\(\int \frac {1}{(d+e x)^2 (a+b x+c x^2)^{5/2}} \, dx\) [2398]

Optimal result

Integrand size = 22, antiderivative size = 473 \[ \int \frac {1}{(d+e x)^2 \left (a+b x+c x^2\right )^{5/2}} \, dx=-\frac {2 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x) \left (a+b x+c x^2\right )^{3/2}}-\frac {2 \left (6 a c e (2 c d-b e)^2-\left (b c d-b^2 e+2 a c e\right ) \left (8 c^2 d^2-5 b^2 e^2-2 c e (b d-8 a e)\right )-c (2 c d-b e) \left (8 c^2 d^2-5 b^2 e^2-4 c e (2 b d-7 a e)\right ) x\right )}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 (d+e x) \sqrt {a+b x+c x^2}}+\frac {e \left (32 c^4 d^4-15 b^4 e^4-16 c^3 d^2 e (4 b d-9 a e)+20 b^2 c e^3 (b d+5 a e)+4 c^2 e^2 \left (3 b^2 d^2-36 a b d e-32 a^2 e^2\right )\right ) \sqrt {a+b x+c x^2}}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^3 (d+e x)}+\frac {5 e^4 (2 c d-b e) \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{2 \left (c d^2-b d e+a e^2\right )^{7/2}} \]

[Out]

-2/3*(b*c*d-b^2*e+2*a*c*e+c*(-b*e+2*c*d)*x)/(-4*a*c+b^2)/(a*e^2-b*d*e+c*d^2)/(e*x+d)/(c*x^2+b*x+a)^(3/2)+5/2*e
^4*(-b*e+2*c*d)*arctanh(1/2*(b*d-2*a*e+(-b*e+2*c*d)*x)/(a*e^2-b*d*e+c*d^2)^(1/2)/(c*x^2+b*x+a)^(1/2))/(a*e^2-b
*d*e+c*d^2)^(7/2)-2/3*(6*a*c*e*(-b*e+2*c*d)^2-(2*a*c*e-b^2*e+b*c*d)*(8*c^2*d^2-5*b^2*e^2-2*c*e*(-8*a*e+b*d))-c
*(-b*e+2*c*d)*(8*c^2*d^2-5*b^2*e^2-4*c*e*(-7*a*e+2*b*d))*x)/(-4*a*c+b^2)^2/(a*e^2-b*d*e+c*d^2)^2/(e*x+d)/(c*x^
2+b*x+a)^(1/2)+1/3*e*(32*c^4*d^4-15*b^4*e^4-16*c^3*d^2*e*(-9*a*e+4*b*d)+20*b^2*c*e^3*(5*a*e+b*d)+4*c^2*e^2*(-3
2*a^2*e^2-36*a*b*d*e+3*b^2*d^2))*(c*x^2+b*x+a)^(1/2)/(-4*a*c+b^2)^2/(a*e^2-b*d*e+c*d^2)^3/(e*x+d)

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 473, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {754, 836, 820, 738, 212} \[ \int \frac {1}{(d+e x)^2 \left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {e \sqrt {a+b x+c x^2} \left (4 c^2 e^2 \left (-32 a^2 e^2-36 a b d e+3 b^2 d^2\right )+20 b^2 c e^3 (5 a e+b d)-16 c^3 d^2 e (4 b d-9 a e)-15 b^4 e^4+32 c^4 d^4\right )}{3 \left (b^2-4 a c\right )^2 (d+e x) \left (a e^2-b d e+c d^2\right )^3}+\frac {5 e^4 (2 c d-b e) \text {arctanh}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{2 \left (a e^2-b d e+c d^2\right )^{7/2}}-\frac {2 \left (-c x (2 c d-b e) \left (-4 c e (2 b d-7 a e)-5 b^2 e^2+8 c^2 d^2\right )-\left (2 a c e+b^2 (-e)+b c d\right ) \left (-2 c e (b d-8 a e)-5 b^2 e^2+8 c^2 d^2\right )+6 a c e (2 c d-b e)^2\right )}{3 \left (b^2-4 a c\right )^2 (d+e x) \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )^2}-\frac {2 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{3 \left (b^2-4 a c\right ) (d+e x) \left (a+b x+c x^2\right )^{3/2} \left (a e^2-b d e+c d^2\right )} \]

[In]

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

[Out]

