3.24.6 \(\int \frac {1}{\sqrt {d+e x} (a+b x+c x^2)^3} \, dx\) [2306]

Optimal. Leaf size=835 \[ -\frac {\sqrt {d+e x} \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^2}-\frac {\sqrt {d+e x} \left (5 a c e (2 c d-b e)^2-\left (b c d-b^2 e+2 a c e\right ) \left (12 c^2 d^2-3 b^2 e^2-7 c e (b d-2 a e)\right )-3 c (2 c d-b e) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right ) x\right )}{4 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 \left (a+b x+c x^2\right )}-\frac {3 \sqrt {c} \left (32 c^4 d^4+b^3 \left (b+\sqrt {b^2-4 a c}\right ) e^4-8 c^3 d^2 e \left (8 b d-\sqrt {b^2-4 a c} d-9 a e\right )+2 b c e^3 \left (b^2 d+b \sqrt {b^2-4 a c} d-5 a b e-4 a \sqrt {b^2-4 a c} e\right )+2 c^2 e^2 \left (15 b^2 d^2-6 b d \left (\sqrt {b^2-4 a c} d+6 a e\right )+4 a e \left (2 \sqrt {b^2-4 a c} d+7 a e\right )\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{4 \sqrt {2} \left (b^2-4 a c\right )^{5/2} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} \left (c d^2-b d e+a e^2\right )^2}+\frac {3 \sqrt {c} \left (32 c^4 d^4+b^3 \left (b-\sqrt {b^2-4 a c}\right ) e^4-8 c^3 d^2 e \left (8 b d+\sqrt {b^2-4 a c} d-9 a e\right )+2 c^2 e^2 \left (15 b^2 d^2-4 a e \left (2 \sqrt {b^2-4 a c} d-7 a e\right )+6 b d \left (\sqrt {b^2-4 a c} d-6 a e\right )\right )+2 b c e^3 \left (b^2 d+4 a \sqrt {b^2-4 a c} e-b \left (\sqrt {b^2-4 a c} d+5 a e\right )\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{4 \sqrt {2} \left (b^2-4 a c\right )^{5/2} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} \left (c d^2-e (b d-a e)\right )^2} \]

[Out]

