\(\int \frac {(a+b x+c x^2)^{5/4}}{(d+e x)^2} \, dx\) [2527]

Optimal result

Integrand size = 22, antiderivative size = 975 \[ \int \frac {\left (a+b x+c x^2\right )^{5/4}}{(d+e x)^2} \, dx=-\frac {5 (3 c d-2 b e-c e x) \sqrt [4]{a+b x+c x^2}}{3 e^3}-\frac {\left (a+b x+c x^2\right )^{5/4}}{e (d+e x)}+\frac {5 \left (-b^2+4 a c\right )^{3/4} (2 c d-b e) \sqrt [4]{c d^2-b d e+a e^2} \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} e^{7/2} \left (a+b x+c x^2\right )^{3/4}}+\frac {5 \left (-b^2+4 a c\right )^{3/4} (2 c d-b e) \sqrt [4]{c d^2-b d e+a e^2} \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} e^{7/2} \left (a+b x+c x^2\right )^{3/4}}+\frac {5 \sqrt [4]{b^2-4 a c} \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right ) \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 )}{6 \sqrt {2} \sqrt [4]{c} e^4 (b+2 c x)}+\frac {5 \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 e^4 (b+2 c x) \left (a+b x+c x^2\right )^{3/4}}+\frac {5 \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 e^4 (b+2 c x) \left (a+b x+c x^2\right )^{3/4}} \]

[Out]

-5/3*(-c*e*x-2*b*e+3*c*d)*(c*x^2+b*x+a)^(1/4)/e^3-(c*x^2+b*x+a)^(5/4)/e/(e*x+d)+5/4*(4*a*c-b^2)^(3/4)*(-b*e+2*
c*d)*(a*e^2-b*d*e+c*d^2)^(1/4)*(-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))/c^(3/4)/e^(7/2)/(c*x^2+b*x+a)^(3/4)+
5/4*(4*a*c-b^2)^(3/4)*(-b*e+2*c*d)*(a*e^2-b*d*e+c*d^2)^(1/4)*(-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))/c^(3/
4)/e^(7/2)/(c*x^2+b*x+a)^(3/4)+5/8*(-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)*EllipticP
i((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/e^4/(2*c*x+b)/(c*x^2+b*x+a)^(3/4)*2^(1/2)+5/8*(-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/e^4/(2*c*x+b)/(c*x^2+b*x+a)^(3/4)*2^(1/2)+5/12*(-4*a
*c+b^2)^(1/4)*(6*c^2*d^2+b^2*e^2-2*c*e*(-a*e+3*b*d))*(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)/cos(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*a
rctan(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)/c^(
1/4)/e^4/(2*c*x+b)*2^(1/2)

Rubi [A] (verified)

Time = 1.50 (sec) , antiderivative size = 975, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.818, Rules used = {746, 828, 857, 637, 226, 763, 762, 761, 410, 109, 418, 1227, 551, 455, 65, 218, 214, 211} \[ \int \frac {\left (a+b x+c x^2\right )^{5/4}}{(d+e x)^2} \, dx=\frac {5 \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 e^4 (b+2 c x) \left (c x^2+b x+a\right )^{3/4}}+\frac {5 \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 e^4 (b+2 c x) \left (c x^2+b x+a\right )^{3/4}}+\frac {5 \left (4 a c-b^2\right )^{3/4} \sqrt [4]{c d^2-b e d+a e^2} \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} e^{7/2} \left (c x^2+b x+a\right )^{3/4}}+\frac {5 \left (4 a c-b^2\right )^{3/4} \sqrt [4]{c d^2-b e d+a e^2} \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} e^{7/2} \left (c x^2+b x+a\right )^{3/4}}-\frac {\left (c x^2+b x+a\right )^{5/4}}{e (d+e x)}+\frac {5 \sqrt [4]{b^2-4 a c} \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right ) \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 )}{6 \sqrt {2} \sqrt [4]{c} e^4 (b+2 c x)}-\frac {5 (3 c d-2 b e-c e x) \sqrt [4]{c x^2+b x+a}}{3 e^3} \]

