\(\int \frac {1}{(d+e x) \sqrt [4]{a+b x+c x^2}} \, dx\) [2532]

Optimal result

Integrand size = 22, antiderivative size = 733 \[ \int \frac {1}{(d+e x) \sqrt [4]{a+b x+c x^2}} \, dx=\frac {\sqrt [4]{-b^2+4 a c} \sqrt [4]{-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \arctan \left (\frac {\sqrt [4]{-b^2+4 a c} \sqrt {e} \sqrt [4]{1-\frac {(b+2 c x)^2}{b^2-4 a c}}}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{c d^2-b d e+a e^2}}\right )}{\sqrt [4]{c} \sqrt {e} \sqrt [4]{c d^2-b d e+a e^2} \sqrt [4]{a+b x+c x^2}}-\frac {\sqrt [4]{-b^2+4 a c} \sqrt [4]{-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \text {arctanh}\left (\frac {\sqrt [4]{-b^2+4 a c} \sqrt {e} \sqrt [4]{1-\frac {(b+2 c x)^2}{b^2-4 a c}}}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{c d^2-b d e+a e^2}}\right )}{\sqrt [4]{c} \sqrt {e} \sqrt [4]{c d^2-b d e+a e^2} \sqrt [4]{a+b x+c x^2}}-\frac {\sqrt {-b^2+4 a c} (2 c d-b e) \sqrt {\frac {(b+2 c x)^2}{b^2-4 a c}} \sqrt [4]{-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {-b^2+4 a c} e}{2 \sqrt {c} \sqrt {c d^2-b d e+a e^2}},\arcsin \left (\sqrt [4]{1-\frac {(b+2 c x)^2}{b^2-4 a c}}\right ),-1\right )}{\sqrt {2} \sqrt {c} e \sqrt {c d^2-b d e+a e^2} (b+2 c x) \sqrt [4]{a+b x+c x^2}}+\frac {\sqrt {-b^2+4 a c} (2 c d-b e) \sqrt {\frac {(b+2 c x)^2}{b^2-4 a c}} \sqrt [4]{-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticPi}\left (\frac {\sqrt {-b^2+4 a c} e}{2 \sqrt {c} \sqrt {c d^2-b d e+a e^2}},\arcsin \left (\sqrt [4]{1-\frac {(b+2 c x)^2}{b^2-4 a c}}\right ),-1\right )}{\sqrt {2} \sqrt {c} e \sqrt {c d^2-b d e+a e^2} (b+2 c x) \sqrt [4]{a+b x+c x^2}} \]

[Out]

(4*a*c-b^2)^(1/4)*(-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^(1/4)*arctan(1/2*(4*a*c-b^2)^(1/4)*(1-(2*c*x+b)^2/(-4*a*c+b^
2))^(1/4)*e^(1/2)/c^(1/4)/(a*e^2-b*d*e+c*d^2)^(1/4)*2^(1/2))/c^(1/4)/(a*e^2-b*d*e+c*d^2)^(1/4)/(c*x^2+b*x+a)^(
1/4)/e^(1/2)-(4*a*c-b^2)^(1/4)*(-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^(1/4)*arctanh(1/2*(4*a*c-b^2)^(1/4)*(1-(2*c*x+b
)^2/(-4*a*c+b^2))^(1/4)*e^(1/2)/c^(1/4)/(a*e^2-b*d*e+c*d^2)^(1/4)*2^(1/2))/c^(1/4)/(a*e^2-b*d*e+c*d^2)^(1/4)/(
c*x^2+b*x+a)^(1/4)/e^(1/2)-1/2*(-b*e+2*c*d)*(-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^(1/4)*EllipticPi((1-(2*c*x+b)^2/(-
4*a*c+b^2))^(1/4),-1/2*e*(4*a*c-b^2)^(1/2)/c^(1/2)/(a*e^2-b*d*e+c*d^2)^(1/2),I)*(4*a*c-b^2)^(1/2)*((2*c*x+b)^2
/(-4*a*c+b^2))^(1/2)/e/(2*c*x+b)/(c*x^2+b*x+a)^(1/4)*2^(1/2)/c^(1/2)/(a*e^2-b*d*e+c*d^2)^(1/2)+1/2*(-b*e+2*c*d
)*(-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^(1/4)*EllipticPi((1-(2*c*x+b)^2/(-4*a*c+b^2))^(1/4),1/2*e*(4*a*c-b^2)^(1/2)/
c^(1/2)/(a*e^2-b*d*e+c*d^2)^(1/2),I)*(4*a*c-b^2)^(1/2)*((2*c*x+b)^2/(-4*a*c+b^2))^(1/2)/e/(2*c*x+b)/(c*x^2+b*x
+a)^(1/4)*2^(1/2)/c^(1/2)/(a*e^2-b*d*e+c*d^2)^(1/2)

