\(\int \frac {1}{(d+e x)^2 (a+b x+c x^2)^{3/4}} \, dx\) [2540]

Optimal result

Integrand size = 22, antiderivative size = 970 \[ \int \frac {1}{(d+e x)^2 \left (a+b x+c x^2\right )^{3/4}} \, dx=-\frac {e \sqrt [4]{a+b x+c x^2}}{\left (c d^2-b d e+a e^2\right ) (d+e x)}-\frac {3 \left (-b^2+4 a c\right )^{3/4} \sqrt {e} (2 c d-b e) \left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4} \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 )}{4 c^{3/4} \left (c d^2-b d e+a e^2\right )^{7/4} \left (a+b x+c x^2\right )^{3/4}}-\frac {3 \left (-b^2+4 a c\right )^{3/4} \sqrt {e} (2 c d-b e) \left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4} \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 )}{4 c^{3/4} \left (c d^2-b d e+a e^2\right )^{7/4} \left (a+b x+c x^2\right )^{3/4}}-\frac {c^{3/4} \sqrt [4]{b^2-4 a c} \sqrt {\frac {(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (1+\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )^2}} \left (1+\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{a+b x+c x^2}}{\sqrt [4]{b^2-4 a c}}\right ),\frac {1}{2}\right )}{\sqrt {2} \left (c d^2-b d e+a e^2\right ) (b+2 c x)}-\frac {3 \left (b^2-4 a c\right ) (2 c d-b e)^2 \sqrt {\frac {(b+2 c x)^2}{b^2-4 a c}} \left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4} \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 )}{4 \sqrt {2} c \left (c d^2-b d e+a e^2\right )^2 (b+2 c x) \left (a+b x+c x^2\right )^{3/4}}-\frac {3 \left (b^2-4 a c\right ) (2 c d-b e)^2 \sqrt {\frac {(b+2 c x)^2}{b^2-4 a c}} \left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4} \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 )}{4 \sqrt {2} c \left (c d^2-b d e+a e^2\right )^2 (b+2 c x) \left (a+b x+c x^2\right )^{3/4}} \]

[Out]

-e*(c*x^2+b*x+a)^(1/4)/(a*e^2-b*d*e+c*d^2)/(e*x+d)-3/4*(4*a*c-b^2)^(3/4)*(-b*e+2*c*d)*(-c*(c*x^2+b*x+a)/(-4*a*
c+b^2))^(3/4)*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))*e^(1/2)/c^(3/4)/(a*e^2-b*d*e+c*d^2)^(7/4)/(c*x^2+b*x+a)^(3/4)-3/4*(4*a*c-b^2)^(3/4)*(-b*e+2
*c*d)*(-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^(3/4)*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))*e^(1/2)/c^(3/4)/(a*e^2-b*d*e+c*d^2)^(7/4)/(c*x^2+b*x+a)^(3/4
)-3/8*(-4*a*c+b^2)*(-b*e+2*c*d)^2*(-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^(3/4)*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)*((2*c*x+b)^2/(-4*a*c+b^2))^(1/2)/c/(a*e^
2-b*d*e+c*d^2)^2/(2*c*x+b)/(c*x^2+b*x+a)^(3/4)*2^(1/2)-3/8*(-4*a*c+b^2)*(-b*e+2*c*d)^2*(-c*(c*x^2+b*x+a)/(-4*a
*c+b^2))^(3/4)*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)*((2*c*x+b)^2/(-4*a*c+b^2))^(1/2)/c/(a*e^2-b*d*e+c*d^2)^2/(2*c*x+b)/(c*x^2+b*x+a)^(3/4)*2^(1/2)-1/2
*c^(3/4)*(-4*a*c+b^2)^(1/4)*(cos(2*arctan(c^(1/4)*(c*x^2+b*x+a)^(1/4)*2^(1/2)/(-4*a*c+b^2)^(1/4)))^2)^(1/2)/co
s(2*arctan(c^(1/4)*(c*x^2+b*x+a)^(1/4)*2^(1/2)/(-4*a*c+b^2)^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*(c*x^2+b*x+
a)^(1/4)*2^(1/2)/(-4*a*c+b^2)^(1/4))),1/2*2^(1/2))*(1+2*c^(1/2)*(c*x^2+b*x+a)^(1/2)/(-4*a*c+b^2)^(1/2))*((2*c*
x+b)^2/(-4*a*c+b^2)/(1+2*c^(1/2)*(c*x^2+b*x+a)^(1/2)/(-4*a*c+b^2)^(1/2))^2)^(1/2)/(a*e^2-b*d*e+c*d^2)/(2*c*x+b
)*2^(1/2)

