3.2547 \(\int \frac{1}{(d+e x)^2 (a+b x+c x^2)^{5/4}} \, dx\)

Optimal. Leaf size=1485 \[ \text{result too large to display} \]

[Out]

(-4*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x))/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*(d + e*x)*(a + b*x +
c*x^2)^(1/4)) - (e*(8*c^2*d^2 + 5*b^2*e^2 - 4*c*e*(2*b*d + 3*a*e))*(a + b*x + c*x^2)^(3/4))/((b^2 - 4*a*c)*(c
*d^2 - b*d*e + a*e^2)^2*(d + e*x)) + (Sqrt[c]*(8*c^2*d^2 + 5*b^2*e^2 - 4*c*e*(2*b*d + 3*a*e))*(b + 2*c*x)*(a +
b*x + c*x^2)^(1/4))/((b^2 - 4*a*c)^(3/2)*(c*d^2 - b*d*e + a*e^2)^2*(1 + (2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sqr
t[b^2 - 4*a*c])) + (5*(-b^2 + 4*a*c)^(1/4)*e^(3/2)*(2*c*d - b*e)*(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))^(1/4
)*ArcTan[((-b^2 + 4*a*c)^(1/4)*Sqrt[e]*(1 - (b + 2*c*x)^2/(b^2 - 4*a*c))^(1/4))/(Sqrt[2]*c^(1/4)*(c*d^2 - b*d*
e + a*e^2)^(1/4))])/(4*c^(1/4)*(c*d^2 - b*d*e + a*e^2)^(9/4)*(a + b*x + c*x^2)^(1/4)) - (5*(-b^2 + 4*a*c)^(1/4
)*e^(3/2)*(2*c*d - b*e)*(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))^(1/4)*ArcTanh[((-b^2 + 4*a*c)^(1/4)*Sqrt[e]*(
1 - (b + 2*c*x)^2/(b^2 - 4*a*c))^(1/4))/(Sqrt[2]*c^(1/4)*(c*d^2 - b*d*e + a*e^2)^(1/4))])/(4*c^(1/4)*(c*d^2 -
b*d*e + a*e^2)^(9/4)*(a + b*x + c*x^2)^(1/4)) - (c^(1/4)*(8*c^2*d^2 + 5*b^2*e^2 - 4*c*e*(2*b*d + 3*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]*Sqr
t[a + b*x + c*x^2])/Sqrt[b^2 - 4*a*c])*EllipticE[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]*(b^2 - 4*a*c)^(1/4)*(c*d^2 - b*d*e + a*e^2)^2*(b + 2*c*x)) + (c^(1/4)*(8*c^2*d^2 +
5*b^2*e^2 - 4*c*e*(2*b*d + 3*a*e))*Sqrt[(b + 2*c*x)^2/((b^2 - 4*a*c)*(1 + (2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sq
rt[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])/(2*Sqrt[2]*(b^2 - 4*a*c)^(1/4)*(c*d^2 - b*d*e + a*e^
2)^2*(b + 2*c*x)) - (5*Sqrt[-b^2 + 4*a*c]*e*(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)))^(1/4)*EllipticPi[-(Sqrt[-b^2 + 4*a*c]*e)/(2*Sqrt[c]*Sqrt[c*d^2 - b*d*e + a*e^2]), Arc
Sin[(1 - (b + 2*c*x)^2/(b^2 - 4*a*c))^(1/4)], -1])/(4*Sqrt[2]*Sqrt[c]*(c*d^2 - b*d*e + a*e^2)^(5/2)*(b + 2*c*x
)*(a + b*x + c*x^2)^(1/4)) + (5*Sqrt[-b^2 + 4*a*c]*e*(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)))^(1/4)*EllipticPi[(Sqrt[-b^2 + 4*a*c]*e)/(2*Sqrt[c]*Sqrt[c*d^2 - b*d*e + a*e^
2]), ArcSin[(1 - (b + 2*c*x)^2/(b^2 - 4*a*c))^(1/4)], -1])/(4*Sqrt[2]*Sqrt[c]*(c*d^2 - b*d*e + a*e^2)^(5/2)*(b
+ 2*c*x)*(a + b*x + c*x^2)^(1/4))