(-2*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x))/(3*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*(d + e*x)*(a + b*x
+ c*x^2)^(3/2)) - (2*(6*a*c*e*(2*c*d - b*e)^2 - (b*c*d - b^2*e + 2*a*c*e)*(8*c^2*d^2 - 5*b^2*e^2 - 2*c*e*(b*d
- 8*a*e)) - c*(2*c*d - b*e)*(8*c^2*d^2 - 5*b^2*e^2 - 4*c*e*(2*b*d - 7*a*e))*x))/(3*(b^2 - 4*a*c)^2*(c*d^2 - b
*d*e + a*e^2)^2*(d + e*x)*Sqrt[a + b*x + c*x^2]) + (e*(32*c^4*d^4 - 15*b^4*e^4 - 16*c^3*d^2*e*(4*b*d - 9*a*e)
+ 20*b^2*c*e^3*(b*d + 5*a*e) + 4*c^2*e^2*(3*b^2*d^2 - 36*a*b*d*e - 32*a^2*e^2))*Sqrt[a + b*x + c*x^2])/(3*(b^2
- 4*a*c)^2*(c*d^2 - b*d*e + a*e^2)^3*(d + e*x)) + (5*e^4*(2*c*d - b*e)*ArcTanh[(b*d - 2*a*e + (2*c*d - b*e)*x
)/(2*Sqrt[c*d^2 - b*d*e + a*e^2]*Sqrt[a + b*x + c*x^2])])/(2*(c*d^2 - b*d*e + a*e^2)^(7/2))

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 738

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

Rule 754

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 820

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)/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2))), x] - Dist[
(b*(e*f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(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] && EqQ[S
implify[m + 2*p + 3], 0]

Rule 836

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[(d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2*a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*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*(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2
*(p + m + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d*m + b*e*m) - b*d*(3*c*d -
b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - f*(2*c*d - b*e))*(m + 2*p + 4)*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] && LtQ[p, -1] && (IntegerQ[m] ||
IntegerQ[p] || IntegersQ[2*m, 2*p])

Rubi steps \begin{align*} \text {integral}& = -\frac {2 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x) \left (a+b x+c x^2\right )^{3/2}}-\frac {2 \int \frac {\frac {1}{2} \left (8 c^2 d^2-5 b^2 e^2-2 c e (b d-8 a e)\right )+3 c e (2 c d-b e) x}{(d+e x)^2 \left (a+b x+c x^2\right )^{3/2}} \, dx}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )} \\ & = -\frac {2 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x) \left (a+b x+c x^2\right )^{3/2}}-\frac {2 \left (6 a c e (2 c d-b e)^2-\left (b c d-b^2 e+2 a c e\right ) \left (8 c^2 d^2-5 b^2 e^2-2 c e (b d-8 a e)\right )-c (2 c d-b e) \left (8 c^2 d^2-5 b^2 e^2-4 c e (2 b d-7 a e)\right ) x\right )}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 (d+e x) \sqrt {a+b x+c x^2}}+\frac {4 \int \frac {-\frac {1}{4} e \left (10 b^3 c d e^2-15 b^4 e^3+32 a c^2 e \left (c d^2-4 a e^2\right )-8 b c^2 d \left (2 c d^2+11 a e^2\right )+4 b^2 c e \left (4 c d^2+25 a e^2\right )\right )+\frac {1}{2} c e (2 c d-b e) \left (8 c^2 d^2-5 b^2 e^2-4 c e (2 b d-7 a e)\right ) x}{(d+e x)^2 \sqrt {a+b x+c x^2}} \, dx}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2} \\ & = -\frac {2 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x) \left (a+b x+c x^2\right )^{3/2}}-\frac {2 \left (6 a c e (2 c d-b e)^2-\left (b c d-b^2 e+2 a c e\right ) \left (8 c^2 d^2-5 b^2 e^2-2 c e (b d-8 a e)\right )-c (2 c d-b e) \left (8 c^2 d^2-5 b^2 e^2-4 c e (2 b d-7 a e)\right ) x\right )}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 (d+e x) \sqrt {a+b x+c x^2}}+\frac {e \left (32 c^4 d^4-15 b^4 e^4-16 c^3 d^2 e (4 b d-9 a e)+20 b^2 c e^3 (b d+5 a e)+4 c^2 e^2 \left (3 b^2 d^2-36 a b d e-32 a^2 e^2\right )\right ) \sqrt {a+b x+c x^2}}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^3 (d+e x)}+\frac {\left (5 e^4 (2 c d-b e)\right ) \int \frac {1}{(d+e x) \sqrt {a+b x+c x^2}} \, dx}{2 \left (c d^2-b d e+a e^2\right )^3} \\ & = -\frac {2 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x) \left (a+b x+c x^2\right )^{3/2}}-\frac {2 \left (6 a c e (2 c d-b e)^2-\left (b c d-b^2 e+2 a c e\right ) \left (8 c^2 d^2-5 b^2 e^2-2 c e (b d-8 a e)\right )-c (2 c d-b e) \left (8 c^2 d^2-5 b^2 e^2-4 c e (2 b d-7 a e)\right ) x\right )}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 (d+e x) \sqrt {a+b x+c x^2}}+\frac {e \left (32 c^4 d^4-15 b^4 e^4-16 c^3 d^2 e (4 b d-9 a e)+20 b^2 c e^3 (b d+5 a e)+4 c^2 e^2 \left (3 b^2 d^2-36 a b d e-32 a^2 e^2\right )\right ) \sqrt {a+b x+c x^2}}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^3 (d+e x)}-\frac {\left (5 e^4 (2 c d-b e)\right ) \text {Subst}\left (\int \frac {1}{4 c d^2-4 b d e+4 a e^2-x^2} \, dx,x,\frac {-b d+2 a e-(2 c d-b e) x}{\sqrt {a+b x+c x^2}}\right )}{\left (c d^2-b d e+a e^2\right )^3} \\ & = -\frac {2 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x) \left (a+b x+c x^2\right )^{3/2}}-\frac {2 \left (6 a c e (2 c d-b e)^2-\left (b c d-b^2 e+2 a c e\right ) \left (8 c^2 d^2-5 b^2 e^2-2 c e (b d-8 a e)\right )-c (2 c d-b e) \left (8 c^2 d^2-5 b^2 e^2-4 c e (2 b d-7 a e)\right ) x\right )}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 (d+e x) \sqrt {a+b x+c x^2}}+\frac {e \left (32 c^4 d^4-15 b^4 e^4-16 c^3 d^2 e (4 b d-9 a e)+20 b^2 c e^3 (b d+5 a e)+4 c^2 e^2 \left (3 b^2 d^2-36 a b d e-32 a^2 e^2\right )\right ) \sqrt {a+b x+c x^2}}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^3 (d+e x)}+\frac {5 e^4 (2 c d-b e) \tanh ^{-1}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{2 \left (c d^2-b d e+a e^2\right )^{7/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 11.10 (sec) , antiderivative size = 482, normalized size of antiderivative = 1.02 \[ \int \frac {1}{(d+e x)^2 \left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {2 \left (\frac {e \left (32 c^4 d^4-15 b^4 e^4-16 c^3 d^2 e (4 b d-9 a e)+20 b^2 c e^3 (b d+5 a e)-4 c^2 e^2 \left (-3 b^2 d^2+36 a b d e+32 a^2 e^2\right )\right ) \sqrt {a+x (b+c x)}}{2 \left (b^2-4 a c\right ) \left (c d^2+e (-b d+a e)\right )^2 (d+e x)}+\frac {b^2 e-2 c (a e+c d x)+b c (-d+e x)}{(d+e x) (a+x (b+c x))^{3/2}}+\frac {-5 b^4 e^3+b^3 c e^2 (3 d-5 e x)-8 c^2 \left (4 a^2 e^3+2 c^2 d^3 x-a c d e (d-7 e x)\right )-4 b c^2 \left (a e^2 (9 d-7 e x)+2 c d^2 (d-3 e x)\right )+2 b^2 c e \left (16 a e^2+c d (5 d+e x)\right )}{\left (b^2-4 a c\right ) \left (-c d^2+e (b d-a e)\right ) (d+e x) \sqrt {a+x (b+c x)}}+\frac {15 \left (b^2-4 a c\right ) e^4 (-2 c d+b e) \text {arctanh}\left (\frac {-b d+2 a e-2 c d x+b e x}{2 \sqrt {c d^2+e (-b d+a e)} \sqrt {a+x (b+c x)}}\right )}{4 \left (c d^2+e (-b d+a e)\right )^{5/2}}\right )}{3 \left (b^2-4 a c\right ) \left (c d^2+e (-b d+a e)\right )} \]