-1/2*(b*c*d-b^2*e+2*a*c*e+c*(-b*e+2*c*d)*x)*(e*x+d)^(1/2)/(-4*a*c+b^2)/(a*e^2-b*d*e+c*d^2)/(c*x^2+b*x+a)^2-1/4
*(5*a*c*e*(-b*e+2*c*d)^2-(2*a*c*e-b^2*e+b*c*d)*(12*c^2*d^2-3*b^2*e^2-7*c*e*(-2*a*e+b*d))-3*c*(-b*e+2*c*d)*(4*c
^2*d^2-b^2*e^2-4*c*e*(-2*a*e+b*d))*x)*(e*x+d)^(1/2)/(-4*a*c+b^2)^2/(a*e^2-b*d*e+c*d^2)^2/(c*x^2+b*x+a)-3/8*arc
tanh(2^(1/2)*c^(1/2)*(e*x+d)^(1/2)/(2*c*d-e*(b-(-4*a*c+b^2)^(1/2)))^(1/2))*c^(1/2)*(32*c^4*d^4+b^3*e^4*(b+(-4*
a*c+b^2)^(1/2))-8*c^3*d^2*e*(8*b*d-9*a*e-d*(-4*a*c+b^2)^(1/2))+2*b*c*e^3*(b^2*d-5*a*b*e+b*d*(-4*a*c+b^2)^(1/2)
-4*a*e*(-4*a*c+b^2)^(1/2))+2*c^2*e^2*(15*b^2*d^2-6*b*d*(6*a*e+d*(-4*a*c+b^2)^(1/2))+4*a*e*(7*a*e+2*d*(-4*a*c+b
^2)^(1/2))))/(-4*a*c+b^2)^(5/2)/(a*e^2-b*d*e+c*d^2)^2*2^(1/2)/(2*c*d-e*(b-(-4*a*c+b^2)^(1/2)))^(1/2)+3/8*arcta
nh(2^(1/2)*c^(1/2)*(e*x+d)^(1/2)/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2)))^(1/2))*c^(1/2)*(32*c^4*d^4+b^3*e^4*(b-(-4*a*
c+b^2)^(1/2))-8*c^3*d^2*e*(8*b*d-9*a*e+d*(-4*a*c+b^2)^(1/2))+2*b*c*e^3*(b^2*d+4*a*e*(-4*a*c+b^2)^(1/2)-b*(5*a*
e+d*(-4*a*c+b^2)^(1/2)))+2*c^2*e^2*(15*b^2*d^2+6*b*d*(-6*a*e+d*(-4*a*c+b^2)^(1/2))-4*a*e*(-7*a*e+2*d*(-4*a*c+b
^2)^(1/2))))/(-4*a*c+b^2)^(5/2)/(a*e^2-b*d*e+c*d^2)^2*2^(1/2)/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2)))^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 10.81, antiderivative size = 834, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {754, 836, 840, 1180, 214} \begin {gather*} -\frac {\sqrt {d+e x} \left (-e b^2+c d b+2 a c e+c (2 c d-b e) x\right )}{2 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) \left (c x^2+b x+a\right )^2}-\frac {3 \sqrt {c} \left (32 c^4 d^4-8 c^3 e \left (8 b d-\sqrt {b^2-4 a c} d-9 a e\right ) d^2+b^3 \left (b+\sqrt {b^2-4 a c}\right ) e^4+2 b c e^3 \left (d b^2+\sqrt {b^2-4 a c} d b-5 a e b-4 a \sqrt {b^2-4 a c} e\right )+2 c^2 e^2 \left (15 b^2 d^2-6 b \left (\sqrt {b^2-4 a c} d+6 a e\right ) d+4 a e \left (2 \sqrt {b^2-4 a c} d+7 a e\right )\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{4 \sqrt {2} \left (b^2-4 a c\right )^{5/2} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} \left (c d^2-b e d+a e^2\right )^2}+\frac {3 \sqrt {c} \left (32 c^4 d^4-8 c^3 e \left (8 b d+\sqrt {b^2-4 a c} d-9 a e\right ) d^2+b^3 \left (b-\sqrt {b^2-4 a c}\right ) e^4+2 c^2 e^2 \left (15 b^2 d^2+6 b \left (\sqrt {b^2-4 a c} d-6 a e\right ) d-4 a e \left (2 \sqrt {b^2-4 a c} d-7 a e\right )\right )+2 b c e^3 \left (d b^2-\left (\sqrt {b^2-4 a c} d+5 a e\right ) b+4 a \sqrt {b^2-4 a c} e\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{4 \sqrt {2} \left (b^2-4 a c\right )^{5/2} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} \left (c d^2-b e d+a e^2\right )^2}-\frac {\sqrt {d+e x} \left (5 a c e (2 c d-b e)^2-3 c \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right ) x (2 c d-b e)-\left (-e b^2+c d b+2 a c e\right ) \left (12 c^2 d^2-3 b^2 e^2-7 c e (b d-2 a e)\right )\right )}{4 \left (b^2-4 a c\right )^2 \left (c d^2-b e d+a e^2\right )^2 \left (c x^2+b x+a\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*(Sqrt[d + e*x]*(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)*(a +
b*x + c*x^2)^2) - (Sqrt[d + e*x]*(5*a*c*e*(2*c*d - b*e)^2 - (b*c*d - b^2*e + 2*a*c*e)*(12*c^2*d^2 - 3*b^2*e^2
- 7*c*e*(b*d - 2*a*e)) - 3*c*(2*c*d - b*e)*(4*c^2*d^2 - b^2*e^2 - 4*c*e*(b*d - 2*a*e))*x))/(4*(b^2 - 4*a*c)^2
*(c*d^2 - b*d*e + a*e^2)^2*(a + b*x + c*x^2)) - (3*Sqrt[c]*(32*c^4*d^4 + b^3*(b + Sqrt[b^2 - 4*a*c])*e^4 - 8*c
^3*d^2*e*(8*b*d - Sqrt[b^2 - 4*a*c]*d - 9*a*e) + 2*b*c*e^3*(b^2*d + b*Sqrt[b^2 - 4*a*c]*d - 5*a*b*e - 4*a*Sqrt
[b^2 - 4*a*c]*e) + 2*c^2*e^2*(15*b^2*d^2 - 6*b*d*(Sqrt[b^2 - 4*a*c]*d + 6*a*e) + 4*a*e*(2*Sqrt[b^2 - 4*a*c]*d
+ 7*a*e)))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]])/(4*Sqrt[2]*(b^2 -
4*a*c)^(5/2)*Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]*(c*d^2 - b*d*e + a*e^2)^2) + (3*Sqrt[c]*(32*c^4*d^4 + b^
3*(b - Sqrt[b^2 - 4*a*c])*e^4 - 8*c^3*d^2*e*(8*b*d + Sqrt[b^2 - 4*a*c]*d - 9*a*e) + 2*c^2*e^2*(15*b^2*d^2 - 4*
a*e*(2*Sqrt[b^2 - 4*a*c]*d - 7*a*e) + 6*b*d*(Sqrt[b^2 - 4*a*c]*d - 6*a*e)) + 2*b*c*e^3*(b^2*d + 4*a*Sqrt[b^2 -
4*a*c]*e - b*(Sqrt[b^2 - 4*a*c]*d + 5*a*e)))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b + Sqrt[b
^2 - 4*a*c])*e]])/(4*Sqrt[2]*(b^2 - 4*a*c)^(5/2)*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]*(c*d^2 - b*d*e + a*e^
2)^2)

Rule 214

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

Rule 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 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])