________________________________________________________________________________________

Rubi [A]  time = 3.17006, antiderivative size = 1485, normalized size of antiderivative = 1., number of steps used = 21, number of rules used = 19, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.864, Rules used = {740, 834, 843, 623, 305, 220, 1196, 749, 748, 746, 399, 490, 1213, 537, 444, 63, 298, 205, 208} \[ -\frac{5 \sqrt{4 a c-b^2} e \sqrt{\frac{(b+2 c x)^2}{b^2-4 a c}} \sqrt [4]{-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}} \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} \sqrt{c} \left (c d^2-b e d+a e^2\right )^{5/2} (b+2 c x) \sqrt [4]{c x^2+b x+a}}+\frac{5 \sqrt{4 a c-b^2} e \sqrt{\frac{(b+2 c x)^2}{b^2-4 a c}} \sqrt [4]{-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}} \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} \sqrt{c} \left (c d^2-b e d+a e^2\right )^{5/2} (b+2 c x) \sqrt [4]{c x^2+b x+a}}+\frac{5 \sqrt [4]{4 a c-b^2} e^{3/2} \sqrt [4]{-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}} \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 \sqrt [4]{c} \left (c d^2-b e d+a e^2\right )^{9/4} \sqrt [4]{c x^2+b x+a}}-\frac{5 \sqrt [4]{4 a c-b^2} e^{3/2} \sqrt [4]{-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}} \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 \sqrt [4]{c} \left (c d^2-b e d+a e^2\right )^{9/4} \sqrt [4]{c x^2+b x+a}}-\frac{\sqrt [4]{c} \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 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 ) E\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 )}{\sqrt{2} \sqrt [4]{b^2-4 a c} \left (c d^2-b e d+a e^2\right )^2 (b+2 c x)}+\frac{\sqrt [4]{c} \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 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 )}{2 \sqrt{2} \sqrt [4]{b^2-4 a c} \left (c d^2-b e d+a e^2\right )^2 (b+2 c x)}-\frac{e \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) \left (c x^2+b x+a\right )^{3/4}}{\left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )^2 (d+e x)}-\frac{4 \left (-e b^2+c d b+2 a c e+c (2 c d-b e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) (d+e x) \sqrt [4]{c x^2+b x+a}}+\frac{\sqrt{c} \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) (b+2 c x) \sqrt [4]{c x^2+b x+a}}{\left (b^2-4 a c\right )^{3/2} \left (c d^2-b e d+a e^2\right )^2 \left (\frac{2 \sqrt{c} \sqrt{c x^2+b x+a}}{\sqrt{b^2-4 a c}}+1\right )} \]

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x)^2*(a + b*x + c*x^2)^(5/4)),x]

[Out]