[In]

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

[Out]

(2*((e*(32*c^4*d^4 - 15*b^4*e^4 - 16*c^3*d^2*e*(4*b*d - 9*a*e) + 20*b^2*c*e^3*(b*d + 5*a*e) - 4*c^2*e^2*(-3*b^
2*d^2 + 36*a*b*d*e + 32*a^2*e^2))*Sqrt[a + x*(b + c*x)])/(2*(b^2 - 4*a*c)*(c*d^2 + e*(-(b*d) + a*e))^2*(d + e*
x)) + (b^2*e - 2*c*(a*e + c*d*x) + b*c*(-d + e*x))/((d + e*x)*(a + x*(b + c*x))^(3/2)) + (-5*b^4*e^3 + b^3*c*e
^2*(3*d - 5*e*x) - 8*c^2*(4*a^2*e^3 + 2*c^2*d^3*x - a*c*d*e*(d - 7*e*x)) - 4*b*c^2*(a*e^2*(9*d - 7*e*x) + 2*c*
d^2*(d - 3*e*x)) + 2*b^2*c*e*(16*a*e^2 + c*d*(5*d + e*x)))/((b^2 - 4*a*c)*(-(c*d^2) + e*(b*d - a*e))*(d + e*x)
*Sqrt[a + x*(b + c*x)]) + (15*(b^2 - 4*a*c)*e^4*(-2*c*d + b*e)*ArcTanh[(-(b*d) + 2*a*e - 2*c*d*x + b*e*x)/(2*S
qrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)])])/(4*(c*d^2 + e*(-(b*d) + a*e))^(5/2))))/(3*(b^2 - 4*a*c)
*(c*d^2 + e*(-(b*d) + a*e)))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1142\) vs. \(2(451)=902\).

