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

Optimal. Leaf size=114 \[ \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{a x^3-b x^2}}{\sqrt {a x^3-b x^2}+x^2}\right )-\sqrt {2} \tan ^{-1}\left (\frac {\frac {\sqrt {a x^3-b x^2}}{\sqrt {2}}-\frac {x^2}{\sqrt {2}}}{x \sqrt [4]{a x^3-b x^2}}\right ) \]

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Rubi [C]  time = 14.51, antiderivative size = 2624, normalized size of antiderivative = 23.02, number of steps used = 21, number of rules used = 8, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2056, 6728, 107, 106, 490, 1217, 220, 1707}

result too large to display

Warning: Unable to verify antiderivative.

[In]

Int[(-2*b + a*x)/((-b + a*x + x^2)*(-(b*x^2) + a*x^3)^(1/4)),x]

[Out]

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

Rule 106

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

Rule 107

Int[1/(((a_.) + (b_.)*(x_))*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(1/4)), x_Symbol] :> Dist[Sqrt[-((f*
(c + d*x))/(d*e - c*f))]/Sqrt[c + d*x], Int[1/((a + b*x)*Sqrt[-((c*f)/(d*e - c*f)) - (d*f*x)/(d*e - c*f)]*(e +
f*x)^(1/4)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] &&  !GtQ[-(f/(d*e - c*f)), 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 490

