\(\int \frac {1}{(d+e x)^4 (a+b x+c x^2)^{3/2}} \, dx\) [2390]

Optimal result

Integrand size = 22, antiderivative size = 519 \[ \int \frac {1}{(d+e x)^4 \left (a+b x+c x^2\right )^{3/2}} \, dx=-\frac {2 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x)^3 \sqrt {a+b x+c x^2}}-\frac {e \left (12 c^2 d^2+7 b^2 e^2-4 c e (3 b d+4 a e)\right ) \sqrt {a+b x+c x^2}}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^3}-\frac {e (2 c d-b e) \left (24 c^2 d^2+35 b^2 e^2-4 c e (6 b d+29 a e)\right ) \sqrt {a+b x+c x^2}}{12 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^3 (d+e x)^2}-\frac {e \left (96 c^4 d^4+105 b^4 e^4-20 b^2 c e^3 (19 b d+23 a e)-16 c^3 d^2 e (12 b d+83 a e)+4 c^2 e^2 \left (119 b^2 d^2+332 a b d e+64 a^2 e^2\right )\right ) \sqrt {a+b x+c x^2}}{24 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^4 (d+e x)}+\frac {5 e^2 (2 c d-b e) \left (16 c^2 d^2+7 b^2 e^2-4 c e (4 b d+3 a e)\right ) \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{16 \left (c d^2-b d e+a e^2\right )^{9/2}} \]

[Out]

5/16*e^2*(-b*e+2*c*d)*(16*c^2*d^2+7*b^2*e^2-4*c*e*(3*a*e+4*b*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)^(9/2)-2*(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)^3/(c*x^2+b*x+a)^(1/2)-1/3*e*(12*c^2*d^2+7*b^2*e^2-4*c*e*(4*a*e+3*b*d))*
(c*x^2+b*x+a)^(1/2)/(-4*a*c+b^2)/(a*e^2-b*d*e+c*d^2)^2/(e*x+d)^3-1/12*e*(-b*e+2*c*d)*(24*c^2*d^2+35*b^2*e^2-4*
c*e*(29*a*e+6*b*d))*(c*x^2+b*x+a)^(1/2)/(-4*a*c+b^2)/(a*e^2-b*d*e+c*d^2)^3/(e*x+d)^2-1/24*e*(96*c^4*d^4+105*b^
4*e^4-20*b^2*c*e^3*(23*a*e+19*b*d)-16*c^3*d^2*e*(83*a*e+12*b*d)+4*c^2*e^2*(64*a^2*e^2+332*a*b*d*e+119*b^2*d^2)
)*(c*x^2+b*x+a)^(1/2)/(-4*a*c+b^2)/(a*e^2-b*d*e+c*d^2)^4/(e*x+d)

Rubi [A] (verified)

Time = 0.56 (sec) , antiderivative size = 519, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {754, 848, 820, 738, 212} \[ \int \frac {1}{(d+e x)^4 \left (a+b x+c x^2\right )^{3/2}} \, dx=-\frac {e \sqrt {a+b x+c x^2} \left (4 c^2 e^2 \left (64 a^2 e^2+332 a b d e+119 b^2 d^2\right )-20 b^2 c e^3 (23 a e+19 b d)-16 c^3 d^2 e (83 a e+12 b d)+105 b^4 e^4+96 c^4 d^4\right )}{24 \left (b^2-4 a c\right ) (d+e x) \left (a e^2-b d e+c d^2\right )^4}+\frac {5 e^2 (2 c d-b e) \left (-4 c e (3 a e+4 b d)+7 b^2 e^2+16 c^2 d^2\right ) \text {arctanh}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{16 \left (a e^2-b d e+c d^2\right )^{9/2}}-\frac {e \sqrt {a+b x+c x^2} (2 c d-b e) \left (-4 c e (29 a e+6 b d)+35 b^2 e^2+24 c^2 d^2\right )}{12 \left (b^2-4 a c\right ) (d+e x)^2 \left (a e^2-b d e+c d^2\right )^3}-\frac {e \sqrt {a+b x+c x^2} \left (-4 c e (4 a e+3 b d)+7 b^2 e^2+12 c^2 d^2\right )}{3 \left (b^2-4 a c\right ) (d+e x)^3 \left (a e^2-b d e+c d^2\right )^2}-\frac {2 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{\left (b^2-4 a c\right ) (d+e x)^3 \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )} \]

