3.6 \(\int \frac{1}{\sqrt{a+b x+c x^2} (d+b x+c x^2)^4} \, dx\)

Optimal. Leaf size=328 \[ -\frac{(b+2 c x) \left (16 c^2 \left (15 a^2-44 a d+44 d^2\right )+8 b^2 c (7 a-22 d)+15 b^4\right ) \sqrt{a+b x+c x^2}}{24 (a-d)^3 \left (b^2-4 c d\right )^3 \left (b x+c x^2+d\right )}+\frac{\left (4 c (a-2 d)+b^2\right ) \left (16 c^2 \left (5 a^2-8 a d+8 d^2\right )-8 b^2 c (a+4 d)+5 b^4\right ) \tanh ^{-1}\left (\frac{\sqrt{a-d} (b+2 c x)}{\sqrt{b^2-4 c d} \sqrt{a+b x+c x^2}}\right )}{8 (a-d)^{7/2} \left (b^2-4 c d\right )^{7/2}}+\frac{5 (b+2 c x) \left (4 c (a-2 d)+b^2\right ) \sqrt{a+b x+c x^2}}{12 (a-d)^2 \left (b^2-4 c d\right )^2 \left (b x+c x^2+d\right )^2}-\frac{(b+2 c x) \sqrt{a+b x+c x^2}}{3 (a-d) \left (b^2-4 c d\right ) \left (b x+c x^2+d\right )^3} \]

[Out]

-((b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(3*(a - d)*(b^2 - 4*c*d)*(d + b*x + c*x^2)^3) + (5*(b^2 + 4*c*(a - 2*d))*
(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(12*(a - d)^2*(b^2 - 4*c*d)^2*(d + b*x + c*x^2)^2) - ((15*b^4 + 8*b^2*c*(7*
a - 22*d) + 16*c^2*(15*a^2 - 44*a*d + 44*d^2))*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(24*(a - d)^3*(b^2 - 4*c*d)^
3*(d + b*x + c*x^2)) + ((b^2 + 4*c*(a - 2*d))*(5*b^4 - 8*b^2*c*(a + 4*d) + 16*c^2*(5*a^2 - 8*a*d + 8*d^2))*Arc
Tanh[(Sqrt[a - d]*(b + 2*c*x))/(Sqrt[b^2 - 4*c*d]*Sqrt[a + b*x + c*x^2])])/(8*(a - d)^(7/2)*(b^2 - 4*c*d)^(7/2
))

________________________________________________________________________________________

Rubi [A]  time = 0.970484, antiderivative size = 328, normalized size of antiderivative = 1., number of steps used = 6, 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} \[ -\frac{(b+2 c x) \left (16 c^2 \left (15 a^2-44 a d+44 d^2\right )+8 b^2 c (7 a-22 d)+15 b^4\right ) \sqrt{a+b x+c x^2}}{24 (a-d)^3 \left (b^2-4 c d\right )^3 \left (b x+c x^2+d\right )}+\frac{\left (4 c (a-2 d)+b^2\right ) \left (16 c^2 \left (5 a^2-8 a d+8 d^2\right )-8 b^2 c (a+4 d)+5 b^4\right ) \tanh ^{-1}\left (\frac{\sqrt{a-d} (b+2 c x)}{\sqrt{b^2-4 c d} \sqrt{a+b x+c x^2}}\right )}{8 (a-d)^{7/2} \left (b^2-4 c d\right )^{7/2}}+\frac{5 (b+2 c x) \left (4 c (a-2 d)+b^2\right ) \sqrt{a+b x+c x^2}}{12 (a-d)^2 \left (b^2-4 c d\right )^2 \left (b x+c x^2+d\right )^2}-\frac{(b+2 c x) \sqrt{a+b x+c x^2}}{3 (a-d) \left (b^2-4 c d\right ) \left (b x+c x^2+d\right )^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(3*(a - d)*(b^2 - 4*c*d)*(d + b*x + c*x^2)^3) + (5*(b^2 + 4*c*(a - 2*d))*
(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(12*(a - d)^2*(b^2 - 4*c*d)^2*(d + b*x + c*x^2)^2) - ((15*b^4 + 8*b^2*c*(7*
a - 22*d) + 16*c^2*(15*a^2 - 44*a*d + 44*d^2))*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(24*(a - d)^3*(b^2 - 4*c*d)^
3*(d + b*x + c*x^2)) + ((b^2 + 4*c*(a - 2*d))*(5*b^4 - 8*b^2*c*(a + 4*d) + 16*c^2*(5*a^2 - 8*a*d + 8*d^2))*Arc
Tanh[(Sqrt[a - d]*(b + 2*c*x))/(Sqrt[b^2 - 4*c*d]*Sqrt[a + b*x + c*x^2])])/(8*(a - d)^(7/2)*(b^2 - 4*c*d)^(7/2
))