Int[(x_)^2/(((a_) + (b_.)*(x_)^4)*Sqrt[(c_) + (d_.)*(x_)^4]), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]],
s = Denominator[Rt[-(a/b), 2]]}, Dist[s/(2*b), Int[1/((r + s*x^2)*Sqrt[c + d*x^4]), x], x] - Dist[s/(2*b), In
t[1/((r - s*x^2)*Sqrt[c + d*x^4]), 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 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 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 {-2 b+a x}{\left (-b+a x+x^2\right ) \sqrt [4]{-b x^2+a x^3}} \, dx &=\frac {\left (\sqrt {x} \sqrt [4]{-b+a x}\right ) \int \frac {-2 b+a x}{\sqrt {x} \sqrt [4]{-b+a x} \left (-b+a x+x^2\right )} \, dx}{\sqrt [4]{-b x^2+a x^3}}\\ &=\frac {\left (\sqrt {x} \sqrt [4]{-b+a x}\right ) \int \left (\frac {a-\sqrt {a^2+4 b}}{\sqrt {x} \left (a-\sqrt {a^2+4 b}+2 x\right ) \sqrt [4]{-b+a x}}+\frac {a+\sqrt {a^2+4 b}}{\sqrt {x} \left (a+\sqrt {a^2+4 b}+2 x\right ) \sqrt [4]{-b+a x}}\right ) \, dx}{\sqrt [4]{-b x^2+a x^3}}\\ &=\frac {\left (\left (a-\sqrt {a^2+4 b}\right ) \sqrt {x} \sqrt [4]{-b+a x}\right ) \int \frac {1}{\sqrt {x} \left (a-\sqrt {a^2+4 b}+2 x\right ) \sqrt [4]{-b+a x}} \, dx}{\sqrt [4]{-b x^2+a x^3}}+\frac {\left (\left (a+\sqrt {a^2+4 b}\right ) \sqrt {x} \sqrt [4]{-b+a x}\right ) \int \frac {1}{\sqrt {x} \left (a+\sqrt {a^2+4 b}+2 x\right ) \sqrt [4]{-b+a x}} \, dx}{\sqrt [4]{-b x^2+a x^3}}\\ &=\frac {\left (\left (a-\sqrt {a^2+4 b}\right ) \sqrt {\frac {a x}{b}} \sqrt [4]{-b+a x}\right ) \int \frac {1}{\sqrt {\frac {a x}{b}} \left (a-\sqrt {a^2+4 b}+2 x\right ) \sqrt [4]{-b+a x}} \, dx}{\sqrt [4]{-b x^2+a x^3}}+\frac {\left (\left (a+\sqrt {a^2+4 b}\right ) \sqrt {\frac {a x}{b}} \sqrt [4]{-b+a x}\right ) \int \frac {1}{\sqrt {\frac {a x}{b}} \left (a+\sqrt {a^2+4 b}+2 x\right ) \sqrt [4]{-b+a x}} \, dx}{\sqrt [4]{-b x^2+a x^3}}\\ &=-\frac {\left (4 \left (a-\sqrt {a^2+4 b}\right ) \sqrt {\frac {a x}{b}} \sqrt [4]{-b+a x}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (-2 b-a \left (a-\sqrt {a^2+4 b}\right )-2 x^4\right ) \sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x}\right )}{\sqrt [4]{-b x^2+a x^3}}-\frac {\left (4 \left (a+\sqrt {a^2+4 b}\right ) \sqrt {\frac {a x}{b}} \sqrt [4]{-b+a x}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (-2 b-a \left (a+\sqrt {a^2+4 b}\right )-2 x^4\right ) \sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x}\right )}{\sqrt [4]{-b x^2+a x^3}}\\ &=-\frac {\left (\sqrt {2} \left (a-\sqrt {a^2+4 b}\right ) \sqrt {\frac {a x}{b}} \sqrt [4]{-b+a x}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\sqrt {-a^2-2 b+a \sqrt {a^2+4 b}}-\sqrt {2} x^2\right ) \sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x}\right )}{\sqrt [4]{-b x^2+a x^3}}+\frac {\left (\sqrt {2} \left (a-\sqrt {a^2+4 b}\right ) \sqrt {\frac {a x}{b}} \sqrt [4]{-b+a x}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\sqrt {-a^2-2 b+a \sqrt {a^2+4 b}}+\sqrt {2} x^2\right ) \sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x}\right )}{\sqrt [4]{-b x^2+a x^3}}-\frac {\left (\sqrt {2} \left (a+\sqrt {a^2+4 b}\right ) \sqrt {\frac {a x}{b}} \sqrt [4]{-b+a x}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\sqrt {-a^2-2 b-a \sqrt {a^2+4 b}}-\sqrt {2} x^2\right ) \sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x}\right )}{\sqrt [4]{-b x^2+a x^3}}+\frac {\left (\sqrt {2} \left (a+\sqrt {a^2+4 b}\right ) \sqrt {\frac {a x}{b}} \sqrt [4]{-b+a x}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\sqrt {-a^2-2 b-a \sqrt {a^2+4 b}}+\sqrt {2} x^2\right ) \sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x}\right )}{\sqrt [4]{-b x^2+a x^3}}\\ &=-\frac {\left (\sqrt {2} \left (a+\sqrt {a^2+4 b}\right ) \left (\sqrt {2} \sqrt {b}-\sqrt {-a^2-2 b-a \sqrt {a^2+4 b}}\right ) \sqrt {\frac {a x}{b}} \sqrt [4]{-b+a x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x}\right )}{\left (a^2+4 b+a \sqrt {a^2+4 b}\right ) \sqrt [4]{-b x^2+a x^3}}-\frac {\left (\sqrt {2} \left (a+\sqrt {a^2+4 b}\right ) \left (\sqrt {2} \sqrt {b}+\sqrt {-a^2-2 b-a \sqrt {a^2+4 b}}\right ) \sqrt {\frac {a x}{b}} \sqrt [4]{-b+a x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x}\right )}{\left (a^2+4 b+a \sqrt {a^2+4 b}\right ) \sqrt [4]{-b x^2+a x^3}}-\frac {\left (\sqrt {2} \sqrt {b} \left (a+\sqrt {a^2+4 b}\right ) \left (2 \sqrt {b}-\sqrt {2} \sqrt {-a^2-2 b-a \sqrt {a^2+4 b}}\right ) \sqrt {\frac {a x}{b}} \sqrt [4]{-b+a x}\right ) \operatorname {Subst}\left (\int \frac {1+\frac {x^2}{\sqrt {b}}}{\left (\sqrt {-a^2-2 b-a \sqrt {a^2+4 b}}-\sqrt {2} x^2\right ) \sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x}\right )}{\left (a^2+4 b+a \sqrt {a^2+4 b}\right ) \sqrt [4]{-b x^2+a x^3}}+\frac {\left (\sqrt {2} \sqrt {b} \left (a+\sqrt {a^2+4 b}\right ) \left (2 \sqrt {b}+\sqrt {2} \sqrt {-a^2-2 b-a \sqrt {a^2+4 b}}\right ) \sqrt {\frac {a x}{b}} \sqrt [4]{-b+a x}\right ) \operatorname {Subst}\left (\int \frac {1+\frac {x^2}{\sqrt {b}}}{\left (\sqrt {-a^2-2 b-a \sqrt {a^2+4 b}}+\sqrt {2} x^2\right ) \sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x}\right )}{\left (a^2+4 b+a \sqrt {a^2+4 b}\right ) \sqrt [4]{-b x^2+a x^3}}-\frac {\left (\sqrt {2} \left (a-\sqrt {a^2+4 b}\right ) \left (\sqrt {2} \sqrt {b}-\sqrt {-a^2-2 b+a \sqrt {a^2+4 b}}\right ) \sqrt {\frac {a x}{b}} \sqrt [4]{-b+a x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x}\right )}{\left (a^2+4 b-a \sqrt {a^2+4 b}\right ) \sqrt [4]{-b x^2+a x^3}}-\frac {\left (\sqrt {2} \left (a-\sqrt {a^2+4 b}\right ) \left (\sqrt {2} \sqrt {b}+\sqrt {-a^2-2 b+a \sqrt {a^2+4 b}}\right ) \sqrt {\frac {a x}{b}} \sqrt [4]{-b+a x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x}\right )}{\left (a^2+4 b-a \sqrt {a^2+4 b}\right ) \sqrt [4]{-b x^2+a x^3}}-\frac {\left (\sqrt {2} \sqrt {b} \left (a-\sqrt {a^2+4 b}\right ) \left (2 \sqrt {b}-\sqrt {2} \sqrt {-a^2-2 b+a \sqrt {a^2+4 b}}\right ) \sqrt {\frac {a x}{b}} \sqrt [4]{-b+a x}\right ) \operatorname {Subst}\left (\int \frac {1+\frac {x^2}{\sqrt {b}}}{\left (\sqrt {-a^2-2 b+a \sqrt {a^2+4 b}}-\sqrt {2} x^2\right ) \sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x}\right )}{\left (a^2+4 b-a \sqrt {a^2+4 b}\right ) \sqrt [4]{-b x^2+a x^3}}+\frac {\left (\sqrt {2} \sqrt {b} \left (a-\sqrt {a^2+4 b}\right ) \left (2 \sqrt {b}+\sqrt {2} \sqrt {-a^2-2 b+a \sqrt {a^2+4 b}}\right ) \sqrt {\frac {a x}{b}} \sqrt [4]{-b+a x}\right ) \operatorname {Subst}\left (\int \frac {1+\frac {x^2}{\sqrt {b}}}{\left (\sqrt {-a^2-2 b+a \sqrt {a^2+4 b}}+\sqrt {2} x^2\right ) \sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x}\right )}{\left (a^2+4 b-a \sqrt {a^2+4 b}\right ) \sqrt [4]{-b x^2+a x^3}}\\ &=\frac {2^{3/4} \sqrt {b} \left (a^3+4 a b+\left (a^2+2 b\right ) \sqrt {a^2+4 b}\right ) \sqrt {\frac {a x}{b}} \sqrt [4]{-b+a x} \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {-a-\sqrt {a^2+4 b}} \sqrt [4]{-b+a x}}{\sqrt [4]{2} \sqrt {b} \sqrt [4]{-a^2-2 b-a \sqrt {a^2+4 b}} \sqrt {\frac {a x}{b}}}\right )}{\sqrt {a} \sqrt {-a-\sqrt {a^2+4 b}} \sqrt [4]{-a^2-2 b-a \sqrt {a^2+4 b}} \left (a^2+4 b+a \sqrt {a^2+4 b}\right ) \sqrt [4]{-b x^2+a x^3}}-\frac {\sqrt {b} \sqrt {a+\sqrt {a^2+4 b}} \sqrt {\frac {a x}{b}} \sqrt [4]{-b+a x} \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {a+\sqrt {a^2+4 b}} \sqrt [4]{-b+a x}}{\sqrt [4]{2} \sqrt {b} \sqrt [4]{-a^2-2 b-a \sqrt {a^2+4 b}} \sqrt {\frac {a x}{b}}}\right )}{\sqrt [4]{2} \sqrt {a} \sqrt [4]{-a^2-2 b-a \sqrt {a^2+4 b}} \sqrt [4]{-b x^2+a x^3}}-\frac {\sqrt {b} \sqrt {a-\sqrt {a^2+4 b}} \sqrt {\frac {a x}{b}} \sqrt [4]{-b+a x} \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {a-\sqrt {a^2+4 b}} \sqrt [4]{-b+a x}}{\sqrt [4]{2} \sqrt {b} \sqrt [4]{-a^2-2 b+a \sqrt {a^2+4 b}} \sqrt {\frac {a x}{b}}}\right )}{\sqrt [4]{2} \sqrt {a} \sqrt [4]{-a^2-2 b+a \sqrt {a^2+4 b}} \sqrt [4]{-b x^2+a x^3}}-\frac {\sqrt {b} \sqrt {-a+\sqrt {a^2+4 b}} \sqrt {\frac {a x}{b}} \sqrt [4]{-b+a x} \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {-a+\sqrt {a^2+4 b}} \sqrt [4]{-b+a x}}{\sqrt [4]{2} \sqrt {b} \sqrt [4]{-a^2-2 b+a \sqrt {a^2+4 b}} \sqrt {\frac {a x}{b}}}\right )}{\sqrt [4]{2} \sqrt {a} \sqrt [4]{-a^2-2 b+a \sqrt {a^2+4 b}} \sqrt [4]{-b x^2+a x^3}}-\frac {\left (a+\sqrt {a^2+4 b}\right ) \left (2 \sqrt {b}-\sqrt {2} \sqrt {-a^2-2 b-a \sqrt {a^2+4 b}}\right ) \sqrt [4]{-b+a x} \sqrt {\frac {a x}{\left (\sqrt {b}+\sqrt {-b+a x}\right )^2}} \left (\sqrt {b}+\sqrt {-b+a x}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{-b+a x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{2 \sqrt [4]{b} \left (a^2+4 b+a \sqrt {a^2+4 b}\right ) \sqrt [4]{-b x^2+a x^3}}-\frac {\left (a+\sqrt {a^2+4 b}\right ) \left (2 \sqrt {b}+\sqrt {2} \sqrt {-a^2-2 b-a \sqrt {a^2+4 b}}\right ) \sqrt [4]{-b+a x} \sqrt {\frac {a x}{\left (\sqrt {b}+\sqrt {-b+a x}\right )^2}} \left (\sqrt {b}+\sqrt {-b+a x}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{-b+a x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{2 \sqrt [4]{b} \left (a^2+4 b+a \sqrt {a^2+4 b}\right ) \sqrt [4]{-b x^2+a x^3}}-\frac {\left (a-\sqrt {a^2+4 b}\right ) \left (2 \sqrt {b}-\sqrt {2} \sqrt {-a^2-2 b+a \sqrt {a^2+4 b}}\right ) \sqrt [4]{-b+a x} \sqrt {\frac {a x}{\left (\sqrt {b}+\sqrt {-b+a x}\right )^2}} \left (\sqrt {b}+\sqrt {-b+a x}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{-b+a x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{2 \sqrt [4]{b} \left (a^2+4 b-a \sqrt {a^2+4 b}\right ) \sqrt [4]{-b x^2+a x^3}}-\frac {\left (a-\sqrt {a^2+4 b}\right ) \left (2 \sqrt {b}+\sqrt {2} \sqrt {-a^2-2 b+a \sqrt {a^2+4 b}}\right ) \sqrt [4]{-b+a x} \sqrt {\frac {a x}{\left (\sqrt {b}+\sqrt {-b+a x}\right )^2}} \left (\sqrt {b}+\sqrt {-b+a x}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{-b+a x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{2 \sqrt [4]{b} \left (a^2+4 b-a \sqrt {a^2+4 b}\right ) \sqrt [4]{-b x^2+a x^3}}+\frac {\left (a+\sqrt {a^2+4 b}\right ) \left (\sqrt {2} \sqrt {b}+\sqrt {-a^2-2 b-a \sqrt {a^2+4 b}}\right )^2 \sqrt [4]{-b+a x} \sqrt {\frac {a x}{\left (\sqrt {b}+\sqrt {-b+a x}\right )^2}} \left (\sqrt {b}+\sqrt {-b+a x}\right ) \Pi \left (-\frac {\left (\sqrt {2} \sqrt {b}-\sqrt {-a^2-2 b-a \sqrt {a^2+4 b}}\right )^2}{4 \sqrt {2} \sqrt {b} \sqrt {-a^2-2 b-a \sqrt {a^2+4 b}}};2 \tan ^{-1}\left (\frac {\sqrt [4]{-b+a x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{2 \sqrt {2} \sqrt [4]{b} \sqrt {-a^2-2 b-a \sqrt {a^2+4 b}} \left (a^2+4 b+a \sqrt {a^2+4 b}\right ) \sqrt [4]{-b x^2+a x^3}}-\frac {\left (a+\sqrt {a^2+4 b}\right ) \left (\sqrt {2} \sqrt {b}-\sqrt {-a^2-2 b-a \sqrt {a^2+4 b}}\right )^2 \sqrt [4]{-b+a x} \sqrt {\frac {a x}{\left (\sqrt {b}+\sqrt {-b+a x}\right )^2}} \left (\sqrt {b}+\sqrt {-b+a x}\right ) \Pi \left (\frac {\left (\sqrt {2} \sqrt {b}+\sqrt {-a^2-2 b-a \sqrt {a^2+4 b}}\right )^2}{4 \sqrt {2} \sqrt {b} \sqrt {-a^2-2 b-a \sqrt {a^2+4 b}}};2 \tan ^{-1}\left (\frac {\sqrt [4]{-b+a x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{2 \sqrt {2} \sqrt [4]{b} \sqrt {-a^2-2 b-a \sqrt {a^2+4 b}} \left (a^2+4 b+a \sqrt {a^2+4 b}\right ) \sqrt [4]{-b x^2+a x^3}}+\frac {\left (a-\sqrt {a^2+4 b}\right ) \left (\sqrt {2} \sqrt {b}+\sqrt {-a^2-2 b+a \sqrt {a^2+4 b}}\right )^2 \sqrt [4]{-b+a x} \sqrt {\frac {a x}{\left (\sqrt {b}+\sqrt {-b+a x}\right )^2}} \left (\sqrt {b}+\sqrt {-b+a x}\right ) \Pi \left (-\frac {\left (\sqrt {2} \sqrt {b}-\sqrt {-a^2-2 b+a \sqrt {a^2+4 b}}\right )^2}{4 \sqrt {2} \sqrt {b} \sqrt {-a^2-2 b+a \sqrt {a^2+4 b}}};2 \tan ^{-1}\left (\frac {\sqrt [4]{-b+a x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{2 \sqrt {2} \sqrt [4]{b} \left (a^2+4 b-a \sqrt {a^2+4 b}\right ) \sqrt {-a^2-2 b+a \sqrt {a^2+4 b}} \sqrt [4]{-b x^2+a x^3}}-\frac {\left (a-\sqrt {a^2+4 b}\right ) \left (\sqrt {2} \sqrt {b}-\sqrt {-a^2-2 b+a \sqrt {a^2+4 b}}\right )^2 \sqrt [4]{-b+a x} \sqrt {\frac {a x}{\left (\sqrt {b}+\sqrt {-b+a x}\right )^2}} \left (\sqrt {b}+\sqrt {-b+a x}\right ) \Pi \left (\frac {\left (\sqrt {2} \sqrt {b}+\sqrt {-a^2-2 b+a \sqrt {a^2+4 b}}\right )^2}{4 \sqrt {2} \sqrt {b} \sqrt {-a^2-2 b+a \sqrt {a^2+4 b}}};2 \tan ^{-1}\left (\frac {\sqrt [4]{-b+a x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{2 \sqrt {2} \sqrt [4]{b} \left (a^2+4 b-a \sqrt {a^2+4 b}\right ) \sqrt {-a^2-2 b+a \sqrt {a^2+4 b}} \sqrt [4]{-b x^2+a x^3}}\\ \end {align*}