[In]

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

[Out]

(-2*(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)^3*Sqrt[a +
b*x + c*x^2]) - (e*(12*c^2*d^2 + 7*b^2*e^2 - 4*c*e*(3*b*d + 4*a*e))*Sqrt[a + b*x + c*x^2])/(3*(b^2 - 4*a*c)*(
c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^3) - (e*(2*c*d - b*e)*(24*c^2*d^2 + 35*b^2*e^2 - 4*c*e*(6*b*d + 29*a*e))*Sq
rt[a + b*x + c*x^2])/(12*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)^3*(d + e*x)^2) - (e*(96*c^4*d^4 + 105*b^4*e^4 -
20*b^2*c*e^3*(19*b*d + 23*a*e) - 16*c^3*d^2*e*(12*b*d + 83*a*e) + 4*c^2*e^2*(119*b^2*d^2 + 332*a*b*d*e + 64*a
^2*e^2))*Sqrt[a + b*x + c*x^2])/(24*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)^4*(d + e*x)) + (5*e^2*(2*c*d - b*e)*
(16*c^2*d^2 + 7*b^2*e^2 - 4*c*e*(4*b*d + 3*a*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])])/(16*(c*d^2 - b*d*e + a*e^2)^(9/2))

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 848

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])

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 )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x)^3 \sqrt {a+b x+c x^2}}-\frac {2 \int \frac {\frac {1}{2} e \left (6 b c d-7 b^2 e+16 a c e\right )+3 c e (2 c d-b e) x}{(d+e x)^4 \sqrt {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 {2 \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)^3 \sqrt {a+b x+c x^2}}-\frac {e \left (12 c^2 d^2+7 b^2 e^2-4 c e (3 b d+4 a e)\right ) \sqrt {a+b x+c x^2}}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^3}+\frac {2 \int \frac {\frac {1}{4} e \left (66 b^2 c d e-168 a c^2 d e-35 b^3 e^2-4 b c \left (6 c d^2-29 a e^2\right )\right )-c e \left (12 c^2 d^2+7 b^2 e^2-4 c e (3 b d+4 a e)\right ) x}{(d+e x)^3 \sqrt {a+b x+c x^2}} \, dx}{3 \left (b^2-4 a c\right ) \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 )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x)^3 \sqrt {a+b x+c x^2}}-\frac {e \left (12 c^2 d^2+7 b^2 e^2-4 c e (3 b d+4 a e)\right ) \sqrt {a+b x+c x^2}}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^3}-\frac {e (2 c d-b e) \left (24 c^2 d^2+35 b^2 e^2-4 c e (6 b d+29 a e)\right ) \sqrt {a+b x+c x^2}}{12 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^3 (d+e x)^2}-\frac {\int \frac {\frac {1}{8} e \left (310 b^3 c d e^2-105 b^4 e^3+8 b c^2 d \left (6 c d^2-137 a e^2\right )-4 b^2 c e \left (72 c d^2-115 a e^2\right )+32 a c^2 e \left (27 c d^2-8 a e^2\right )\right )+\frac {1}{4} c e (2 c d-b e) \left (24 c^2 d^2+35 b^2 e^2-4 c e (6 b d+29 a e)\right ) x}{(d+e x)^2 \sqrt {a+b x+c x^2}} \, dx}{3 \left (b^2-4 a c\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 )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x)^3 \sqrt {a+b x+c x^2}}-\frac {e \left (12 c^2 d^2+7 b^2 e^2-4 c e (3 b d+4 a e)\right ) \sqrt {a+b x+c x^2}}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^3}-\frac {e (2 c d-b e) \left (24 c^2 d^2+35 b^2 e^2-4 c e (6 b d+29 a e)\right ) \sqrt {a+b x+c x^2}}{12 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^3 (d+e x)^2}-\frac {e \left (96 c^4 d^4+105 b^4 e^4-20 b^2 c e^3 (19 b d+23 a e)-16 c^3 d^2 e (12 b d+83 a e)+4 c^2 e^2 \left (119 b^2 d^2+332 a b d