\(\int \frac {b+2 a x}{(-b+a x) (2 b+a x) \sqrt [4]{-1+b x+a x^2}} \, dx\) [2397]

Optimal result

Integrand size = 39, antiderivative size = 193 \[ \int \frac {b+2 a x}{(-b+a x) (2 b+a x) \sqrt [4]{-1+b x+a x^2}} \, dx=\frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{a-2 b^2} \sqrt [4]{-1+b x+a x^2}}{\sqrt {a-2 b^2}-\sqrt {a} \sqrt {-1+b x+a x^2}}\right )}{a^{3/4} \sqrt [4]{a-2 b^2}}-\frac {\sqrt {2} \text {arctanh}\left (\frac {\frac {\sqrt [4]{a-2 b^2}}{\sqrt {2} \sqrt [4]{a}}+\frac {\sqrt [4]{a} \sqrt {-1+b x+a x^2}}{\sqrt {2} \sqrt [4]{a-2 b^2}}}{\sqrt [4]{-1+b x+a x^2}}\right )}{a^{3/4} \sqrt [4]{a-2 b^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 1.47 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.27, number of steps used = 30, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6874, 763, 762, 760, 408, 504, 1227, 551, 455, 65, 304, 211, 214} \[ \int \frac {b+2 a x}{(-b+a x) (2 b+a x) \sqrt [4]{-1+b x+a x^2}} \, dx=\frac {2 \sqrt [4]{4 a+b^2} \sqrt [4]{\frac {a \left (-a x^2-b x+1\right )}{4 a+b^2}} \arctan \left (\frac {\sqrt [4]{4 a+b^2} \sqrt [4]{1-\frac {(2 a x+b)^2}{4 a+b^2}}}{\sqrt {2} \sqrt [4]{a-2 b^2}}\right )}{a \sqrt [4]{a-2 b^2} \sqrt [4]{a x^2+b x-1}}-\frac {2 \sqrt [4]{4 a+b^2} \sqrt [4]{\frac {a \left (-a x^2-b x+1\right )}{4 a+b^2}} \text {arctanh}\left (\frac {\sqrt [4]{4 a+b^2} \sqrt [4]{1-\frac {(2 a x+b)^2}{4 a+b^2}}}{\sqrt {2} \sqrt [4]{a-2 b^2}}\right )}{a \sqrt [4]{a-2 b^2} \sqrt [4]{a x^2+b x-1}} \]

[In]

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

[Out]

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

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 211

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

Rule 214

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

Rule 304

Int[(x_)^2/((a_) + (b_.)*(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), x], x] - Dist[s/(2*b), Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !
GtQ[a/b, 0]

Rule 408

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

Rule 455

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[m
- n + 1, 0]

Rule 504

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), Int[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 551

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(1/(a*Sqr
t[c]*Sqrt[e]*Rt[-d/c, 2]))*EllipticPi[b*(c/(a*d)), ArcSin[Rt[-d/c, 2]*x], c*(f/(d*e))], x] /; FreeQ[{a, b, c,
d, e, f}, x] &&  !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] &&  !( !GtQ[f/e, 0] && SimplerSqrtQ[-f/e, -d/c])

Rule 760

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

Rule 762

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

Rule 763

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

Rule 1227

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[(-a)*c, 2]}, Dist[Sqrt[-c],
Int[1/((d + e*x^2)*Sqrt[q + c*x^2]*Sqrt[q - c*x^2]), x], x]] /; FreeQ[{a, c, d, e}, x] && GtQ[a, 0] && LtQ[c,
0]