Rule 840

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

Rule 1180

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

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {d+e x} \left (a+b x+c x^2\right )^3} \, dx &=-\frac {\sqrt {d+e x} \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^2}-\frac {\int \frac {\frac {1}{2} \left (12 c^2 d^2-3 b^2 e^2-7 c e (b d-2 a e)\right )+\frac {5}{2} c e (2 c d-b e) x}{\sqrt {d+e x} \left (a+b x+c x^2\right )^2} \, dx}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )}\\ &=-\frac {\sqrt {d+e x} \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^2}-\frac {\sqrt {d+e x} \left (5 a c e (2 c d-b e)^2-\left (b c d-b^2 e+2 a c e\right ) \left (12 c^2 d^2-3 b^2 e^2-7 c e (b d-2 a e)\right )-3 c (2 c d-b e) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right ) x\right )}{4 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 \left (a+b x+c x^2\right )}+\frac {\int \frac {\frac {3}{4} \left (16 c^4 d^4+b^4 e^4+b^2 c e^3 (2 b d-9 a e)-4 c^3 d^2 e (7 b d-9 a e)+c^2 e^2 \left (9 b^2 d^2-28 a b d e+28 a^2 e^2\right )\right )+\frac {3}{4} c e (2 c d-b e) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right ) x}{\sqrt {d+e x} \left (a+b x+c x^2\right )} \, dx}{2 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2}\\ &=-\frac {\sqrt {d+e x} \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^2}-\frac {\sqrt {d+e x} \left (5 a c e (2 c d-b e)^2-\left (b c d-b^2 e+2 a c e\right ) \left (12 c^2 d^2-3 b^2 e^2-7 c e (b d-2 a e)\right )-3 c (2 c d-b e) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right ) x\right )}{4 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 \left (a+b x+c x^2\right )}+\frac {\text {Subst}\left (\int \frac {-\frac {3}{4} c d e (2 c d-b e) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right )+\frac {3}{4} e \left (16 c^4 d^4+b^4 e^4+b^2 c e^3 (2 b d-9 a e)-4 c^3 d^2 e (7 b d-9 a e)+c^2 e^2 \left (9 b^2 d^2-28 a b d e+28 a^2 e^2\right )\right )+\frac {3}{4} c e (2 c d-b e) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right ) x^2}{c d^2-b d e+a e^2+(-2 c d+b e) x^2+c x^4} \, dx,x,\sqrt {d+e x}\right )}{\left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2}\\ &=-\frac {\sqrt {d+e x} \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^2}-\frac {\sqrt {d+e x} \left (5 a c e (2 c d-b e)^2-\left (b c d-b^2 e+2 a c e\right ) \left (12 c^2 d^2-3 b^2 e^2-7 c e (b d-2 a e)\right )-3 c (2 c d-b e) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right ) x\right )}{4 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 \left (a+b x+c x^2\right )}+\frac {\left (3 c \left (32 c^4 d^4+b^3 \left (b+\sqrt {b^2-4 a c}\right ) e^4-8 c^3 d^2 e \left (8 b d-\sqrt {b^2-4 a c} d-9 a e\right )+2 b c e^3 \left (b^2 d+b \sqrt {b^2-4 a c} d-5 a b e-4 a \sqrt {b^2-4 a c} e\right )+2 c^2 e^2 \left (15 b^2 d^2-6 b d \left (\sqrt {b^2-4 a c} d+6 a e\right )+4 a e \left (2 \sqrt {b^2-4 a c} d+7 a e\right )\right )\right )\right ) \text {Subst}\left (\int \frac {1}{-\frac {1}{2} \sqrt {b^2-4 a c} e+\frac {1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt {d+e x}\right )}{8 \left (b^2-4 a c\right )^{5/2} \left (c d^2-b d e+a e^2\right )^2}+\frac {\left (\frac {3}{8} c e (2 c d-b e) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right )-\frac {-\frac {3}{4} c e (2 c d-b e) (-2 c d+b e) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right )+2 c \left (-\frac {3}{4} c d e (2 c d-b e) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right )+\frac {3}{4} e \left (16 c^4 d^4+b^4 e^4+b^2 c e^3 (2 b d-9 a e)-4 c^3 d^2 e (7 b d-9 a e)+c^2 e^2 \left (9 b^2 d^2-28 a b d e+28 a^2 e^2\right )\right )\right )}{2 \sqrt {b^2-4 a c} e}\right ) \text {Subst}\left (\int \frac {1}{\frac {1}{2} \sqrt {b^2-4 a c} e+\frac {1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt {d+e x}\right )}{\left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2}\\ &=-\frac {\sqrt {d+e x} \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^2}-\frac {\sqrt {d+e x} \left (5 a c e (2 c d-b e)^2-\left (b c d-b^2 e+2 a c e\right ) \left (12 c^2 d^2-3 b^2 e^2-7 c e (b d-2 a e)\right )-3 c (2 c d-b e) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right ) x\right )}{4 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 \left (a+b x+c x^2\right )}-\frac {3 \sqrt {c} \left (32 c^4 d^4+b^3 \left (b+\sqrt {b^2-4 a c}\right ) e^4-8 c^3 d^2 e \left (8 b d-\sqrt {b^2-4 a c} d-9 a e\right )+2 b c e^3 \left (b^2 d+b \sqrt {b^2-4 a c} d-5 a b e-4 a \sqrt {b^2-4 a c} e\right )+2 c^2 e^2 \left (15 b^2 d^2-6 b d \left (\sqrt {b^2-4 a c} d+6 a e\right )+4 a e \left (2 \sqrt {b^2-4 a c} d+7 a e\right )\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{4 \sqrt {2} \left (b^2-4 a c\right )^{5/2} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} \left (c d^2-e (b d-a e)\right )^2}+\frac {3 \sqrt {c} \left (32 c^4 d^4+b^3 \left (b-\sqrt {b^2-4 a c}\right ) e^4-8 c^3 d^2 e \left (8 b d+\sqrt {b^2-4 a c} d-9 a e\right )+2 b c e^3 \left (b^2 d-b \sqrt {b^2-4 a c} d-5 a b e+4 a \sqrt {b^2-4 a c} e\right )+2 c^2 e^2 \left (15 b^2 d^2-4 a e \left (2 \sqrt {b^2-4 a c} d-7 a e\right )+6 b d \left (\sqrt {b^2-4 a c} d-6 a e\right )\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{4 \sqrt {2} \left (b^2-4 a c\right )^{5/2} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} \left (c d^2-e (b d-a e)\right )^2}\\ \end {align*}