[In]

Int[(a + b*x + c*x^2)^(5/4)/(d + e*x)^2,x]

[Out]

(-5*(3*c*d - 2*b*e - c*e*x)*(a + b*x + c*x^2)^(1/4))/(3*e^3) - (a + b*x + c*x^2)^(5/4)/(e*(d + e*x)) + (5*(-b^
2 + 4*a*c)^(3/4)*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^(1/4)*(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))^(3/4)*Ar
cTan[((-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)*e^(7/2)*(a + b*x + c*x^2)^(3/4)) + (5*(-b^2 + 4*a*c)^(3/4)*(2*c*d - b*e)*(c*d^2 - b
*d*e + a*e^2)^(1/4)*(-((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)*e^(7/2)*(a +
b*x + c*x^2)^(3/4)) + (5*(b^2 - 4*a*c)^(1/4)*(6*c^2*d^2 + b^2*e^2 - 2*c*e*(3*b*d - a*e))*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]
)/(6*Sqrt[2]*c^(1/4)*e^4*(b + 2*c*x)) + (5*(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*(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])/(4*Sqrt[2]*c*e^4*(b + 2*c*x)*(a + b*x + c*x^
2)^(3/4)) + (5*(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*Sqrt[2]*c*e^4*(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 746

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

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 828

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*(c*d + 2*c*d*p - b*e*p) + g*c*e*(m + 2*p + 1)*x)*((a + b*x + c*x^
2)^p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), x] - Dist[p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)), Int[(d + e*x)^m*(a
+ b*x + c*x^2)^(p - 1)*Simp[c*e*f*(b*d - 2*a*e)*(m + 2*p + 2) + g*(a*e*(b*e - 2*c*d*m + b*e*m) + b*d*(b*e*p -
c*d - 2*c*d*p)) + (c*e*f*(2*c*d - b*e)*(m + 2*p + 2) + g*(b^2*e^2*(p + m + 1) - 2*c^2*d^2*(1 + 2*p) - c*e*(b*
d*(m - 2*p) + 2*a*e*(m + 2*p + 1))))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0
] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && GtQ[p, 0] && (IntegerQ[p] ||  !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])
) &&  !ILtQ[m + 2*p, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*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 {\left (a+b x+c x^2\right )^{5/4}}{e (d+e x)}+\frac {5 \int \frac {(b+2 c x) \sqrt [4]{a+b x+c x^2}}{d+e x} \, dx}{4 e} \\ & = -\frac {5 (3 c d-2 b e-c e x) \sqrt [4]{a+b x+c x^2}}{3 e^3}-\frac {\left (a+b x+c x^2\right )^{5/4}}{e (d+e x)}-\frac {5 \int \frac {c \left (2 b^2 d e+4 a c d e-3 b \left (c d^2+a e^2\right )\right )-c \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right ) x}{(d+e x) \left (a+b x+c x^2\right )^{3/4}} \, dx}{12 c e^3} \\ & = -\frac {5 (3 c d-2 b e-c e x) \sqrt [4]{a+b x+c x^2}}{3 e^3}-\frac {\left (a+b x+c x^2\right )^{5/4}}{e (d+e x)}-\frac {\left (5 (2 c d-b e) \left (c d^2-b d e+a e^2\right )\right ) \int \frac {1}{(d+e x) \left (a+b x+c x^2\right )^{3/4}} \, dx}{4 e^4}+\frac {\left (5 \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right )\right ) \int \frac {1}{\left (a+b x+c x^2\right )^{3/4}} \, dx}{12 e^4} \\ & = -\frac {5 (3 c d-2 b e-c e x) \sqrt [4]{a+b x+c x^2}}{3 e^3}-\frac {\left (a+b x+c x^2\right )^{5/4}}{e (d+e x)}+\frac {\left (5 \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right ) \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 )}{3 e^4 (b+2 c x)}-\frac {\left (5 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \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 e^4 \left (a+b x+c x^2\right )^{3/4}} \\ & = -\frac {5 (3 c d-2 b e-c e x) \sqrt [4]{a+b x+c x^2}}{3 e^3}-\frac {\left (a+b x+c x^2\right )^{5/4}}{e (d+e x)}+\frac {5 \sqrt [4]{b^2-4 a c} \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right ) \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 )}{6 \sqrt {2} \sqrt [4]{c} e^4 (b+2 c x)}-\frac {\left (5 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \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} e^4 \left (a+b x+c x^2\right )^{3/4}} \\ & = -\frac {5 (3 c d-2 b e-c e x) \sqrt [4]{a+b x+c x^2}}{3 e^3}-\frac {\left (a+b x+c x^2\right )^{5/4}}{e (d+e x)}+\frac {5 \sqrt [4]{b^2-4 a c} \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right ) \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 )}{6 \sqrt {2} \sqrt [4]{c} e^4 (b+2 c x)}+\frac {\left (5 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \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} e^3 \left (a+b x+c x^2\right )^{3/4}}+\frac {\left (5 c (2 c d-b e)^2 \left (c d^2-b d e+a e^2\right ) \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 ) e^4 \left (a+b x+c x^2\right )^{3/4}} \\ & = -\frac {5 (3 c d-2 b e-c e x) \sqrt [4]{a+b x+c x^2}}{3 e^3}-\frac {\left (a+b x+c x^2\right )^{5/4}}{e (d+e x)}+\frac {5 \sqrt [4]{b^2-4 a c} \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right ) \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 )}{6 \sqrt {2} \sqrt [4]{c} e^4 (b+2 c x)}+\frac {\left (5 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \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} e^3 \left (a+b x+c x^2\right )^{3/4}}+\frac {\left (5 c (2 c d-b e)^2 \left (c d^2-b d e+a e^2\right ) \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 ) e^4 \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 {5 (3 c d-2 b e-c e x) \sqrt [4]{a+b x+c x^2}}{3 e^3}-\frac {\left (a+b x+c x^2\right )^{5/4}}{e (d+e x)}+\frac {5 \sqrt [4]{b^2-4 a c} \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right ) \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 )}{6 \sqrt {2} \sqrt [4]{c} e^4 (b+2 c x)}-\frac {\left (5 \sqrt {2} c^2 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \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 ) e^3 \left (a+b x+c x^2\right )^{3/4}}-\frac {\left (5 \sqrt {2} c (2 c d-b e)^2 \left (c d^2-b d e+a e^2\right ) \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 ) e^4 \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 {5 (3 c d-2 b e-c e x) \sqrt [4]{a+b x+c x^2}}{3 e^3}-\frac {\left (a+b x+c x^2\right )^{5/4}}{e (d+e x)}+\frac {5 \sqrt [4]{b^2-4 a c} \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right ) \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 )}{6 \sqrt {2} \sqrt [4]{c} e^4 (b+2 c x)}-\frac {\left (5 \left (b^2-4 a c\right ) (2 c d-b e) \sqrt {c d^2-b d e+a 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}{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} e^3 \left (a+b x+c x^2\right )^{3/4}}-\frac {\left (5 \left (b^2-4 a c\right ) (2 c d-b e) \sqrt {c d^2-b d e+a 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}{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} e^3 \left (a+b x+c x^2\right )^{3/4}}-\frac {\left (5 c (2 c d-b e)^2 \left (c d^2-b d e+a e^2\right ) \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 ) e^4 \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 (5 c (2 c d-b e)^2 \left (c d^2-b d e+a e^2\right ) \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 ) e^4 \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 {5 (3 c d-2 b e-c e x) \sqrt [4]{a+b x+c x^2}}{3 e^3}-\frac {\left (a+b x+c x^2\right )^{5/4}}{e (d+e x)}+\frac {5 \left (-b^2+4 a c\right )^{3/4} (2 c d-b e) \sqrt [4]{c d^2-b d e+a e^2} \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} e^{7/2} \left (a+b x+c x^2\right )^{3/4}}+\frac {5 \left (-b^2+4 a c\right )^{3/4} (2 c d-b e) \sqrt [4]{c d^2-b d e+a e^2} \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} e^{7/2} \left (a+b x+c x^2\right )^{3/4}}+\frac {5 \sqrt [4]{b^2-4 a c} \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right ) \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 )}{6 \sqrt {2} \sqrt [4]{c} e^4 (b+2 c x)}-\frac {\left (5 c (2 c d-b e)^2 \left (c d^2-b d e+a e^2\right ) \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 ) e^4 \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 (5 c (2 c d-b e)^2 \left (c d^2-b d e+a e^2\right ) \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 ) e^4 \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 {5 (3 c d-2 b e-c e x) \sqrt [4]{a+b x+c x^2}}{3 e^3}-\frac {\left (a+b x+c x^2\right )^{5/4}}{e (d+e x)}+\frac {5 \left (-b^2+4 a c\right )^{3/4} (2 c d-b e) \sqrt [4]{c d^2-b d e+a e^2} \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} e^{7/2} \left (a+b x+c x^2\right )^{3/4}}+\frac {5 \left (-b^2+4 a c\right )^{3/4} (2 c d-b e) \sqrt [4]{c d^2-b d e+a e^2} \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} e^{7/2} \left (a+b x+c x^2\right )^{3/4}}+\frac {5 \sqrt [4]{b^2-4 a c} \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right ) \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 )}{6 \sqrt {2} \sqrt [4]{c} e^4 (b+2 c x)}+\frac {5 \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 e^4 (b+2 c x) \left (a+b x+c x^2\right )^{3/4}}+\frac {5 \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 e^4 (b+2 c x) \left (a+b x+c x^2\right )^{3/4}} \\ \end{align*}

