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

Optimal. Leaf size=224 \[ -\frac{\left (16 c^2 \left (3 a^2-8 a d+8 d^2\right )+8 b^2 c (a-4 d)+3 b^4\right ) \tanh ^{-1}\left (\frac{\sqrt{a-d} (b+2 c x)}{\sqrt{b^2-4 c d} \sqrt{a+b x+c x^2}}\right )}{4 (a-d)^{5/2} \left (b^2-4 c d\right )^{5/2}}+\frac{3 (b+2 c x) \left (4 c (a-2 d)+b^2\right ) \sqrt{a+b x+c x^2}}{4 (a-d)^2 \left (b^2-4 c d\right )^2 \left (b x+c x^2+d\right )}-\frac{(b+2 c x) \sqrt{a+b x+c x^2}}{2 (a-d) \left (b^2-4 c d\right ) \left (b x+c x^2+d\right )^2} \]

[Out]

-((b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(2*(a - d)*(b^2 - 4*c*d)*(d + b*x + c*x^2)^2) + (3*(b^2 + 4*c*(a - 2*d))*
(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(4*(a - d)^2*(b^2 - 4*c*d)^2*(d + b*x + c*x^2)) - ((3*b^4 + 8*b^2*c*(a - 4*
d) + 16*c^2*(3*a^2 - 8*a*d + 8*d^2))*ArcTanh[(Sqrt[a - d]*(b + 2*c*x))/(Sqrt[b^2 - 4*c*d]*Sqrt[a + b*x + c*x^2
])])/(4*(a - d)^(5/2)*(b^2 - 4*c*d)^(5/2))

________________________________________________________________________________________

Rubi [A]  time = 0.426872, antiderivative size = 224, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {974, 1060, 12, 982, 208} \[ -\frac{\left (16 c^2 \left (3 a^2-8 a d+8 d^2\right )+8 b^2 c (a-4 d)+3 b^4\right ) \tanh ^{-1}\left (\frac{\sqrt{a-d} (b+2 c x)}{\sqrt{b^2-4 c d} \sqrt{a+b x+c x^2}}\right )}{4 (a-d)^{5/2} \left (b^2-4 c d\right )^{5/2}}+\frac{3 (b+2 c x) \left (4 c (a-2 d)+b^2\right ) \sqrt{a+b x+c x^2}}{4 (a-d)^2 \left (b^2-4 c d\right )^2 \left (b x+c x^2+d\right )}-\frac{(b+2 c x) \sqrt{a+b x+c x^2}}{2 (a-d) \left (b^2-4 c d\right ) \left (b x+c x^2+d\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(2*(a - d)*(b^2 - 4*c*d)*(d + b*x + c*x^2)^2) + (3*(b^2 + 4*c*(a - 2*d))*
(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(4*(a - d)^2*(b^2 - 4*c*d)^2*(d + b*x + c*x^2)) - ((3*b^4 + 8*b^2*c*(a - 4*
d) + 16*c^2*(3*a^2 - 8*a*d + 8*d^2))*ArcTanh[(Sqrt[a - d]*(b + 2*c*x))/(Sqrt[b^2 - 4*c*d]*Sqrt[a + b*x + c*x^2
])])/(4*(a - d)^(5/2)*(b^2 - 4*c*d)^(5/2))

Rule 974

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_.) + (e_.)*(x_) + (f_.)*(x_)^2)^(q_), x_Symbol] :> Simp[((2*a
*c^2*e - b^2*c*e + b^3*f + b*c*(c*d - 3*a*f) + c*(2*c^2*d + b^2*f - c*(b*e + 2*a*f))*x)*(a + b*x + c*x^2)^(p +
1)*(d + e*x + f*x^2)^(q + 1))/((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1)), x] - Dist[1/
((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1)), Int[(a + b*x + c*x^2)^(p + 1)*(d + e*x + f*
x^2)^q*Simp[2*c*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1) - (2*c^2*d + b^2*f - c*(b*e + 2*a*f))*(a*f*(
p + 1) - c*d*(p + 2)) - e*(b^2*c*e - 2*a*c^2*e - b^3*f - b*c*(c*d - 3*a*f))*(p + q + 2) + (2*f*(2*a*c^2*e - b^
2*c*e + b^3*f + b*c*(c*d - 3*a*f))*(p + q + 2) - (2*c^2*d + b^2*f - c*(b*e + 2*a*f))*(b*f*(p + 1) - c*e*(2*p +
q + 4)))*x + c*f*(2*c^2*d + b^2*f - c*(b*e + 2*a*f))*(2*p + 2*q + 5)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e,
f, q}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && LtQ[p, -1] && NeQ[(c*d - a*f)^2 - (b*d - a*e)*(c*e
- b*f), 0] &&  !( !IntegerQ[p] && ILtQ[q, -1]) &&  !IGtQ[q, 0]

