^(5/4)

Rubi [C] (verified)

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

Time = 1.10 (sec) , antiderivative size = 503, normalized size of antiderivative = 3.01, number of steps used = 18, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {2081, 6847, 6857, 226, 1469, 541, 537, 418, 1225, 1713, 212, 209} \[ \int \frac {b^2+c x^2+a^2 x^4}{\sqrt {b x+a x^3} \left (-b^2+a^2 x^4\right )} \, dx=-\frac {\sqrt {x} (2 a b+c) \sqrt {a x^2+b} \arctan \left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a x^2+b}}\right )}{4 \sqrt {2} a^{5/4} b^{5/4} \sqrt {a x^3+b x}}-\frac {\sqrt {x} (2 a b-c) \left (\sqrt {a} x+\sqrt {b}\right ) \sqrt {\frac {a x^2+b}{\left (\sqrt {a} x+\sqrt {b}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{4 a^{5/4} b^{5/4} \sqrt {a x^3+b x}}-\frac {\sqrt {x} (2 a b+c) \left (\sqrt {a} x+\sqrt {b}\right ) \sqrt {\frac {a x^2+b}{\left (\sqrt {a} x+\sqrt {b}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{4 a^{5/4} b^{5/4} \sqrt {a x^3+b x}}-\frac {\sqrt {x} (2 a b+c) \sqrt {a x^2+b} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a x^2+b}}\right )}{4 \sqrt {2} a^{5/4} b^{5/4} \sqrt {a x^3+b x}}+\frac {\sqrt {x} \left (\sqrt {a} x+\sqrt {b}\right ) \sqrt {\frac {a x^2+b}{\left (\sqrt {a} x+\sqrt {b}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{\sqrt [4]{a} \sqrt [4]{b} \sqrt {a x^3+b x}}-\frac {x (2 a b-c)}{2 a b \sqrt {a x^3+b x}} \]

[In]

Int[(b^2 + c*x^2 + a^2*x^4)/(Sqrt[b*x + a*x^3]*(-b^2 + a^2*x^4)),x]

[Out]

-1/2*((2*a*b - c)*x)/(a*b*Sqrt[b*x + a*x^3]) - ((2*a*b + c)*Sqrt[x]*Sqrt[b + a*x^2]*ArcTan[(Sqrt[2]*a^(1/4)*b^
(1/4)*Sqrt[x])/Sqrt[b + a*x^2]])/(4*Sqrt[2]*a^(5/4)*b^(5/4)*Sqrt[b*x + a*x^3]) - ((2*a*b + c)*Sqrt[x]*Sqrt[b +
a*x^2]*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/Sqrt[b + a*x^2]])/(4*Sqrt[2]*a^(5/4)*b^(5/4)*Sqrt[b*x + a*x^
3]) + (Sqrt[x]*(Sqrt[b] + Sqrt[a]*x)*Sqrt[(b + a*x^2)/(Sqrt[b] + Sqrt[a]*x)^2]*EllipticF[2*ArcTan[(a^(1/4)*Sqr
t[x])/b^(1/4)], 1/2])/(a^(1/4)*b^(1/4)*Sqrt[b*x + a*x^3]) - ((2*a*b - c)*Sqrt[x]*(Sqrt[b] + Sqrt[a]*x)*Sqrt[(b
+ a*x^2)/(Sqrt[b] + Sqrt[a]*x)^2]*EllipticF[2*ArcTan[(a^(1/4)*Sqrt[x])/b^(1/4)], 1/2])/(4*a^(5/4)*b^(5/4)*Sqr
t[b*x + a*x^3]) - ((2*a*b + c)*Sqrt[x]*(Sqrt[b] + Sqrt[a]*x)*Sqrt[(b + a*x^2)/(Sqrt[b] + Sqrt[a]*x)^2]*Ellipti
cF[2*ArcTan[(a^(1/4)*Sqrt[x])/b^(1/4)], 1/2])/(4*a^(5/4)*b^(5/4)*Sqrt[b*x + a*x^3])