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Mathematica [A]
time = 13.76, size = 799, normalized size = 0.96 \begin {gather*} \frac {\frac {\sqrt {d+e x} \left (b^2 e-2 c (a e+c d x)+b c (-d+e x)\right )}{(a+x (b+c x))^2}-\frac {\sqrt {d+e x} \left (3 b^4 e^3+b^3 c e^2 (4 d+3 e x)+4 b c^2 \left (a e^2 (5 d-6 e x)+3 c d^2 (d-3 e x)\right )+b^2 c e \left (-25 a e^2+c d (-19 d+6 e x)\right )+4 c^2 \left (7 a^2 e^3+6 c^2 d^3 x+a c d e (d+12 e x)\right )\right )}{2 \left (b^2-4 a c\right ) \left (-c d^2+e (b d-a e)\right ) (a+x (b+c x))}+\frac {3 \sqrt {c} \left (-\frac {\left (32 c^4 d^4+b^3 \left (b+\sqrt {b^2-4 a c}\right ) e^4+8 c^3 d^2 e \left (-8 b d+\sqrt {b^2-4 a c} d+9 a e\right )+2 b c e^3 \left (b^2 d+b \sqrt {b^2-4 a c} d-5 a b e-4 a \sqrt {b^2-4 a c} e\right )+2 c^2 e^2 \left (15 b^2 d^2-6 b d \left (\sqrt {b^2-4 a c} d+6 a e\right )+4 a e \left (2 \sqrt {b^2-4 a c} d+7 a e\right )\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-b e+\sqrt {b^2-4 a c} e}}\right )}{\sqrt {2 c d+\left (-b+\sqrt {b^2-4 a c}\right ) e}}+\frac {\left (32 c^4 d^4+b^3 \left (b-\sqrt {b^2-4 a c}\right ) e^4-8 c^3 d^2 e \left (8 b d+\sqrt {b^2-4 a c} d-9 a e\right )+2 b c e^3 \left (b^2 d+4 a \sqrt {b^2-4 a c} e-b \left (\sqrt {b^2-4 a c} d+5 a e\right )\right )+2 c^2 e^2 \left (15 b^2 d^2+6 b d \left (\sqrt {b^2-4 a c} d-6 a e\right )+4 a e \left (-2 \sqrt {b^2-4 a c} d+7 a e\right )\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{2 \sqrt {2} \left (b^2-4 a c\right )^{3/2} \left (c d^2+e (-b d+a e)\right )}}{2 \left (b^2-4 a c\right ) \left (c d^2+e (-b d+a e)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((Sqrt[d + e*x]*(b^2*e - 2*c*(a*e + c*d*x) + b*c*(-d + e*x)))/(a + x*(b + c*x))^2 - (Sqrt[d + e*x]*(3*b^4*e^3
+ b^3*c*e^2*(4*d + 3*e*x) + 4*b*c^2*(a*e^2*(5*d - 6*e*x) + 3*c*d^2*(d - 3*e*x)) + b^2*c*e*(-25*a*e^2 + c*d*(-1
9*d + 6*e*x)) + 4*c^2*(7*a^2*e^3 + 6*c^2*d^3*x + a*c*d*e*(d + 12*e*x))))/(2*(b^2 - 4*a*c)*(-(c*d^2) + e*(b*d -
a*e))*(a + x*(b + c*x))) + (3*Sqrt[c]*(-(((32*c^4*d^4 + b^3*(b + Sqrt[b^2 - 4*a*c])*e^4 + 8*c^3*d^2*e*(-8*b*d
+ Sqrt[b^2 - 4*a*c]*d + 9*a*e) + 2*b*c*e^3*(b^2*d + b*Sqrt[b^2 - 4*a*c]*d - 5*a*b*e - 4*a*Sqrt[b^2 - 4*a*c]*e
) + 2*c^2*e^2*(15*b^2*d^2 - 6*b*d*(Sqrt[b^2 - 4*a*c]*d + 6*a*e) + 4*a*e*(2*Sqrt[b^2 - 4*a*c]*d + 7*a*e)))*ArcT
anh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e]])/Sqrt[2*c*d + (-b + Sqrt[b^2 - 4*
a*c])*e]) + ((32*c^4*d^4 + b^3*(b - Sqrt[b^2 - 4*a*c])*e^4 - 8*c^3*d^2*e*(8*b*d + Sqrt[b^2 - 4*a*c]*d - 9*a*e)
+ 2*b*c*e^3*(b^2*d + 4*a*Sqrt[b^2 - 4*a*c]*e - b*(Sqrt[b^2 - 4*a*c]*d + 5*a*e)) + 2*c^2*e^2*(15*b^2*d^2 + 6*b
*d*(Sqrt[b^2 - 4*a*c]*d - 6*a*e) + 4*a*e*(-2*Sqrt[b^2 - 4*a*c]*d + 7*a*e)))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d +
e*x])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]))/(2*Sqrt[2]*(b^2 - 4*
a*c)^(3/2)*(c*d^2 + e*(-(b*d) + a*e))))/(2*(b^2 - 4*a*c)*(c*d^2 + e*(-(b*d) + a*e)))

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Maple [A]
time = 1.07, size = 1027, normalized size = 1.23 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