Rule 1060

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((A_.) + (B_.)*(x_) + (C_.)*(x_)^2)*((d_) + (e_.)*(x_) + (f_.)*(x_
)^2)^(q_), x_Symbol] :> Simp[((a + b*x + c*x^2)^(p + 1)*(d + e*x + f*x^2)^(q + 1)*((A*c - a*C)*(2*a*c*e - b*(c
*d + a*f)) + (A*b - a*B)*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)) + c*(A*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)) - B*(b
*c*d - 2*a*c*e + a*b*f) + C*(b^2*d - a*b*e - 2*a*(c*d - a*f)))*x))/((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)
*(c*e - b*f))*(p + 1)), x] + Dist[1/((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1)), Int[(a
+ b*x + c*x^2)^(p + 1)*(d + e*x + f*x^2)^q*Simp[(b*B - 2*A*c - 2*a*C)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f)
)*(p + 1) + (b^2*(C*d + A*f) - b*(B*c*d + A*c*e + a*C*e + a*B*f) + 2*(A*c*(c*d - a*f) - a*(c*C*d - B*c*e - a*C
*f)))*(a*f*(p + 1) - c*d*(p + 2)) - e*((A*c - a*C)*(2*a*c*e - b*(c*d + a*f)) + (A*b - a*B)*(2*c^2*d + b^2*f -
c*(b*e + 2*a*f)))*(p + q + 2) - (2*f*((A*c - a*C)*(2*a*c*e - b*(c*d + a*f)) + (A*b - a*B)*(2*c^2*d + b^2*f - c
*(b*e + 2*a*f)))*(p + q + 2) - (b^2*(C*d + A*f) - b*(B*c*d + A*c*e + a*C*e + a*B*f) + 2*(A*c*(c*d - a*f) - a*(
c*C*d - B*c*e - a*C*f)))*(b*f*(p + 1) - c*e*(2*p + q + 4)))*x - c*f*(b^2*(C*d + A*f) - b*(B*c*d + A*c*e + a*C*
e + a*B*f) + 2*(A*c*(c*d - a*f) - a*(c*C*d - B*c*e - a*C*f)))*(2*p + 2*q + 5)*x^2, x], x], x] /; FreeQ[{a, b,
c, d, e, f, A, B, C, q}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && LtQ[p, -1] && NeQ[(c*d - a*f)^2 -
(b*d - a*e)*(c*e - b*f), 0] &&  !( !IntegerQ[p] && ILtQ[q, -1]) &&  !IGtQ[q, 0]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 982

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

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{a+b x+c x^2} \left (d+b x+c x^2\right )^3} \, dx &=-\frac{(b+2 c x) \sqrt{a+b x+c x^2}}{2 (a-d) \left (b^2-4 c d\right ) \left (d+b x+c x^2\right )^2}+\frac{\int \frac{-\frac{1}{2} c^2 (a-d) \left (3 b^2+12 a c-16 c d\right )-4 b c^3 (a-d) x-4 c^4 (a-d) x^2}{\sqrt{a+b x+c x^2} \left (d+b x+c x^2\right )^2} \, dx}{2 c^2 (a-d)^2 \left (b^2-4 c d\right )}\\ &=-\frac{(b+2 c x) \sqrt{a+b x+c x^2}}{2 (a-d) \left (b^2-4 c d\right ) \left (d+b x+c x^2\right )^2}+\frac{3 \left (b^2+4 c (a-2 d)\right ) (b+2 c x) \sqrt{a+b x+c x^2}}{4 (a-d)^2 \left (b^2-4 c d\right )^2 \left (d+b x+c x^2\right )}-\frac{\int -\frac{c^4 (a-d)^2 \left (3 b^4+8 b^2 c (a-4 d)+16 c^2 \left (3 a^2-8 a d+8 d^2\right )\right )}{4 \sqrt{a+b x+c x^2} \left (d+b x+c x^2\right )} \, dx}{2 c^4 (a-d)^4 \left (b^2-4 c d\right )^2}\\ &=-\frac{(b+2 c x) \sqrt{a+b x+c x^2}}{2 (a-d) \left (b^2-4 c d\right ) \left (d+b x+c x^2\right )^2}+\frac{3 \left (b^2+4 c (a-2 d)\right ) (b+2 c x) \sqrt{a+b x+c x^2}}{4 (a-d)^2 \left (b^2-4 c d\right )^2 \left (d+b x+c x^2\right )}+\frac{\left (3 b^4+8 b^2 c (a-4 d)+16 c^2 \left (3 a^2-8 a d+8 d^2\right )\right ) \int \frac{1}{\sqrt{a+b x+c x^2} \left (d+b x+c x^2\right )} \, dx}{8 (a-d)^2 \left (b^2-4 c d\right )^2}\\ &=-\frac{(b+2 c x) \sqrt{a+b x+c x^2}}{2 (a-d) \left (b^2-4 c d\right ) \left (d+b x+c x^2\right )^2}+\frac{3 \left (b^2+4 c (a-2 d)\right ) (b+2 c x) \sqrt{a+b x+c x^2}}{4 (a-d)^2 \left (b^2-4 c d\right )^2 \left (d+b x+c x^2\right )}-\frac{\left (b \left (3 b^4+8 b^2 c (a-4 d)+16 c^2 \left (3 a^2-8 a d+8 d^2\right )\right )\right ) \operatorname{Subst}\left (\int \frac{1}{b \left (b^2-4 c d\right )-(a b-b d) x^2} \, dx,x,\frac{b+2 c x}{\sqrt{a+b x+c x^2}}\right )}{4 (a-d)^2 \left (b^2-4 c d\right )^2}\\ &=-\frac{(b+2 c x) \sqrt{a+b x+c x^2}}{2 (a-d) \left (b^2-4 c d\right ) \left (d+b x+c x^2\right )^2}+\frac{3 \left (b^2+4 c (a-2 d)\right ) (b+2 c x) \sqrt{a+b x+c x^2}}{4 (a-d)^2 \left (b^2-4 c d\right )^2 \left (d+b x+c x^2\right )}-\frac{\left (3 b^4+8 b^2 c (a-4 d)+16 c^2 \left (3 a^2-8 a d+8 d^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{a-d} (b+2 c x)}{\sqrt{b^2-4 c d} \sqrt{a+b x+c x^2}}\right )}{4 (a-d)^{5/2} \left (b^2-4 c d\right )^{5/2}}\\ \end{align*}