(-4*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x))/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*(d + e*x)*(a + b*x +
c*x^2)^(1/4)) - (e*(8*c^2*d^2 + 5*b^2*e^2 - 4*c*e*(2*b*d + 3*a*e))*(a + b*x + c*x^2)^(3/4))/((b^2 - 4*a*c)*(c
*d^2 - b*d*e + a*e^2)^2*(d + e*x)) + (Sqrt[c]*(8*c^2*d^2 + 5*b^2*e^2 - 4*c*e*(2*b*d + 3*a*e))*(b + 2*c*x)*(a +
b*x + c*x^2)^(1/4))/((b^2 - 4*a*c)^(3/2)*(c*d^2 - b*d*e + a*e^2)^2*(1 + (2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sqr
t[b^2 - 4*a*c])) + (5*(-b^2 + 4*a*c)^(1/4)*e^(3/2)*(2*c*d - b*e)*(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))^(1/4
)*ArcTan[((-b^2 + 4*a*c)^(1/4)*Sqrt[e]*(1 - (b + 2*c*x)^2/(b^2 - 4*a*c))^(1/4))/(Sqrt[2]*c^(1/4)*(c*d^2 - b*d*
e + a*e^2)^(1/4))])/(4*c^(1/4)*(c*d^2 - b*d*e + a*e^2)^(9/4)*(a + b*x + c*x^2)^(1/4)) - (5*(-b^2 + 4*a*c)^(1/4
)*e^(3/2)*(2*c*d - b*e)*(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))^(1/4)*ArcTanh[((-b^2 + 4*a*c)^(1/4)*Sqrt[e]*(
1 - (b + 2*c*x)^2/(b^2 - 4*a*c))^(1/4))/(Sqrt[2]*c^(1/4)*(c*d^2 - b*d*e + a*e^2)^(1/4))])/(4*c^(1/4)*(c*d^2 -
b*d*e + a*e^2)^(9/4)*(a + b*x + c*x^2)^(1/4)) - (c^(1/4)*(8*c^2*d^2 + 5*b^2*e^2 - 4*c*e*(2*b*d + 3*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]*Sqr
t[a + b*x + c*x^2])/Sqrt[b^2 - 4*a*c])*EllipticE[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]*(b^2 - 4*a*c)^(1/4)*(c*d^2 - b*d*e + a*e^2)^2*(b + 2*c*x)) + (c^(1/4)*(8*c^2*d^2 +
5*b^2*e^2 - 4*c*e*(2*b*d + 3*a*e))*Sqrt[(b + 2*c*x)^2/((b^2 - 4*a*c)*(1 + (2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sq
rt[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])/(2*Sqrt[2]*(b^2 - 4*a*c)^(1/4)*(c*d^2 - b*d*e + a*e^
2)^2*(b + 2*c*x)) - (5*Sqrt[-b^2 + 4*a*c]*e*(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)))^(1/4)*EllipticPi[-(Sqrt[-b^2 + 4*a*c]*e)/(2*Sqrt[c]*Sqrt[c*d^2 - b*d*e + a*e^2]), Arc
Sin[(1 - (b + 2*c*x)^2/(b^2 - 4*a*c))^(1/4)], -1])/(4*Sqrt[2]*Sqrt[c]*(c*d^2 - b*d*e + a*e^2)^(5/2)*(b + 2*c*x
)*(a + b*x + c*x^2)^(1/4)) + (5*Sqrt[-b^2 + 4*a*c]*e*(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)))^(1/4)*EllipticPi[(Sqrt[-b^2 + 4*a*c]*e)/(2*Sqrt[c]*Sqrt[c*d^2 - b*d*e + a*e^
2]), ArcSin[(1 - (b + 2*c*x)^2/(b^2 - 4*a*c))^(1/4)], -1])/(4*Sqrt[2]*Sqrt[c]*(c*d^2 - b*d*e + a*e^2)^(5/2)*(b
+ 2*c*x)*(a + b*x + c*x^2)^(1/4))

Rule 740

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m + 1)*(
b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e
+ a*e^2)), x] + Dist[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*Simp[b*c*d*e*(2*p - m
+ 2) + b^2*e^2*(m + p + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3) - c*e*(2*c*d - b*e)*(m + 2*p + 4)*x
, 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] && LtQ[p, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]

Rule 834

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[((e*f - d*g)*(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)*(a + b*x + c*x^2)^p*Simp[(c*d*f - f*b*e + a*e*g)*(m + 1)
+ b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] &&
NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, -1] && (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 305

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

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 1196

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 4]}, -Simp[(d*x*Sqrt[a + c
*x^4])/(a*(1 + q^2*x^2)), x] + Simp[(d*(1 + q^2*x^2)*Sqrt[(a + c*x^4)/(a*(1 + q^2*x^2)^2)]*EllipticE[2*ArcTan[
q*x], 1/2])/(q*Sqrt[a + c*x^4]), x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, c, d, e}, x] && PosQ[c/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 746

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

Rule 399

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

Rule 490

Int[(x_)^2/(((a_) + (b_.)*(x_)^4)*Sqrt[(c_) + (d_.)*(x_)^4]), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]],
s = Denominator[Rt[-(a/b), 2]]}, Dist[s/(2*b), Int[1/((r + s*x^2)*Sqrt[c + d*x^4]), x], x] - Dist[s/(2*b), In
t[1/((r - s*x^2)*Sqrt[c + d*x^4]), x], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 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 298

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(a/b),
2]]}, Dist[s/(2*b), Int[1/(r + s*x^2), x], x] - Dist[s/(2*b), Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] &
&  !GtQ[a/b, 0]

Rule 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]

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]