Rule 209

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

Rule 212

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

Rule 226

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

Rule 418

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

Rule 537

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

Rule 541

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

Rule 1225

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 1469

Int[((d_) + (e_.)*(x_)^(n_))^(q_.)*((f_) + (g_.)*(x_)^(n_))^(r_.)*((a_) + (c_.)*(x_)^(n2_))^(p_.), x_Symbol] :
> Int[(d + e*x^n)^(p + q)*(f + g*x^n)^r*(a/d + (c/e)*x^n)^p, x] /; FreeQ[{a, c, d, e, f, g, n, q, r}, x] && Eq
Q[n2, 2*n] && EqQ[c*d^2 + a*e^2, 0] && IntegerQ[p]

Rule 1713

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 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 6857

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

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

Mathematica [A] (verified)

Time = 1.58 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.02 \[ \int \frac {b^2+c x^2+a^2 x^4}{\sqrt {b x+a x^3} \left (-b^2+a^2 x^4\right )} \, dx=-\frac {\sqrt {x} \left (4 \sqrt [4]{a} \sqrt [4]{b} (2 a b-c) \sqrt {x}+\sqrt {2} (2 a b+c) \sqrt {b+a x^2} \arctan \left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {b+a x^2}}\right )+\sqrt {2} (2 a b+c) \sqrt {b+a x^2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {b+a x^2}}\right )\right )}{8 a^{5/4} b^{5/4} \sqrt {x \left (b+a x^2\right )}} \]

[In]

Integrate[(b^2 + c*x^2 + a^2*x^4)/(Sqrt[b*x + a*x^3]*(-b^2 + a^2*x^4)),x]

[Out]

-1/8*(Sqrt[x]*(4*a^(1/4)*b^(1/4)*(2*a*b - c)*Sqrt[x] + Sqrt[2]*(2*a*b + c)*Sqrt[b + a*x^2]*ArcTan[(Sqrt[2]*a^(
1/4)*b^(1/4)*Sqrt[x])/Sqrt[b + a*x^2]] + Sqrt[2]*(2*a*b + c)*Sqrt[b + a*x^2]*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*
Sqrt[x])/Sqrt[b + a*x^2]]))/(a^(5/4)*b^(5/4)*Sqrt[x*(b + a*x^2)])

Maple [A] (verified)