Mathematica [B]  time = 6.37555, size = 1748, normalized size = 7.8 \[ \text{result too large to display} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*c^2*(a + b*x + c*x^2))/((a - d)*(b^2 - 4*c*d)^(3/2)*(b - Sqrt[b^2 - 4*c*d] + 2*c*x)^2*Sqrt[a + x*(b + c*x)
]) + (6*c^2*(a + b*x + c*x^2))/((a - d)*(b^2 - 4*c*d)^2*(b - Sqrt[b^2 - 4*c*d] + 2*c*x)*Sqrt[a + x*(b + c*x)])
+ (2*c^2*(a + b*x + c*x^2))/((a - d)*(b^2 - 4*c*d)^(3/2)*(b + Sqrt[b^2 - 4*c*d] + 2*c*x)^2*Sqrt[a + x*(b + c*
x)]) + (6*c^2*(a + b*x + c*x^2))/((a - d)*(b^2 - 4*c*d)^2*(b + Sqrt[b^2 - 4*c*d] + 2*c*x)*Sqrt[a + x*(b + c*x)
]) + (6*c^2*Sqrt[a + b*x + c*x^2]*ArcTanh[(b^2 - 4*a*c - b*Sqrt[b^2 - 4*c*d] - 2*c*Sqrt[b^2 - 4*c*d]*x)/(4*c*S
qrt[a - d]*Sqrt[a + b*x + c*x^2])])/(Sqrt[a - d]*(b^2 - 4*c*d)^(5/2)*Sqrt[a + x*(b + c*x)]) + (3*c*Sqrt[a + b*
x + c*x^2]*ArcTanh[(b^2 - 4*a*c - b*Sqrt[b^2 - 4*c*d] - 2*c*Sqrt[b^2 - 4*c*d]*x)/(4*c*Sqrt[a - d]*Sqrt[a + b*x
+ c*x^2])])/(2*(a - d)^(3/2)*(b^2 - 4*c*d)^(3/2)*Sqrt[a + x*(b + c*x)]) + (6*c^2*Sqrt[a + b*x + c*x^2]*ArcTan
h[(4*a*c - b*(b + Sqrt[b^2 - 4*c*d]) - 2*c*Sqrt[b^2 - 4*c*d]*x)/(4*c*Sqrt[a - d]*Sqrt[a + b*x + c*x^2])])/(Sqr
t[a - d]*(b^2 - 4*c*d)^(5/2)*Sqrt[a + x*(b + c*x)]) + (3*c*Sqrt[a + b*x + c*x^2]*ArcTanh[(4*a*c - b*(b + Sqrt[
b^2 - 4*c*d]) - 2*c*Sqrt[b^2 - 4*c*d]*x)/(4*c*Sqrt[a - d]*Sqrt[a + b*x + c*x^2])])/(2*(a - d)^(3/2)*(b^2 - 4*c
*d)^(3/2)*Sqrt[a + x*(b + c*x)]) + (4*c^3*Sqrt[a + b*x + c*x^2]*(-(((2*c^2*(-b + Sqrt[b^2 - 4*c*d]) + 2*c^2*(b
+ 2*Sqrt[b^2 - 4*c*d]))*Sqrt[a + b*x + c*x^2])/((4*a*c^2 + 2*b*c*(-b + Sqrt[b^2 - 4*c*d]) + c*(-b + Sqrt[b^2
- 4*c*d])^2)*(-b + Sqrt[b^2 - 4*c*d] - 2*c*x))) + (4*c*Sqrt[a - d]*(b*(-2*c^2*(-b + Sqrt[b^2 - 4*c*d]) + 2*c^2
*(b + 2*Sqrt[b^2 - 4*c*d])) - 2*(4*a*c^3 - c^2*(-b + Sqrt[b^2 - 4*c*d])*(b + 2*Sqrt[b^2 - 4*c*d])))*ArcTanh[(-
4*a*c - b*(-b + Sqrt[b^2 - 4*c*d]) - (2*b*c + 2*c*(-b + Sqrt[b^2 - 4*c*d]))*x)/(4*c*Sqrt[a - d]*Sqrt[a + b*x +
c*x^2])])/((4*a*c^2 + 2*b*c*(-b + Sqrt[b^2 - 4*c*d]) + c*(-b + Sqrt[b^2 - 4*c*d])^2)*(16*a*c^2 + 8*b*c*(-b +
Sqrt[b^2 - 4*c*d]) + 4*c*(-b + Sqrt[b^2 - 4*c*d])^2))))/((b^2 - 4*c*d)^(3/2)*(4*a*c^2 + 2*b*c*(-b + Sqrt[b^2 -
4*c*d]) + c*(-b + Sqrt[b^2 - 4*c*d])^2)*Sqrt[a + x*(b + c*x)]) + (4*c^3*Sqrt[a + b*x + c*x^2]*(-(((2*c^2*(b -
2*Sqrt[b^2 - 4*c*d]) - 2*c^2*(b + Sqrt[b^2 - 4*c*d]))*Sqrt[a + b*x + c*x^2])/((4*a*c^2 - 2*b*c*(b + Sqrt[b^2
- 4*c*d]) + c*(b + Sqrt[b^2 - 4*c*d])^2)*(b + Sqrt[b^2 - 4*c*d] + 2*c*x))) + (4*c*Sqrt[a - d]*(b*(2*c^2*(b - 2
*Sqrt[b^2 - 4*c*d]) + 2*c^2*(b + Sqrt[b^2 - 4*c*d])) - 2*(4*a*c^3 + c^2*(b - 2*Sqrt[b^2 - 4*c*d])*(b + Sqrt[b^
2 - 4*c*d])))*ArcTanh[(4*a*c - b*(b + Sqrt[b^2 - 4*c*d]) - (-2*b*c + 2*c*(b + Sqrt[b^2 - 4*c*d]))*x)/(4*c*Sqrt
[a - d]*Sqrt[a + b*x + c*x^2])])/((4*a*c^2 - 2*b*c*(b + Sqrt[b^2 - 4*c*d]) + c*(b + Sqrt[b^2 - 4*c*d])^2)*(16*
a*c^2 - 8*b*c*(b + Sqrt[b^2 - 4*c*d]) + 4*c*(b + Sqrt[b^2 - 4*c*d])^2))))/((b^2 - 4*c*d)^(3/2)*(4*a*c^2 - 2*b*
c*(b + Sqrt[b^2 - 4*c*d]) + c*(b + Sqrt[b^2 - 4*c*d])^2)*Sqrt[a + x*(b + c*x)])

