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

Optimal. Leaf size=167 \[ -\frac {(2 a b+c) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {a x^3+b x}}{a x^2+b}\right )}{4 \sqrt {2} a^{5/4} b^{5/4}}-\frac {(2 a b+c) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {a x^3+b x}}{a x^2+b}\right )}{4 \sqrt {2} a^{5/4} b^{5/4}}+\frac {(c-2 a b) \sqrt {a x^3+b x}}{2 a b \left (a x^2+b\right )} \]

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Rubi [C]  time = 1.74, antiderivative size = 503, normalized size of antiderivative = 3.01, number of steps used = 18, number of rules used = 12, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {2056, 6715, 6725, 220, 1455, 527, 523, 409, 1211, 1699, 206, 203} \begin {gather*} -\frac {\sqrt {x} (2 a b+c) \sqrt {a x^2+b} \tan ^{-1}\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) \sqrt {a x^2+b} \tanh ^{-1}\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}} F\left (2 \tan ^{-1}\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}} F\left (2 \tan ^{-1}\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 {x (2 a b-c)}{2 a b \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}} F\left (2 \tan ^{-1}\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}} \end {gather*}

Antiderivative was successfully verified.

[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 203

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

Rule 206

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

Rule 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 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 523

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 527

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 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 1455

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*x^n)/e)^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 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 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 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]

Rubi steps

\begin {align*} \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 {\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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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}} F\left (2 \tan ^{-1}\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 ) \operatorname {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}} F\left (2 \tan ^{-1}\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 ) \operatorname {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}} F\left (2 \tan ^{-1}\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 ) \operatorname {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 ) \operatorname {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}} F\left (2 \tan ^{-1}\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}} F\left (2 \tan ^{-1}\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 ) \operatorname {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 ) \operatorname {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}} F\left (2 \tan ^{-1}\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}} F\left (2 \tan ^{-1}\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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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}} F\left (2 \tan ^{-1}\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}} F\left (2 \tan ^{-1}\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}} F\left (2 \tan ^{-1}\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 ) \operatorname {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 ) \operatorname {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} \tan ^{-1}\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} \tanh ^{-1}\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}} F\left (2 \tan ^{-1}\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}} F\left (2 \tan ^{-1}\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}} F\left (2 \tan ^{-1}\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*}

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Mathematica [C]  time = 0.70, size = 105, normalized size = 0.63 \begin {gather*} -\frac {x \left ((2 a b+c) \left (a x^2+b\right ) F_1\left (\frac {1}{4};-\frac {1}{2},1;\frac {5}{4};-\frac {a x^2}{b},\frac {a x^2}{b}\right )+b (2 a b-c) \sqrt {\frac {a x^2}{b}+1}\right )}{2 a b^2 \sqrt {x \left (a x^2+b\right )} \sqrt {\frac {a x^2}{b}+1}} \end {gather*}

Warning: Unable to verify antiderivative.

[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]

-1/2*(x*(b*(2*a*b - c)*Sqrt[1 + (a*x^2)/b] + (2*a*b + c)*(b + a*x^2)*AppellF1[1/4, -1/2, 1, 5/4, -((a*x^2)/b),
(a*x^2)/b]))/(a*b^2*Sqrt[x*(b + a*x^2)]*Sqrt[1 + (a*x^2)/b])

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IntegrateAlgebraic [A]  time = 0.66, size = 167, normalized size = 1.00 \begin {gather*} \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) \tan ^{-1}\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) \tanh ^{-1}\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}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-2*a*b + c)*Sqrt[b*x + a*x^3])/(2*a*b*(b + a*x^2)) - ((2*a*b + c)*ArcTan[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[b*x +
a*x^3])/(b + a*x^2)])/(4*Sqrt[2]*a^(5/4)*b^(5/4)) - ((2*a*b + c)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[b*x +
a*x^3])/(b + a*x^2)])/(4*Sqrt[2]*a^(5/4)*b^(5/4))

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fricas [B]  time = 1.05, size = 1675, normalized size = 10.03