128*e^5*c^3*(-1/16/(-e^2*(4*a*c-b^2))^(1/2)/e^4/(4*a*c-b^2)^2*((-3/16*(-2*b*e+4*c*d-3*(-4*a*c*e^2+b^2*e^2)^(1/
2))*(-4*a*c*e^2+b^2*e^2)^(1/2)/c/(-2*a*c*e^2+b^2*e^2-2*b*c*d*e+2*c^2*d^2+b*e*(-4*a*c*e^2+b^2*e^2)^(1/2)-2*d*(-
4*a*c*e^2+b^2*e^2)^(1/2)*c)*(e*x+d)^(3/2)+1/16*(-6*b*e+12*c*d-11*(-4*a*c*e^2+b^2*e^2)^(1/2))*(-4*a*c*e^2+b^2*e
^2)^(1/2)/(-b*e+2*c*d-(-4*a*c*e^2+b^2*e^2)^(1/2))/c^2*(e*x+d)^(1/2))/(-e*x-1/2*b*e/c-1/2/c*(e^2*(-4*a*c+b^2))^
(1/2))^2+3/4*(-28*a*c*e^2+11*b^2*e^2-16*b*c*d*e+16*c^2*d^2+10*b*e*(-4*a*c*e^2+b^2*e^2)^(1/2)-20*d*(-4*a*c*e^2+
b^2*e^2)^(1/2)*c)/(-8*a*c*e^2+4*b^2*e^2-8*b*c*d*e+8*c^2*d^2+4*b*e*(-4*a*c*e^2+b^2*e^2)^(1/2)-8*d*(-4*a*c*e^2+b
^2*e^2)^(1/2)*c)*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctan(c*(e*x+d)^(1/2)*2^(1/2)/((b*e-2
*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)))+1/16/(-e^2*(4*a*c-b^2))^(1/2)/e^4/(4*a*c-b^2)^2*((3/16*(-2*b*e+4*c*d
+3*(-4*a*c*e^2+b^2*e^2)^(1/2))*(-4*a*c*e^2+b^2*e^2)^(1/2)/c/(-2*a*c*e^2+b^2*e^2-2*b*c*d*e+2*c^2*d^2-b*e*(-4*a*
c*e^2+b^2*e^2)^(1/2)+2*d*(-4*a*c*e^2+b^2*e^2)^(1/2)*c)*(e*x+d)^(3/2)-1/16*(-6*b*e+12*c*d+11*(-4*a*c*e^2+b^2*e^
2)^(1/2))*(-4*a*c*e^2+b^2*e^2)^(1/2)/(-b*e+2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2))/c^2*(e*x+d)^(1/2))/(-e*x-1/2*b*e/
c+1/2/c*(e^2*(-4*a*c+b^2))^(1/2))^2-3/4*(28*a*c*e^2-11*b^2*e^2+16*b*c*d*e-16*c^2*d^2+10*b*e*(-4*a*c*e^2+b^2*e^
2)^(1/2)-20*d*(-4*a*c*e^2+b^2*e^2)^(1/2)*c)/(8*a*c*e^2-4*b^2*e^2+8*b*c*d*e-8*c^2*d^2+4*b*e*(-4*a*c*e^2+b^2*e^2
)^(1/2)-8*d*(-4*a*c*e^2+b^2*e^2)^(1/2)*c)*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctanh(c*(e
*x+d)^(1/2)*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 4233 vs. \(2 (785) = 1570\).
time = 5.27, size = 4233, normalized size = 5.07 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/4*(16*c^5*d^2 - 4*(4*b*c^4 - 5*sqrt(b^2 - 4*a*c)*c^4)*d*e + (11*b^2*c^3 - 28*a*c^4 - 10*sqrt(b^2 - 4*a*c)*b
*c^3)*e^2)*sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e)*arctan(2*sqrt(1/2)*sqrt(x*e + d)/sqrt(-(2*b^4*c^3*
d^5 - 16*a*b^2*c^4*d^5 + 32*a^2*c^5*d^5 - 5*b^5*c^2*d^4*e + 40*a*b^3*c^3*d^4*e - 80*a^2*b*c^4*d^4*e + 4*b^6*c*
d^3*e^2 - 28*a*b^4*c^2*d^3*e^2 + 32*a^2*b^2*c^3*d^3*e^2 + 64*a^3*c^4*d^3*e^2 - b^7*d^2*e^3 + 2*a*b^5*c*d^2*e^3
+ 32*a^2*b^3*c^2*d^2*e^3 - 96*a^3*b*c^3*d^2*e^3 + 2*a*b^6*d*e^4 - 14*a^2*b^4*c*d*e^4 + 16*a^3*b^2*c^2*d*e^4 +
32*a^4*c^3*d*e^4 - a^2*b^5*e^5 + 8*a^3*b^3*c*e^5 - 16*a^4*b*c^2*e^5 + sqrt((2*b^4*c^3*d^5 - 16*a*b^2*c^4*d^5
+ 32*a^2*c^5*d^5 - 5*b^5*c^2*d^4*e + 40*a*b^3*c^3*d^4*e - 80*a^2*b*c^4*d^4*e + 4*b^6*c*d^3*e^2 - 28*a*b^4*c^2*
d^3*e^2 + 32*a^2*b^2*c^3*d^3*e^2 + 64*a^3*c^4*d^3*e^2 - b^7*d^2*e^3 + 2*a*b^5*c*d^2*e^3 + 32*a^2*b^3*c^2*d^2*e
^3 - 96*a^3*b*c^3*d^2*e^3 + 2*a*b^6*d*e^4 - 14*a^2*b^4*c*d*e^4 + 16*a^3*b^2*c^2*d*e^4 + 32*a^4*c^3*d*e^4 - a^2
*b^5*e^5 + 8*a^3*b^3*c*e^5 - 16*a^4*b*c^2*e^5)^2 - 4*(b^4*c^3*d^6 - 8*a*b^2*c^4*d^6 + 16*a^2*c^5*d^6 - 3*b^5*c
^2*d^5*e + 24*a*b^3*c^3*d^5*e - 48*a^2*b*c^4*d^5*e + 3*b^6*c*d^4*e^2 - 21*a*b^4*c^2*d^4*e^2 + 24*a^2*b^2*c^3*d
^4*e^2 + 48*a^3*c^4*d^4*e^2 - b^7*d^3*e^3 + 2*a*b^5*c*d^3*e^3 + 32*a^2*b^3*c^2*d^3*e^3 - 96*a^3*b*c^3*d^3*e^3
+ 3*a*b^6*d^2*e^4 - 21*a^2*b^4*c*d^2*e^4 + 24*a^3*b^2*c^2*d^2*e^4 + 48*a^4*c^3*d^2*e^4 - 3*a^2*b^5*d*e^5 + 24*
a^3*b^3*c*d*e^5 - 48*a^4*b*c^2*d*e^5 + a^3*b^4*e^6 - 8*a^4*b^2*c*e^6 + 16*a^5*c^2*e^6)*(b^4*c^3*d^4 - 8*a*b^2*
c^4*d^4 + 16*a^2*c^5*d^4 - 2*b^5*c^2*d^3*e + 16*a*b^3*c^3*d^3*e - 32*a^2*b*c^4*d^3*e + b^6*c*d^2*e^2 - 6*a*b^4