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Mathematica [C]  time = 7.43, size = 611, normalized size = 5.36 \begin {gather*} \frac {i \sqrt {2} a \sqrt {\frac {a x}{a x-b}} (a x-b)^{3/4} \left (\left (-a^2+\sqrt {a^4+4 a^2 b}-4 b\right ) \sqrt {-\frac {a^2+\sqrt {a^4+4 a^2 b}+2 b}{b^2}} \Pi \left (-\frac {i \sqrt {2}}{\sqrt {b} \sqrt {\frac {-a^2-2 b+\sqrt {a^4+4 b a^2}}{b^2}}};\left .i \sinh ^{-1}\left (\frac {\sqrt {i \sqrt {b}}}{\sqrt [4]{a x-b}}\right )\right |-1\right )+\left (a^2-\sqrt {a^4+4 a^2 b}+4 b\right ) \sqrt {-\frac {a^2+\sqrt {a^4+4 a^2 b}+2 b}{b^2}} \Pi \left (\frac {i \sqrt {2}}{\sqrt {b} \sqrt {\frac {-a^2-2 b+\sqrt {a^4+4 b a^2}}{b^2}}};\left .i \sinh ^{-1}\left (\frac {\sqrt {i \sqrt {b}}}{\sqrt [4]{a x-b}}\right )\right |-1\right )+\sqrt {\frac {-a^2+\sqrt {a^4+4 a^2 b}-2 b}{b^2}} \left (a^2+\sqrt {a^4+4 a^2 b}+4 b\right ) \left (\Pi \left (-\frac {i \sqrt {2}}{\sqrt {b} \sqrt {-\frac {a^2+2 b+\sqrt {a^4+4 b a^2}}{b^2}}};\left .i \sinh ^{-1}\left (\frac {\sqrt {i \sqrt {b}}}{\sqrt [4]{a x-b}}\right )\right |-1\right )-\Pi \left (\frac {i \sqrt {2}}{\sqrt {b} \sqrt {-\frac {a^2+2 b+\sqrt {a^4+4 b a^2}}{b^2}}};\left .i \sinh ^{-1}\left (\frac {\sqrt {i \sqrt {b}}}{\sqrt [4]{a x-b}}\right )\right |-1\right )\right )\right )}{\left (i \sqrt {b}\right )^{5/2} \sqrt {a^4+4 a^2 b} \sqrt {\frac {-a^2+\sqrt {a^4+4 a^2 b}-2 b}{b^2}} \sqrt {-\frac {a^2+\sqrt {a^4+4 a^2 b}+2 b}{b^2}} \sqrt [4]{x^2 (a x-b)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-2*b + a*x)/((-b + a*x + x^2)*(-(b*x^2) + a*x^3)^(1/4)),x]

