3.2306 \(\int \frac{1}{\sqrt{d+e x} (a+b x+c x^2)^3} \, dx\)

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

[Out]

-(Sqrt[d + e*x]*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x))/(2*(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 - e*(b*d - a*e))
^2)

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Rubi [A]  time = 13.9194, antiderivative size = 834, normalized size of antiderivative = 1., 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 = {740, 822, 826, 1166, 208} \[ -\frac{\sqrt{d+e x} \left (-e b^2+c d b+2 a c e+c (2 c d-b e) x\right )}{2 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) \left (c x^2+b x+a\right )^2}-\frac{3 \sqrt{c} \left (32 c^4 d^4-8 c^3 e \left (8 b d-\sqrt{b^2-4 a c} d-9 a e\right ) d^2+b^3 \left (b+\sqrt{b^2-4 a c}\right ) e^4+2 b c e^3 \left (d b^2+\sqrt{b^2-4 a c} d b-5 a e b-4 a \sqrt{b^2-4 a c} e\right )+2 c^2 e^2 \left (15 b^2 d^2-6 b \left (\sqrt{b^2-4 a c} d+6 a e\right ) d+4 a e \left (2 \sqrt{b^2-4 a c} d+7 a e\right )\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e}}\right )}{4 \sqrt{2} \left (b^2-4 a c\right )^{5/2} \sqrt{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e} \left (c d^2-b e d+a e^2\right )^2}+\frac{3 \sqrt{c} \left (32 c^4 d^4-8 c^3 e \left (8 b d+\sqrt{b^2-4 a c} d-9 a e\right ) d^2+b^3 \left (b-\sqrt{b^2-4 a c}\right ) e^4+2 c^2 e^2 \left (15 b^2 d^2+6 b \left (\sqrt{b^2-4 a c} d-6 a e\right ) d-4 a e \left (2 \sqrt{b^2-4 a c} d-7 a e\right )\right )+2 b c e^3 \left (d b^2-\left (\sqrt{b^2-4 a c} d+5 a e\right ) b+4 a \sqrt{b^2-4 a c} e\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}}\right )}{4 \sqrt{2} \left (b^2-4 a c\right )^{5/2} \sqrt{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e} \left (c d^2-b e d+a e^2\right )^2}-\frac{\sqrt{d+e x} \left (5 a c e (2 c d-b e)^2-3 c \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right ) x (2 c d-b e)-\left (-e b^2+c d b+2 a c e\right ) \left (12 c^2 d^2-3 b^2 e^2-7 c e (b d-2 a e)\right )\right )}{4 \left (b^2-4 a c\right )^2 \left (c d^2-b e d+a e^2\right )^2 \left (c x^2+b x+a\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[d + e*x]*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x))/(2*(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 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 822

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 826

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 1166

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]

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}{\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{\operatorname{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 ) \operatorname{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 ) \operatorname{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*}

Mathematica [A]  time = 6.20855, size = 1056, normalized size = 1.26 \[ -\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{-\frac{\sqrt{d+e x} \left (-\frac{5}{2} a c e (2 c d-b e)^2+\frac{1}{2} \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 )+c \left (\frac{1}{2} (2 c d-b e) \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 (b d-2 a e) (2 c d-b e)\right ) x\right )}{\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 )}-\frac{2 \left (\frac{\sqrt{2 c d-b e-\sqrt{b^2-4 a c} e} \left (\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 )-\frac{2 c \left (\frac{3}{4} e \left (16 c^4 d^4-4 c^3 e (7 b d-9 a e) d^2+b^4 e^4+b^2 c e^3 (2 b d-9 a e)+c^2 e^2 \left (9 b^2 d^2-28 a b e d+28 a^2 e^2\right )\right )-\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 )\right )-\frac{3}{4} c e (2 c d-b e) (b e-2 c d) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right )}{\sqrt{b^2-4 a c} e}\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} \sqrt{c} \left (-2 c d+b e+\sqrt{b^2-4 a c} e\right )}+\frac{\sqrt{2 c d-b e+\sqrt{b^2-4 a c} e} \left (\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 )+\frac{2 c \left (\frac{3}{4} e \left (16 c^4 d^4-4 c^3 e (7 b d-9 a e) d^2+b^4 e^4+b^2 c e^3 (2 b d-9 a e)+c^2 e^2 \left (9 b^2 d^2-28 a b e d+28 a^2 e^2\right )\right )-\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 )\right )-\frac{3}{4} c e (2 c d-b e) (b e-2 c d) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right )}{\sqrt{b^2-4 a c} e}\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} \sqrt{c} \left (-2 c d+b e-\sqrt{b^2-4 a c} e\right )}\right )}{\left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )}}{2 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )} \]