Rubi [A] (verified)

Time = 1.33 (sec) , antiderivative size = 970, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.773, Rules used = {758, 857, 637, 226, 763, 762, 761, 410, 109, 418, 1227, 551, 455, 65, 218, 214, 211} \[ \int \frac {1}{(d+e x)^2 \left (a+b x+c x^2\right )^{3/4}} \, dx=-\frac {3 \left (b^2-4 a c\right ) \sqrt {\frac {(b+2 c x)^2}{b^2-4 a c}} \left (-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}\right )^{3/4} \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 ) (2 c d-b e)^2}{4 \sqrt {2} c \left (c d^2-b e d+a e^2\right )^2 (b+2 c x) \left (c x^2+b x+a\right )^{3/4}}-\frac {3 \left (b^2-4 a c\right ) \sqrt {\frac {(b+2 c x)^2}{b^2-4 a c}} \left (-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}\right )^{3/4} \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 ) (2 c d-b e)^2}{4 \sqrt {2} c \left (c d^2-b e d+a e^2\right )^2 (b+2 c x) \left (c x^2+b x+a\right )^{3/4}}-\frac {3 \left (4 a c-b^2\right )^{3/4} \sqrt {e} \left (-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}\right )^{3/4} \arctan \left (\frac {\sqrt [4]{4 a c-b^2} \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 e d+a e^2}}\right ) (2 c d-b e)}{4 c^{3/4} \left (c d^2-b e d+a e^2\right )^{7/4} \left (c x^2+b x+a\right )^{3/4}}-\frac {3 \left (4 a c-b^2\right )^{3/4} \sqrt {e} \left (-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}\right )^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{4 a c-b^2} \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 e d+a e^2}}\right ) (2 c d-b e)}{4 c^{3/4} \left (c d^2-b e d+a e^2\right )^{7/4} \left (c x^2+b x+a\right )^{3/4}}-\frac {c^{3/4} \sqrt [4]{b^2-4 a c} \sqrt {\frac {(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right )^2}} \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right ),\frac {1}{2}\right )}{\sqrt {2} \left (c d^2-b e d+a e^2\right ) (b+2 c x)}-\frac {e \sqrt [4]{c x^2+b x+a}}{\left (c d^2-b e d+a e^2\right ) (d+e x)} \]

[In]

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

[Out]

-((e*(a + b*x + c*x^2)^(1/4))/((c*d^2 - b*d*e + a*e^2)*(d + e*x))) - (3*(-b^2 + 4*a*c)^(3/4)*Sqrt[e]*(2*c*d -
b*e)*(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))^(3/4)*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))])/(4*c^(3/4)*(c*d^2 - b*d*e + a*e^2)^(7/4)
*(a + b*x + c*x^2)^(3/4)) - (3*(-b^2 + 4*a*c)^(3/4)*Sqrt[e]*(2*c*d - b*e)*(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*
c)))^(3/4)*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))])/(4*c^(3/4)*(c*d^2 - b*d*e + a*e^2)^(7/4)*(a + b*x + c*x^2)^(3/4)) - (c^(3/4)*(b^
2 - 4*a*c)^(1/4)*Sqrt[(b + 2*c*x)^2/((b^2 - 4*a*c)*(1 + (2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sqrt[b^2 - 4*a*c])^2
)]*(1 + (2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sqrt[b^2 - 4*a*c])*EllipticF[2*ArcTan[(Sqrt[2]*c^(1/4)*(a + b*x + c*
x^2)^(1/4))/(b^2 - 4*a*c)^(1/4)], 1/2])/(Sqrt[2]*(c*d^2 - b*d*e + a*e^2)*(b + 2*c*x)) - (3*(b^2 - 4*a*c)*(2*c*
d - b*e)^2*Sqrt[(b + 2*c*x)^2/(b^2 - 4*a*c)]*(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))^(3/4)*EllipticPi[-1/2*(S
qrt[-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])/(4*Sqrt[2]*c*(c*d^2 - b*d*e + a*e^2)^2*(b + 2*c*x)*(a + b*x + c*x^2)^(3/4)) - (3*(b^2 - 4*a*c)*(2*c*d - b*
e)^2*Sqrt[(b + 2*c*x)^2/(b^2 - 4*a*c)]*(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))^(3/4)*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])/(4*Sq
rt[2]*c*(c*d^2 - b*d*e + a*e^2)^2*(b + 2*c*x)*(a + b*x + c*x^2)^(3/4))