e+64 a^2 e^2\right )\right ) \sqrt {a+b x+c x^2}}{24 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^4 (d+e x)}+\frac {\left (5 e^2 (2 c d-b e) \left (16 c^2 d^2+7 b^2 e^2-4 c e (4 b d+3 a e)\right )\right ) \int \frac {1}{(d+e x) \sqrt {a+b x+c x^2}} \, dx}{16 \left (c d^2-b d e+a e^2\right )^4} \\ & = -\frac {2 \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)^3 \sqrt {a+b x+c x^2}}-\frac {e \left (12 c^2 d^2+7 b^2 e^2-4 c e (3 b d+4 a e)\right ) \sqrt {a+b x+c x^2}}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^3}-\frac {e (2 c d-b e) \left (24 c^2 d^2+35 b^2 e^2-4 c e (6 b d+29 a e)\right ) \sqrt {a+b x+c x^2}}{12 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^3 (d+e x)^2}-\frac {e \left (96 c^4 d^4+105 b^4 e^4-20 b^2 c e^3 (19 b d+23 a e)-16 c^3 d^2 e (12 b d+83 a e)+4 c^2 e^2 \left (119 b^2 d^2+332 a b d e+64 a^2 e^2\right )\right ) \sqrt {a+b x+c x^2}}{24 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^4 (d+e x)}-\frac {\left (5 e^2 (2 c d-b e) \left (16 c^2 d^2+7 b^2 e^2-4 c e (4 b d+3 a e)\right )\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 )}{8 \left (c d^2-b d e+a e^2\right )^4} \\ & = -\frac {2 \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)^3 \sqrt {a+b x+c x^2}}-\frac {e \left (12 c^2 d^2+7 b^2 e^2-4 c e (3 b d+4 a e)\right ) \sqrt {a+b x+c x^2}}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^3}-\frac {e (2 c d-b e) \left (24 c^2 d^2+35 b^2 e^2-4 c e (6 b d+29 a e)\right ) \sqrt {a+b x+c x^2}}{12 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^3 (d+e x)^2}-\frac {e \left (96 c^4 d^4+105 b^4 e^4-20 b^2 c e^3 (19 b d+23 a e)-16 c^3 d^2 e (12 b d+83 a e)+4 c^2 e^2 \left (119 b^2 d^2+332 a b d e+64 a^2 e^2\right )\right ) \sqrt {a+b x+c x^2}}{24 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^4 (d+e x)}+\frac {5 e^2 (2 c d-b e) \left (16 c^2 d^2+7 b^2 e^2-4 c e (4 b d+3 a e)\right ) \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 )}{16 \left (c d^2-b d e+a e^2\right )^{9/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 11.37 (sec) , antiderivative size = 486, normalized size of antiderivative = 0.94 \[ \int \frac {1}{(d+e x)^4 \left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {2 \left (-\frac {e \left (12 c^2 d^2+7 b^2 e^2-4 c e (3 b d+4 a e)\right ) \sqrt {a+x (b+c x)}}{6 \left (c d^2+e (-b d+a e)\right ) (d+e x)^3}+\frac {b^2 e-2 c (a e+c d x)+b c (-d+e x)}{(d+e x)^3 \sqrt {a+x (b+c x)}}+\frac {e \left (-\frac {4 (2 c d-b e) \left (24 c^2 d^2+35 b^2 e^2-4 c e (6 b d+29 a e)\right ) \sqrt {a+x (b+c x)}}{(d+e x)^2}-\frac {2 \left (96 c^4 d^4+105 b^4 e^4-20 b^2 c e^3 (19 b d+23 a e)-16 c^3 d^2 e (12 b d+83 a e)+4 c^2 e^2 \left (119 b^2 d^2+332 a b d e+64 a^2 e^2\right )\right ) \sqrt {a+x (b+c x)}}{\left (c d^2+e (-b d+a e)\right ) (d+e x)}+\frac {15 \left (b^2-4 a c\right ) e (-2 c d+b e) \left (16 c^2 d^2+7 b^2 e^2-4 c e (4 b d+3 a e)\right ) \text {arctanh}\left (\frac {-b d+2 a e-2 c d x+b e x}{2 \sqrt {c d^2+e (-b d+a e)} \sqrt {a+x (b+c x)}}\right )}{\left (c d^2+e (-b d+a e)\right )^{3/2}}\right )}{96 \left (c d^2+e (-b d+a e)\right )^2}\right )}{\left (b^2-4 a c\right ) \left (c d^2+e (-b d+a e)\right )} \]