[Out]

(I*Sqrt[2]*a*Sqrt[(a*x)/(-b + a*x)]*(-b + a*x)^(3/4)*((-a^2 - 4*b + Sqrt[a^4 + 4*a^2*b])*Sqrt[-((a^2 + 2*b + S
qrt[a^4 + 4*a^2*b])/b^2)]*EllipticPi[((-I)*Sqrt[2])/(Sqrt[b]*Sqrt[(-a^2 - 2*b + Sqrt[a^4 + 4*a^2*b])/b^2]), I*
ArcSinh[Sqrt[I*Sqrt[b]]/(-b + a*x)^(1/4)], -1] + (a^2 + 4*b - Sqrt[a^4 + 4*a^2*b])*Sqrt[-((a^2 + 2*b + Sqrt[a^
4 + 4*a^2*b])/b^2)]*EllipticPi[(I*Sqrt[2])/(Sqrt[b]*Sqrt[(-a^2 - 2*b + Sqrt[a^4 + 4*a^2*b])/b^2]), I*ArcSinh[S
qrt[I*Sqrt[b]]/(-b + a*x)^(1/4)], -1] + Sqrt[(-a^2 - 2*b + Sqrt[a^4 + 4*a^2*b])/b^2]*(a^2 + 4*b + Sqrt[a^4 + 4
*a^2*b])*(EllipticPi[((-I)*Sqrt[2])/(Sqrt[b]*Sqrt[-((a^2 + 2*b + Sqrt[a^4 + 4*a^2*b])/b^2)]), I*ArcSinh[Sqrt[I
*Sqrt[b]]/(-b + a*x)^(1/4)], -1] - EllipticPi[(I*Sqrt[2])/(Sqrt[b]*Sqrt[-((a^2 + 2*b + Sqrt[a^4 + 4*a^2*b])/b^
2)]), I*ArcSinh[Sqrt[I*Sqrt[b]]/(-b + a*x)^(1/4)], -1])))/((I*Sqrt[b])^(5/2)*Sqrt[a^4 + 4*a^2*b]*Sqrt[(-a^2 -
2*b + Sqrt[a^4 + 4*a^2*b])/b^2]*Sqrt[-((a^2 + 2*b + Sqrt[a^4 + 4*a^2*b])/b^2)]*(x^2*(-b + a*x))^(1/4))