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 109

Int[1/(((a_.) + (b_.)*(x_))*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(3/4)), x_Symbol] :> Dist[-4, Subst[
Int[1/((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 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 218

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]},
Dist[r/(2*a), Int[1/(r - s*x^2), x], x] + Dist[r/(2*a), Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !Gt
Q[a/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 410

Int[1/(((a_) + (b_.)*(x_)^2)^(3/4)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Dist[Sqrt[(-b)*(x^2/a)]/(2*x), Subst[I
nt[1/(Sqrt[(-b)*(x/a)]*(a + b*x)^(3/4)*(c + d*x)), x], x, x^2], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d,
0]

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 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 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 637

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{d = Denominator[p]}, Dist[d*(Sqrt[(b + 2*c*x)
^2]/(b + 2*c*x)), Subst[Int[x^(d*(p + 1) - 1)/Sqrt[b^2 - 4*a*c + 4*c*x^d], x], x, (a + b*x + c*x^2)^(1/d)], x]
/; 3 <= d <= 4] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && RationalQ[p]

Rule 758

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*(d + e*x)^(m + 1)*
((a + b*x + c*x^2)^(p + 1)/((m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Dist[1/((m + 1)*(c*d^2 - b*d*e + a*e^2)),
Int[(d + e*x)^(m + 1)*Simp[c*d*(m + 1) - b*e*(m + p + 2) - c*e*(m + 2*p + 3)*x, x]*(a + b*x + c*x^2)^p, x], x]
/; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e
, 0] && NeQ[m, -1] && ((LtQ[m, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]) || (SumSimplerQ[m, 1] && IntegerQ
[p]) || ILtQ[Simplify[m + 2*p + 3], 0])

Rule 761

Int[1/(((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(3/4)), x_Symbol] :> Dist[d, Int[1/((d^2 - e^2*x^2)*(a + c*x^
2)^(3/4)), x], x] - Dist[e, Int[x/((d^2 - e^2*x^2)*(a + c*x^2)^(3/4)), 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 857

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dis
t[g/e, Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + b*x + c*x^
2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
&&  !IGtQ[m, 0]