Rubi steps

\begin{align*} \int \frac{1}{(d+e x)^2 \left (a+b x+c x^2\right )^{5/4}} \, dx &=-\frac{4 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x) \sqrt [4]{a+b x+c x^2}}-\frac{4 \int \frac{\frac{1}{4} \left (-4 c^2 d^2-5 b^2 e^2+6 c e (b d+2 a e)\right )+\frac{1}{2} c e (2 c d-b e) x}{(d+e x)^2 \sqrt [4]{a+b x+c x^2}} \, dx}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )}\\ &=-\frac{4 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x) \sqrt [4]{a+b x+c x^2}}-\frac{e \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) \left (a+b x+c x^2\right )^{3/4}}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 (d+e x)}+\frac{4 \int \frac{\frac{1}{16} \left (16 c^3 d^3-5 b^3 e^3+20 b c e^2 (b d+a e)-16 c^2 d e (b d+4 a e)\right )+\frac{1}{8} c e \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) x}{(d+e x) \sqrt [4]{a+b x+c x^2}} \, dx}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2}\\ &=-\frac{4 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x) \sqrt [4]{a+b x+c x^2}}-\frac{e \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) \left (a+b x+c x^2\right )^{3/4}}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 (d+e x)}+\frac{\left (5 e^2 (2 c d-b e)\right ) \int \frac{1}{(d+e x) \sqrt [4]{a+b x+c x^2}} \, dx}{4 \left (c d^2-b d e+a e^2\right )^2}+\frac{\left (c \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right )\right ) \int \frac{1}{\sqrt [4]{a+b x+c x^2}} \, dx}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2}\\ &=-\frac{4 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x) \sqrt [4]{a+b x+c x^2}}-\frac{e \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) \left (a+b x+c x^2\right )^{3/4}}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 (d+e x)}+\frac{\left (2 c \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) \sqrt{(b+2 c x)^2}\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{b^2-4 a c+4 c x^4}} \, dx,x,\sqrt [4]{a+b x+c x^2}\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 (b+2 c x)}+\frac{\left (5 e^2 (2 c d-b e) \sqrt [4]{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \int \frac{1}{(d+e x) \sqrt [4]{-\frac{a c}{b^2-4 a c}-\frac{b c x}{b^2-4 a c}-\frac{c^2 x^2}{b^2-4 a c}}} \, dx}{4 \left (c d^2-b d e+a e^2\right )^2 \sqrt [4]{a+b x+c x^2}}\\ &=-\frac{4 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x) \sqrt [4]{a+b x+c x^2}}-\frac{e \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) \left (a+b x+c x^2\right )^{3/4}}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 (d+e x)}+\frac{\left (\sqrt{c} \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 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 )}{\sqrt{b^2-4 a c} \left (c d^2-b d e+a e^2\right )^2 (b+2 c x)}-\frac{\left (\sqrt{c} \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) \sqrt{(b+2 c x)^2}\right ) \operatorname{Subst}\left (\int \frac{1-\frac{2 \sqrt{c} x^2}{\sqrt{b^2-4 a c}}}{\sqrt{b^2-4 a c+4 c x^4}} \, dx,x,\sqrt [4]{a+b x+c x^2}\right )}{\sqrt{b^2-4 a c} \left (c d^2-b d e+a e^2\right )^2 (b+2 c x)}+\frac{\left (5 e^2 (2 c d-b e) \sqrt [4]{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (-\frac{c (2 c d-b e)}{b^2-4 a c}+e x\right ) \sqrt [4]{1-\frac{\left (b^2-4 a c\right ) x^2}{c^2}}} \, dx,x,-\frac{b c}{b^2-4 a c}-\frac{2 c^2 x}{b^2-4 a c}\right )}{2 \sqrt{2} \left (c d^2-b d e+a e^2\right )^2 \sqrt [4]{a+b x+c x^2}}\\ &=-\frac{4 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x) \sqrt [4]{a+b x+c x^2}}-\frac{e \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) \left (a+b x+c x^2\right )^{3/4}}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 (d+e x)}+\frac{\sqrt{c} \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) (b+2 c x) \sqrt [4]{a+b x+c x^2}}{\left (b^2-4 a c\right )^{3/2} \left (c d^2-b d e+a e^2\right )^2 \left (1+\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )}-\frac{\sqrt [4]{c} \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 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 ) E\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} \sqrt [4]{b^2-4 a c} \left (c d^2-b d e+a e^2\right )^2 (b+2 c x)}+\frac{\sqrt [4]{c} \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 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 )}{2 \sqrt{2} \sqrt [4]{b^2-4 a c} \left (c d^2-b d e+a e^2\right )^2 (b+2 c x)}-\frac{\left (5 e^3 (2 c d-b e) \sqrt [4]{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt [4]{1-\frac{\left (b^2-4 a c\right ) x^2}{c^2}} \left (\frac{c^2 (2 c d-b e)^2}{\left (b^2-4 a c\right )^2}-e^2 x^2\right )} \, dx,x,-\frac{b c}{b^2-4 a c}-\frac{2 c^2 x}{b^2-4 a c}\right )}{2 \sqrt{2} \left (c d^2-b d e+a e^2\right )^2 \sqrt [4]{a+b x+c x^2}}-\frac{\left (5 c e^2 (2 c d-b e)^2 \sqrt [4]{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt [4]{1-\frac{\left (b^2-4 a c\right ) x^2}{c^2}} \left (\frac{c^2 (2 c d-b e)^2}{\left (b^2-4 a c\right )^2}-e^2 x^2\right )} \, dx,x,-\frac{b c}{b^2-4 a c}-\frac{2 c^2 x}{b^2-4 a c}\right )}{2 \sqrt{2} \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 \sqrt [4]{a+b x+c x^2}}\\ &=-\frac{4 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x) \sqrt [4]{a+b x+c x^2}}-\frac{e \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) \left (a+b x+c x^2\right )^{3/4}}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 (d+e x)}+\frac{\sqrt{c} \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) (b+2 c x) \sqrt [4]{a+b x+c x^2}}{\left (b^2-4 a c\right )^{3/2} \left (c d^2-b d e+a e^2\right )^2 \left (1+\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )}-\frac{\sqrt [4]{c} \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 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 ) E\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} \sqrt [4]{b^2-4 a c} \left (c d^2-b d e+a e^2\right )^2 (b+2 c x)}+\frac{\sqrt [4]{c} \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 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 )}{2 \sqrt{2} \sqrt [4]{b^2-4 a c} \left (c d^2-b d e+a e^2\right )^2 (b+2 c x)}-\frac{\left (5 e^3 (2 c d-b e) \sqrt [4]{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt [4]{1-\frac{\left (b^2-4 a c\right ) x}{c^2}} \left (\frac{c^2 (2 c d-b e)^2}{\left (b^2-4 a c\right )^2}-e^2 x\right )} \, dx,x,\left (-\frac{b c}{b^2-4 a c}-\frac{2 c^2 x}{b^2-4 a c}\right )^2\right )}{4 \sqrt{2} \left (c d^2-b d e+a e^2\right )^2 \sqrt [4]{a+b x+c x^2}}-\frac{\left (5 c e^2 (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}} \sqrt [4]{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{1-x^4} \left (e^2-\frac{(2 c d-b e)^2}{b^2-4 a c}-e^2 x^4\right )} \, dx,x,\sqrt [4]{1-\frac{\left (b^2-4 a c\right ) \left (-\frac{b c}{b^2-4 a c}-\frac{2 c^2 x}{b^2-4 a c}\right )^2}{c^2}}\right )}{\sqrt{2} \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 \left (-\frac{b c}{b^2-4 a c}-\frac{2 c^2 x}{b^2-4 a c}\right ) \sqrt [4]{a+b x+c x^2}}\\ &=-\frac{4 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x) \sqrt [4]{a+b x+c x^2}}-\frac{e \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) \left (a+b x+c x^2\right )^{3/4}}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 (d+e x)}+\frac{\sqrt{c} \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) (b+2 c x) \sqrt [4]{a+b x+c x^2}}{\left (b^2-4 a c\right )^{3/2} \left (c d^2-b d e+a e^2\right )^2 \left (1+\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )}-\frac{\sqrt [4]{c} \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 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 ) E\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} \sqrt [4]{b^2-4 a c} \left (c d^2-b d e+a e^2\right )^2 (b+2 c x)}+\frac{\sqrt [4]{c} \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 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 )}{2 \sqrt{2} \sqrt [4]{b^2-4 a c} \left (c d^2-b d e+a e^2\right )^2 (b+2 c x)}+\frac{\left (5 c^2 e^3 (2 c d-b e) \sqrt [4]{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \operatorname{Subst}\left (\int \frac{x^2}{-\frac{c^2 e^2}{b^2-4 a c}+\frac{c^2 (2 c d-b e)^2}{\left (b^2-4 a c\right )^2}+\frac{c^2 e^2 x^4}{b^2-4 a c}} \, dx,x,\sqrt [4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}\right )}{\sqrt{2} \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 \sqrt [4]{a+b x+c x^2}}-\frac{\left (5 c \sqrt{-b^2+4 a c} e (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}} \sqrt [4]{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (2 \sqrt{c} \sqrt{c d^2-b d e+a e^2}-\sqrt{-b^2+4 a c} e x^2\right ) \sqrt{1-x^4}} \, dx,x,\sqrt [4]{1-\frac{\left (b^2-4 a c\right ) \left (-\frac{b c}{b^2-4 a c}-\frac{2 c^2 x}{b^2-4 a c}\right )^2}{c^2}}\right )}{2 \sqrt{2} \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 \left (-\frac{b c}{b^2-4 a c}-\frac{2 c^2 x}{b^2-4 a c}\right ) \sqrt [4]{a+b x+c x^2}}+\frac{\left (5 c \sqrt{-b^2+4 a c} e (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}} \sqrt [4]{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (2 \sqrt{c} \sqrt{c d^2-b d e+a e^2}+\sqrt{-b^2+4 a c} e x^2\right ) \sqrt{1-x^4}} \, dx,x,\sqrt [4]{1-\frac{\left (b^2-4 a c\right ) \left (-\frac{b c}{b^2-4 a c}-\frac{2 c^2 x}{b^2-4 a c}\right )^2}{c^2}}\right )}{2 \sqrt{2} \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 \left (-\frac{b c}{b^2-4 a c}-\frac{2 c^2 x}{b^2-4 a c}\right ) \sqrt [4]{a+b x+c x^2}}\\ &=-\frac{4 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x) \sqrt [4]{a+b x+c x^2}}-\frac{e \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) \left (a+b x+c