Mathematica [A] (verified)

Time = 12.42 (sec) , antiderivative size = 658, normalized size of antiderivative = 0.67 \[ \int \frac {\left (a+b x+c x^2\right )^{5/4}}{(d+e x)^2} \, dx=\frac {5 (-3 c d+2 b e+c e x) \sqrt [4]{a+x (b+c x)}}{3 e^3}-\frac {(a+x (b+c x))^{5/4}}{e (d+e x)}+\frac {5 \left (\frac {c (a+x (b+c x))}{-b^2+4 a c}\right )^{3/4} \left (\sqrt {b^2-4 a c} \left (6 c^2 d^2+b^2 e^2+2 c e (-3 b d+a e)\right ) \operatorname {EllipticF}\left (\frac {1}{2} \arcsin \left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ),2\right )-\frac {3 \left (-b^2+4 a c\right )^{3/4} (-2 c d+b e) \left (\sqrt {2} \sqrt [4]{c} \sqrt {e} \sqrt [4]{c d^2+e (-b d+a e)} (b+2 c x) \left (\arctan \left (\frac {\sqrt [4]{-b^2+4 a c} \sqrt {e} \sqrt [4]{\frac {c (a+x (b+c x))}{-b^2+4 a c}}}{\sqrt [4]{c} \sqrt [4]{c d^2+e (-b d+a e)}}\right )+\text {arctanh}\left (\frac {\sqrt [4]{-b^2+4 a c} \sqrt {e} \sqrt [4]{\frac {c (a+x (b+c x))}{-b^2+4 a c}}}{\sqrt [4]{c} \sqrt [4]{c d^2+e (-b d+a e)}}\right )\right )+\sqrt [4]{-b^2+4 a c} (-2 c d+b e) \sqrt {\frac {(b+2 c x)^2}{b^2-4 a c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {-b^2+4 a c} e}{2 \sqrt {c} \sqrt {c d^2+e (-b d+a e)}},\arcsin \left (\sqrt {2} \sqrt [4]{\frac {c (a+x (b+c x))}{-b^2+4 a c}}\right ),-1\right )+\sqrt [4]{-b^2+4 a c} (-2 c d+b e) \sqrt {\frac {(b+2 c x)^2}{b^2-4 a c}} \operatorname {EllipticPi}\left (\frac {\sqrt {-b^2+4 a c} e}{2 \sqrt {c} \sqrt {c d^2+e (-b d+a e)}},\arcsin \left (\sqrt {2} \sqrt [4]{\frac {c (a+x (b+c x))}{-b^2+4 a c}}\right ),-1\right )\right )}{4 (b+2 c x)}\right )}{3 \sqrt {2} c e^4 (a+x (b+c x))^{3/4}} \]