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 {e \sqrt [4]{a+b x+c x^2}}{\left (c d^2-b d e+a e^2\right ) (d+e x)}-\frac {\int \frac {\frac {1}{4} (-4 c d+3 b e)+\frac {c e x}{2}}{(d+e x) \left (a+b x+c x^2\right )^{3/4}} \, dx}{c d^2-b d e+a e^2} \\ & = -\frac {e \sqrt [4]{a+b x+c x^2}}{\left (c d^2-b d e+a e^2\right ) (d+e x)}-\frac {c \int \frac {1}{\left (a+b x+c x^2\right )^{3/4}} \, dx}{2 \left (c d^2-b d e+a e^2\right )}+\frac {(3 (2 c d-b e)) \int \frac {1}{(d+e x) \left (a+b x+c x^2\right )^{3/4}} \, dx}{4 \left (c d^2-b d e+a e^2\right )} \\ & = -\frac {e \sqrt [4]{a+b x+c x^2}}{\left (c d^2-b d e+a e^2\right ) (d+e x)}-\frac {\left (2 c \sqrt {(b+2 c x)^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b^2-4 a c+4 c x^4}} \, dx,x,\sqrt [4]{a+b x+c x^2}\right )}{\left (c d^2-b d e+a e^2\right ) (b+2 c x)}+\frac {\left (3 (2 c d-b e) \left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4}\right ) \int \frac {1}{(d+e x) \left (-\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}\right )^{3/4}} \, dx}{4 \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^{3/4}} \\ & = -\frac {e \sqrt [4]{a+b x+c x^2}}{\left (c d^2-b d e+a e^2\right ) (d+e x)}-\frac {c^{3/4} \sqrt [4]{b^2-4 a c} \sqrt {\frac {(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (1+\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )^2}} \left (1+\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{a+b x+c x^2}}{\sqrt [4]{b^2-4 a c}}\right )|\frac {1}{2}\right )}{\sqrt {2} \left (c d^2-b d e+a e^2\right ) (b+2 c x)}+\frac {\left (3 (2 c d-b e) \left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4}\right ) \text {Subst}\left (\int \frac {1}{\left (-\frac {c (2 c d-b e)}{b^2-4 a c}+e x\right ) \left (1-\frac {\left (b^2-4 a c\right ) x^2}{c^2}\right )^{3/4}} \, dx,x,-\frac {b c}{b^2-4 a c}-\frac {2 c^2 x}{b^2-4 a c}\right )}{\sqrt {2} \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^{3/4}} \\ & = -\frac {e \sqrt [4]{a+b x+c x^2}}{\left (c d^2-b d e+a e^2\right ) (d+e x)}-\frac {c^{3/4} \sqrt [4]{b^2-4 a c} \sqrt {\frac {(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (1+\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )^2}} \left (1+\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{a+b x+c x^2}}{\sqrt [4]{b^2-4 a c}}\right )|\frac {1}{2}\right )}{\sqrt {2} \left (c d^2-b d e+a e^2\right ) (b+2 c x)}-\frac {\left (3 e (2 c d-b e) \left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4}\right ) \text {Subst}\left (\int \frac {x}{\left (1-\frac {\left (b^2-4 a c\right ) x^2}{c^2}\right )^{3/4} \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 {2} \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^{3/4}}-\frac {\left (3 c (2 c d-b e)^2 \left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4}\right ) \text {Subst}\left (\int \frac {1}{\left (1-\frac {\left (b^2-4 a c\right ) x^2}{c^2}\right )^{3/4} \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 {2} \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^{3/4}} \\ & = -\frac {e \sqrt [4]{a+b x+c x^2}}{\left (c d^2-b d e+a e^2\right ) (d+e x)}-\frac {c^{3/4} \sqrt [4]{b^2-4 a c} \sqrt {\frac {(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (1+\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )^2}} \left (1+\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{a+b x+c x^2}}{\sqrt [4]{b^2-4 a c}}\right )|\frac {1}{2}\right )}{\sqrt {2} \left (c d^2-b d e+a e^2\right ) (b+2 c x)}-\frac {\left (3 e (2 c d-b e) \left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4}\right ) \text {Subst}\left (\int \frac {1}{\left (1-\frac {\left (b^2-4 a c\right ) x}{c^2}\right )^{3/4} \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 )}{2 \sqrt {2} \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^{3/4}}-\frac {\left (3 c (2 c d-b e)^2 \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}} \left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {\frac {\left (b^2-4 a c\right ) x}{c^2}} \left (1-\frac {\left (b^2-4 a c\right ) x}{c^2}\right )^{3/4} \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 )}{2 \sqrt {2} \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (-\frac {b c}{b^2-4 a c}-\frac {2 c^2 x}{b^2-4 a c}\right ) \left (a+b x+c x^2\right )^{3/4}} \\ & = -\frac {e \sqrt [4]{a+b x+c x^2}}{\left (c d^2-b d e+a e^2\right ) (d+e x)}-\frac {c^{3/4} \sqrt [4]{b^2-4 a c} \sqrt {\frac {(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (1+\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )^2}} \left (1+\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{a+b x+c x^2}}{\sqrt [4]{b^2-4 a c}}\right )|\frac {1}{2}\right )}{\sqrt {2} \left (c d^2-b d e+a e^2\right ) (b+2 c x)}+\frac {\left (3 \sqrt {2} c^2 e (2 c d-b e) \left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4}\right ) \text {Subst}\left (\int \frac {1}{-\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 ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^{3/4}}+\frac {\left (3 \sqrt {2} c (2 c d-b e)^2 \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}} \left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4}\right ) \text {Subst}\left (\int \frac {1}{\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 {(b+2 c x)^2}{b^2-4 a c}}\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (-\frac {b c}{b^2-4 a c}-\frac {2 c^2 x}{b^2-4 a c}\right ) \left (a+b x+c x^2\right )^{3/4}} \\ & = -\frac {e \sqrt [4]{a+b x+c x^2}}{\left (c d^2-b d e+a e^2\right ) (d+e x)}-\frac {c^{3/4} \sqrt [4]{b^2-4 a c} \sqrt {\frac {(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (1+\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )^2}} \left (1+\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{a+b x+c x^2}}{\sqrt [4]{b^2-4 a c}}\right )|\frac {1}{2}\right )}{\sqrt {2} \left (c d^2-b d e+a e^2\right ) (b+2 c x)}+\frac {\left (3 \left (b^2-4 a c\right ) e (2 c d-b e) \left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4}\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 )}{2 \sqrt {2} \sqrt {c} \left (c d^2-b d e+a e^2\right )^{3/2} \left (a+b x+c x^2\right )^{3/4}}+\frac {\left (3 \left (b^2-4 a c\right ) e (2 c d-b e) \left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4}\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 )}{2 \sqrt {2} \sqrt {c} \left (c d^2-b d e+a e^2\right )^{3/2} \left (a+b x+c x^2\right )^{3/4}}+\frac {\left (3 c (2 c d-b e)^2 \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}} \left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4}\right ) \text {Subst}\left (\int \frac {1}{\left (1-\frac {\sqrt {-b^2+4 a c} e x^2}{2 \sqrt {c} \sqrt {c d^2-b d e+a e^2}}\right ) \sqrt {1-x^4}} \, dx,x,\sqrt [4]{1-\frac {(b+2 c x)^2}{b^2-4 a c}}\right )}{\sqrt {2} \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (-e^2+\frac {(2 c d-b e)^2}{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 ) \left (a+b x+c x^2\right )^{3/4}}+\frac {\left (3 c (2 c d-b e)^2 \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}} \left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4}\right ) \text {Subst}\left (\int \frac {1}{\left (1+\frac {\sqrt {-b^2+4 a c} e x^2}{2 \sqrt {c} \sqrt {c d^2-b d e+a e^2}}\right ) \sqrt {1-x^4}} \, dx,x,\sqrt [4]{1-\frac {(b+2 c x)^2}{b^2-4 a c}}\right )}{\sqrt {2} \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (-e^2+\frac {(2 c d-b e)^2}{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 ) \left (a+b x+c x^2\right )^{3/4}} \\ & = -\frac {e \sqrt [4]{a+b x+c x^2}}{\left (c d^2-b d e+a e^2\right ) (d+e x)}-\frac {3 \left (-b^2+4 a c\right )^{3/4} \sqrt {e} (2 c d-b e) \left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4} \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 )}{4 c^{3/4} \left (c d^2-b d e+a e^2\right )^{7/4} \left (a+b x+c x^2\right )^{3/4}}-\frac {3 \left (-b^2+4 a c\right )^{3/4} \sqrt {e} (2 c d-b e) \left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4} \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 )}{4 c^{3/4} \left (c d^2-b d e+a e^2\right )^{7/4} \left (a+b x+c x^2\right )^{3/4}}-\frac {c^{3/4} \sqrt [4]{b^2-4 a c} \sqrt {\frac {(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (1+\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )^2}} \left (1+\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{a+b x+c x^2}}{\sqrt [4]{b^2-4 a c}}\right )|\frac {1}{2}\right )}{\sqrt {2} \left (c d^2-b d e+a e^2\right ) (b+2 c x)}+\frac {\left (3 c (2 c d-b e)^2 \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}} \left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+x^2} \left (1-\frac {\sqrt {-b^2+4 a c} e x^2}{2 \sqrt {c} \sqrt {c d^2-b d e+a e^2}}\right )} \, dx,x,\sqrt [4]{1-\frac {(b+2 c x)^2}{b^2-4 a c}}\right )}{\sqrt {2} \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (-e^2+\frac {(2 c d-b e)^2}{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 ) \left (a+b x+c x^2\right )^{3/4}}+\frac {\left (3 c (2 c d-b e)^2 \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}} \left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+x^2} \left (1+\frac {\sqrt {-b^2+4 a c} e x^2}{2 \sqrt {c} \sqrt {c d^2-b d e+a e^2}}\right )} \, dx,x,\sqrt [4]{1-\frac {(b+2 c x)^2}{b^2-4 a c}}\right )}{\sqrt {2} \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (-e^2+\frac {(2 c d-b e)^2}{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 ) \left (a+b x+c x^2\right )^{3/4}} \\ & = -\frac {e \sqrt [4]{a+b x+c x^2}}{\left (c d^2-b d e+a e^2\right ) (d+e x)}-\frac {3 \left (-b^2+4 a c\right )^{3/4} \sqrt {e} (2 c d-b e) \left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4} \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 )}{4 c^{3/4} \left (c d^2-b d e+a e^2\right )^{7/4} \left (a+b x+c x^2\right )^{3/4}}-\frac {3 \left (-b^2+4 a c\right )^{3/4} \sqrt {e} (2 c d-b e) \left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4} \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 )}{4 c^{3/4} \left (c d^2-b d e+a e^2\right )^{7/4} \left (a+b x+c x^2\right )^{3/4}}-\frac {c^{3/4} \sqrt [4]{b^2-4 a c} \sqrt {\frac {(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (1+\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )^2}} \left (1+\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{a+b x+c x^2}}{\sqrt [4]{b^2-4 a c}}\right )|\frac {1}{2}\right )}{\sqrt {2} \left (c d^2-b d e+a e^2\right ) (b+2 c x)}-\frac {3 \left (b^2-4 a c\right ) (2 c d-b e)^2 \sqrt {\frac {(b+2 c x)^2}{b^2-4 a c}} \left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4} \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 )}{4 \sqrt {2} c \left (c d^2-b d e+a e^2\right )^2 (b+2 c x) \left (a+b x+c x^2\right )^{3/4}}-\frac {3 \left (b^2-4 a c\right ) (2 c d-b e)^2 \sqrt {\frac {(b+2 c x)^2}{b^2-4 a c}} \left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4} \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 )}{4 \sqrt {2} c \left (c d^2-b d e+a e^2\right )^2 (b+2 c x) \left (a+b x+c x^2\right )^{3/4}} \\ \end{align*}