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 )^4} \, dx &=-\frac{(b+2 c x) \sqrt{a+b x+c x^2}}{3 (a-d) \left (b^2-4 c d\right ) \left (d+b x+c x^2\right )^3}+\frac{\int \frac{-\frac{1}{2} c^2 (a-d) \left (5 b^2+20 a c-24 c d\right )-8 b c^3 (a-d) x-8 c^4 (a-d) x^2}{\sqrt{a+b x+c x^2} \left (d+b x+c x^2\right )^3} \, dx}{3 c^2 (a-d)^2 \left (b^2-4 c d\right )}\\ &=-\frac{(b+2 c x) \sqrt{a+b x+c x^2}}{3 (a-d) \left (b^2-4 c d\right ) \left (d+b x+c x^2\right )^3}+\frac{5 \left (b^2+4 c (a-2 d)\right ) (b+2 c x) \sqrt{a+b x+c x^2}}{12 (a-d)^2 \left (b^2-4 c d\right )^2 \left (d+b x+c x^2\right )^2}-\frac{\int \frac{-\frac{1}{4} c^4 (a-d)^2 \left (15 b^4+8 b^2 c (7 a-17 d)+16 c^2 \left (15 a^2-34 a d+24 d^2\right )\right )-10 b c^5 \left (b^2+4 c (a-2 d)\right ) (a-d)^2 x-10 c^6 \left (b^2+4 c (a-2 d)\right ) (a-d)^2 x^2}{\sqrt{a+b x+c x^2} \left (d+b x+c x^2\right )^2} \, dx}{6 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}}{3 (a-d) \left (b^2-4 c d\right ) \left (d+b x+c x^2\right )^3}+\frac{5 \left (b^2+4 c (a-2 d)\right ) (b+2 c x) \sqrt{a+b x+c x^2}}{12 (a-d)^2 \left (b^2-4 c d\right )^2 \left (d+b x+c x^2\right )^2}-\frac{\left (15 b^4+8 b^2 c (7 a-22 d)+16 c^2 \left (15 a^2-44 a d+44 d^2\right )\right ) (b+2 c x) \sqrt{a+b x+c x^2}}{24 (a-d)^3 \left (b^2-4 c d\right )^3 \left (d+b x+c x^2\right )}+\frac{\int -\frac{3 c^6 (a-d)^3 \left (b^2+4 a c-8 c d\right ) \left (5 b^4-8 a b^2 c+80 a^2 c^2-32 b^2 c d-128 a c^2 d+128 c^2 d^2\right )}{8 \sqrt{a+b x+c x^2} \left (d+b x+c x^2\right )} \, dx}{6 c^6 (a-d)^6 \left (b^2-4 c d\right )^3}\\ &=-\frac{(b+2 c x) \sqrt{a+b x+c x^2}}{3 (a-d) \left (b^2-4 c d\right ) \left (d+b x+c x^2\right )^3}+\frac{5 \left (b^2+4 c (a-2 d)\right ) (b+2 c x) \sqrt{a+b x+c x^2}}{12 (a-d)^2 \left (b^2-4 c d\right )^2 \left (d+b x+c x^2\right )^2}-\frac{\left (15 b^4+8 b^2 c (7 a-22 d)+16 c^2 \left (15 a^2-44 a d+44 d^2\right )\right ) (b+2 c x) \sqrt{a+b x+c x^2}}{24 (a-d)^3 \left (b^2-4 c d\right )^3 \left (d+b x+c x^2\right )}-\frac{\left (\left (b^2+4 c (a-2 d)\right ) \left (5 b^4-8 b^2 c (a+4 d)+16 c^2 \left (5 a^2-8 a d+8 d^2\right )\right )\right ) \int \frac{1}{\sqrt{a+b x+c x^2} \left (d+b x+c x^2\right )} \, dx}{16 (a-d)^3 \left (b^2-4 c d\right )^3}\\ &=-\frac{(b+2 c x) \sqrt{a+b x+c x^2}}{3 (a-d) \left (b^2-4 c d\right ) \left (d+b x+c x^2\right )^3}+\frac{5 \left (b^2+4 c (a-2 d)\right ) (b+2 c x) \sqrt{a+b x+c x^2}}{12 (a-d)^2 \left (b^2-4 c d\right )^2 \left (d+b x+c x^2\right )^2}-\frac{\left (15 b^4+8 b^2 c (7 a-22 d)+16 c^2 \left (15 a^2-44 a d+44 d^2\right )\right ) (b+2 c x) \sqrt{a+b x+c x^2}}{24 (a-d)^3 \left (b^2-4 c d\right )^3 \left (d+b x+c x^2\right )}+\frac{\left (b \left (b^2+4 c (a-2 d)\right ) \left (5 b^4-8 b^2 c (a+4 d)+16 c^2 \left (5 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 )}{8 (a-d)^3 \left (b^2-4 c d\right )^3}\\ &=-\frac{(b+2 c x) \sqrt{a+b x+c x^2}}{3 (a-d) \left (b^2-4 c d\right ) \left (d+b x+c x^2\right )^3}+\frac{5 \left (b^2+4 c (a-2 d)\right ) (b+2 c x) \sqrt{a+b x+c x^2}}{12 (a-d)^2 \left (b^2-4 c d\right )^2 \left (d+b x+c x^2\right )^2}-\frac{\left (15 b^4+8 b^2 c (7 a-22 d)+16 c^2 \left (15 a^2-44 a d+44 d^2\right )\right ) (b+2 c x) \sqrt{a+b x+c x^2}}{24 (a-d)^3 \left (b^2-4 c d\right )^3 \left (d+b x+c x^2\right )}+\frac{\left (b^2+4 c (a-2 d)\right ) \left (5 b^4-8 b^2 c (a+4 d)+16 c^2 \left (5 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 )}{8 (a-d)^{7/2} \left (b^2-4 c d\right )^{7/2}}\\ \end{align*}