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*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x))/(2*(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)/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)))/2 + c*((-5*c*e*(b*d - 2*a*e)*(2*c*d - b*e))/2 + ((2*c*d - b*e)*(12*c^2*d^2 - 3
*b^2*e^2 - 7*c*e*(b*d - 2*a*e)))/2)*x))/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*(a + b*x + c*x^2))) - (2*((Sqrt
[2*c*d - b*e - Sqrt[b^2 - 4*a*c]*e]*((3*c*e*(2*c*d - b*e)*(4*c^2*d^2 - b^2*e^2 - 4*c*e*(b*d - 2*a*e)))/4 - ((-
3*c*e*(2*c*d - b*e)*(-2*c*d + b*e)*(4*c^2*d^2 - b^2*e^2 - 4*c*e*(b*d - 2*a*e)))/4 + 2*c*((-3*c*d*e*(2*c*d - b*
e)*(4*c^2*d^2 - b^2*e^2 - 4*c*e*(b*d - 2*a*e)))/4 + (3*e*(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*(9*b^2*d^2 - 28*a*b*d*e + 28*a^2*e^2)))/4))/(Sqrt[b^2 - 4*a*c]*e))*ArcTan
h[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - b*e - Sqrt[b^2 - 4*a*c]*e]])/(Sqrt[2]*Sqrt[c]*(-2*c*d + b*e + S
qrt[b^2 - 4*a*c]*e)) + (Sqrt[2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e]*((3*c*e*(2*c*d - b*e)*(4*c^2*d^2 - b^2*e^2 - 4
*c*e*(b*d - 2*a*e)))/4 + ((-3*c*e*(2*c*d - b*e)*(-2*c*d + b*e)*(4*c^2*d^2 - b^2*e^2 - 4*c*e*(b*d - 2*a*e)))/4
+ 2*c*((-3*c*d*e*(2*c*d - b*e)*(4*c^2*d^2 - b^2*e^2 - 4*c*e*(b*d - 2*a*e)))/4 + (3*e*(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*(9*b^2*d^2 - 28*a*b*d*e + 28*a^2*e^2)))/4))/(
Sqrt[b^2 - 4*a*c]*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e]])/(Sqrt[
2]*Sqrt[c]*(-2*c*d + b*e - Sqrt[b^2 - 4*a*c]*e))))/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)))/(2*(b^2 - 4*a*c)*(
c*d^2 - b*d*e + a*e^2))

________________________________________________________________________________________

Maple [B]  time = 0.341, size = 4488, normalized size = 5.4 \begin{align*} \text{output too large to display} \end{align*}

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)

[Out]