________________________________________________________________________________________

Maple [B]  time = 0.242, size = 1884, normalized size = 8.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/(b^2-4*c*d)^(3/2)/(a-d)/(x-1/2/c*(b^2-4*c*d)^(1/2)+1/2*b/c)^2*((x-1/2*(-b+(b^2-4*c*d)^(1/2))/c)^2*c+(b^2-
4*c*d)^(1/2)*(x-1/2*(-b+(b^2-4*c*d)^(1/2))/c)+a-d)^(1/2)+3/4/(b^2-4*c*d)/(a-d)^2/(x-1/2/c*(b^2-4*c*d)^(1/2)+1/
2*b/c)*((x-1/2*(-b+(b^2-4*c*d)^(1/2))/c)^2*c+(b^2-4*c*d)^(1/2)*(x-1/2*(-b+(b^2-4*c*d)^(1/2))/c)+a-d)^(1/2)-3/8
/(b^2-4*c*d)^(3/2)/(a-d)^(5/2)*ln((2*a-2*d+(b^2-4*c*d)^(1/2)*(x-1/2*(-b+(b^2-4*c*d)^(1/2))/c)+2*(a-d)^(1/2)*((
x-1/2*(-b+(b^2-4*c*d)^(1/2))/c)^2*c+(b^2-4*c*d)^(1/2)*(x-1/2*(-b+(b^2-4*c*d)^(1/2))/c)+a-d)^(1/2))/(x-1/2*(-b+
(b^2-4*c*d)^(1/2))/c))*b^2+3/2/(b^2-4*c*d)^(3/2)/(a-d)^(5/2)*ln((2*a-2*d+(b^2-4*c*d)^(1/2)*(x-1/2*(-b+(b^2-4*c
*d)^(1/2))/c)+2*(a-d)^(1/2)*((x-1/2*(-b+(b^2-4*c*d)^(1/2))/c)^2*c+(b^2-4*c*d)^(1/2)*(x-1/2*(-b+(b^2-4*c*d)^(1/
2))/c)+a-d)^(1/2))/(x-1/2*(-b+(b^2-4*c*d)^(1/2))/c))*c*d-1/(b^2-4*c*d)^(3/2)*c/(a-d)^(3/2)*ln((2*a-2*d+(b^2-4*
c*d)^(1/2)*(x-1/2*(-b+(b^2-4*c*d)^(1/2))/c)+2*(a-d)^(1/2)*((x-1/2*(-b+(b^2-4*c*d)^(1/2))/c)^2*c+(b^2-4*c*d)^(1
/2)*(x-1/2*(-b+(b^2-4*c*d)^(1/2))/c)+a-d)^(1/2))/(x-1/2*(-b+(b^2-4*c*d)^(1/2))/c))+6*c^2/(b^2-4*c*d)^(5/2)/(a-
d)^(1/2)*ln((2*a-2*d-(b^2-4*c*d)^(1/2)*(x+1/2*((b^2-4*c*d)^(1/2)+b)/c)+2*(a-d)^(1/2)*((x+1/2*((b^2-4*c*d)^(1/2
)+b)/c)^2*c-(b^2-4*c*d)^(1/2)*(x+1/2*((b^2-4*c*d)^(1/2)+b)/c)+a-d)^(1/2))/(x+1/2*((b^2-4*c*d)^(1/2)+b)/c))+3/(
b^2-4*c*d)^2*c/(a-d)/(x-1/2/c*(b^2-4*c*d)^(1/2)+1/2*b/c)*((x-1/2*(-b+(b^2-4*c*d)^(1/2))/c)^2*c+(b^2-4*c*d)^(1/
2)*(x-1/2*(-b+(b^2-4*c*d)^(1/2))/c)+a-d)^(1/2)+1/2/(b^2-4*c*d)^(3/2)/(a-d)/(x+1/2/c*(b^2-4*c*d)^(1/2)+1/2*b/c)
^2*((x+1/2*((b^2-4*c*d)^(1/2)+b)/c)^2*c-(b^2-4*c*d)^(1/2)*(x+1/2*((b^2-4*c*d)^(1/2)+b)/c)+a-d)^(1/2)+3/4/(b^2-