Rubi [A] (verified)

Time = 1.00 (sec) , antiderivative size = 733, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {763, 762, 760, 408, 504, 1227, 551, 455, 65, 304, 211, 214} \[ \int \frac {1}{(d+e x) \sqrt [4]{a+b x+c x^2}} \, dx=-\frac {\sqrt {4 a c-b^2} \sqrt {\frac {(b+2 c x)^2}{b^2-4 a c}} \sqrt [4]{-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \operatorname {EllipticPi}\left (-\frac {\sqrt {4 a c-b^2} e}{2 \sqrt {c} \sqrt {c d^2-b e d+a e^2}},\arcsin \left (\sqrt [4]{1-\frac {(b+2 c x)^2}{b^2-4 a c}}\right ),-1\right )}{\sqrt {2} \sqrt {c} e (b+2 c x) \sqrt [4]{a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}+\frac {\sqrt {4 a c-b^2} \sqrt {\frac {(b+2 c x)^2}{b^2-4 a c}} \sqrt [4]{-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \operatorname {EllipticPi}\left (\frac {\sqrt {4 a c-b^2} e}{2 \sqrt {c} \sqrt {c d^2-b e d+a e^2}},\arcsin \left (\sqrt [4]{1-\frac {(b+2 c x)^2}{b^2-4 a c}}\right ),-1\right )}{\sqrt {2} \sqrt {c} e (b+2 c x) \sqrt [4]{a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}+\frac {\sqrt [4]{4 a c-b^2} \sqrt [4]{-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \arctan \left (\frac {\sqrt {e} \sqrt [4]{4 a c-b^2} \sqrt [4]{1-\frac {(b+2 c x)^2}{b^2-4 a c}}}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{a e^2-b d e+c d^2}}\right )}{\sqrt [4]{c} \sqrt {e} \sqrt [4]{a+b x+c x^2} \sqrt [4]{a e^2-b d e+c d^2}}-\frac {\sqrt [4]{4 a c-b^2} \sqrt [4]{-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \text {arctanh}\left (\frac {\sqrt {e} \sqrt [4]{4 a c-b^2} \sqrt [4]{1-\frac {(b+2 c x)^2}{b^2-4 a c}}}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{a e^2-b d e+c d^2}}\right )}{\sqrt [4]{c} \sqrt {e} \sqrt [4]{a+b x+c x^2} \sqrt [4]{a e^2-b d e+c d^2}} \]

[In]

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

[Out]

((-b^2 + 4*a*c)^(1/4)*(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))^(1/4)*ArcTan[((-b^2 + 4*a*c)^(1/4)*Sqrt[e]*(1 -
(b + 2*c*x)^2/(b^2 - 4*a*c))^(1/4))/(Sqrt[2]*c^(1/4)*(c*d^2 - b*d*e + a*e^2)^(1/4))])/(c^(1/4)*Sqrt[e]*(c*d^2
- b*d*e + a*e^2)^(1/4)*(a + b*x + c*x^2)^(1/4)) - ((-b^2 + 4*a*c)^(1/4)*(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c
)))^(1/4)*ArcTanh[((-b^2 + 4*a*c)^(1/4)*Sqrt[e]*(1 - (b + 2*c*x)^2/(b^2 - 4*a*c))^(1/4))/(Sqrt[2]*c^(1/4)*(c*d
^2 - b*d*e + a*e^2)^(1/4))])/(c^(1/4)*Sqrt[e]*(c*d^2 - b*d*e + a*e^2)^(1/4)*(a + b*x + c*x^2)^(1/4)) - (Sqrt[-
b^2 + 4*a*c]*(2*c*d - b*e)*Sqrt[(b + 2*c*x)^2/(b^2 - 4*a*c)]*(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))^(1/4)*El
lipticPi[-1/2*(Sqrt[-b^2 + 4*a*c]*e)/(Sqrt[c]*Sqrt[c*d^2 - b*d*e + a*e^2]), ArcSin[(1 - (b + 2*c*x)^2/(b^2 - 4
*a*c))^(1/4)], -1])/(Sqrt[2]*Sqrt[c]*e*Sqrt[c*d^2 - b*d*e + a*e^2]*(b + 2*c*x)*(a + b*x + c*x^2)^(1/4)) + (Sqr
t[-b^2 + 4*a*c]*(2*c*d - b*e)*Sqrt[(b + 2*c*x)^2/(b^2 - 4*a*c)]*(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))^(1/4)
*EllipticPi[(Sqrt[-b^2 + 4*a*c]*e)/(2*Sqrt[c]*Sqrt[c*d^2 - b*d*e + a*e^2]), ArcSin[(1 - (b + 2*c*x)^2/(b^2 - 4
*a*c))^(1/4)], -1])/(Sqrt[2]*Sqrt[c]*e*Sqrt[c*d^2 - b*d*e + a*e^2]*(b + 2*c*x)*(a + b*x + c*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]