Rule 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{(-b+a x) \sqrt [4]{-1+b x+a x^2}}+\frac {1}{(2 b+a x) \sqrt [4]{-1+b x+a x^2}}\right ) \, dx \\ & = \int \frac {1}{(-b+a x) \sqrt [4]{-1+b x+a x^2}} \, dx+\int \frac {1}{(2 b+a x) \sqrt [4]{-1+b x+a x^2}} \, dx \\ & = \frac {\sqrt [4]{-\frac {a \left (-1+b x+a x^2\right )}{4 a+b^2}} \int \frac {1}{(-b+a x) \sqrt [4]{\frac {a}{4 a+b^2}-\frac {a b x}{4 a+b^2}-\frac {a^2 x^2}{4 a+b^2}}} \, dx}{\sqrt [4]{-1+b x+a x^2}}+\frac {\sqrt [4]{-\frac {a \left (-1+b x+a x^2\right )}{4 a+b^2}} \int \frac {1}{(2 b+a x) \sqrt [4]{\frac {a}{4 a+b^2}-\frac {a b x}{4 a+b^2}-\frac {a^2 x^2}{4 a+b^2}}} \, dx}{\sqrt [4]{-1+b x+a x^2}} \\ & = \frac {\left (\sqrt {2} \sqrt [4]{-\frac {a \left (-1+b x+a x^2\right )}{4 a+b^2}}\right ) \text {Subst}\left (\int \frac {1}{\left (-\frac {3 a^2 b}{4 a+b^2}+a x\right ) \sqrt [4]{1-\frac {\left (4 a+b^2\right ) x^2}{a^2}}} \, dx,x,-\frac {a b}{4 a+b^2}-\frac {2 a^2 x}{4 a+b^2}\right )}{\sqrt [4]{-1+b x+a x^2}}+\frac {\left (\sqrt {2} \sqrt [4]{-\frac {a \left (-1+b x+a x^2\right )}{4 a+b^2}}\right ) \text {Subst}\left (\int \frac {1}{\left (\frac {3 a^2 b}{4 a+b^2}+a x\right ) \sqrt [4]{1-\frac {\left (4 a+b^2\right ) x^2}{a^2}}} \, dx,x,-\frac {a b}{4 a+b^2}-\frac {2 a^2 x}{4 a+b^2}\right )}{\sqrt [4]{-1+b x+a x^2}} \\ & = -2 \frac {\left (\sqrt {2} a \sqrt [4]{-\frac {a \left (-1+b x+a x^2\right )}{4 a+b^2}}\right ) \text {Subst}\left (\int \frac {x}{\left (\frac {9 a^4 b^2}{\left (4 a+b^2\right )^2}-a^2 x^2\right ) \sqrt [4]{1-\frac {\left (4 a+b^2\right ) x^2}{a^2}}} \, dx,x,-\frac {a b}{4 a+b^2}-\frac {2 a^2 x}{4 a+b^2}\right )}{\sqrt [4]{-1+b x+a x^2}} \\ & = -2 \frac {\left (a \sqrt [4]{-\frac {a \left (-1+b x+a x^2\right )}{4 a+b^2}}\right ) \text {Subst}\left (\int \frac {1}{\left (\frac {9 a^4 b^2}{\left (4 a+b^2\right )^2}-a^2 x\right ) \sqrt [4]{1-\frac {\left (4 a+b^2\right ) x}{a^2}}} \, dx,x,\left (-\frac {a b}{4 a+b^2}-\frac {2 a^2 x}{4 a+b^2}\right )^2\right )}{\sqrt {2} \sqrt [4]{-1+b x+a x^2}} \\ & = 2 \frac {\left (2 \sqrt {2} a^3 \sqrt [4]{-\frac {a \left (-1+b x+a x^2\right )}{4 a+b^2}}\right ) \text {Subst}\left (\int \frac {x^2}{\frac {9 a^4 b^2}{\left (4 a+b^2\right )^2}-\frac {a^4}{4 a+b^2}+\frac {a^4 x^4}{4 a+b^2}} \, dx,x,\sqrt [4]{1-\frac {(b+2 a x)^2}{4 a+b^2}}\right )}{\left (4 a+b^2\right ) \sqrt [4]{-1+b x+a x^2}} \\ & = 2 \left (-\frac {\left (\sqrt {2} \sqrt {4 a+b^2} \sqrt [4]{-\frac {a \left (-1+b x+a x^2\right )}{4 a+b^2}}\right ) \text {Subst}\left (\int \frac {1}{2 \sqrt {a-2 b^2}-\sqrt {4 a+b^2} x^2} \, dx,x,\sqrt [4]{1-\frac {(b+2 a x)^2}{4 a+b^2}}\right )}{a \sqrt [4]{-1+b x+a x^2}}+\frac {\left (\sqrt {2} \sqrt {4 a+b^2} \sqrt [4]{-\frac {a \left (-1+b x+a x^2\right )}{4 a+b^2}}\right ) \text {Subst}\left (\int \frac {1}{2 \sqrt {a-2 b^2}+\sqrt {4 a+b^2} x^2} \, dx,x,\sqrt [4]{1-\frac {(b+2 a x)^2}{4 a+b^2}}\right )}{a \sqrt [4]{-1+b x+a x^2}}\right ) \\ & = 2 \left (\frac {\sqrt [4]{4 a+b^2} \sqrt [4]{\frac {a \left (1-b x-a x^2\right )}{4 a+b^2}} \arctan \left (\frac {\sqrt [4]{4 a+b^2} \sqrt [4]{1-\frac {(b+2 a x)^2}{4 a+b^2}}}{\sqrt {2} \sqrt [4]{a-2 b^2}}\right )}{a \sqrt [4]{a-2 b^2} \sqrt [4]{-1+b x+a x^2}}-\frac {\sqrt [4]{4 a+b^2} \sqrt [4]{\frac {a \left (1-b x-a x^2\right )}{4 a+b^2}} \text {arctanh}\left (\frac {\sqrt [4]{4 a+b^2} \sqrt [4]{1-\frac {(b+2 a x)^2}{4 a+b^2}}}{\sqrt {2} \sqrt [4]{a-2 b^2}}\right )}{a \sqrt [4]{a-2 b^2} \sqrt [4]{-1+b x+a x^2}}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.43 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.86 \[ \int \frac {b+2 a x}{(-b+a x) (2 b+a x) \sqrt [4]{-1+b x+a x^2}} \, dx=\frac {\sqrt {2} \left (\arctan \left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{a-2 b^2} \sqrt [4]{-1+b x+a x^2}}{\sqrt {a-2 b^2}-\sqrt {a} \sqrt {-1+b x+a x^2}}\right )-\text {arctanh}\left (\frac {\sqrt {a-2 b^2}+\sqrt {a} \sqrt {-1+b x+a x^2}}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{a-2 b^2} \sqrt [4]{-1+b x+a x^2}}\right )\right )}{a^{3/4} \sqrt [4]{a-2 b^2}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.66 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.60