Mathematica [B]  time = 6.58038, size = 3386, normalized size = 10.32 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-8*c^3*(a + b*x + c*x^2))/(3*(a - d)*(b^2 - 4*c*d)^2*(b - Sqrt[b^2 - 4*c*d] + 2*c*x)^3*Sqrt[a + x*(b + c*x)])
+ (8*c^3*(a + b*x + c*x^2))/((a - d)*(b^2 - 4*c*d)^(5/2)*(b - Sqrt[b^2 - 4*c*d] + 2*c*x)^2*Sqrt[a + x*(b + c*
x)]) - (20*c^3*(a + b*x + c*x^2))/((a - d)*(b^2 - 4*c*d)^3*(b - Sqrt[b^2 - 4*c*d] + 2*c*x)*Sqrt[a + x*(b + c*x
)]) - (8*c^3*(a + b*x + c*x^2))/(3*(a - d)*(b^2 - 4*c*d)^2*(b + Sqrt[b^2 - 4*c*d] + 2*c*x)^3*Sqrt[a + x*(b + c
*x)]) - (8*c^3*(a + b*x + c*x^2))/((a - d)*(b^2 - 4*c*d)^(5/2)*(b + Sqrt[b^2 - 4*c*d] + 2*c*x)^2*Sqrt[a + x*(b
+ c*x)]) - (20*c^3*(a + b*x + c*x^2))/((a - d)*(b^2 - 4*c*d)^3*(b + Sqrt[b^2 - 4*c*d] + 2*c*x)*Sqrt[a + x*(b
+ c*x)]) - (20*c^3*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])])/(Sqrt[a - d]*(b^2 - 4*c*d)^(7/2)*Sqrt[a + x*(b + c*x)]) - (5*c^2*Sq
rt[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]*Sqr
t[a + b*x + c*x^2])])/((a - d)^(3/2)*(b^2 - 4*c*d)^(5/2)*Sqrt[a + x*(b + c*x)]) - (20*c^3*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]
)])/(Sqrt[a - d]*(b^2 - 4*c*d)^(7/2)*Sqrt[a + x*(b + c*x)]) - (5*c^2*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])])/((a - d)^(3/2)*(b
^2 - 4*c*d)^(5/2)*Sqrt[a + x*(b + c*x)]) - (16*c^4*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])))*A
rcTanh[(-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)^(5/2)*(4*a*c^2 + 2*b*c*(-b + S
qrt[b^2 - 4*c*d]) + c*(-b + Sqrt[b^2 - 4*c*d])^2)*Sqrt[a + x*(b + c*x)]) - (16*c^4*Sqrt[a + b*x + c*x^2]*(-((4
*c^2*(-b + Sqrt[b^2 - 4*c*d]) + 2*c^2*(2*b + 3*Sqrt[b^2 - 4*c*d]))*Sqrt[a + b*x + c*x^2])/(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)^2) - (-(((-10*c^3*Sq
rt[b^2 - 4*c*d]*(-b + Sqrt[b^2 - 4*c*d]) - 2*c^3*(10*b^2 - 16*a*c - 24*c*d + 5*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*(10*c^3*Sqrt[b^2 - 4*c*d]*(-b + Sqrt[b^2 - 4*c*d]) - 2*c^3*(10*b^2 - 16*a*
c - 24*c*d + 5*b*Sqrt[b^2 - 4*c*d])) - 2*(-20*a*c^4*Sqrt[b^2 - 4*c*d] + c^3*(-b + Sqrt[b^2 - 4*c*d])*(10*b^2 -
16*a*c - 24*c*d + 5*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)))/(
2*(4*a*c^2 + 2*b*c*(-b + Sqrt[b^2 - 4*c*d]) + c*(-b + Sqrt[b^2 - 4*c*d])^2))))/(3*(b^2 - 4*c*d)^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)]) - (16*c^4*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)
^(5/2)*(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)]) - (16*c^
4*Sqrt[a + b*x + c*x^2]*(-((2*c^2*(2*b - 3*Sqrt[b^2 - 4*c*d]) - 4*c^2*(b + Sqrt[b^2 - 4*c*d]))*Sqrt[a + b*x +
c*x^2])/(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)^2) - (-(((10*c^3*Sqrt[b^2 - 4*c*d]*(b + Sqrt[b^2 - 4*c*d]) + 2*c^3*(10*b^2 - 16*a*c - 24*c*d - 5*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*(-10*c^3*Sqrt[b^2 - 4*c*d]*(b + Sqrt[b^2 - 4*c*d]) + 2
*c^3*(10*b^2 - 16*a*c - 24*c*d - 5*b*Sqrt[b^2 - 4*c*d])) - 2*(-20*a*c^4*Sqrt[b^2 - 4*c*d] + c^3*(b + Sqrt[b^2
- 4*c*d])*(10*b^2 - 16*a*c - 24*c*d - 5*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)))/(2*(4*a*c^2 - 2*b*c*(b + Sqrt[b^2 - 4*c*d]) + c*(b + Sqrt[b^2 - 4*c*d])^2))))/(3*(b^2 - 4*c*d)^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)])