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt [4]{-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \int \frac {1}{(d+e x) \sqrt [4]{-\frac {a c}{b^2-4 a c}-\frac {b c x}{b^2-4 a c}-\frac {c^2 x^2}{b^2-4 a c}}} \, dx}{\sqrt [4]{a+b x+c x^2}} \\ & = \frac {\left (\sqrt {2} \sqrt [4]{-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \text {Subst}\left (\int \frac {1}{\left (-\frac {c (2 c d-b e)}{b^2-4 a c}+e x\right ) \sqrt [4]{1-\frac {\left (b^2-4 a c\right ) x^2}{c^2}}} \, dx,x,-\frac {b c}{b^2-4 a c}-\frac {2 c^2 x}{b^2-4 a c}\right )}{\sqrt [4]{a+b x+c x^2}} \\ & = -\frac {\left (\sqrt {2} e \sqrt [4]{-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \text {Subst}\left (\int \frac {x}{\sqrt [4]{1-\frac {\left (b^2-4 a c\right ) x^2}{c^2}} \left (\frac {c^2 (2 c d-b e)^2}{\left (b^2-4 a c\right )^2}-e^2 x^2\right )} \, dx,x,-\frac {b c}{b^2-4 a c}-\frac {2 c^2 x}{b^2-4 a c}\right )}{\sqrt [4]{a+b x+c x^2}}-\frac {\left (\sqrt {2} c (2 c d-b e) \sqrt [4]{-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{1-\frac {\left (b^2-4 a c\right ) x^2}{c^2}} \left (\frac {c^2 (2 c d-b e)^2}{\left (b^2-4 a c\right )^2}-e^2 x^2\right )} \, dx,x,-\frac {b c}{b^2-4 a c}-\frac {2 c^2 x}{b^2-4 a c}\right )}{\left (b^2-4 a c\right ) \sqrt [4]{a+b x+c x^2}} \\ & = -\frac {\left (e \sqrt [4]{-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{1-\frac {\left (b^2-4 a c\right ) x}{c^2}} \left (\frac {c^2 (2 c d-b e)^2}{\left (b^2-4 a c\right )^2}-e^2 x\right )} \, dx,x,\left (-\frac {b c}{b^2-4 a c}-\frac {2 c^2 x}{b^2-4 a c}\right )^2\right )}{\sqrt {2} \sqrt [4]{a+b x+c x^2}}-\frac {\left (2 \sqrt {2} c (2 c d-b e) \sqrt {\frac {\left (b^2-4 a c\right ) \left (-\frac {b c}{b^2-4 a c}-\frac {2 c^2 x}{b^2-4 a c}\right )^2}{c^2}} \sqrt [4]{-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {1-x^4} \left (e^2-\frac {(2 c d-b e)^2}{b^2-4 a c}-e^2 x^4\right )} \, dx,x,\sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {b c}{b^2-4 a c}-\frac {2 c^2 x}{b^2-4 a c}\right )^2}{c^2}}\right )}{\left (b^2-4 a c\right ) \left (-\frac {b c}{b^2-4 a c}-\frac {2 c^2 x}{b^2-4 a c}\right ) \sqrt [4]{a+b x+c x^2}} \\ & = \frac {\left (2 \sqrt {2} c^2 e \sqrt [4]{-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \text {Subst}\left (\int \frac {x^2}{-\frac {c^2 e^2}{b^2-4 a c}+\frac {c^2 (2 c d-b e)^2}{\left (b^2-4 a c\right )^2}+\frac {c^2 e^2 x^4}{b^2-4 a c}} \, dx,x,\sqrt [4]{1-\frac {(b+2 c x)^2}{b^2-4 a c}}\right )}{\left (b^2-4 a c\right ) \sqrt [4]{a+b x+c x^2}}-\frac {\left (\sqrt {2} c \sqrt {-b^2+4 a c} (2 c d-b e) \sqrt {\frac {\left (b^2-4 a c\right ) \left (-\frac {b c}{b^2-4 a c}-\frac {2 c^2 x}{b^2-4 a c}\right )^2}{c^2}} \sqrt [4]{-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \text {Subst}\left (\int \frac {1}{\left (2 \sqrt {c} \sqrt {c d^2-b d e+a e^2}-\sqrt {-b^2+4 a c} e x^2\right ) \sqrt {1-x^4}} \, dx,x,\sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {b c}{b^2-4 a c}-\frac {2 c^2 x}{b^2-4 a c}\right )^2}{c^2}}\right )}{\left (b^2-4 a c\right ) e \left (-\frac {b c}{b^2-4 a c}-\frac {2 c^2 x}{b^2-4 a c}\right ) \sqrt [4]{a+b x+c x^2}}+\frac {\left (\sqrt {2} c \sqrt {-b^2+4 a c} (2 c d-b e) \sqrt {\frac {\left (b^2-4 a c\right ) \left (-\frac {b c}{b^2-4 a c}-\frac {2 c^2 x}{b^2-4 a c}\right )^2}{c^2}} \sqrt [4]{-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \text {Subst}\left (\int \frac {1}{\left (2 \sqrt {c} \sqrt {c d^2-b d e+a e^2}+\sqrt {-b^2+4 a c} e x^2\right ) \sqrt {1-x^4}} \, dx,x,\sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {b c}{b^2-4 a c}-\frac {2 