Time = 0.62 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.86

method	result	size
default	\(-\frac {8 \left (a b \right )^{\frac {1}{4}} x \left (a b -\frac {c}{2}\right )+\sqrt {2}\, \left (a b +\frac {c}{2}\right ) \left (\ln \left (\frac {-\sqrt {2}\, \left (a b \right )^{\frac {1}{4}} x -\sqrt {\left (a \,x^{2}+b \right ) x}}{\sqrt {2}\, \left (a b \right )^{\frac {1}{4}} x -\sqrt {\left (a \,x^{2}+b \right ) x}}\right )-2 \arctan \left (\frac {\sqrt {\left (a \,x^{2}+b \right ) x}\, \sqrt {2}}{2 x \left (a b \right )^{\frac {1}{4}}}\right )\right ) \sqrt {\left (a \,x^{2}+b \right ) x}}{8 \sqrt {\left (a \,x^{2}+b \right ) x}\, \left (a b \right )^{\frac {1}{4}} b a}\)	\(144\)
pseudoelliptic	\(-\frac {8 \left (a b \right )^{\frac {1}{4}} x \left (a b -\frac {c}{2}\right )+\sqrt {2}\, \left (a b +\frac {c}{2}\right ) \left (\ln \left (\frac {-\sqrt {2}\, \left (a b \right )^{\frac {1}{4}} x -\sqrt {\left (a \,x^{2}+b \right ) x}}{\sqrt {2}\, \left (a b \right )^{\frac {1}{4}} x -\sqrt {\left (a \,x^{2}+b \right ) x}}\right )-2 \arctan \left (\frac {\sqrt {\left (a \,x^{2}+b \right ) x}\, \sqrt {2}}{2 x \left (a b \right )^{\frac {1}{4}}}\right )\right ) \sqrt {\left (a \,x^{2}+b \right ) x}}{8 \sqrt {\left (a \,x^{2}+b \right ) x}\, \left (a b \right )^{\frac {1}{4}} b a}\)	\(144\)
elliptic	\(-\frac {x \left (2 a b -c \right )}{2 a b \sqrt {\left (x^{2}+\frac {b}{a}\right ) a x}}+\frac {\sqrt {-a b}\, \sqrt {\frac {x a}{\sqrt {-a b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {-a b}}+2}\, \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 )}{2 a \sqrt {a \,x^{3}+b x}}+\frac {\sqrt {-a b}\, \sqrt {\frac {x a}{\sqrt {-a b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {-a b}}+2}\, \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 ) c}{4 a^{2} \sqrt {a \,x^{3}+b x}\, b}+\frac {b \sqrt {-a b}\, \sqrt {\frac {x a}{\sqrt {-a b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {-a b}}+2}\, \sqrt {-\frac {x a}{\sqrt {-a b}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, -\frac {\sqrt {-a b}}{a \left (-\frac {\sqrt {-a b}}{a}-\frac {\sqrt {a b}}{a}\right )}, \frac {\sqrt {2}}{2}\right )}{2 \sqrt {a b}\, a \sqrt {a \,x^{3}+b x}\, \left (-\frac {\sqrt {-a b}}{a}-\frac {\sqrt {a b}}{a}\right )}+\frac {\sqrt {-a b}\, \sqrt {\frac {x a}{\sqrt {-a b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {-a b}}+2}\, \sqrt {-\frac {x a}{\sqrt {-a b}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, -\frac {\sqrt {-a b}}{a \left (-\frac {\sqrt {-a b}}{a}-\frac {\sqrt {a b}}{a}\right )}, \frac {\sqrt {2}}{2}\right ) c}{4 a^{2} \sqrt {a b}\, \sqrt {a \,x^{3}+b x}\, \left (-\frac {\sqrt {-a b}}{a}-\frac {\sqrt {a b}}{a}\right )}-\frac {b \sqrt {-a b}\, \sqrt {\frac {x a}{\sqrt {-a b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {-a b}}+2}\, \sqrt {-\frac {x a}{\sqrt {-a b}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, -\frac {\sqrt {-a b}}{a \left (-\frac {\sqrt {-a b}}{a}+\frac {\sqrt {a b}}{a}\right )}, \frac {\sqrt {2}}{2}\right )}{2 \sqrt {a b}\, a \sqrt {a \,x^{3}+b x}\, \left (-\frac {\sqrt {-a b}}{a}+\frac {\sqrt {a b}}{a}\right )}-\frac {\sqrt {-a b}\, \sqrt {\frac {x a}{\sqrt {-a b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {-a b}}+2}\, \sqrt {-\frac {x a}{\sqrt {-a b}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, -\frac {\sqrt {-a b}}{a \left (-\frac {\sqrt {-a b}}{a}+\frac {\sqrt {a b}}{a}\right )}, \frac {\sqrt {2}}{2}\right ) c}{4 a^{2} \sqrt {a b}\, \sqrt {a \,x^{3}+b x}\, \left (-\frac {\sqrt {-a b}}{a}+\frac {\sqrt {a b}}{a}\right )}\)	\(827\)

[In]

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

[Out]

-1/8*(8*(a*b)^(1/4)*x*(a*b-1/2*c)+2^(1/2)*(a*b+1/2*c)*(ln((-2^(1/2)*(a*b)^(1/4)*x-((a*x^2+b)*x)^(1/2))/(2^(1/2
)*(a*b)^(1/4)*x-((a*x^2+b)*x)^(1/2)))-2*arctan(1/2*((a*x^2+b)*x)^(1/2)/x*2^(1/2)/(a*b)^(1/4)))*((a*x^2+b)*x)^(
1/2))/((a*x^2+b)*x)^(1/2)/(a*b)^(1/4)/b/a