[In]

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

[Out]

(2*(-1/6*(e*(12*c^2*d^2 + 7*b^2*e^2 - 4*c*e*(3*b*d + 4*a*e))*Sqrt[a + x*(b + c*x)])/((c*d^2 + e*(-(b*d) + a*e)
)*(d + e*x)^3) + (b^2*e - 2*c*(a*e + c*d*x) + b*c*(-d + e*x))/((d + e*x)^3*Sqrt[a + x*(b + c*x)]) + (e*((-4*(2
*c*d - b*e)*(24*c^2*d^2 + 35*b^2*e^2 - 4*c*e*(6*b*d + 29*a*e))*Sqrt[a + x*(b + c*x)])/(d + e*x)^2 - (2*(96*c^4
*d^4 + 105*b^4*e^4 - 20*b^2*c*e^3*(19*b*d + 23*a*e) - 16*c^3*d^2*e*(12*b*d + 83*a*e) + 4*c^2*e^2*(119*b^2*d^2
+ 332*a*b*d*e + 64*a^2*e^2))*Sqrt[a + x*(b + c*x)])/((c*d^2 + e*(-(b*d) + a*e))*(d + e*x)) + (15*(b^2 - 4*a*c)
*e*(-2*c*d + b*e)*(16*c^2*d^2 + 7*b^2*e^2 - 4*c*e*(4*b*d + 3*a*e))*ArcTanh[(-(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)])])/(c*d^2 + e*(-(b*d) + a*e))^(3/2)))/(96*(c*d^2 + e*(
-(b*d) + a*e))^2)))/((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. \(1977\) vs. \(2(495)=990\).

Time = 0.55 (sec) , antiderivative size = 1978, normalized size of antiderivative = 3.81

method	result	size
default	\(\text {Expression too large to display}\)	\(1978\)

[In]

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

[Out]

1/e^4*(-1/3/(a*e^2-b*d*e+c*d^2)*e^2/(x+d/e)^3/((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
)-7/6*(b*e-2*c*d)*e/(a*e^2-b*d*e+c*d^2)*(-1/2/(a*e^2-b*d*e+c*d^2)*e^2/(x+d/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)-5/4*(b*e-2*c*d)*e/(a*e^2-b*d*e+c*d^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)^(1/2)-3/2*(b*e-2*c*d)*e/(a*e^2-b*d*e+c*d^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*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))-3/2*c
/(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))))-4/3*c/(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)/((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)-3/2*(b*e-2*c
*d)*e/(a*e^2-b*d*e+c*d^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*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)))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4683 vs. \(2 (495) = 990\).

Time = 28.62 (sec) , antiderivative size = 9408, normalized size of antiderivative = 18.13 \[ \int \frac {1}{(d+e x)^4 \left (a+b x+c x^2\right )^{3/2}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Too large to include

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

integrate(1/(e*x+d)^4/(c*x^2+b*x+a)^(3/2),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. 5445 vs. \(2 (495) = 990\).