c^2 x}{b^2-4 a c}\right )^2}{c^2}}\right )}{\left (b^2-4 a c\right ) e \left (-\frac {b c}{b^2-4 a c}-\frac {2 c^2 x}{b^2-4 a c}\right ) \sqrt [4]{a+b x+c x^2}} \\ & = \frac {\left (\sqrt {2} \left (-b^2+4 a c\right )^{3/2} \sqrt [4]{-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \text {Subst}\left (\int \frac {1}{2 \sqrt {c} \sqrt {c d^2-b d e+a e^2}-\sqrt {-b^2+4 a c} e x^2} \, dx,x,\sqrt [4]{1-\frac {(b+2 c x)^2}{b^2-4 a c}}\right )}{\left (b^2-4 a c\right ) \sqrt [4]{a+b x+c x^2}}-\frac {\left (\sqrt {2} \left (-b^2+4 a c\right )^{3/2} \sqrt [4]{-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \text {Subst}\left (\int \frac {1}{2 \sqrt {c} \sqrt {c d^2-b d e+a e^2}+\sqrt {-b^2+4 a c} e x^2} \, dx,x,\sqrt [4]{1-\frac {(b+2 c x)^2}{b^2-4 a c}}\right )}{\left (b^2-4 a c\right ) \sqrt [4]{a+b x+c x^2}}-\frac {\left (\sqrt {2} c \sqrt {-b^2+4 a c} (2 c d-b e) \sqrt {\frac {\left (b^2-4 a c\right ) \left (-\frac {b c}{b^2-4 a c}-\frac {2 c^2 x}{b^2-4 a c}\right )^2}{c^2}} \sqrt [4]{-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+x^2} \left (2 \sqrt {c} \sqrt {c d^2-b d e+a e^2}-\sqrt {-b^2+4 a c} e x^2\right )} \, dx,x,\sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {b c}{b^2-4 a c}-\frac {2 c^2 x}{b^2-4 a c}\right )^2}{c^2}}\right )}{\left (b^2-4 a c\right ) e \left (-\frac {b c}{b^2-4 a c}-\frac {2 c^2 x}{b^2-4 a c}\right ) \sqrt [4]{a+b x+c x^2}}+\frac {\left (\sqrt {2} c \sqrt {-b^2+4 a c} (2 c d-b e) \sqrt {\frac {\left (b^2-4 a c\right ) \left (-\frac {b c}{b^2-4 a c}-\frac {2 c^2 x}{b^2-4 a c}\right )^2}{c^2}} \sqrt [4]{-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+x^2} \left (2 \sqrt {c} \sqrt {c d^2-b d e+a e^2}+\sqrt {-b^2+4 a c} e x^2\right )} \, dx,x,\sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {b c}{b^2-4 a c}-\frac {2 c^2 x}{b^2-4 a c}\right )^2}{c^2}}\right )}{\left (b^2-4 a c\right ) e \left (-\frac {b c}{b^2-4 a c}-\frac {2 c^2 x}{b^2-4 a c}\right ) \sqrt [4]{a+b x+c x^2}} \\ & = \frac {\sqrt [4]{-b^2+4 a c} \sqrt [4]{-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \tan ^{-1}\left (\frac {\sqrt [4]{-b^2+4 a c} \sqrt {e} \sqrt [4]{1-\frac {(b+2 c x)^2}{b^2-4 a c}}}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{c d^2-b d e+a e^2}}\right )}{\sqrt [4]{c} \sqrt {e} \sqrt [4]{c d^2-b d e+a e^2} \sqrt [4]{a+b x+c x^2}}-\frac {\sqrt [4]{-b^2+4 a c} \sqrt [4]{-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \tanh ^{-1}\left (\frac {\sqrt [4]{-b^2+4 a c} \sqrt {e} \sqrt [4]{1-\frac {(b+2 c x)^2}{b^2-4 a c}}}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{c d^2-b d e+a e^2}}\right )}{\sqrt [4]{c} \sqrt {e} \sqrt [4]{c d^2-b d e+a e^2} \sqrt [4]{a+b x+c x^2}}+\frac {\left (b^2-4 a c\right ) (2 c d-b e) \sqrt {\frac {(b+2 c x)^2}{b^2-4 a c}} \sqrt [4]{-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \Pi \left (-\frac {\sqrt {-b^2+4 a c} e}{2 \sqrt {c} \sqrt {c d^2-b d e+a e^2}};\left .\sin ^{-1}\left (\sqrt [4]{1-\frac {(b+2 c x)^2}{b^2-4 a c}}\right )\right |-1\right )}{\sqrt {2} \sqrt {c} \sqrt {-b^2+4 a c} e \sqrt {c d^2-b d e+a e^2} (b+2 c x) \sqrt [4]{a+b x+c x^2}}-\frac {\left (b^2-4 a c\right ) (2 c d-b e) \sqrt {\frac {(b+2 c x)^2}{b^2-4 a c}} \sqrt [4]{-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \Pi \left (\frac {\sqrt {-b^2+4 a c} e}{2 \sqrt {c} \sqrt {c d^2-b d e+a e^2}};\left .\sin ^{-1}\left (\sqrt [4]{1-\frac {(b+2 c x)^2}{b^2-4 a c}}\right )\right |-1\right )}{\sqrt {2} \sqrt {c} \sqrt {-b^2+4 a c} e \sqrt {c d^2-b d e+a e^2} (b+2 c x) \sqrt [4]{a+b x+c x^2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.