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.46 (sec) , antiderivative size = 1877, normalized size of antiderivative = 11.24 \[ \int \frac {b^2+c x^2+a^2 x^4}{\sqrt {b x+a x^3} \left (-b^2+a^2 x^4\right )} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/16*((1/4)^(1/4)*(a^2*b*x^2 + a*b^2)*((16*a^4*b^4 + 32*a^3*b^3*c + 24*a^2*b^2*c^2 + 8*a*b*c^3 + c^4)/(a^5*b^
5))^(1/4)*log((8*a^3*b^5 + 12*a^2*b^4*c + 6*a*b^3*c^2 + b^2*c^3 + (8*a^5*b^3 + 12*a^4*b^2*c + 6*a^3*b*c^2 + a^
2*c^3)*x^4 + 6*(8*a^4*b^4 + 12*a^3*b^3*c + 6*a^2*b^2*c^2 + a*b*c^3)*x^2 + 8*sqrt(a*x^3 + b*x)*((1/4)^(1/4)*(4*
a^4*b^4 + 4*a^3*b^3*c + a^2*b^2*c^2)*x*((16*a^4*b^4 + 32*a^3*b^3*c + 24*a^2*b^2*c^2 + 8*a*b*c^3 + c^4)/(a^5*b^
5))^(1/4) + (1/4)^(3/4)*(a^5*b^4*x^2 + a^4*b^5)*((16*a^4*b^4 + 32*a^3*b^3*c + 24*a^2*b^2*c^2 + 8*a*b*c^3 + c^4
)/(a^5*b^5))^(3/4)) + 4*((2*a^5*b^4 + a^4*b^3*c)*x^3 + (2*a^4*b^5 + a^3*b^4*c)*x)*sqrt((16*a^4*b^4 + 32*a^3*b^
3*c + 24*a^2*b^2*c^2 + 8*a*b*c^3 + c^4)/(a^5*b^5)))/(a^2*x^4 - 2*a*b*x^2 + b^2)) - (1/4)^(1/4)*(a^2*b*x^2 + a*
b^2)*((16*a^4*b^4 + 32*a^3*b^3*c + 24*a^2*b^2*c^2 + 8*a*b*c^3 + c^4)/(a^5*b^5))^(1/4)*log((8*a^3*b^5 + 12*a^2*
b^4*c + 6*a*b^3*c^2 + b^2*c^3 + (8*a^5*b^3 + 12*a^4*b^2*c + 6*a^3*b*c^2 + a^2*c^3)*x^4 + 6*(8*a^4*b^4 + 12*a^3
*b^3*c + 6*a^2*b^2*c^2 + a*b*c^3)*x^2 - 8*sqrt(a*x^3 + b*x)*((1/4)^(1/4)*(4*a^4*b^4 + 4*a^3*b^3*c + a^2*b^2*c^
2)*x*((16*a^4*b^4 + 32*a^3*b^3*c + 24*a^2*b^2*c^2 + 8*a*b*c^3 + c^4)/(a^5*b^5))^(1/4) + (1/4)^(3/4)*(a^5*b^4*x
^2 + a^4*b^5)*((16*a^4*b^4 + 32*a^3*b^3*c + 24*a^2*b^2*c^2 + 8*a*b*c^3 + c^4)/(a^5*b^5))^(3/4)) + 4*((2*a^5*b^
4 + a^4*b^3*c)*x^3 + (2*a^4*b^5 + a^3*b^4*c)*x)*sqrt((16*a^4*b^4 + 32*a^3*b^3*c + 24*a^2*b^2*c^2 + 8*a*b*c^3 +
c^4)/(a^5*b^5)))/(a^2*x^4 - 2*a*b*x^2 + b^2)) - (1/4)^(1/4)*(I*a^2*b*x^2 + I*a*b^2)*((16*a^4*b^4 + 32*a^3*b^3
*c + 24*a^2*b^2*c^2 + 8*a*b*c^3 + c^4)/(a^5*b^5))^(1/4)*log((8*a^3*b^5 + 12*a^2*b^4*c + 6*a*b^3*c^2 + b^2*c^3