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[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*(4*(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)*arctan(1/2*sqrt(a*x^3 + b*x)*(4*(1/4)^(3/4)*(8*a^7*b^7 + 12*a^6*b^6*c + 6*a^5*b^5*c^2 + a^4*b^4*c^
3)*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))^(3/4) + sqrt(64*a^6*b^6 + 192*
a^5*b^5*c + 240*a^4*b^4*c^2 + 160*a^3*b^3*c^3 + 60*a^2*b^2*c^4 + 12*a*b*c^5 + c^6)*(4*(1/4)^(3/4)*a^4*b^4*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))^(3/4) + (1/4)^(1/4)*(4*a^3*b^4 + 4*a^
2*b^3*c + a*b^2*c^2 + (4*a^4*b^3 + 4*a^3*b^2*c + a^2*b*c^2)*x^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)) - (1/4)^(1/4)*(32*a^6*b^7 + 80*a^5*b^6*c + 80*a^4*b^5*c^2 + 40*a^3*b^4*c^
3 + 10*a^2*b^3*c^4 + a*b^2*c^5 + (32*a^7*b^6 + 80*a^6*b^5*c + 80*a^5*b^4*c^2 + 40*a^4*b^3*c^3 + 10*a^3*b^2*c^4
+ a^2*b*c^5)*x^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))/((64*a^7*
b^6 + 192*a^6*b^5*c + 240*a^5*b^4*c^2 + 160*a^4*b^3*c^3 + 60*a^3*b^2*c^4 + 12*a^2*b*c^5 + a*c^6)*x^3 + (64*a^6
*b^7 + 192*a^5*b^6*c + 240*a^4*b^5*c^2 + 160*a^3*b^4*c^3 + 60*a^2*b^3*c^4 + 12*a*b^2*c^5 + b*c^6)*x)) + (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)) + 8*sqrt(a*x^3 + b*x)*(2*a*b - c))/(a^2*b*x^2 + a*b^2)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [C]  time = 0.17, size = 567, normalized size = 3.40

method	result	size
default	\(\frac {\sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {x a}{\sqrt {-a b}}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{a \sqrt {a \,x^{3}+b x}}+\frac {\left (-2 a b +c \right ) \left (\frac {x}{b \sqrt {\left (x^{2}+\frac {b}{a}\right ) a x}}+\frac {\sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {x a}{\sqrt {-a b}}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{2 b a \sqrt {a \,x^{3}+b x}}\right )}{2 a}+\frac {\left (2 a b +c \right ) \left (\frac {\sqrt {-a b}\, \sqrt {\frac {x a}{\sqrt {-a b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {-a b}}+2}\, \sqrt {-\frac {x a}{\sqrt {-a b}}}\, \EllipticPi \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, -\frac {\sqrt {-a b}}{a \left (-\frac {\sqrt {-a b}}{a}-\frac {\sqrt {a b}}{a}\right )}, \frac {\sqrt {2}}{2}\right )}{2 \sqrt {a b}\, a \sqrt {a \,x^{3}+b x}\, \left (-\frac {\sqrt {-a b}}{a}-\frac {\sqrt {a b}}{a}\right )}-\frac {\sqrt {-a b}\, \sqrt {\frac {x a}{\sqrt {-a b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {-a b}}+2}\, \sqrt {-\frac {x a}{\sqrt {-a b}}}\, \EllipticPi \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, -\frac {\sqrt {-a b}}{a \left (-\frac {\sqrt {-a b}}{a}+\frac {\sqrt {a b}}{a}\right )}, \frac {\sqrt {2}}{2}\right )}{2 \sqrt {a b}\, a \sqrt {a \,x^{3}+b x}\, \left (-\frac {\sqrt {-a b}}{a}+\frac {\sqrt {a b}}{a}\right )}\right )}{2 a}\)	\(567\)
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}}}\, \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}}}\, \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}}}\, \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}}}\, \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}}}\, \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}}}\, \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 )}\)	\(827\)

Verification of antiderivative is not currently implemented for this CAS.

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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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(-(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}

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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