-96*e*c^5/(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^2)^2/(-8*a*c*e^2+4*b^2*e^2-8*b*c*d*e+8*c^2*d^2+4*(-4*a*c*e^2+b^2*e
^2)^(1/2)*b*e-8*(-4*a*c*e^2+b^2*e^2)^(1/2)*c*d)*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctan(
(e*x+d)^(1/2)*c*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*d^2-3*e^2*c^2/(-e^2*(4*a*c-b^2))^(1/2)
/(4*a*c-b^2)^2/(e*x+1/2*b*e/c+1/2/c*(e^2*(-4*a*c+b^2))^(1/2))^2*(-4*a*c*e^2+b^2*e^2)^(1/2)/(-2*a*c*e^2+b^2*e^2
-2*b*c*d*e+2*c^2*d^2+(-4*a*c*e^2+b^2*e^2)^(1/2)*b*e-2*(-4*a*c*e^2+b^2*e^2)^(1/2)*c*d)*(e*x+d)^(3/2)*b+3*e^2*c/
(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^2)^2/(e*x+1/2*b*e/c-1/2/c*(e^2*(-4*a*c+b^2))^(1/2))^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))*(e*x+d)^(1/2)*b+6*e*c^3/(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^2)^2/(e
*x+1/2*b*e/c-1/2/c*(e^2*(-4*a*c+b^2))^(1/2))^2*(-4*a*c*e^2+b^2*e^2)^(1/2)/(-2*a*c*e^2+b^2*e^2-2*b*c*d*e+2*c^2*
d^2-(-4*a*c*e^2+b^2*e^2)^(1/2)*b*e+2*(-4*a*c*e^2+b^2*e^2)^(1/2)*c*d)*(e*x+d)^(3/2)*d+3*e^2*c/(-e^2*(4*a*c-b^2)
)^(1/2)/(4*a*c-b^2)^2/(e*x+1/2*b*e/c+1/2/c*(e^2*(-4*a*c+b^2))^(1/2))^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))*(e*x+d)^(1/2)*b-6*e*c^2/(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^2)^2/(e*x+1/2*b*e/c+1/2/
c*(e^2*(-4*a*c+b^2))^(1/2))^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))*(e*x+d)^(1/2)
*d+168*e^3*c^4/(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^2)^2/(-8*a*c*e^2+4*b^2*e^2-8*b*c*d*e+8*c^2*d^2+4*(-4*a*c*e^2+
b^2*e^2)^(1/2)*b*e-8*(-4*a*c*e^2+b^2*e^2)^(1/2)*c*d)*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*ar
ctan((e*x+d)^(1/2)*c*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*a-6*e*c^2/(-e^2*(4*a*c-b^2))^(1/2
)/(4*a*c-b^2)^2/(e*x+1/2*b*e/c-1/2/c*(e^2*(-4*a*c+b^2))^(1/2))^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))*(e*x+d)^(1/2)*d-168*e^3*c^4/(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^2)^2/(8*a*c*e^2-4*b^2*e^2+
8*b*c*d*e-8*c^2*d^2+4*(-4*a*c*e^2+b^2*e^2)^(1/2)*b*e-8*(-4*a*c*e^2+b^2*e^2)^(1/2)*c*d)*2^(1/2)/((-b*e+2*c*d+(-
e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)*c*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/
2))*a+96*e*c^5/(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^2)^2/(8*a*c*e^2-4*b^2*e^2+8*b*c*d*e-8*c^2*d^2+4*(-4*a*c*e^2+b
^2*e^2)^(1/2)*b*e-8*(-4*a*c*e^2+b^2*e^2)^(1/2)*c*d)*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*ar
ctanh((e*x+d)^(1/2)*c*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*d^2-3*e^2*c^2/(-e^2*(4*a*c-b^2)
)^(1/2)/(4*a*c-b^2)^2/(e*x+1/2*b*e/c-1/2/c*(e^2*(-4*a*c+b^2))^(1/2))^2*(-4*a*c*e^2+b^2*e^2)^(1/2)/(-2*a*c*e^2+
b^2*e^2-2*b*c*d*e+2*c^2*d^2-(-4*a*c*e^2+b^2*e^2)^(1/2)*b*e+2*(-4*a*c*e^2+b^2*e^2)^(1/2)*c*d)*(e*x+d)^(3/2)*b+6
6*e^3*c^3/(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^2)^2/(8*a*c*e^2-4*b^2*e^2+8*b*c*d*e-8*c^2*d^2+4*(-4*a*c*e^2+b^2*e^
2)^(1/2)*b*e-8*(-4*a*c*e^2+b^2*e^2)^(1/2)*c*d)*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctanh
((e*x+d)^(1/2)*c*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*b^2+6*e*c^3/(-e^2*(4*a*c-b^2))^(1/2)
/(4*a*c-b^2)^2/(e*x+1/2*b*e/c+1/2/c*(e^2*(-4*a*c+b^2))^(1/2))^2*(-4*a*c*e^2+b^2*e^2)^(1/2)/(-2*a*c*e^2+b^2*e^2
-2*b*c*d*e+2*c^2*d^2+(-4*a*c*e^2+b^2*e^2)^(1/2)*b*e-2*(-4*a*c*e^2+b^2*e^2)^(1/2)*c*d)*(e*x+d)^(3/2)*d-66*e^3*c
^3/(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^2)^2/(-8*a*c*e^2+4*b^2*e^2-8*b*c*d*e+8*c^2*d^2+4*(-4*a*c*e^2+b^2*e^2)^(1/
2)*b*e-8*(-4*a*c*e^2+b^2*e^2)^(1/2)*c*d)*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctan((e*x+d)
^(1/2)*c*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*b^2+9/2*e^3*c^2/(-e^2*(4*a*c-b^2))^(1/2)/(4*a
*c-b^2)^2/(e*x+1/2*b*e/c-1/2/c*(e^2*(-4*a*c+b^2))^(1/2))^2/(-2*a*c*e^2+b^2*e^2-2*b*c*d*e+2*c^2*d^2-(-4*a*c*e^2
+b^2*e^2)^(1/2)*b*e+2*(-4*a*c*e^2+b^2*e^2)^(1/2)*c*d)*(e*x+d)^(3/2)*b^2+22*e^3*c^2/(-e^2*(4*a*c-b^2))^(1/2)/(4