*c^2*d^2*e^2 + 32*a^3*c^4*d^2*e^2 - 2*a*b^5*c*d*e^3 + 16*a^2*b^3*c^2*d*e^3 - 32*a^3*b*c^3*d*e^3 + a^2*b^4*c*e^
4 - 8*a^3*b^2*c^2*e^4 + 16*a^4*c^3*e^4)))/(b^4*c^3*d^4 - 8*a*b^2*c^4*d^4 + 16*a^2*c^5*d^4 - 2*b^5*c^2*d^3*e +
16*a*b^3*c^3*d^3*e - 32*a^2*b*c^4*d^3*e + b^6*c*d^2*e^2 - 6*a*b^4*c^2*d^2*e^2 + 32*a^3*c^4*d^2*e^2 - 2*a*b^5*c
*d*e^3 + 16*a^2*b^3*c^2*d*e^3 - 32*a^3*b*c^3*d*e^3 + a^2*b^4*c*e^4 - 8*a^3*b^2*c^2*e^4 + 16*a^4*c^3*e^4)))/((2
*(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*sqrt(b^2 - 4*a*c)*d^3 + 3*(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*
a^3*c^5 - (b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*sqrt(b^2 - 4*a*c))*d^2*e - 3*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b
^3*c^3 - 64*a^3*b*c^4 - (b^6*c - 10*a*b^4*c^2 + 32*a^2*b^2*c^3 - 32*a^3*c^4)*sqrt(b^2 - 4*a*c))*d*e^2 + (b^8 -
13*a*b^6*c + 60*a^2*b^4*c^2 - 112*a^3*b^2*c^3 + 64*a^4*c^4 - (b^7 - 11*a*b^5*c + 40*a^2*b^3*c^2 - 48*a^3*b*c^
3)*sqrt(b^2 - 4*a*c))*e^3)*abs(c)) + 3/4*(16*c^5*d^2 - 4*(4*b*c^4 + 5*sqrt(b^2 - 4*a*c)*c^4)*d*e + (11*b^2*c^3
- 28*a*c^4 + 10*sqrt(b^2 - 4*a*c)*b*c^3)*e^2)*sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e)*arctan(2*sqrt(
1/2)*sqrt(x*e + d)/sqrt(-(2*b^4*c^3*d^5 - 16*a*b^2*c^4*d^5 + 32*a^2*c^5*d^5 - 5*b^5*c^2*d^4*e + 40*a*b^3*c^3*d
^4*e - 80*a^2*b*c^4*d^4*e + 4*b^6*c*d^3*e^2 - 28*a*b^4*c^2*d^3*e^2 + 32*a^2*b^2*c^3*d^3*e^2 + 64*a^3*c^4*d^3*e
^2 - b^7*d^2*e^3 + 2*a*b^5*c*d^2*e^3 + 32*a^2*b^3*c^2*d^2*e^3 - 96*a^3*b*c^3*d^2*e^3 + 2*a*b^6*d*e^4 - 14*a^2*
b^4*c*d*e^4 + 16*a^3*b^2*c^2*d*e^4 + 32*a^4*c^3*d*e^4 - a^2*b^5*e^5 + 8*a^3*b^3*c*e^5 - 16*a^4*b*c^2*e^5 - sqr
t((2*b^4*c^3*d^5 - 16*a*b^2*c^4*d^5 + 32*a^2*c^5*d^5 - 5*b^5*c^2*d^4*e + 40*a*b^3*c^3*d^4*e - 80*a^2*b*c^4*d^4
*e + 4*b^6*c*d^3*e^2 - 28*a*b^4*c^2*d^3*e^2 + 32*a^2*b^2*c^3*d^3*e^2 + 64*a^3*c^4*d^3*e^2 - b^7*d^2*e^3 + 2*a*
b^5*c*d^2*e^3 + 32*a^2*b^3*c^2*d^2*e^3 - 96*a^3*b*c^3*d^2*e^3 + 2*a*b^6*d*e^4 - 14*a^2*b^4*c*d*e^4 + 16*a^3*b^
2*c^2*d*e^4 + 32*a^4*c^3*d*e^4 - a^2*b^5*e^5 + 8*a^3*b^3*c*e^5 - 16*a^4*b*c^2*e^5)^2 - 4*(b^4*c^3*d^6 - 8*a*b^
2*c^4*d^6 + 16*a^2*c^5*d^6 - 3*b^5*c^2*d^5*e + 24*a*b^3*c^3*d^5*e - 48*a^2*b*c^4*d^5*e + 3*b^6*c*d^4*e^2 - 21*
a*b^4*c^2*d^4*e^2 + 24*a^2*b^2*c^3*d^4*e^2 + 48*a^3*c^4*d^4*e^2 - b^7*d^3*e^3 + 2*a*b^5*c*d^3*e^3 + 32*a^2*b^3
*c^2*d^3*e^3 - 96*a^3*b*c^3*d^3*e^3 + 3*a*b^6*d^2*e^4 - 21*a^2*b^4*c*d^2*e^4 + 24*a^3*b^2*c^2*d^2*e^4 + 48*a^4
*c^3*d^2*e^4 - 3*a^2*b^5*d*e^5 + 24*a^3*b^3*c*d*e^5 - 48*a^4*b*c^2*d*e^5 + a^3*b^4*e^6 - 8*a^4*b^2*c*e^6 + 16*
a^5*c^2*e^6)*(b^4*c^3*d^4 - 8*a*b^2*c^4*d^4 + 16*a^2*c^5*d^4 - 2*b^5*c^2*d^3*e + 16*a*b^3*c^3*d^3*e - 32*a^2*b
*c^4*d^3*e + b^6*c*d^2*e^2 - 6*a*b^4*c^2*d^2*e^2 + 32*a^3*c^4*d^2*e^2 - 2*a*b^5*c*d*e^3 + 16*a^2*b^3*c^2*d*e^3
- 32*a^3*b*c^3*d*e^3 + a^2*b^4*c*e^4 - 8*a^3*b^2*c^2*e^4 + 16*a^4*c^3*e^4)))/(b^4*c^3*d^4 - 8*a*b^2*c^4*d^4 +
16*a^2*c^5*d^4 - 2*b^5*c^2*d^3*e + 16*a*b^3*c^3*d^3*e - 32*a^2*b*c^4*d^3*e + b^6*c*d^2*e^2 - 6*a*b^4*c^2*d^2*
e^2 + 32*a^3*c^4*d^2*e^2 - 2*a*b^5*c*d*e^3 + 16*a^2*b^3*c^2*d*e^3 - 32*a^3*b*c^3*d*e^3 + a^2*b^4*c*e^4 - 8*a^3
*b^2*c^2*e^4 + 16*a^4*c^3*e^4)))/((2*(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*sqrt(b^2 - 4*a*c)*d^3 - 3*(b^6*c^2 -
12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5 + (b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*sqrt(b^2 - 4*a*c))*d^2*e
+ 3*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4 + (b^6*c - 10*a*b^4*c^2 + 32*a^2*b^2*c^3 - 32*a^3*c^
4)*sqrt(b^2 - 4*a*c))*d*e^2 - (b^8 - 13*a*b^6*c...