Time = 0.39 (sec) , antiderivative size = 1143, normalized size of antiderivative = 2.42

method	result	size
default	\(\text {Expression too large to display}\)	\(1143\)

[In]

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

[Out]

1/e^2*(-1/(a*e^2-b*d*e+c*d^2)*e^2/(x+d/e)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)-5/
2*(b*e-2*c*d)*e/(a*e^2-b*d*e+c*d^2)*(1/3/(a*e^2-b*d*e+c*d^2)*e^2/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d
*e+c*d^2)/e^2)^(3/2)-1/2*(b*e-2*c*d)*e/(a*e^2-b*d*e+c*d^2)*(2/3*(2*c*(x+d/e)+(b*e-2*c*d)/e)/(4*c*(a*e^2-b*d*e+
c*d^2)/e^2-(b*e-2*c*d)^2/e^2)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)+16/3*c/(4*c*(a
*e^2-b*d*e+c*d^2)/e^2-(b*e-2*c*d)^2/e^2)^2*(2*c*(x+d/e)+(b*e-2*c*d)/e)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e
^2-b*d*e+c*d^2)/e^2)^(1/2))+1/(a*e^2-b*d*e+c*d^2)*e^2*(1/(a*e^2-b*d*e+c*d^2)*e^2/((x+d/e)^2*c+(b*e-2*c*d)/e*(x
+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)-(b*e-2*c*d)*e/(a*e^2-b*d*e+c*d^2)*(2*c*(x+d/e)+(b*e-2*c*d)/e)/(4*c*(a*e^2
-b*d*e+c*d^2)/e^2-(b*e-2*c*d)^2/e^2)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)-1/(a*e^
2-b*d*e+c*d^2)*e^2/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(x+d/e)+2*((a*e
^2-b*d*e+c*d^2)/e^2)^(1/2)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(x+d/e))))-4*c/(
a*e^2-b*d*e+c*d^2)*e^2*(2/3*(2*c*(x+d/e)+(b*e-2*c*d)/e)/(4*c*(a*e^2-b*d*e+c*d^2)/e^2-(b*e-2*c*d)^2/e^2)/((x+d/
e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)+16/3*c/(4*c*(a*e^2-b*d*e+c*d^2)/e^2-(b*e-2*c*d)^2/
e^2)^2*(2*c*(x+d/e)+(b*e-2*c*d)/e)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4416 vs. \(2 (451) = 902\).

Time = 7.45 (sec) , antiderivative size = 8874, normalized size of antiderivative = 18.76 \[ \int \frac {1}{(d+e x)^2 \left (a+b x+c x^2\right )^{5/2}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Too large to include

Sympy [F]

\[ \int \frac {1}{(d+e x)^2 \left (a+b x+c x^2\right )^{5/2}} \, dx=\int \frac {1}{\left (d + e x\right )^{2} \left (a + b x + c x^{2}\right )^{\frac {5}{2}}}\, dx \]

[In]

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

[Out]

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{(d+e x)^2 \left (a+b x+c x^2\right )^{5/2}} \, dx=\text {Exception raised: ValueError} \]

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(a*e^2-b*d*e>0)', see `assume?`
for more de

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 5572 vs. \(2 (451) = 902\).