4*c*d)/(a-d)^2/(x+1/2/c*(b^2-4*c*d)^(1/2)+1/2*b/c)*((x+1/2*((b^2-4*c*d)^(1/2)+b)/c)^2*c-(b^2-4*c*d)^(1/2)*(x+1
/2*((b^2-4*c*d)^(1/2)+b)/c)+a-d)^(1/2)+3/8/(b^2-4*c*d)^(3/2)/(a-d)^(5/2)*ln((2*a-2*d-(b^2-4*c*d)^(1/2)*(x+1/2*
((b^2-4*c*d)^(1/2)+b)/c)+2*(a-d)^(1/2)*((x+1/2*((b^2-4*c*d)^(1/2)+b)/c)^2*c-(b^2-4*c*d)^(1/2)*(x+1/2*((b^2-4*c
*d)^(1/2)+b)/c)+a-d)^(1/2))/(x+1/2*((b^2-4*c*d)^(1/2)+b)/c))*b^2-3/2/(b^2-4*c*d)^(3/2)/(a-d)^(5/2)*ln((2*a-2*d
-(b^2-4*c*d)^(1/2)*(x+1/2*((b^2-4*c*d)^(1/2)+b)/c)+2*(a-d)^(1/2)*((x+1/2*((b^2-4*c*d)^(1/2)+b)/c)^2*c-(b^2-4*c
*d)^(1/2)*(x+1/2*((b^2-4*c*d)^(1/2)+b)/c)+a-d)^(1/2))/(x+1/2*((b^2-4*c*d)^(1/2)+b)/c))*c*d+1/(b^2-4*c*d)^(3/2)
*c/(a-d)^(3/2)*ln((2*a-2*d-(b^2-4*c*d)^(1/2)*(x+1/2*((b^2-4*c*d)^(1/2)+b)/c)+2*(a-d)^(1/2)*((x+1/2*((b^2-4*c*d
)^(1/2)+b)/c)^2*c-(b^2-4*c*d)^(1/2)*(x+1/2*((b^2-4*c*d)^(1/2)+b)/c)+a-d)^(1/2))/(x+1/2*((b^2-4*c*d)^(1/2)+b)/c
))-6*c^2/(b^2-4*c*d)^(5/2)/(a-d)^(1/2)*ln((2*a-2*d+(b^2-4*c*d)^(1/2)*(x-1/2*(-b+(b^2-4*c*d)^(1/2))/c)+2*(a-d)^
(1/2)*((x-1/2*(-b+(b^2-4*c*d)^(1/2))/c)^2*c+(b^2-4*c*d)^(1/2)*(x-1/2*(-b+(b^2-4*c*d)^(1/2))/c)+a-d)^(1/2))/(x-
1/2*(-b+(b^2-4*c*d)^(1/2))/c))+3/(b^2-4*c*d)^2*c/(a-d)/(x+1/2/c*(b^2-4*c*d)^(1/2)+1/2*b/c)*((x+1/2*((b^2-4*c*d
)^(1/2)+b)/c)^2*c-(b^2-4*c*d)^(1/2)*(x+1/2*((b^2-4*c*d)^(1/2)+b)/c)+a-d)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{c x^{2} + b x + a}{\left (c x^{2} + b x + d\right )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 19.0583, size = 8058, normalized size = 35.97 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/16*((128*c^2*d^4 + (3*b^4*c^2 + 8*a*b^2*c^3 + 48*a^2*c^4 + 128*c^4*d^2 - 32*(b^2*c^3 + 4*a*c^4)*d)*x^4 - 32
*(b^2*c + 4*a*c^2)*d^3 + 2*(3*b^5*c + 8*a*b^3*c^2 + 48*a^2*b*c^3 + 128*b*c^3*d^2 - 32*(b^3*c^2 + 4*a*b*c^3)*d)
*x^3 + (3*b^4 + 8*a*b^2*c + 48*a^2*c^2)*d^2 + (3*b^6 + 8*a*b^4*c + 48*a^2*b^2*c^2 + 256*c^3*d^3 + 64*(b^2*c^2