method	result	size
pseudoelliptic	\(\frac {2 \arctan \left (\frac {\left (a \,x^{2}+b x -1\right )^{\frac {1}{4}}}{\left (\frac {2 b^{2}-a}{a}\right )^{\frac {1}{4}}}\right )-\ln \left (\frac {\left (a \,x^{2}+b x -1\right )^{\frac {1}{4}}+\left (\frac {2 b^{2}-a}{a}\right )^{\frac {1}{4}}}{\left (a \,x^{2}+b x -1\right )^{\frac {1}{4}}-\left (\frac {2 b^{2}-a}{a}\right )^{\frac {1}{4}}}\right )}{\left (\frac {2 b^{2}-a}{a}\right )^{\frac {1}{4}} a}\)	\(116\)

[In]

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

[Out]

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

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.30 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.31 \[ \int \frac {b+2 a x}{(-b+a x) (2 b+a x) \sqrt [4]{-1+b x+a x^2}} \, dx=-\frac {\log \left (\frac {2 \, a^{2} b^{2} - a^{3}}{{\left (2 \, a^{3} b^{2} - a^{4}\right )}^{\frac {3}{4}}} + {\left (a x^{2} + b x - 1\right )}^{\frac {1}{4}}\right )}{{\left (2 \, a^{3} b^{2} - a^{4}\right )}^{\frac {1}{4}}} + \frac {\log \left (-\frac {2 \, a^{2} b^{2} - a^{3}}{{\left (2 \, a^{3} b^{2} - a^{4}\right )}^{\frac {3}{4}}} + {\left (a x^{2} + b x - 1\right )}^{\frac {1}{4}}\right )}{{\left (2 \, a^{3} b^{2} - a^{4}\right )}^{\frac {1}{4}}} + \frac {i \, \log \left (\frac {2 i \, a^{2} b^{2} - i \, a^{3}}{{\left (2 \, a^{3} b^{2} - a^{4}\right )}^{\frac {3}{4}}} + {\left (a x^{2} + b x - 1\right )}^{\frac {1}{4}}\right )}{{\left (2 \, a^{3} b^{2} - a^{4}\right )}^{\frac {1}{4}}} - \frac {i \, \log \left (\frac {-2 i \, a^{2} b^{2} + i \, a^{3}}{{\left (2 \, a^{3} b^{2} - a^{4}\right )}^{\frac {3}{4}}} + {\left (a x^{2} + b x - 1\right )}^{\frac {1}{4}}\right )}{{\left (2 \, a^{3} b^{2} - a^{4}\right )}^{\frac {1}{4}}} \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {b+2 a x}{(-b+a x) (2 b+a x) \sqrt [4]{-1+b x+a x^2}} \, dx=\int \frac {2 a x + b}{\left (a x - b\right ) \left (a x + 2 b\right ) \sqrt [4]{a x^{2} + b x - 1}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {b+2 a x}{(-b+a x) (2 b+a x) \sqrt [4]{-1+b x+a x^2}} \, dx=\int { \frac {2 \, a x + b}{{\left (a x^{2} + b x - 1\right )}^{\frac {1}{4}} {\left (a x + 2 \, b\right )} {\left (a x - b\right )}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {b+2 a x}{(-b+a x) (2 b+a x) \sqrt [4]{-1+b x+a x^2}} \, dx=\int { \frac {2 \, a x + b}{{\left (a x^{2} + b x - 1\right )}^{\frac {1}{4}} {\left (a x + 2 \, b\right )} {\left (a x - b\right )}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {b+2 a x}{(-b+a x) (2 b+a x) \sqrt [4]{-1+b x+a x^2}} \, dx=-\int \frac {b+2\,a\,x}{\left (2\,b+a\,x\right )\,\left (b-a\,x\right )\,{\left (a\,x^2+b\,x-1\right )}^{1/4}} \,d x \]

[In]

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

[Out]

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