Time = 0.84 (sec) , antiderivative size = 5572, normalized size of antiderivative = 11.78 \[ \int \frac {1}{(d+e x)^2 \left (a+b x+c x^2\right )^{5/2}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/6*((30*b^4*c^(3/2)*d*e^7*log(abs(2*c*d*e - b*e^2 - 2*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(c)*abs(e))) - 240*a*b^
2*c^(5/2)*d*e^7*log(abs(2*c*d*e - b*e^2 - 2*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(c)*abs(e))) + 480*a^2*c^(7/2)*d*e
^7*log(abs(2*c*d*e - b*e^2 - 2*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(c)*abs(e))) - 15*b^5*sqrt(c)*e^8*log(abs(2*c*d
*e - b*e^2 - 2*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(c)*abs(e))) + 120*a*b^3*c^(3/2)*e^8*log(abs(2*c*d*e - b*e^2 -
2*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(c)*abs(e))) - 240*a^2*b*c^(5/2)*e^8*log(abs(2*c*d*e - b*e^2 - 2*sqrt(c*d^2
- b*d*e + a*e^2)*sqrt(c)*abs(e))) - 64*sqrt(c*d^2 - b*d*e + a*e^2)*c^5*d^4*e^2*abs(e) + 128*sqrt(c*d^2 - b*d*e
+ a*e^2)*b*c^4*d^3*e^3*abs(e) - 24*sqrt(c*d^2 - b*d*e + a*e^2)*b^2*c^3*d^2*e^4*abs(e) - 288*sqrt(c*d^2 - b*d*
e + a*e^2)*a*c^4*d^2*e^4*abs(e) - 40*sqrt(c*d^2 - b*d*e + a*e^2)*b^3*c^2*d*e^5*abs(e) + 288*sqrt(c*d^2 - b*d*e
+ a*e^2)*a*b*c^3*d*e^5*abs(e) + 30*sqrt(c*d^2 - b*d*e + a*e^2)*b^4*c*e^6*abs(e) - 200*sqrt(c*d^2 - b*d*e + a*
e^2)*a*b^2*c^2*e^6*abs(e) + 256*sqrt(c*d^2 - b*d*e + a*e^2)*a^2*c^3*e^6*abs(e))*sgn(1/(e*x + d))*sgn(e)/(sqrt(
c*d^2 - b*d*e + a*e^2)*b^4*c^(7/2)*d^6*abs(e) - 8*sqrt(c*d^2 - b*d*e + a*e^2)*a*b^2*c^(9/2)*d^6*abs(e) + 16*sq
rt(c*d^2 - b*d*e + a*e^2)*a^2*c^(11/2)*d^6*abs(e) - 3*sqrt(c*d^2 - b*d*e + a*e^2)*b^5*c^(5/2)*d^5*e*abs(e) + 2
4*sqrt(c*d^2 - b*d*e + a*e^2)*a*b^3*c^(7/2)*d^5*e*abs(e) - 48*sqrt(c*d^2 - b*d*e + a*e^2)*a^2*b*c^(9/2)*d^5*e*
abs(e) + 3*sqrt(c*d^2 - b*d*e + a*e^2)*b^6*c^(3/2)*d^4*e^2*abs(e) - 21*sqrt(c*d^2 - b*d*e + a*e^2)*a*b^4*c^(5/
2)*d^4*e^2*abs(e) + 24*sqrt(c*d^2 - b*d*e + a*e^2)*a^2*b^2*c^(7/2)*d^4*e^2*abs(e) + 48*sqrt(c*d^2 - b*d*e + a*
e^2)*a^3*c^(9/2)*d^4*e^2*abs(e) - sqrt(c*d^2 - b*d*e + a*e^2)*b^7*sqrt(c)*d^3*e^3*abs(e) + 2*sqrt(c*d^2 - b*d*
e + a*e^2)*a*b^5*c^(3/2)*d^3*e^3*abs(e) + 32*sqrt(c*d^2 - b*d*e + a*e^2)*a^2*b^3*c^(5/2)*d^3*e^3*abs(e) - 96*s
qrt(c*d^2 - b*d*e + a*e^2)*a^3*b*c^(7/2)*d^3*e^3*abs(e) + 3*sqrt(c*d^2 - b*d*e + a*e^2)*a*b^6*sqrt(c)*d^2*e^4*
abs(e) - 21*sqrt(c*d^2 - b*d*e + a*e^2)*a^2*b^4*c^(3/2)*d^2*e^4*abs(e) + 24*sqrt(c*d^2 - b*d*e + a*e^2)*a^3*b^
2*c^(5/2)*d^2*e^4*abs(e) + 48*sqrt(c*d^2 - b*d*e + a*e^2)*a^4*c^(7/2)*d^2*e^4*abs(e) - 3*sqrt(c*d^2 - b*d*e +
a*e^2)*a^2*b^5*sqrt(c)*d*e^5*abs(e) + 24*sqrt(c*d^2 - b*d*e + a*e^2)*a^3*b^3*c^(3/2)*d*e^5*abs(e) - 48*sqrt(c*
d^2 - b*d*e + a*e^2)*a^4*b*c^(5/2)*d*e^5*abs(e) + sqrt(c*d^2 - b*d*e + a*e^2)*a^3*b^4*sqrt(c)*e^6*abs(e) - 8*s