- 4*a*c^3)*d^2 - 2*(13*b^4*c + 56*a*b^2*c^2 - 48*a^2*c^3)*d)*x^2 + 2*(128*b*c^2*d^3 - 32*(b^3*c + 4*a*b*c^2)*d
^2 + (3*b^5 + 8*a*b^3*c + 48*a^2*b*c^2)*d)*x)*sqrt(a*b^2 + 4*c*d^2 - (b^2 + 4*a*c)*d)*log((8*a^2*b^4 + (b^4*c^
2 + 24*a*b^2*c^3 + 16*a^2*c^4 + 128*c^4*d^2 - 32*(b^2*c^3 + 4*a*c^4)*d)*x^4 + 2*(b^5*c + 24*a*b^3*c^2 + 16*a^2
*b*c^3 + 128*b*c^3*d^2 - 32*(b^3*c^2 + 4*a*b*c^3)*d)*x^3 + (b^4 + 24*a*b^2*c + 16*a^2*c^2)*d^2 + (b^6 + 32*a*b
^4*c + 48*a^2*b^2*c^2 + 32*(5*b^2*c^2 + 4*a*c^3)*d^2 - 2*(19*b^4*c + 104*a*b^2*c^2 + 48*a^2*c^3)*d)*x^2 - 4*(2
*a*b^3 + 2*(b^2*c^2 + 4*a*c^3 - 8*c^3*d)*x^3 + 3*(b^3*c + 4*a*b*c^2 - 8*b*c^2*d)*x^2 - (b^3 + 4*a*b*c)*d + (b^
4 + 8*a*b^2*c - 2*(5*b^2*c + 4*a*c^2)*d)*x)*sqrt(a*b^2 + 4*c*d^2 - (b^2 + 4*a*c)*d)*sqrt(c*x^2 + b*x + a) - 8*
(a*b^4 + 4*a^2*b^2*c)*d + 2*(4*a*b^5 + 16*a^2*b^3*c + 16*(b^3*c + 4*a*b*c^2)*d^2 - (3*b^5 + 40*a*b^3*c + 48*a^
2*b*c^2)*d)*x)/(c^2*x^4 + 2*b*c*x^3 + 2*b*d*x + (b^2 + 2*c*d)*x^2 + d^2)) - 4*(2*a^2*b^5 + 128*b*c^2*d^4 - 52*
(b^3*c + 4*a*b*c^2)*d^3 - 6*(a*b^4*c^2 + 4*a^2*b^2*c^3 - 32*c^4*d^3 + 12*(b^2*c^3 + 4*a*c^4)*d^2 - (b^4*c^2 +
16*a*b^2*c^3 + 16*a^2*c^4)*d)*x^3 + 5*(b^5 + 16*a*b^3*c + 16*a^2*b*c^2)*d^2 - 9*(a*b^5*c + 4*a^2*b^3*c^2 - 32*
b*c^3*d^3 + 12*(b^3*c^2 + 4*a*b*c^3)*d^2 - (b^5*c + 16*a*b^3*c^2 + 16*a^2*b*c^3)*d)*x^2 - 7*(a*b^5 + 4*a^2*b^3
*c)*d - (3*a*b^6 + 8*a^2*b^4*c - 256*c^3*d^4 + 8*(b^2*c^2 + 52*a*c^3)*d^3 + 2*(13*b^4*c - 8*a*b^2*c^2 - 80*a^2
*c^3)*d^2 - (3*b^6 + 34*a*b^4*c - 8*a^2*b^2*c^2)*d)*x)*sqrt(c*x^2 + b*x + a))/(a^3*b^6*d^2 + 64*c^3*d^8 - 48*(
b^2*c^2 + 4*a*c^3)*d^7 + 12*(b^4*c + 12*a*b^2*c^2 + 16*a^2*c^3)*d^6 - (b^6 + 36*a*b^4*c + 144*a^2*b^2*c^2 + 64
*a^3*c^3)*d^5 + 3*(a*b^6 + 12*a^2*b^4*c + 16*a^3*b^2*c^2)*d^4 + (a^3*b^6*c^2 + 64*c^5*d^6 - 48*(b^2*c^4 + 4*a*
c^5)*d^5 + 12*(b^4*c^3 + 12*a*b^2*c^4 + 16*a^2*c^5)*d^4 - (b^6*c^2 + 36*a*b^4*c^3 + 144*a^2*b^2*c^4 + 64*a^3*c
^5)*d^3 + 3*(a*b^6*c^2 + 12*a^2*b^4*c^3 + 16*a^3*b^2*c^4)*d^2 - 3*(a^2*b^6*c^2 + 4*a^3*b^4*c^3)*d)*x^4 - 3*(a^