________________________________________________________________________________________

Mupad [B]
time = 18.17, size = 2500, normalized size = 2.99 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((e*(d + e*x)^(5/2)*(6*b^4*c*e^4 - 72*c^5*d^4 + 28*a^2*c^3*e^4 - 49*a*b^2*c^2*e^4 - 140*a*c^4*d^2*e^2 + b^3*c^
2*d*e^3 - 73*b^2*c^3*d^2*e^2 + 144*b*c^4*d^3*e + 140*a*b*c^3*d*e^3))/(4*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)*(a*e^2
+ c*d^2 - b*d*e)^2) - ((d + e*x)^(1/2)*(24*c^4*d^4*e - 5*b^4*e^5 - 44*a^2*c^2*e^5 + 60*a*c^3*d^2*e^3 - 48*b*c^
3*d^3*e^2 + 21*b^2*c^2*d^2*e^3 + 37*a*b^2*c*e^5 + 3*b^3*c*d*e^4 - 60*a*b*c^2*d*e^4))/(4*(b^4 + 16*a^2*c^2 - 8*
a*b^2*c)*(a*e^2 + c*d^2 - b*d*e)) + (e*(d + e*x)^(3/2)*(3*b^5*e^5 + 72*c^5*d^5 - 4*a^2*b*c^2*e^5 + 176*a*c^4*d
^3*e^2 + 8*a^2*c^3*d*e^4 + 136*b^2*c^3*d^3*e^2 - 24*b^3*c^2*d^2*e^3 - 20*a*b^3*c*e^5 - 180*b*c^4*d^4*e - 10*b^
4*c*d*e^4 - 264*a*b*c^3*d^2*e^3 + 128*a*b^2*c^2*d*e^4))/(4*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)*(a*e^2 + c*d^2 - b*d
*e)^2) + (3*c*e*(d + e*x)^(7/2)*(8*c^4*d^3 + b^3*c*e^3 + 2*b^2*c^2*d*e^2 - 8*a*b*c^2*e^3 + 16*a*c^3*d*e^2 - 12
*b*c^3*d^2*e))/(4*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)*(a*e^2 + c*d^2 - b*d*e)^2))/(c^2*(d + e*x)^4 - (d + e*x)*(4*c
^2*d^3 + 2*b^2*d*e^2 - 2*a*b*e^3 + 4*a*c*d*e^2 - 6*b*c*d^2*e) - (4*c^2*d - 2*b*c*e)*(d + e*x)^3 + (d + e*x)^2*
(b^2*e^2 + 6*c^2*d^2 + 2*a*c*e^2 - 6*b*c*d*e) + a^2*e^4 + c^2*d^4 + b^2*d^2*e^2 - 2*a*b*d*e^3 - 2*b*c*d^3*e +
2*a*c*d^2*e^2) + atan(((((3*(1835008*a^9*c^9*e^11 - 64*a^2*b^14*c^2*e^11 + 1856*a^3*b^12*c^3*e^11 - 23552*a^4*
b^10*c^4*e^11 + 168960*a^5*b^8*c^5*e^11 - 737280*a^6*b^6*c^6*e^11 + 1949696*a^7*b^4*c^7*e^11 - 2883584*a^8*b^2
*c^8*e^11 + 524288*a^5*c^13*d^8*e^3 + 2359296*a^6*c^12*d^6*e^5 + 4980736*a^7*c^11*d^4*e^7 + 4980736*a^8*c^10*d
^2*e^9 - 512*b^10*c^8*d^8*e^3 + 2048*b^11*c^7*d^7*e^4 - 3008*b^12*c^6*d^6*e^5 + 1856*b^13*c^5*d^5*e^6 - 384*b^
14*c^4*d^4*e^7 + 64*b^15*c^3*d^3*e^8 - 64*b^16*c^2*d^2*e^9 - 81920*a^2*b^6*c^10*d^8*e^3 + 327680*a^2*b^7*c^9*d
^7*e^4 - 435200*a^2*b^8*c^8*d^6*e^5 + 158720*a^2*b^9*c^7*d^5*e^6 + 57856*a^2*b^10*c^6*d^4*e^7 + 2048*a^2*b^11*
c^5*d^3*e^8 - 25408*a^2*b^12*c^4*d^2*e^9 + 327680*a^3*b^4*c^11*d^8*e^3 - 1310720*a^3*b^5*c^10*d^7*e^4 + 155648
0*a^3*b^6*c^9*d^6*e^5 - 81920*a^3*b^7*c^8*d^5*e^6 - 650240*a^3*b^8*c^7*d^4*e^7 - 92160*a^3*b^9*c^6*d^3*e^8 + 2
01216*a^3*b^10*c^5*d^2*e^9 - 655360*a^4*b^2*c^12*d^8*e^3 + 2621440*a^4*b^3*c^11*d^7*e^4 - 2375680*a^4*b^4*c^10
*d^6*e^5 - 2048000*a^4*b^5*c^9*d^5*e^6 + 2703360*a^4*b^6*c^8*d^4*e^7 + 1064960*a^4*b^7*c^7*d^3*e^8 - 936960*a^
4*b^8*c^6*d^2*e^9 + 131072*a^5*b^2*c^11*d^6*e^5 + 6946816*a^5*b^3*c^10*d^5*e^6 - 4440064*a^5*b^4*c^9*d^4*e^7 -