*a*c-b^2)^2/(e*x+1/2*b*e/c-1/2/c*(e^2*(-4*a*c+b^2))^(1/2))^2/(-b*e+2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2))*(e*x+d)^(
1/2)*a-11/2*e^3*c/(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^2)^2/(e*x+1/2*b*e/c-1/2/c*(e^2*(-4*a*c+b^2))^(1/2))^2/(-b*
e+2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2))*(e*x+d)^(1/2)*b^2-9/2*e^3*c^2/(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^2)^2/(e*x+
1/2*b*e/c+1/2/c*(e^2*(-4*a*c+b^2))^(1/2))^2/(-2*a*c*e^2+b^2*e^2-2*b*c*d*e+2*c^2*d^2+(-4*a*c*e^2+b^2*e^2)^(1/2)
*b*e-2*(-4*a*c*e^2+b^2*e^2)^(1/2)*c*d)*(e*x+d)^(3/2)*b^2-22*e^3*c^2/(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^2)^2/(e*
x+1/2*b*e/c+1/2/c*(e^2*(-4*a*c+b^2))^(1/2))^2/(-b*e+2*c*d-(-4*a*c*e^2+b^2*e^2)^(1/2))*(e*x+d)^(1/2)*a+11/2*e^3
*c/(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^2)^2/(e*x+1/2*b*e/c+1/2/c*(e^2*(-4*a*c+b^2))^(1/2))^2/(-b*e+2*c*d-(-4*a*c
*e^2+b^2*e^2)^(1/2))*(e*x+d)^(1/2)*b^2-18*e^3*c^3/(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^2)^2/(e*x+1/2*b*e/c-1/2/c*
(e^2*(-4*a*c+b^2))^(1/2))^2/(-2*a*c*e^2+b^2*e^2-2*b*c*d*e+2*c^2*d^2-(-4*a*c*e^2+b^2*e^2)^(1/2)*b*e+2*(-4*a*c*e
^2+b^2*e^2)^(1/2)*c*d)*(e*x+d)^(3/2)*a+18*e^3*c^3/(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^2)^2/(e*x+1/2*b*e/c+1/2/c*
(e^2*(-4*a*c+b^2))^(1/2))^2/(-2*a*c*e^2+b^2*e^2-2*b*c*d*e+2*c^2*d^2+(-4*a*c*e^2+b^2*e^2)^(1/2)*b*e-2*(-4*a*c*e
^2+b^2*e^2)^(1/2)*c*d)*(e*x+d)^(3/2)*a-60*e^2*c^3/(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^2)^2/(-8*a*c*e^2+4*b^2*e^2
-8*b*c*d*e+8*c^2*d^2+4*(-4*a*c*e^2+b^2*e^2)^(1/2)*b*e-8*(-4*a*c*e^2+b^2*e^2)^(1/2)*c*d)*2^(1/2)/((b*e-2*c*d+(-
e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)*c*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)
)*(-4*a*c*e^2+b^2*e^2)^(1/2)*b-60*e^2*c^3/(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^2)^2/(8*a*c*e^2-4*b^2*e^2+8*b*c*d*
e-8*c^2*d^2+4*(-4*a*c*e^2+b^2*e^2)^(1/2)*b*e-8*(-4*a*c*e^2+b^2*e^2)^(1/2)*c*d)*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a
*c-b^2))^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)*c*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*(-4*
a*c*e^2+b^2*e^2)^(1/2)*b-96*e^2*c^4/(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^2)^2/(8*a*c*e^2-4*b^2*e^2+8*b*c*d*e-8*c^
2*d^2+4*(-4*a*c*e^2+b^2*e^2)^(1/2)*b*e-8*(-4*a*c*e^2+b^2*e^2)^(1/2)*c*d)*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2
))^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)*c*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*b*d+120*e*
c^4/(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^2)^2/(8*a*c*e^2-4*b^2*e^2+8*b*c*d*e-8*c^2*d^2+4*(-4*a*c*e^2+b^2*e^2)^(1/
2)*b*e-8*(-4*a*c*e^2+b^2*e^2)^(1/2)*c*d)*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctanh((e*x+
d)^(1/2)*c*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*(-4*a*c*e^2+b^2*e^2)^(1/2)*d+96*e^2*c^4/(-
e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^2)^2/(-8*a*c*e^2+4*b^2*e^2-8*b*c*d*e+8*c^2*d^2+4*(-4*a*c*e^2+b^2*e^2)^(1/2)*b*
e-8*(-4*a*c*e^2+b^2*e^2)^(1/2)*c*d)*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2
)*c*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*b*d+120*e*c^4/(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^2)
^2/(-8*a*c*e^2+4*b^2*e^2-8*b*c*d*e+8*c^2*d^2+4*(-4*a*c*e^2+b^2*e^2)^(1/2)*b*e-8*(-4*a*c*e^2+b^2*e^2)^(1/2)*c*d
)*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)*c*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a
*c-b^2))^(1/2))*c)^(1/2))*(-4*a*c*e^2+b^2*e^2)^(1/2)*d

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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 )}^{3} \sqrt{e x + d}}\,{d x} \end{align*}

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [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)^(1/2)/(c*x^2+b*x+a)^3,x, algorithm="giac")

[Out]

Timed out