2*b^6 + 4*a^3*b^4*c)*d^3 + 2*(a^3*b^7*c + 64*b*c^4*d^6 - 48*(b^3*c^3 + 4*a*b*c^4)*d^5 + 12*(b^5*c^2 + 12*a*b^3
*c^3 + 16*a^2*b*c^4)*d^4 - (b^7*c + 36*a*b^5*c^2 + 144*a^2*b^3*c^3 + 64*a^3*b*c^4)*d^3 + 3*(a*b^7*c + 12*a^2*b
^5*c^2 + 16*a^3*b^3*c^3)*d^2 - 3*(a^2*b^7*c + 4*a^3*b^5*c^2)*d)*x^3 + (a^3*b^8 + 128*c^4*d^7 - 32*(b^2*c^3 + 1
2*a*c^4)*d^6 - 24*(b^4*c^2 - 4*a*b^2*c^3 - 16*a^2*c^4)*d^5 + 2*(5*b^6*c + 36*a*b^4*c^2 - 48*a^2*b^2*c^3 - 64*a
^3*c^4)*d^4 - (b^8 + 30*a*b^6*c + 72*a^2*b^4*c^2 - 32*a^3*b^2*c^3)*d^3 + 3*(a*b^8 + 10*a^2*b^6*c + 8*a^3*b^4*c
^2)*d^2 - (3*a^2*b^8 + 10*a^3*b^6*c)*d)*x^2 + 2*(a^3*b^7*d + 64*b*c^3*d^7 - 48*(b^3*c^2 + 4*a*b*c^3)*d^6 + 12*
(b^5*c + 12*a*b^3*c^2 + 16*a^2*b*c^3)*d^5 - (b^7 + 36*a*b^5*c + 144*a^2*b^3*c^2 + 64*a^3*b*c^3)*d^4 + 3*(a*b^7
+ 12*a^2*b^5*c + 16*a^3*b^3*c^2)*d^3 - 3*(a^2*b^7 + 4*a^3*b^5*c)*d^2)*x), -1/8*((128*c^2*d^4 + (3*b^4*c^2 + 8
*a*b^2*c^3 + 48*a^2*c^4 + 128*c^4*d^2 - 32*(b^2*c^3 + 4*a*c^4)*d)*x^4 - 32*(b^2*c + 4*a*c^2)*d^3 + 2*(3*b^5*c
+ 8*a*b^3*c^2 + 48*a^2*b*c^3 + 128*b*c^3*d^2 - 32*(b^3*c^2 + 4*a*b*c^3)*d)*x^3 + (3*b^4 + 8*a*b^2*c + 48*a^2*c
^2)*d^2 + (3*b^6 + 8*a*b^4*c + 48*a^2*b^2*c^2 + 256*c^3*d^3 + 64*(b^2*c^2 - 4*a*c^3)*d^2 - 2*(13*b^4*c + 56*a*
b^2*c^2 - 48*a^2*c^3)*d)*x^2 + 2*(128*b*c^2*d^3 - 32*(b^3*c + 4*a*b*c^2)*d^2 + (3*b^5 + 8*a*b^3*c + 48*a^2*b*c
^2)*d)*x)*sqrt(-a*b^2 - 4*c*d^2 + (b^2 + 4*a*c)*d)*arctan(-1/2*(2*a*b^2 + (b^2*c + 4*a*c^2 - 8*c^2*d)*x^2 - (b
^2 + 4*a*c)*d + (b^3 + 4*a*b*c - 8*b*c*d)*x)*sqrt(-a*b^2 - 4*c*d^2 + (b^2 + 4*a*c)*d)*sqrt(c*x^2 + b*x + a)/(a
^2*b^3 + 4*a*b*c*d^2 + 2*(a*b^2*c^2 + 4*c^3*d^2 - (b^2*c^2 + 4*a*c^3)*d)*x^3 + 3*(a*b^3*c + 4*b*c^2*d^2 - (b^3
*c + 4*a*b*c^2)*d)*x^2 - (a*b^3 + 4*a^2*b*c)*d + (a*b^4 + 2*a^2*b^2*c + 4*(b^2*c + 2*a*c^2)*d^2 - (b^4 + 6*a*b
^2*c + 8*a^2*c^2)*d)*x)) + 2*(2*a^2*b^5 + 128*b*c^2*d^4 - 52*(b^3*c + 4*a*b*c^2)*d^3 - 6*(a*b^4*c^2 + 4*a^2*b^