qrt(c*d^2 - b*d*e + a*e^2)*a^4*b^2*c^(3/2)*e^6*abs(e) + 16*sqrt(c*d^2 - b*d*e + a*e^2)*a^5*c^(5/2)*e^6*abs(e))
+ 2*((32*c^6*d^4*e^13 - 64*b*c^5*d^3*e^14 + 12*b^2*c^4*d^2*e^15 + 144*a*c^5*d^2*e^15 + 20*b^3*c^3*d*e^16 - 14
4*a*b*c^4*d*e^16 - 15*b^4*c^2*e^17 + 100*a*b^2*c^3*e^17 - 128*a^2*c^4*e^17)/(b^4*c^3*d^6*e^11*sgn(1/(e*x + d))
*sgn(e) - 8*a*b^2*c^4*d^6*e^11*sgn(1/(e*x + d))*sgn(e) + 16*a^2*c^5*d^6*e^11*sgn(1/(e*x + d))*sgn(e) - 3*b^5*c
^2*d^5*e^12*sgn(1/(e*x + d))*sgn(e) + 24*a*b^3*c^3*d^5*e^12*sgn(1/(e*x + d))*sgn(e) - 48*a^2*b*c^4*d^5*e^12*sg
n(1/(e*x + d))*sgn(e) + 3*b^6*c*d^4*e^13*sgn(1/(e*x + d))*sgn(e) - 21*a*b^4*c^2*d^4*e^13*sgn(1/(e*x + d))*sgn(
e) + 24*a^2*b^2*c^3*d^4*e^13*sgn(1/(e*x + d))*sgn(e) + 48*a^3*c^4*d^4*e^13*sgn(1/(e*x + d))*sgn(e) - b^7*d^3*e
^14*sgn(1/(e*x + d))*sgn(e) + 2*a*b^5*c*d^3*e^14*sgn(1/(e*x + d))*sgn(e) + 32*a^2*b^3*c^2*d^3*e^14*sgn(1/(e*x
+ d))*sgn(e) - 96*a^3*b*c^3*d^3*e^14*sgn(1/(e*x + d))*sgn(e) + 3*a*b^6*d^2*e^15*sgn(1/(e*x + d))*sgn(e) - 21*a
^2*b^4*c*d^2*e^15*sgn(1/(e*x + d))*sgn(e) + 24*a^3*b^2*c^2*d^2*e^15*sgn(1/(e*x + d))*sgn(e) + 48*a^4*c^3*d^2*e
^15*sgn(1/(e*x + d))*sgn(e) - 3*a^2*b^5*d*e^16*sgn(1/(e*x + d))*sgn(e) + 24*a^3*b^3*c*d*e^16*sgn(1/(e*x + d))*
sgn(e) - 48*a^4*b*c^2*d*e^16*sgn(1/(e*x + d))*sgn(e) + a^3*b^4*e^17*sgn(1/(e*x + d))*sgn(e) - 8*a^4*b^2*c*e^17
*sgn(1/(e*x + d))*sgn(e) + 16*a^5*c^2*e^17*sgn(1/(e*x + d))*sgn(e)) - (6*(16*c^6*d^5*e^14 - 40*b*c^5*d^4*e^15
+ 22*b^2*c^4*d^3*e^16 + 72*a*c^5*d^3*e^16 + 7*b^3*c^3*d^2*e^17 - 108*a*b*c^4*d^2*e^17 - 15*b^4*c^2*d*e^18 + 10
6*a*b^2*c^3*d*e^18 - 104*a^2*c^4*d*e^18 + 5*b^5*c*e^19 - 35*a*b^3*c^2*e^19 + 52*a^2*b*c^3*e^19)/(b^4*c^3*d^6*e
^11*sgn(1/(e*x + d))*sgn(e) - 8*a*b^2*c^4*d^6*e^11*sgn(1/(e*x + d))*sgn(e) + 16*a^2*c^5*d^6*e^11*sgn(1/(e*x +
d))*sgn(e) - 3*b^5*c^2*d^5*e^12*sgn(1/(e*x + d))*sgn(e) + 24*a*b^3*c^3*d^5*e^12*sgn(1/(e*x + d))*sgn(e) - 48*a
^2*b*c^4*d^5*e^12*sgn(1/(e*x + d))*sgn(e) + 3*b^6*c*d^4*e^13*sgn(1/(e*x + d))*sgn(e) - 21*a*b^4*c^2*d^4*e^13*s
gn(1/(e*x + d))*sgn(e) + 24*a^2*b^2*c^3*d^4*e^13*sgn(1/(e*x + d))*sgn(e) + 48*a^3*c^4*d^4*e^13*sgn(1/(e*x + d)
)*sgn(e) - b^7*d^3*e^14*sgn(1/(e*x + d))*sgn(e) + 2*a*b^5*c*d^3*e^14*sgn(1/(e*x + d))*sgn(e) + 32*a^2*b^3*c^2*
d^3*e^14*sgn(1/(e*x + d))*sgn(e) - 96*a^3*b*c^3*d^3*e^14*sgn(1/(e*x + d))*sgn(e) + 3*a*b^6*d^2*e^15*sgn(1/(e*x
+ d))*sgn(e) - 21*a^2*b^4*c*d^2*e^15*sgn(1/(e*x + d))*sgn(e) + 24*a^3*b^2*c^2*d^2*e^15*sgn(1/(e*x + d))*sgn(e
) + 48*a^4*c^3*d^2*e^15*sgn(1/(e*x + d))*sgn(e) - 3*a^2*b^5*d*e^16*sgn(1/(e*x + d))*sgn(e) + 24*a^3*b^3*c*d*e^
16*sgn(1/(e*x + d))*sgn(e) - 48*a^4*b*c^2*d*e^16*sgn(1/(e*x + d))*sgn(e) + a^3*b^4*e^17*sgn(1/(e*x + d))*sgn(e