Time = 0.60 (sec) , antiderivative size = 5445, normalized size of antiderivative = 10.49 \[ \int \frac {1}{(d+e x)^4 \left (a+b x+c x^2\right )^{3/2}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-2*((2*c^9*d^12 - 12*b*c^8*d^11*e + 34*b^2*c^7*d^10*e^2 - 4*a*c^8*d^10*e^2 - 60*b^3*c^6*d^9*e^3 + 20*a*b*c^7*d
^9*e^3 + 71*b^4*c^5*d^8*e^4 - 28*a*b^2*c^6*d^8*e^4 - 34*a^2*c^7*d^8*e^4 - 56*b^5*c^4*d^7*e^5 - 8*a*b^3*c^5*d^7
*e^5 + 136*a^2*b*c^6*d^7*e^5 + 28*b^6*c^3*d^6*e^6 + 56*a*b^4*c^4*d^6*e^6 - 196*a^2*b^2*c^5*d^6*e^6 - 56*a^3*c^
6*d^6*e^6 - 8*b^7*c^2*d^5*e^7 - 56*a*b^5*c^3*d^5*e^7 + 112*a^2*b^3*c^4*d^5*e^7 + 168*a^3*b*c^5*d^5*e^7 + b^8*c
*d^4*e^8 + 24*a*b^6*c^2*d^4*e^8 - 4*a^2*b^4*c^3*d^4*e^8 - 176*a^3*b^2*c^4*d^4*e^8 - 34*a^4*c^5*d^4*e^8 - 4*a*b
^7*c*d^3*e^9 - 20*a^2*b^5*c^2*d^3*e^9 + 72*a^3*b^3*c^3*d^3*e^9 + 68*a^4*b*c^4*d^3*e^9 + 6*a^2*b^6*c*d^2*e^10 -
4*a^3*b^4*c^2*d^2*e^10 - 46*a^4*b^2*c^3*d^2*e^10 - 4*a^5*c^4*d^2*e^10 - 4*a^3*b^5*c*d*e^11 + 12*a^4*b^3*c^2*d
*e^11 + 4*a^5*b*c^3*d*e^11 + a^4*b^4*c*e^12 - 4*a^5*b^2*c^2*e^12 + 2*a^6*c^3*e^12)*x/(b^2*c^8*d^16 - 4*a*c^9*d
^16 - 8*b^3*c^7*d^15*e + 32*a*b*c^8*d^15*e + 28*b^4*c^6*d^14*e^2 - 104*a*b^2*c^7*d^14*e^2 - 32*a^2*c^8*d^14*e^
2 - 56*b^5*c^5*d^13*e^3 + 168*a*b^3*c^6*d^13*e^3 + 224*a^2*b*c^7*d^13*e^3 + 70*b^6*c^4*d^12*e^4 - 112*a*b^4*c^
5*d^12*e^4 - 644*a^2*b^2*c^6*d^12*e^4 - 112*a^3*c^7*d^12*e^4 - 56*b^7*c^3*d^11*e^5 - 56*a*b^5*c^4*d^11*e^5 + 9
52*a^2*b^3*c^5*d^11*e^5 + 672*a^3*b*c^6*d^11*e^5 + 28*b^8*c^2*d^10*e^6 + 168*a*b^6*c^3*d^10*e^6 - 700*a^2*b^4*
c^4*d^10*e^6 - 1624*a^3*b^2*c^5*d^10*e^6 - 224*a^4*c^6*d^10*e^6 - 8*b^9*c*d^9*e^7 - 136*a*b^7*c^2*d^9*e^7 + 11
2*a^2*b^5*c^3*d^9*e^7 + 1960*a^3*b^3*c^4*d^9*e^7 + 1120*a^4*b*c^5*d^9*e^7 + b^10*d^8*e^8 + 52*a*b^8*c*d^8*e^8
+ 196*a^2*b^6*c^2*d^8*e^8 - 1120*a^3*b^4*c^3*d^8*e^8 - 2170*a^4*b^2*c^4*d^8*e^8 - 280*a^5*c^5*d^8*e^8 - 8*a*b^
9*d^7*e^9 - 136*a^2*b^7*c*d^7*e^9 + 112*a^3*b^5*c^2*d^7*e^9 + 1960*a^4*b^3*c^3*d^7*e^9 + 1120*a^5*b*c^4*d^7*e^
9 + 28*a^2*b^8*d^6*e^10 + 168*a^3*b^6*c*d^6*e^10 - 700*a^4*b^4*c^2*d^6*e^10 - 1624*a^5*b^2*c^3*d^6*e^10 - 224*
a^6*c^4*d^6*e^10 - 56*a^3*b^7*d^5*e^11 - 56*a^4*b^5*c*d^5*e^11 + 952*a^5*b^3*c^2*d^5*e^11 + 672*a^6*b*c^3*d^5*
e^11 + 70*a^4*b^6*d^4*e^12 - 112*a^5*b^4*c*d^4*e^12 - 644*a^6*b^2*c^2*d^4*e^12 - 112*a^7*c^3*d^4*e^12 - 56*a^5
*b^5*d^3*e^13 + 168*a^6*b^3*c*d^3*e^13 + 224*a^7*b*c^2*d^3*e^13 + 28*a^6*b^4*d^2*e^14 - 104*a^7*b^2*c*d^2*e^14
- 32*a^8*c^2*d^2*e^14 - 8*a^7*b^3*d*e^15 + 32*a^8*b*c*d*e^15 + a^8*b^2*e^16 - 4*a^9*c*e^16) + (b*c^8*d^12 - 8