x^2\right )^{3/4}}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 (d+e x)}+\frac{\sqrt{c} \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) (b+2 c x) \sqrt [4]{a+b x+c x^2}}{\left (b^2-4 a c\right )^{3/2} \left (c d^2-b d e+a e^2\right )^2 \left (1+\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )}-\frac{\sqrt [4]{c} \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 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 ) E\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} \sqrt [4]{b^2-4 a c} \left (c d^2-b d e+a e^2\right )^2 (b+2 c x)}+\frac{\sqrt [4]{c} \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 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 )}{2 \sqrt{2} \sqrt [4]{b^2-4 a c} \left (c d^2-b d e+a e^2\right )^2 (b+2 c x)}+\frac{\left (5 \left (-b^2+4 a c\right )^{3/2} e^2 (2 c d-b e) \sqrt [4]{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}}\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} \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 \sqrt [4]{a+b x+c x^2}}-\frac{\left (5 \left (-b^2+4 a c\right )^{3/2} e^2 (2 c d-b e) \sqrt [4]{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}}\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} \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 \sqrt [4]{a+b x+c x^2}}-\frac{\left (5 c \sqrt{-b^2+4 a c} e (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}} \sqrt [4]{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2} \sqrt{1+x^2} \left (2 \sqrt{c} \sqrt{c d^2-b d e+a e^2}-\sqrt{-b^2+4 a c} e x^2\right )} \, dx,x,\sqrt [4]{1-\frac{\left (b^2-4 a c\right ) \left (-\frac{b c}{b^2-4 a c}-\frac{2 c^2 x}{b^2-4 a c}\right )^2}{c^2}}\right )}{2 \sqrt{2} \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 \left (-\frac{b c}{b^2-4 a c}-\frac{2 c^2 x}{b^2-4 a c}\right ) \sqrt [4]{a+b x+c x^2}}+\frac{\left (5 c \sqrt{-b^2+4 a c} e (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}} \sqrt [4]{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2} \sqrt{1+x^2} \left (2 \sqrt{c} \sqrt{c d^2-b d e+a e^2}+\sqrt{-b^2+4 a c} e x^2\right )} \, dx,x,\sqrt [4]{1-\frac{\left (b^2-4 a c\right ) \left (-\frac{b c}{b^2-4 a c}-\frac{2 c^2 x}{b^2-4 a c}\right )^2}{c^2}}\right )}{2 \sqrt{2} \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 \left (-\frac{b c}{b^2-4 a c}-\frac{2 c^2 x}{b^2-4 a c}\right ) \sqrt [4]{a+b x+c x^2}}\\ &=-\frac{4 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x) \sqrt [4]{a+b x+c x^2}}-\frac{e \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) \left (a+b x+c x^2\right )^{3/4}}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 (d+e x)}+\frac{\sqrt{c} \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) (b+2 c x) \sqrt [4]{a+b x+c x^2}}{\left (b^2-4 a c\right )^{3/2} \left (c d^2-b d e+a e^2\right )^2 \left (1+\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )}+\frac{5 \sqrt [4]{-b^2+4 a c} e^{3/2} (2 c d-b e) \sqrt [4]{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \tan ^{-1}\left (\frac{\sqrt [4]{-b^2+4 a c} \sqrt{e} \sqrt [4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}}{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c d^2-b d e+a e^2}}\right )}{4 \sqrt [4]{c} \left (c d^2-b d e+a e^2\right )^{9/4} \sqrt [4]{a+b x+c x^2}}-\frac{5 \sqrt [4]{-b^2+4 a c} e^{3/2} (2 c d-b e) \sqrt [4]{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \tanh ^{-1}\left (\frac{\sqrt [4]{-b^2+4 a c} \sqrt{e} \sqrt [4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}}{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c d^2-b d e+a e^2}}\right )}{4 \sqrt [4]{c} \left (c d^2-b d e+a e^2\right )^{9/4} \sqrt [4]{a+b x+c x^2}}-\frac{\sqrt [4]{c} \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 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 ) E\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} \sqrt [4]{b^2-4 a c} \left (c d^2-b d e+a e^2\right )^2 (b+2 c x)}+\frac{\sqrt [4]{c} \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 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 )}{2 \sqrt{2} \sqrt [4]{b^2-4 a c} \left (c d^2-b d e+a e^2\right )^2 (b+2 c x)}+\frac{5 \left (b^2-4 a c\right ) e (2 c d-b e)^2 \sqrt{\frac{(b+2 c x)^2}{b^2-4 a c}} \sqrt [4]{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \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} \sqrt{c} \sqrt{-b^2+4 a c} \left (c d^2-b d e+a e^2\right )^{5/2} (b+2 c x) \sqrt [4]{a+b x+c x^2}}-\frac{5 \left (b^2-4 a c\right ) e (2 c d-b e)^2 \sqrt{\frac{(b+2 c x)^2}{b^2-4 a c}} \sqrt [4]{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \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} \sqrt{c} \sqrt{-b^2+4 a c} \left (c d^2-b d e+a e^2\right )^{5/2} (b+2 c x) \sqrt [4]{a+b x+c x^2}}\\ \end{align*}