) - 8*a^4*b^2*c*e^17*sgn(1/(e*x + d))*sgn(e) + 16*a^5*c^2*e^17*sgn(1/(e*x + d))*sgn(e)) - (3*(32*c^6*d^6*e^15
- 96*b*c^5*d^5*e^16 + 80*b^2*c^4*d^4*e^17 + 160*a*c^5*d^4*e^17 - 320*a*b*c^4*d^3*e^18 - 46*b^4*c^2*d^2*e^19 +
368*a*b^2*c^3*d^2*e^19 - 256*a^2*c^4*d^2*e^19 + 30*b^5*c*d*e^20 - 208*a*b^3*c^2*d*e^20 + 256*a^2*b*c^3*d*e^20
- 5*b^6*e^21 + 30*a*b^4*c*e^21 - 16*a^2*b^2*c^2*e^21 - 64*a^3*c^3*e^21)/(b^4*c^3*d^6*e^11*sgn(1/(e*x + d))*sgn
(e) - 8*a*b^2*c^4*d^6*e^11*sgn(1/(e*x + d))*sgn(e) + 16*a^2*c^5*d^6*e^11*sgn(1/(e*x + d))*sgn(e) - 3*b^5*c^2*d
^5*e^12*sgn(1/(e*x + d))*sgn(e) + 24*a*b^3*c^3*d^5*e^12*sgn(1/(e*x + d))*sgn(e) - 48*a^2*b*c^4*d^5*e^12*sgn(1/
(e*x + d))*sgn(e) + 3*b^6*c*d^4*e^13*sgn(1/(e*x + d))*sgn(e) - 21*a*b^4*c^2*d^4*e^13*sgn(1/(e*x + d))*sgn(e) +
24*a^2*b^2*c^3*d^4*e^13*sgn(1/(e*x + d))*sgn(e) + 48*a^3*c^4*d^4*e^13*sgn(1/(e*x + d))*sgn(e) - b^7*d^3*e^14*
sgn(1/(e*x + d))*sgn(e) + 2*a*b^5*c*d^3*e^14*sgn(1/(e*x + d))*sgn(e) + 32*a^2*b^3*c^2*d^3*e^14*sgn(1/(e*x + d)
)*sgn(e) - 96*a^3*b*c^3*d^3*e^14*sgn(1/(e*x + d))*sgn(e) + 3*a*b^6*d^2*e^15*sgn(1/(e*x + d))*sgn(e) - 21*a^2*b
^4*c*d^2*e^15*sgn(1/(e*x + d))*sgn(e) + 24*a^3*b^2*c^2*d^2*e^15*sgn(1/(e*x + d))*sgn(e) + 48*a^4*c^3*d^2*e^15*
sgn(1/(e*x + d))*sgn(e) - 3*a^2*b^5*d*e^16*sgn(1/(e*x + d))*sgn(e) + 24*a^3*b^3*c*d*e^16*sgn(1/(e*x + d))*sgn(
e) - 48*a^4*b*c^2*d*e^16*sgn(1/(e*x + d))*sgn(e) + a^3*b^4*e^17*sgn(1/(e*x + d))*sgn(e) - 8*a^4*b^2*c*e^17*sgn
(1/(e*x + d))*sgn(e) + 16*a^5*c^2*e^17*sgn(1/(e*x + d))*sgn(e)) - (4*(8*c^6*d^7*e^16 - 28*b*c^5*d^6*e^17 + 30*
b^2*c^4*d^5*e^18 + 48*a*c^5*d^5*e^18 - 5*b^3*c^3*d^4*e^19 - 120*a*b*c^4*d^4*e^19 - 18*b^4*c^2*d^3*e^20 + 164*a
*b^2*c^3*d^3*e^20 - 88*a^2*c^4*d^3*e^20 + 18*b^5*c*d^2*e^21 - 126*a*b^3*c^2*d^2*e^21 + 132*a^2*b*c^3*d^2*e^21
- 5*b^6*d*e^22 + 24*a*b^4*c*d*e^22 + 30*a^2*b^2*c^2*d*e^22 - 128*a^3*c^3*d*e^22 + 5*a*b^5*e^23 - 37*a^2*b^3*c*
e^23 + 64*a^3*b*c^2*e^23)/(b^4*c^3*d^6*e^11*sgn(1/(e*x + d))*sgn(e) - 8*a*b^2*c^4*d^6*e^11*sgn(1/(e*x + d))*sg
n(e) + 16*a^2*c^5*d^6*e^11*sgn(1/(e*x + d))*sgn(e) - 3*b^5*c^2*d^5*e^12*sgn(1/(e*x + d))*sgn(e) + 24*a*b^3*c^3
*d^5*e^12*sgn(1/(e*x + d))*sgn(e) - 48*a^2*b*c^4*d^5*e^12*sgn(1/(e*x + d))*sgn(e) + 3*b^6*c*d^4*e^13*sgn(1/(e*
x + d))*sgn(e) - 21*a*b^4*c^2*d^4*e^13*sgn(1/(e*x + d))*sgn(e) + 24*a^2*b^2*c^3*d^4*e^13*sgn(1/(e*x + d))*sgn(
e) + 48*a^3*c^4*d^4*e^13*sgn(1/(e*x + d))*sgn(e) - b^7*d^3*e^14*sgn(1/(e*x + d))*sgn(e) + 2*a*b^5*c*d^3*e^14*s
gn(1/(e*x + d))*sgn(e) + 32*a^2*b^3*c^2*d^3*e^14*sgn(1/(e*x + d))*sgn(e) - 96*a^3*b*c^3*d^3*e^14*sgn(1/(e*x +
d))*sgn(e) + 3*a*b^6*d^2*e^15*sgn(1/(e*x + d))*sgn(e) - 21*a^2*b^4*c*d^2*e^15*sgn(1/(e*x + d))*sgn(e) + 24*a^3