Mathematica [A] (verified)

Time = 10.99 (sec) , antiderivative size = 652, normalized size of antiderivative = 0.67 \[ \int \frac {1}{(d+e x)^2 \left (a+b x+c x^2\right )^{3/4}} \, dx=\frac {\frac {e (a+x (b+c x))}{d+e x}+\sqrt {2} \sqrt {b^2-4 a c} \left (\frac {c (a+x (b+c x))}{-b^2+4 a c}\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arcsin \left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ),2\right )-\frac {3 \left (-b^2+4 a c\right )^{3/4} (-2 c d+b e) \left (\frac {c (a+x (b+c x))}{-b^2+4 a c}\right )^{3/4} \left (\sqrt {2} \sqrt [4]{c} \sqrt {e} \sqrt [4]{c d^2+e (-b d+a e)} (b+2 c x) \left (\arctan \left (\frac {\sqrt [4]{-b^2+4 a c} \sqrt {e} \sqrt [4]{\frac {c (a+x (b+c x))}{-b^2+4 a c}}}{\sqrt [4]{c} \sqrt [4]{c d^2+e (-b d+a e)}}\right )+\text {arctanh}\left (\frac {\sqrt [4]{-b^2+4 a c} \sqrt {e} \sqrt [4]{\frac {c (a+x (b+c x))}{-b^2+4 a c}}}{\sqrt [4]{c} \sqrt [4]{c d^2+e (-b d+a e)}}\right )\right )+\sqrt [4]{-b^2+4 a c} (-2 c d+b e) \sqrt {\frac {(b+2 c x)^2}{b^2-4 a c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {-b^2+4 a c} e}{2 \sqrt {c} \sqrt {c d^2+e (-b d+a e)}},\arcsin \left (\sqrt {2} \sqrt [4]{\frac {c (a+x (b+c x))}{-b^2+4 a c}}\right ),-1\right )+\sqrt [4]{-b^2+4 a c} (-2 c d+b e) \sqrt {\frac {(b+2 c x)^2}{b^2-4 a c}} \operatorname {EllipticPi}\left (\frac {\sqrt {-b^2+4 a c} e}{2 \sqrt {c} \sqrt {c d^2+e (-b d+a e)}},\arcsin \left (\sqrt {2} \sqrt [4]{\frac {c (a+x (b+c x))}{-b^2+4 a c}}\right ),-1\right )\right )}{4 \sqrt {2} c \left (c d^2+e (-b d+a e)\right ) (b+2 c x)}}{\left (-c d^2+e (b d-a e)\right ) (a+x (b+c x))^{3/4}} \]