Mathematica [C]  time = 0.937171, size = 187, normalized size = 0.13 \[ -\frac{\left (\frac{e \left (-\sqrt{b^2-4 a c}+b+2 c x\right )}{c (d+e x)}\right )^{5/4} \left (\frac{e \left (\sqrt{b^2-4 a c}+b+2 c x\right )}{c (d+e x)}\right )^{5/4} F_1\left (\frac{7}{2};\frac{5}{4},\frac{5}{4};\frac{9}{2};\frac{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}{2 c (d+e x)},\frac{2 c d-b e+\sqrt{b^2-4 a c} e}{2 c d+2 c e x}\right )}{14 \sqrt{2} e (d+e x) (a+x (b+c x))^{5/4}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/((d + e*x)^2*(a + b*x + c*x^2)^(5/4)),x]

[Out]

-(((e*(b - Sqrt[b^2 - 4*a*c] + 2*c*x))/(c*(d + e*x)))^(5/4)*((e*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/(c*(d + e*x))
)^(5/4)*AppellF1[7/2, 5/4, 5/4, 9/2, (2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)/(2*c*(d + e*x)), (2*c*d - b*e + Sqrt[
b^2 - 4*a*c]*e)/(2*c*d + 2*c*e*x)])/(14*Sqrt[2]*e*(d + e*x)*(a + x*(b + c*x))^(5/4))

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Maple [F]  time = 1.247, 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(1/(e*x+d)^2/(c*x^2+b*x+a)^(5/4),x)

[Out]

int(1/(e*x+d)^2/(c*x^2+b*x+a)^(5/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\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(1/(e*x+d)^2/(c*x^2+b*x+a)^(5/4),x, algorithm="maxima")

[Out]

integrate(1/((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(1/(e*x+d)^2/(c*x^2+b*x+a)^(5/4),x, algorithm="fricas")

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)**2/(c*x**2+b*x+a)**(5/4),x)

[Out]

Integral(1/((d + e*x)**2*(a + b*x + c*x**2)**(5/4)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\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(1/(e*x+d)^2/(c*x^2+b*x+a)^(5/4),x, algorithm="giac")

[Out]

integrate(1/((c*x^2 + b*x + a)^(5/4)*(e*x + d)^2), x)