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IntegrateAlgebraic [A]  time = 0.55, size = 114, normalized size = 1.00 \begin {gather*} -\sqrt {2} \tan ^{-1}\left (\frac {-\frac {x^2}{\sqrt {2}}+\frac {\sqrt {-b x^2+a x^3}}{\sqrt {2}}}{x \sqrt [4]{-b x^2+a x^3}}\right )+\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{-b x^2+a x^3}}{x^2+\sqrt {-b x^2+a x^3}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(-2*b + a*x)/((-b + a*x + x^2)*(-(b*x^2) + a*x^3)^(1/4)),x]

[Out]

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

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a x - 2 \, b}{{\left (a x^{3} - b x^{2}\right )}^{\frac {1}{4}} {\left (a x + x^{2} - b\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.09, size = 0, normalized size = 0.00 \[\int \frac {a x -2 b}{\left (a x +x^{2}-b \right ) \left (a \,x^{3}-b \,x^{2}\right )^{\frac {1}{4}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a x - 2 \, b}{{\left (a x^{3} - b x^{2}\right )}^{\frac {1}{4}} {\left (a x + x^{2} - b\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int -\frac {2\,b-a\,x}{{\left (a\,x^3-b\,x^2\right )}^{1/4}\,\left (x^2+a\,x-b\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a*x - 2*b)/((x**2*(a*x - b))**(1/4)*(a*x - b + x**2)), x)

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