[In]

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

[Out]

((e*(a + x*(b + c*x)))/(d + e*x) + Sqrt[2]*Sqrt[b^2 - 4*a*c]*((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))^(3/4)*Elli
pticF[ArcSin[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]]/2, 2] - (3*(-b^2 + 4*a*c)^(3/4)*(-2*c*d + b*e)*((c*(a + x*(b + c*x
)))/(-b^2 + 4*a*c))^(3/4)*(Sqrt[2]*c^(1/4)*Sqrt[e]*(c*d^2 + e*(-(b*d) + a*e))^(1/4)*(b + 2*c*x)*(ArcTan[((-b^2
+ 4*a*c)^(1/4)*Sqrt[e]*((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))^(1/4))/(c^(1/4)*(c*d^2 + e*(-(b*d) + a*e))^(1/4
))] + ArcTanh[((-b^2 + 4*a*c)^(1/4)*Sqrt[e]*((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))^(1/4))/(c^(1/4)*(c*d^2 + e*
(-(b*d) + a*e))^(1/4))]) + (-b^2 + 4*a*c)^(1/4)*(-2*c*d + b*e)*Sqrt[(b + 2*c*x)^2/(b^2 - 4*a*c)]*EllipticPi[-1
/2*(Sqrt[-b^2 + 4*a*c]*e)/(Sqrt[c]*Sqrt[c*d^2 + e*(-(b*d) + a*e)]), ArcSin[Sqrt[2]*((c*(a + x*(b + c*x)))/(-b^
2 + 4*a*c))^(1/4)], -1] + (-b^2 + 4*a*c)^(1/4)*(-2*c*d + b*e)*Sqrt[(b + 2*c*x)^2/(b^2 - 4*a*c)]*EllipticPi[(Sq
rt[-b^2 + 4*a*c]*e)/(2*Sqrt[c]*Sqrt[c*d^2 + e*(-(b*d) + a*e)]), ArcSin[Sqrt[2]*((c*(a + x*(b + c*x)))/(-b^2 +
4*a*c))^(1/4)], -1]))/(4*Sqrt[2]*c*(c*d^2 + e*(-(b*d) + a*e))*(b + 2*c*x)))/((-(c*d^2) + e*(b*d - a*e))*(a + x
*(b + c*x))^(3/4))

Maple [F]

\[\int \frac {1}{\left (e x +d \right )^{2} \left (c \,x^{2}+b x +a \right )^{\frac {3}{4}}}d x\]

[In]

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

[Out]

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

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{(d+e x)^2 \left (a+b x+c x^2\right )^{3/4}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Sympy [F]

\[ \int \frac {1}{(d+e x)^2 \left (a+b x+c x^2\right )^{3/4}} \, dx=\int \frac {1}{\left (d + e x\right )^{2} \left (a + b x + c x^{2}\right )^{\frac {3}{4}}}\, dx \]

[In]

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

[Out]

Integral(1/((d + e*x)**2*(a + b*x + c*x**2)**(3/4)), x)

Maxima [F]

\[ \int \frac {1}{(d+e x)^2 \left (a+b x+c x^2\right )^{3/4}} \, dx=\int { \frac {1}{{\left (c x^{2} + b x + a\right )}^{\frac {3}{4}} {\left (e x + d\right )}^{2}} \,d x } \]

[In]

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

[Out]

integrate(1/((c*x^2 + b*x + a)^(3/4)*(e*x + d)^2), x)

Giac [F]

\[ \int \frac {1}{(d+e x)^2 \left (a+b x+c x^2\right )^{3/4}} \, dx=\int { \frac {1}{{\left (c x^{2} + b x + a\right )}^{\frac {3}{4}} {\left (e x + d\right )}^{2}} \,d x } \]

[In]

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

[Out]

integrate(1/((c*x^2 + b*x + a)^(3/4)*(e*x + d)^2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(d+e x)^2 \left (a+b x+c x^2\right )^{3/4}} \, dx=\int \frac {1}{{\left (d+e\,x\right )}^2\,{\left (c\,x^2+b\,x+a\right )}^{3/4}} \,d x \]

[In]

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

[Out]

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