5144576*a^5*b^5*c^8*d^3*e^8 + 2359296*a^5*b^6*c^7*d^2*e^9 + 131072*a^6*b^2*c^10*d^4*e^7 + 11534336*a^6*b^3*c^
9*d^3*e^8 - 2146304*a^6*b^4*c^8*d^2*e^9 - 2490368*a^7*b^2*c^9*d^2*e^9 + 128*a*b^15*c^2*d*e^10 - 4980736*a^8*b*
c^9*d*e^10 + 10240*a*b^8*c^9*d^8*e^3 - 40960*a*b^9*c^8*d^7*e^4 + 57856*a*b^10*c^7*d^6*e^5 - 30208*a*b^11*c^6*d
^5*e^6 + 1472*a*b^12*c^5*d^4*e^7 - 384*a*b^13*c^4*d^3*e^8 + 1856*a*b^14*c^3*d^2*e^9 - 3776*a^2*b^13*c^3*d*e^10
+ 49664*a^3*b^11*c^4*d*e^10 - 373760*a^4*b^9*c^5*d*e^10 - 2097152*a^5*b*c^12*d^7*e^4 + 1720320*a^5*b^7*c^6*d*
e^10 - 7077888*a^6*b*c^11*d^5*e^6 - 4800512*a^6*b^5*c^7*d*e^10 - 9961472*a^7*b*c^10*d^3*e^8 + 7471104*a^7*b^3*
c^8*d*e^10))/(64*(a^4*b^12*e^8 + 4096*a^6*c^10*d^8 + 4096*a^10*c^6*e^8 + b^12*c^4*d^8 + b^16*d^4*e^4 - 24*a*b^
10*c^5*d^8 - 24*a^5*b^10*c*e^8 - 4*a*b^15*d^3*e^5 - 4*a^3*b^13*d*e^7 - 4*b^13*c^3*d^7*e - 4*b^15*c*d^5*e^3 + 2
40*a^2*b^8*c^6*d^8 - 1280*a^3*b^6*c^7*d^8 + 3840*a^4*b^4*c^8*d^8 - 6144*a^5*b^2*c^9*d^8 + 240*a^6*b^8*c^2*e^8
- 1280*a^7*b^6*c^3*e^8 + 3840*a^8*b^4*c^4*e^8 - 6144*a^9*b^2*c^5*e^8 + 6*a^2*b^14*d^2*e^6 + 16384*a^7*c^9*d^6*
e^2 + 24576*a^8*c^8*d^4*e^4 + 16384*a^9*c^7*d^2*e^6 + 6*b^14*c^2*d^6*e^2 + 1344*a^2*b^10*c^4*d^6*e^2 - 672*a^2
*b^11*c^3*d^5*e^3 - 42*a^2*b^12*c^2*d^4*e^4 - 6720*a^3*b^8*c^5*d^6*e^2 + 2240*a^3*b^9*c^4*d^5*e^3 + 1456*a^3*b
^10*c^3*d^4*e^4 - 672*a^3*b^11*c^2*d^3*e^5 + 17920*a^4*b^6*c^6*d^6*e^2 - 10080*a^4*b^8*c^4*d^4*e^4 + 2240*a^4*
b^9*c^3*d^3*e^5 + 1344*a^4*b^10*c^2*d^2*e^6 - 21504*a^5*b^4*c^7*d^6*e^2 - 21504*a^5*b^5*c^6*d^5*e^3 + 32256*a^
5*b^6*c^5*d^4*e^4 - 6720*a^5*b^8*c^3*d^2*e^6 + 57344*a^6*b^3*c^7*d^5*e^3 - 46592*a^6*b^4*c^6*d^4*e^4 - 21504*a
^6*b^5*c^5*d^3*e^5 + 17920*a^6*b^6*c^4*d^2*e^6 + 12288*a^7*b^2*c^7*d^4*e^4 + 57344*a^7*b^3*c^6*d^3*e^5 - 21504
*a^7*b^4*c^5*d^2*e^6 + 96*a*b^11*c^4*d^7*e - 12*a*b^14*c*d^4*e^4 + 96*a^4*b^11*c*d*e^7 - 16384*a^6*b*c^9*d^7*e
- 16384*a^9*b*c^6*d*e^7 - 140*a*b^12*c^3*d^6*e^2 + 84*a*b^13*c^2*d^5*e^3 - 960*a^2*b^9*c^5*d^7*e + 84*a^2*b^1
3*c*d^3*e^5 + 5120*a^3*b^7*c^6*d^7*e - 140*a^3*b^12*c*d^2*e^6 - 15360*a^4*b^5*c^7*d^7*e + 24576*a^5*b^3*c^8*d^
7*e - 960*a^5*b^9*c^2*d*e^7 + 5120*a^6*b^7*c^3*d*e^7 - 49152*a^7*b*c^8*d^5*e^3 - 15360*a^7*b^5*c^4*d*e^7 - 491
52*a^8*b*c^7*d^3*e^5 + 24576*a^8*b^3*c^5*d*e^7)) - ((d + e*x)^(1/2)*(-(9*(b^19*e^9 + 524288*a^5*c^14*d^9 - 512
*b^10*c^9*d^9 + b^4*e^9*(-(4*a*c - b^2)^15)^(1/2) + 10240*a*b^8*c^10*d^9 - 1720320*a^9*b*c^9*e^9 + 3440640*a^9
*c^10*d*e^8 + 2304*b^11*c^8*d^8*e - 81920*a^2*b^6*c^11*d^9 + 327680*a^3*b^4*c^12*d^9 - 655360*a^4*b^2*c^13*d^9
+ 769*a^2*b^15*c^2*e^9 - 8620*a^3*b^13*c^3*e^9...

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