*b^2*c^7*d^11*e + 8*a*c^8*d^11*e + 28*b^3*c^6*d^10*e^2 - 46*a*b*c^7*d^10*e^2 - 56*b^4*c^5*d^9*e^3 + 108*a*b^2*
c^6*d^9*e^3 + 24*a^2*c^7*d^9*e^3 + 70*b^5*c^4*d^8*e^4 - 125*a*b^3*c^5*d^8*e^4 - 125*a^2*b*c^6*d^8*e^4 - 56*b^6
*c^3*d^7*e^5 + 56*a*b^4*c^4*d^7*e^5 + 272*a^2*b^2*c^5*d^7*e^5 + 16*a^3*c^6*d^7*e^5 + 28*b^7*c^2*d^6*e^6 + 28*a
*b^5*c^3*d^6*e^6 - 308*a^2*b^3*c^4*d^6*e^6 - 84*a^3*b*c^5*d^6*e^6 - 8*b^8*c*d^5*e^7 - 48*a*b^6*c^2*d^5*e^7 + 1
76*a^2*b^4*c^3*d^5*e^7 + 184*a^3*b^2*c^4*d^5*e^7 - 16*a^4*c^5*d^5*e^7 + b^9*d^4*e^8 + 23*a*b^7*c*d^4*e^8 - 29*
a^2*b^5*c^2*d^4*e^8 - 198*a^3*b^3*c^3*d^4*e^8 + 23*a^4*b*c^4*d^4*e^8 - 4*a*b^8*d^3*e^9 - 16*a^2*b^6*c*d^3*e^9
+ 96*a^3*b^4*c^2*d^3*e^9 + 24*a^4*b^2*c^3*d^3*e^9 - 24*a^5*c^4*d^3*e^9 + 6*a^2*b^7*d^2*e^10 - 10*a^3*b^5*c*d^2
*e^10 - 48*a^4*b^3*c^2*d^2*e^10 + 34*a^5*b*c^3*d^2*e^10 - 4*a^3*b^6*d*e^11 + 16*a^4*b^4*c*d*e^11 - 4*a^5*b^2*c
^2*d*e^11 - 8*a^6*c^3*d*e^11 + a^4*b^5*e^12 - 5*a^5*b^3*c*e^12 + 5*a^6*b*c^2*e^12)/(b^2*c^8*d^16 - 4*a*c^9*d^1
6 - 8*b^3*c^7*d^15*e + 32*a*b*c^8*d^15*e + 28*b^4*c^6*d^14*e^2 - 104*a*b^2*c^7*d^14*e^2 - 32*a^2*c^8*d^14*e^2
- 56*b^5*c^5*d^13*e^3 + 168*a*b^3*c^6*d^13*e^3 + 224*a^2*b*c^7*d^13*e^3 + 70*b^6*c^4*d^12*e^4 - 112*a*b^4*c^5*
d^12*e^4 - 644*a^2*b^2*c^6*d^12*e^4 - 112*a^3*c^7*d^12*e^4 - 56*b^7*c^3*d^11*e^5 - 56*a*b^5*c^4*d^11*e^5 + 952
*a^2*b^3*c^5*d^11*e^5 + 672*a^3*b*c^6*d^11*e^5 + 28*b^8*c^2*d^10*e^6 + 168*a*b^6*c^3*d^10*e^6 - 700*a^2*b^4*c^
4*d^10*e^6 - 1624*a^3*b^2*c^5*d^10*e^6 - 224*a^4*c^6*d^10*e^6 - 8*b^9*c*d^9*e^7 - 136*a*b^7*c^2*d^9*e^7 + 112*
a^2*b^5*c^3*d^9*e^7 + 1960*a^3*b^3*c^4*d^9*e^7 + 1120*a^4*b*c^5*d^9*e^7 + b^10*d^8*e^8 + 52*a*b^8*c*d^8*e^8 +
196*a^2*b^6*c^2*d^8*e^8 - 1120*a^3*b^4*c^3*d^8*e^8 - 2170*a^4*b^2*c^4*d^8*e^8 - 280*a^5*c^5*d^8*e^8 - 8*a*b^9*
d^7*e^9 - 136*a^2*b^7*c*d^7*e^9 + 112*a^3*b^5*c^2*d^7*e^9 + 1960*a^4*b^3*c^3*d^7*e^9 + 1120*a^5*b*c^4*d^7*e^9
+ 28*a^2*b^8*d^6*e^10 + 168*a^3*b^6*c*d^6*e^10 - 700*a^4*b^4*c^2*d^6*e^10 - 1624*a^5*b^2*c^3*d^6*e^10 - 224*a^
6*c^4*d^6*e^10 - 56*a^3*b^7*d^5*e^11 - 56*a^4*b^5*c*d^5*e^11 + 952*a^5*b^3*c^2*d^5*e^11 + 672*a^6*b*c^3*d^5*e^
11 + 70*a^4*b^6*d^4*e^12 - 112*a^5*b^4*c*d^4*e^12 - 644*a^6*b^2*c^2*d^4*e^12 - 112*a^7*c^3*d^4*e^12 - 56*a^5*b
^5*d^3*e^13 + 168*a^6*b^3*c*d^3*e^13 + 224*a^7*b*c^2*d^3*e^13 + 28*a^6*b^4*d^2*e^14 - 104*a^7*b^2*c*d^2*e^14 -
32*a^8*c^2*d^2*e^14 - 8*a^7*b^3*d*e^15 + 32*a^8*b*c*d*e^15 + a^8*b^2*e^16 - 4*a^9*c*e^16))/sqrt(c*x^2 + b*x +
a) + 5/8*(32*c^3*d^3*e^2 - 48*b*c^2*d^2*e^3 + 30*b^2*c*d*e^4 - 24*a*c^2*d*e^4 - 7*b^3*e^5 + 12*a*b*c*e^5)*arc