*b^2*c^2*d^2*e^15*sgn(1/(e*x + d))*sgn(e) + 48*a^4*c^3*d^2*e^15*sgn(1/(e*x + d))*sgn(e) - 3*a^2*b^5*d*e^16*sgn
(1/(e*x + d))*sgn(e) + 24*a^3*b^3*c*d*e^16*sgn(1/(e*x + d))*sgn(e) - 48*a^4*b*c^2*d*e^16*sgn(1/(e*x + d))*sgn(
e) + a^3*b^4*e^17*sgn(1/(e*x + d))*sgn(e) - 8*a^4*b^2*c*e^17*sgn(1/(e*x + d))*sgn(e) + 16*a^5*c^2*e^17*sgn(1/(
e*x + d))*sgn(e)) + 3*(b^4*c^2*d^4*e^21 - 8*a*b^2*c^3*d^4*e^21 + 16*a^2*c^4*d^4*e^21 - 2*b^5*c*d^3*e^22 + 16*a
*b^3*c^2*d^3*e^22 - 32*a^2*b*c^3*d^3*e^22 + b^6*d^2*e^23 - 6*a*b^4*c*d^2*e^23 + 32*a^3*c^3*d^2*e^23 - 2*a*b^5*
d*e^24 + 16*a^2*b^3*c*d*e^24 - 32*a^3*b*c^2*d*e^24 + a^2*b^4*e^25 - 8*a^3*b^2*c*e^25 + 16*a^4*c^2*e^25)/((b^4*
c^3*d^6*e^11*sgn(1/(e*x + d))*sgn(e) - 8*a*b^2*c^4*d^6*e^11*sgn(1/(e*x + d))*sgn(e) + 16*a^2*c^5*d^6*e^11*sgn(
1/(e*x + d))*sgn(e) - 3*b^5*c^2*d^5*e^12*sgn(1/(e*x + d))*sgn(e) + 24*a*b^3*c^3*d^5*e^12*sgn(1/(e*x + d))*sgn(
e) - 48*a^2*b*c^4*d^5*e^12*sgn(1/(e*x + d))*sgn(e) + 3*b^6*c*d^4*e^13*sgn(1/(e*x + d))*sgn(e) - 21*a*b^4*c^2*d
^4*e^13*sgn(1/(e*x + d))*sgn(e) + 24*a^2*b^2*c^3*d^4*e^13*sgn(1/(e*x + d))*sgn(e) + 48*a^3*c^4*d^4*e^13*sgn(1/
(e*x + d))*sgn(e) - b^7*d^3*e^14*sgn(1/(e*x + d))*sgn(e) + 2*a*b^5*c*d^3*e^14*sgn(1/(e*x + d))*sgn(e) + 32*a^2
*b^3*c^2*d^3*e^14*sgn(1/(e*x + d))*sgn(e) - 96*a^3*b*c^3*d^3*e^14*sgn(1/(e*x + d))*sgn(e) + 3*a*b^6*d^2*e^15*s
gn(1/(e*x + d))*sgn(e) - 21*a^2*b^4*c*d^2*e^15*sgn(1/(e*x + d))*sgn(e) + 24*a^3*b^2*c^2*d^2*e^15*sgn(1/(e*x +
d))*sgn(e) + 48*a^4*c^3*d^2*e^15*sgn(1/(e*x + d))*sgn(e) - 3*a^2*b^5*d*e^16*sgn(1/(e*x + d))*sgn(e) + 24*a^3*b
^3*c*d*e^16*sgn(1/(e*x + d))*sgn(e) - 48*a^4*b*c^2*d*e^16*sgn(1/(e*x + d))*sgn(e) + a^3*b^4*e^17*sgn(1/(e*x +
d))*sgn(e) - 8*a^4*b^2*c*e^17*sgn(1/(e*x + d))*sgn(e) + 16*a^5*c^2*e^17*sgn(1/(e*x + d))*sgn(e))*(e*x + d)*e))
/((e*x + d)*e))/((e*x + d)*e))/((e*x + d)*e))/(c - 2*c*d/(e*x + d) + c*d^2/(e*x + d)^2 + b*e/(e*x + d) - b*d*e
/(e*x + d)^2 + a*e^2/(e*x + d)^2)^(3/2) - 15*(2*c*d*e^7 - b*e^8)*log(abs(2*c*d*e - b*e^2 - 2*sqrt(c*d^2 - b*d*
e + a*e^2)*(sqrt(c - 2*c*d/(e*x + d) + c*d^2/(e*x + d)^2 + b*e/(e*x + d) - b*d*e/(e*x + d)^2 + a*e^2/(e*x + d)
^2) + sqrt(c*d^2*e^2 - b*d*e^3 + a*e^4)/((e*x + d)*e))*abs(e)))/((c^3*d^6 - 3*b*c^2*d^5*e + 3*b^2*c*d^4*e^2 +
3*a*c^2*d^4*e^2 - b^3*d^3*e^3 - 6*a*b*c*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a^2*c*d^2*e^4 - 3*a^2*b*d*e^5 + a^3*e^6)
*sqrt(c*d^2 - b*d*e + a*e^2)*abs(e)*sgn(1/(e*x + d))*sgn(e)))/e^2

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(d+e x)^2 \left (a+b x+c x^2\right )^{5/2}} \, dx=\int \frac {1}{{\left (d+e\,x\right )}^2\,{\left (c\,x^2+b\,x+a\right )}^{5/2}} \,d x \]

[In]

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

[Out]

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