[In]

Integrate[(a + b*x + c*x^2)^(5/4)/(d + e*x)^2,x]

[Out]

(5*(-3*c*d + 2*b*e + c*e*x)*(a + x*(b + c*x))^(1/4))/(3*e^3) - (a + x*(b + c*x))^(5/4)/(e*(d + e*x)) + (5*((c*
(a + x*(b + c*x)))/(-b^2 + 4*a*c))^(3/4)*(Sqrt[b^2 - 4*a*c]*(6*c^2*d^2 + b^2*e^2 + 2*c*e*(-3*b*d + a*e))*Ellip
ticF[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)*(Sqrt[2]*c^(1/4)*Sqr
t[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[(Sqrt[-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*(b + 2*c*x))))/(3*Sqr
t[2]*c*e^4*(a + x*(b + c*x))^(3/4))

Maple [F]

\[\int \frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{4}}}{\left (e x +d \right )^{2}}d x\]

[In]

int((c*x^2+b*x+a)^(5/4)/(e*x+d)^2,x)

[Out]

int((c*x^2+b*x+a)^(5/4)/(e*x+d)^2,x)

Fricas [F(-1)]

Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{5/4}}{(d+e x)^2} \, dx=\text {Timed out} \]

[In]

integrate((c*x^2+b*x+a)^(5/4)/(e*x+d)^2,x, algorithm="fricas")

[Out]

Timed out

Sympy [F]

\[ \int \frac {\left (a+b x+c x^2\right )^{5/4}}{(d+e x)^2} \, dx=\int \frac {\left (a + b x + c x^{2}\right )^{\frac {5}{4}}}{\left (d + e x\right )^{2}}\, dx \]

[In]

integrate((c*x**2+b*x+a)**(5/4)/(e*x+d)**2,x)

[Out]

Integral((a + b*x + c*x**2)**(5/4)/(d + e*x)**2, x)

Maxima [F]

\[ \int \frac {\left (a+b x+c x^2\right )^{5/4}}{(d+e x)^2} \, dx=\int { \frac {{\left (c x^{2} + b x + a\right )}^{\frac {5}{4}}}{{\left (e x + d\right )}^{2}} \,d x } \]

[In]

integrate((c*x^2+b*x+a)^(5/4)/(e*x+d)^2,x, algorithm="maxima")

[Out]

integrate((c*x^2 + b*x + a)^(5/4)/(e*x + d)^2, x)

Giac [F]

\[ \int \frac {\left (a+b x+c x^2\right )^{5/4}}{(d+e x)^2} \, dx=\int { \frac {{\left (c x^{2} + b x + a\right )}^{\frac {5}{4}}}{{\left (e x + d\right )}^{2}} \,d x } \]

[In]

integrate((c*x^2+b*x+a)^(5/4)/(e*x+d)^2,x, algorithm="giac")

[Out]

integrate((c*x^2 + b*x + a)^(5/4)/(e*x + d)^2, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{5/4}}{(d+e x)^2} \, dx=\int \frac {{\left (c\,x^2+b\,x+a\right )}^{5/4}}{{\left (d+e\,x\right )}^2} \,d x \]

[In]

int((a + b*x + c*x^2)^(5/4)/(d + e*x)^2,x)

[Out]

int((a + b*x + c*x^2)^(5/4)/(d + e*x)^2, x)