2*c^3 - 32*c^4*d^3 + 12*(b^2*c^3 + 4*a*c^4)*d^2 - (b^4*c^2 + 16*a*b^2*c^3 + 16*a^2*c^4)*d)*x^3 + 5*(b^5 + 16*a
*b^3*c + 16*a^2*b*c^2)*d^2 - 9*(a*b^5*c + 4*a^2*b^3*c^2 - 32*b*c^3*d^3 + 12*(b^3*c^2 + 4*a*b*c^3)*d^2 - (b^5*c
+ 16*a*b^3*c^2 + 16*a^2*b*c^3)*d)*x^2 - 7*(a*b^5 + 4*a^2*b^3*c)*d - (3*a*b^6 + 8*a^2*b^4*c - 256*c^3*d^4 + 8*
(b^2*c^2 + 52*a*c^3)*d^3 + 2*(13*b^4*c - 8*a*b^2*c^2 - 80*a^2*c^3)*d^2 - (3*b^6 + 34*a*b^4*c - 8*a^2*b^2*c^2)*
d)*x)*sqrt(c*x^2 + b*x + a))/(a^3*b^6*d^2 + 64*c^3*d^8 - 48*(b^2*c^2 + 4*a*c^3)*d^7 + 12*(b^4*c + 12*a*b^2*c^2
+ 16*a^2*c^3)*d^6 - (b^6 + 36*a*b^4*c + 144*a^2*b^2*c^2 + 64*a^3*c^3)*d^5 + 3*(a*b^6 + 12*a^2*b^4*c + 16*a^3*
b^2*c^2)*d^4 + (a^3*b^6*c^2 + 64*c^5*d^6 - 48*(b^2*c^4 + 4*a*c^5)*d^5 + 12*(b^4*c^3 + 12*a*b^2*c^4 + 16*a^2*c^
5)*d^4 - (b^6*c^2 + 36*a*b^4*c^3 + 144*a^2*b^2*c^4 + 64*a^3*c^5)*d^3 + 3*(a*b^6*c^2 + 12*a^2*b^4*c^3 + 16*a^3*
b^2*c^4)*d^2 - 3*(a^2*b^6*c^2 + 4*a^3*b^4*c^3)*d)*x^4 - 3*(a^2*b^6 + 4*a^3*b^4*c)*d^3 + 2*(a^3*b^7*c + 64*b*c^
4*d^6 - 48*(b^3*c^3 + 4*a*b*c^4)*d^5 + 12*(b^5*c^2 + 12*a*b^3*c^3 + 16*a^2*b*c^4)*d^4 - (b^7*c + 36*a*b^5*c^2
+ 144*a^2*b^3*c^3 + 64*a^3*b*c^4)*d^3 + 3*(a*b^7*c + 12*a^2*b^5*c^2 + 16*a^3*b^3*c^3)*d^2 - 3*(a^2*b^7*c + 4*a
^3*b^5*c^2)*d)*x^3 + (a^3*b^8 + 128*c^4*d^7 - 32*(b^2*c^3 + 12*a*c^4)*d^6 - 24*(b^4*c^2 - 4*a*b^2*c^3 - 16*a^2
*c^4)*d^5 + 2*(5*b^6*c + 36*a*b^4*c^2 - 48*a^2*b^2*c^3 - 64*a^3*c^4)*d^4 - (b^8 + 30*a*b^6*c + 72*a^2*b^4*c^2
- 32*a^3*b^2*c^3)*d^3 + 3*(a*b^8 + 10*a^2*b^6*c + 8*a^3*b^4*c^2)*d^2 - (3*a^2*b^8 + 10*a^3*b^6*c)*d)*x^2 + 2*(
a^3*b^7*d + 64*b*c^3*d^7 - 48*(b^3*c^2 + 4*a*b*c^3)*d^6 + 12*(b^5*c + 12*a*b^3*c^2 + 16*a^2*b*c^3)*d^5 - (b^7
+ 36*a*b^5*c + 144*a^2*b^3*c^2 + 64*a^3*b*c^3)*d^4 + 3*(a*b^7 + 12*a^2*b^5*c + 16*a^3*b^3*c^2)*d^3 - 3*(a^2*b^
7 + 4*a^3*b^5*c)*d^2)*x)]

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

[Out]

Timed out