tan(-((sqrt(c)*x - sqrt(c*x^2 + b*x + a))*e + sqrt(c)*d)/sqrt(-c*d^2 + b*d*e - a*e^2))/((c^4*d^8 - 4*b*c^3*d^7
*e + 6*b^2*c^2*d^6*e^2 + 4*a*c^3*d^6*e^2 - 4*b^3*c*d^5*e^3 - 12*a*b*c^2*d^5*e^3 + b^4*d^4*e^4 + 12*a*b^2*c*d^4
*e^4 + 6*a^2*c^2*d^4*e^4 - 4*a*b^3*d^3*e^5 - 12*a^2*b*c*d^3*e^5 + 6*a^2*b^2*d^2*e^6 + 4*a^3*c*d^2*e^6 - 4*a^3*
b*d*e^7 + a^4*e^8)*sqrt(-c*d^2 + b*d*e - a*e^2)) - 1/24*(288*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*c^3*d^3*e^4
- 432*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*b*c^2*d^2*e^5 + 258*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*b^2*c*d
*e^6 - 168*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a*c^2*d*e^6 - 57*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*b^3*e^
7 + 84*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a*b*c*e^7 + 1296*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*c^(7/2)*d^
4*e^3 - 1872*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*b*c^(5/2)*d^3*e^4 + 1098*(sqrt(c)*x - sqrt(c*x^2 + b*x + a)
)^4*b^2*c^(3/2)*d^2*e^5 - 936*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a*c^(5/2)*d^2*e^5 - 237*(sqrt(c)*x - sqrt(
c*x^2 + b*x + a))^4*b^3*sqrt(c)*d*e^6 + 516*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a*b*c^(3/2)*d*e^6 - 48*(sqrt
(c)*x - sqrt(c*x^2 + b*x + a))^4*a*b^2*sqrt(c)*e^7 + 48*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^2*c^(3/2)*e^7
+ 1504*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*c^4*d^5*e^2 - 1168*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b*c^3*d^
4*e^3 - 60*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b^2*c^2*d^3*e^4 - 2576*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*
a*c^3*d^3*e^4 + 430*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b^3*c*d^2*e^5 + 1992*(sqrt(c)*x - sqrt(c*x^2 + b*x +
a))^3*a*b*c^2*d^2*e^5 - 136*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b^4*d*e^6 - 720*(sqrt(c)*x - sqrt(c*x^2 + b
*x + a))^3*a*b^2*c*d*e^6 + 480*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^2*c^2*d*e^6 + 136*(sqrt(c)*x - sqrt(c*x
^2 + b*x + a))^3*a*b^3*e^7 - 144*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^2*b*c*e^7 + 2256*(sqrt(c)*x - sqrt(c*
x^2 + b*x + a))^2*b*c^(7/2)*d^5*e^2 - 2892*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b^2*c^(5/2)*d^4*e^3 - 3216*(s
qrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*c^(7/2)*d^4*e^3 + 1470*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b^3*c^(3/2)