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Maple [B]  time = 0.259, size = 3695, normalized size = 11.3 \begin{align*} \text{output 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)^4/(c*x^2+b*x+a)^(1/2),x)

[Out]

-5/8/(b^2-4*c*d)^2/(a-d)^3/(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)*b^2-7/3/(b^2-4*c*d)^2*c/(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)-2/(
b^2-4*c*d)^(5/2)*c/(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)-3/2/(b^2-4*c*d)^(5/2)*c/(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+6/(b^2-4*c*d)^(5/2)*c^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))*d+5/4/(
b^2-4*c*d)^(3/2)/(a-d)^(7/2)*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-5/4/(b^2-4*c*d)^(3/2)/(a-d)^(7/2)*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+2/(b^2-4*c*d)^(5/2)*c/(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/(
b^2-4*c*d)^(5/2)*c^2/(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))-20*c^3/(b^2-4*c*d)^(7/2)/(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))+5/12/(b^2-4*c*d)^(3/2)/(a-d)^2/(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)+5/16/
(b^2-4*c*d)^(3/2)/(a-d)^(7/2)*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^2-3/4/(b^2-4*c*d)^(3/2)/(a-d)^(5/2)*c*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))-1/3/(b^2-4*c*d)^2/(a-d)/(x-1/2/c*(b^2-4*c*d)^(1/2)+1/2*b
/c)^3*((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/(b
^2-4*c*d)^(5/2)*c^2/(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))-5/12/(b^2-4*c*d)^(3/2)/(a-d)^2/(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)-5/16/(b^2-4*c*d)^(3/2)/(a-d)^
(7/2)*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/
4/(b^2-4*c*d)^(3/2)/(a-d)^(5/2)*c*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))-1/3/(b^2-4*c*d)^2/(a-d)/(x+1/2/c*(b^2-4*c*d)^(1/2)+1/2*b/c)^3*((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)+20*c^3/(b^2-4*c*d)^(7/2)/(a-d)^(1/2)*l
n((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))-5/8/(b^2-
4*c*d)^2/(a-d)^3/(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)*b^2-10/(b^2-4*c*d)^3*c^2/(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)+5/2/(b^2-4
*c*d)^2/(a-d)^3/(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)*c*d+5/2/(b^2-4*c*d)^2/(a-d)^3/(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)*c*d-7/3/(b^2-4
*c*d)^2*c/(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)+3/2/(b^2-4*c*d)^(5/2)*c/(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^2-6/(b^2-4*c*d)^(5/2)*c^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))*d-10/(
b^2-4*c*d)^3*c^2/(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 )}^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out