### 3.2527 $$\int \frac{(a+b x+c x^2)^{5/4}}{(d+e x)^2} \, dx$$

Optimal. Leaf size=975 $\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} \Pi \left (-\frac{\sqrt{4 a c-b^2} e}{2 \sqrt{c} \sqrt{c d^2-b e d+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 ) (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} \Pi \left (\frac{\sqrt{4 a c-b^2} e}{2 \sqrt{c} \sqrt{c d^2-b e d+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 ) (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} \tan ^{-1}\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} \tanh ^{-1}\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 ) \text{EllipticF}\left (2 \tan ^{-1}\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}$

[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[-(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)) + (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))

________________________________________________________________________________________

Rubi [A]  time = 2.32835, antiderivative size = 975, normalized size of antiderivative = 1., number of steps used = 20, number of rules used = 18, integrand size = 22, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.818, Rules used = {732, 814, 843, 623, 220, 749, 748, 747, 401, 108, 409, 1213, 537, 444, 63, 212, 208, 205} $\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} \Pi \left (-\frac{\sqrt{4 a c-b^2} e}{2 \sqrt{c} \sqrt{c d^2-b e d+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 ) (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} \Pi \left (\frac{\sqrt{4 a c-b^2} e}{2 \sqrt{c} \sqrt{c d^2-b e d+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 ) (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} \tan ^{-1}\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} \tanh ^{-1}\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 ) F\left (2 \tan ^{-1}\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}$

Antiderivative was successfully veriﬁed.

[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[-(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)) + (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 732

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 814

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 843

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 623

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 220

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[((1 + q^2*x^2)*Sqrt[(a + b*x^4)/(a*(
1 + q^2*x^2)^2)]*EllipticF[2*ArcTan[q*x], 1/2])/(2*q*Sqrt[a + b*x^4]), x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rule 749

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 748

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 747

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 401

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 108

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 409

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 1213

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 537

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(1*Ellipt
icPi[(b*c)/(a*d), ArcSin[Rt[-(d/c), 2]*x], (c*f)/(d*e)])/(a*Sqrt[c]*Sqrt[e]*Rt[-(d/c), 2]), 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 444

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 63

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 212

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] &&
!GtQ[a/b, 0]

Rule 208

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

Rule 205

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

Rubi steps

\begin{align*} \int \frac{\left (a+b x+c x^2\right )^{5/4}}{(d+e x)^2} \, dx &=-\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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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]  time = 4.71552, size = 663, normalized size = 0.68 $\frac{5 \left (\frac{c (a+x (b+c x))}{4 a c-b^2}\right )^{3/4} \left (\sqrt{b^2-4 a c} \left (2 c e (a e-3 b d)+b^2 e^2+6 c^2 d^2\right ) \text{EllipticF}\left (\frac{1}{2} \sin ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right ),2\right )+\frac{3 \left (4 a c-b^2\right )^{3/4} (b e-2 c d) \left (-\sqrt{2} \sqrt [4]{c} \sqrt{e} (b+2 c x) \sqrt [4]{e (a e-b d)+c d^2} \left (\tan ^{-1}\left (\frac{\sqrt{e} \sqrt [4]{4 a c-b^2} \sqrt [4]{\frac{c (a+x (b+c x))}{4 a c-b^2}}}{\sqrt [4]{c} \sqrt [4]{e (a e-b d)+c d^2}}\right )+\tanh ^{-1}\left (\frac{\sqrt{e} \sqrt [4]{4 a c-b^2} \sqrt [4]{\frac{c (a+x (b+c x))}{4 a c-b^2}}}{\sqrt [4]{c} \sqrt [4]{e (a e-b d)+c d^2}}\right )\right )+\sqrt [4]{4 a c-b^2} \sqrt{\frac{(b+2 c x)^2}{b^2-4 a c}} (b e-2 c d) \Pi \left (-\frac{\sqrt{4 a c-b^2} e}{2 \sqrt{c} \sqrt{c d^2+e (a e-b d)}};\left .-\sin ^{-1}\left (\sqrt{2} \sqrt [4]{\frac{c (a+x (b+c x))}{4 a c-b^2}}\right )\right |-1\right )+\sqrt [4]{4 a c-b^2} \sqrt{\frac{(b+2 c x)^2}{b^2-4 a c}} (b e-2 c d) \Pi \left (\frac{\sqrt{4 a c-b^2} e}{2 \sqrt{c} \sqrt{c d^2+e (a e-b d)}};\left .-\sin ^{-1}\left (\sqrt{2} \sqrt [4]{\frac{c (a+x (b+c x))}{4 a c-b^2}}\right )\right |-1\right )\right )}{4 (b+2 c x)}\right )}{3 \sqrt{2} c e^4 (a+x (b+c x))^{3/4}}+\frac{5 \sqrt [4]{a+x (b+c x)} (2 b e-3 c d+c e x)}{3 e^3}-\frac{(a+x (b+c x))^{5/4}}{e (d+e x)}$

Antiderivative was successfully veriﬁed.

[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)*S
qrt[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[-(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] + (-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*
Sqrt[2]*c*e^4*(a + x*(b + c*x))^(3/4))

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Maple [F]  time = 1.625, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( ex+d \right ) ^{2}} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{5}{4}}}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (c x^{2} + b x + a\right )}^{\frac{5}{4}}}{{\left (e x + d\right )}^{2}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (c x^{2} + b x + a\right )}^{\frac{5}{4}}}{{\left (e x + d\right )}^{2}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)