+ (8*a^5*b^3 + 12*a^4*b^2*c + 6*a^3*b*c^2 + a^2*c^3)*x^4 + 6*(8*a^4*b^4 + 12*a^3*b^3*c + 6*a^2*b^2*c^2 + a*b*c
^3)*x^2 - 8*sqrt(a*x^3 + b*x)*(I*(1/4)^(1/4)*(4*a^4*b^4 + 4*a^3*b^3*c + a^2*b^2*c^2)*x*((16*a^4*b^4 + 32*a^3*b
^3*c + 24*a^2*b^2*c^2 + 8*a*b*c^3 + c^4)/(a^5*b^5))^(1/4) + (1/4)^(3/4)*(-I*a^5*b^4*x^2 - I*a^4*b^5)*((16*a^4*
b^4 + 32*a^3*b^3*c + 24*a^2*b^2*c^2 + 8*a*b*c^3 + c^4)/(a^5*b^5))^(3/4)) - 4*((2*a^5*b^4 + a^4*b^3*c)*x^3 + (2
*a^4*b^5 + a^3*b^4*c)*x)*sqrt((16*a^4*b^4 + 32*a^3*b^3*c + 24*a^2*b^2*c^2 + 8*a*b*c^3 + c^4)/(a^5*b^5)))/(a^2*
x^4 - 2*a*b*x^2 + b^2)) - (1/4)^(1/4)*(-I*a^2*b*x^2 - I*a*b^2)*((16*a^4*b^4 + 32*a^3*b^3*c + 24*a^2*b^2*c^2 +
8*a*b*c^3 + c^4)/(a^5*b^5))^(1/4)*log((8*a^3*b^5 + 12*a^2*b^4*c + 6*a*b^3*c^2 + b^2*c^3 + (8*a^5*b^3 + 12*a^4*
b^2*c + 6*a^3*b*c^2 + a^2*c^3)*x^4 + 6*(8*a^4*b^4 + 12*a^3*b^3*c + 6*a^2*b^2*c^2 + a*b*c^3)*x^2 - 8*sqrt(a*x^3
+ b*x)*(-I*(1/4)^(1/4)*(4*a^4*b^4 + 4*a^3*b^3*c + a^2*b^2*c^2)*x*((16*a^4*b^4 + 32*a^3*b^3*c + 24*a^2*b^2*c^2
+ 8*a*b*c^3 + c^4)/(a^5*b^5))^(1/4) + (1/4)^(3/4)*(I*a^5*b^4*x^2 + I*a^4*b^5)*((16*a^4*b^4 + 32*a^3*b^3*c + 2
4*a^2*b^2*c^2 + 8*a*b*c^3 + c^4)/(a^5*b^5))^(3/4)) - 4*((2*a^5*b^4 + a^4*b^3*c)*x^3 + (2*a^4*b^5 + a^3*b^4*c)*
x)*sqrt((16*a^4*b^4 + 32*a^3*b^3*c + 24*a^2*b^2*c^2 + 8*a*b*c^3 + c^4)/(a^5*b^5)))/(a^2*x^4 - 2*a*b*x^2 + b^2)
) + 8*sqrt(a*x^3 + b*x)*(2*a*b - c))/(a^2*b*x^2 + a*b^2)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

integrate((a^2*x^4 + c*x^2 + b^2)/((a^2*x^4 - b^2)*sqrt(a*x^3 + b*x)), x)

Giac [F]

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

[In]

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

[Out]

integrate((a^2*x^4 + c*x^2 + b^2)/((a^2*x^4 - b^2)*sqrt(a*x^3 + b*x)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {b^2+c x^2+a^2 x^4}{\sqrt {b x+a x^3} \left (-b^2+a^2 x^4\right )} \, dx=\text {Hanged} \]

[In]

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

[Out]

\text{Hanged}