Time = 9.31 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.24 \[ \int \frac {1}{(d+e x) \sqrt [4]{a+b x+c x^2}} \, dx=-\frac {\sqrt {2} \sqrt [4]{\frac {e \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{c (d+e x)}} \sqrt [4]{\frac {e \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{c (d+e x)}} \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{4},\frac {1}{4},\frac {3}{2},\frac {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}{2 c (d+e x)},\frac {2 c d-b e+\sqrt {b^2-4 a c} e}{2 c d+2 c e x}\right )}{e \sqrt [4]{a+x (b+c x)}} \]

[In]

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

[Out]

-((Sqrt[2]*((e*(b - Sqrt[b^2 - 4*a*c] + 2*c*x))/(c*(d + e*x)))^(1/4)*((e*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/(c*(
d + e*x)))^(1/4)*AppellF1[1/2, 1/4, 1/4, 3/2, (2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)/(2*c*(d + e*x)), (2*c*d - b*
e + Sqrt[b^2 - 4*a*c]*e)/(2*c*d + 2*c*e*x)])/(e*(a + x*(b + c*x))^(1/4)))

Maple [F]

\[\int \frac {1}{\left (e x +d \right ) \left (c \,x^{2}+b x +a \right )^{\frac {1}{4}}}d x\]

[In]

int(1/(e*x+d)/(c*x^2+b*x+a)^(1/4),x)

[Out]

int(1/(e*x+d)/(c*x^2+b*x+a)^(1/4),x)

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{(d+e x) \sqrt [4]{a+b x+c x^2}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Sympy [F]

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

[In]

integrate(1/(e*x+d)/(c*x**2+b*x+a)**(1/4),x)

[Out]

Integral(1/((d + e*x)*(a + b*x + c*x**2)**(1/4)), x)

Maxima [F]

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

[In]

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

[Out]

integrate(1/((c*x^2 + b*x + a)^(1/4)*(e*x + d)), x)

Giac [F]

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

[In]

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

[Out]

integrate(1/((c*x^2 + b*x + a)^(1/4)*(e*x + d)), x)

Mupad [F(-1)]

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

[In]

int(1/((d + e*x)*(a + b*x + c*x^2)^(1/4)),x)

[Out]

int(1/((d + e*x)*(a + b*x + c*x^2)^(1/4)), x)