*d^3*e^4 + 2568*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*b*c^(5/2)*d^3*e^4 - 264*(sqrt(c)*x - sqrt(c*x^2 + b*x
+ a))^2*b^4*sqrt(c)*d^2*e^5 - 1008*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*b^2*c^(3/2)*d^2*e^5 + 1152*(sqrt(c)
*x - sqrt(c*x^2 + b*x + a))^2*a^2*c^(5/2)*d^2*e^5 + 120*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*b^3*sqrt(c)*d*
e^6 - 432*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^2*b*c^(3/2)*d*e^6 + 144*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^
2*a^2*b^2*sqrt(c)*e^7 - 192*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^3*c^(3/2)*e^7 + 1128*(sqrt(c)*x - sqrt(c*x
^2 + b*x + a))*b^2*c^3*d^5*e^2 - 1368*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^3*c^2*d^4*e^3 - 3216*(sqrt(c)*x -
sqrt(c*x^2 + b*x + a))*a*b*c^3*d^4*e^3 + 612*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^4*c*d^3*e^4 + 3516*(sqrt(c)
*x - sqrt(c*x^2 + b*x + a))*a*b^2*c^2*d^3*e^4 + 1968*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^2*c^3*d^3*e^4 - 87*
(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^5*d^2*e^5 - 1494*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b^3*c*d^2*e^5 - 1
800*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^2*b*c^2*d^2*e^5 + 174*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b^4*d*e^
6 + 918*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^2*b^2*c*d*e^6 - 312*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^3*c^2*
d*e^6 - 87*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^2*b^3*e^7 - 36*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^3*b*c*e^
7 + 188*b^3*c^(5/2)*d^5*e^2 - 188*b^4*c^(3/2)*d^4*e^3 - 804*a*b^2*c^(5/2)*d^4*e^3 + 57*b^5*sqrt(c)*d^3*e^4 + 7
94*a*b^3*c^(3/2)*d^3*e^4 + 984*a^2*b*c^(5/2)*d^3*e^4 - 258*a*b^4*sqrt(c)*d^2*e^5 - 906*a^2*b^2*c^(3/2)*d^2*e^5
- 376*a^3*c^(5/2)*d^2*e^5 + 345*a^2*b^3*sqrt(c)*d*e^6 + 220*a^3*b*c^(3/2)*d*e^6 - 144*a^3*b^2*sqrt(c)*e^7 + 8
0*a^4*c^(3/2)*e^7)/((c^4*d^8 - 4*b*c^3*d^7*e + 6*b^2*c^2*d^6*e^2 + 4*a*c^3*d^6*e^2 - 4*b^3*c*d^5*e^3 - 12*a*b*
c^2*d^5*e^3 + b^4*d^4*e^4 + 12*a*b^2*c*d^4*e^4 + 6*a^2*c^2*d^4*e^4 - 4*a*b^3*d^3*e^5 - 12*a^2*b*c*d^3*e^5 + 6*
a^2*b^2*d^2*e^6 + 4*a^3*c*d^2*e^6 - 4*a^3*b*d*e^7 + a^4*e^8)*((sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*e + 2*(sqr
t(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c)*d + b*d - a*e)^3)

Mupad [F(-1)]

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

[In]

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

[Out]

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