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]

1/8*(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))*arctanh((2*c*x+b)*(a-d)^(1/2)/(b^2-4*
c*d)^(1/2)/(c*x^2+b*x+a)^(1/2))/(a-d)^(7/2)/(b^2-4*c*d)^(7/2)-1/3*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/(a-d)/(b^2-4*c
*d)/(c*x^2+b*x+d)^3+5/12*(b^2+4*c*(a-2*d))*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/(a-d)^2/(b^2-4*c*d)^2/(c*x^2+b*x+d)^2
-1/24*(15*b^4+8*b^2*c*(7*a-22*d)+16*c^2*(15*a^2-44*a*d+44*d^2))*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/(a-d)^3/(b^2-4*c
*d)^3/(c*x^2+b*x+d)

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Rubi [A]  time = 0.97, antiderivative size = 328, normalized size of antiderivative = 1.00, 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 12

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

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]

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 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 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]

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*}

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Mathematica [B]  time = 6.61, size = 3382, normalized size = 10.31 \[ \text {Result too large to show} \]

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 + S
qrt[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])))*Arc
Tanh[(-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 + Sqr
t[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]*(-1/2*(
(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])/((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)]) - (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]*(-1/2*((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])/((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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fricas [B]  time = 40.66, size = 8134, normalized size = 24.80 \[ \text {result too large to display} \]

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]

[1/96*(3*(1024*c^3*d^6 - (5*b^6*c^3 + 12*a*b^4*c^4 + 48*a^2*b^2*c^5 + 320*a^3*c^6 - 1024*c^6*d^3 + 384*(b^2*c^
5 + 4*a*c^6)*d^2 - 24*(3*b^4*c^4 + 8*a*b^2*c^5 + 48*a^2*c^6)*d)*x^6 - 384*(b^2*c^2 + 4*a*c^3)*d^5 - 3*(5*b^7*c
^2 + 12*a*b^5*c^3 + 48*a^2*b^3*c^4 + 320*a^3*b*c^5 - 1024*b*c^5*d^3 + 384*(b^3*c^4 + 4*a*b*c^5)*d^2 - 24*(3*b^
5*c^3 + 8*a*b^3*c^4 + 48*a^2*b*c^5)*d)*x^5 + 24*(3*b^4*c + 8*a*b^2*c^2 + 48*a^2*c^3)*d^4 - 3*(5*b^8*c + 12*a*b
^6*c^2 + 48*a^2*b^4*c^3 + 320*a^3*b^2*c^4 - 1024*c^5*d^4 - 128*(5*b^2*c^4 - 12*a*c^5)*d^3 + 24*(13*b^4*c^3 + 5
6*a*b^2*c^4 - 48*a^2*c^5)*d^2 - (67*b^6*c^2 + 180*a*b^4*c^3 + 1104*a^2*b^2*c^4 - 320*a^3*c^5)*d)*x^4 - (5*b^6
+ 12*a*b^4*c + 48*a^2*b^2*c^2 + 320*a^3*c^3)*d^3 - (5*b^9 + 12*a*b^7*c + 48*a^2*b^5*c^2 + 320*a^3*b^3*c^3 - 61
44*b*c^4*d^4 + 256*(5*b^3*c^3 + 36*a*b*c^4)*d^3 - 48*(b^5*c^2 - 8*a*b^3*c^3 + 144*a^2*b*c^4)*d^2 - 6*(7*b^7*c
+ 20*a*b^5*c^2 + 144*a^2*b^3*c^3 - 320*a^3*b*c^4)*d)*x^3 + 3*(1024*c^4*d^5 + 128*(5*b^2*c^3 - 12*a*c^4)*d^4 -
24*(13*b^4*c^2 + 56*a*b^2*c^3 - 48*a^2*c^4)*d^3 + (67*b^6*c + 180*a*b^4*c^2 + 1104*a^2*b^2*c^3 - 320*a^3*c^4)*
d^2 - (5*b^8 + 12*a*b^6*c + 48*a^2*b^4*c^2 + 320*a^3*b^2*c^3)*d)*x^2 + 3*(1024*b*c^3*d^5 - 384*(b^3*c^2 + 4*a*
b*c^3)*d^4 + 24*(3*b^5*c + 8*a*b^3*c^2 + 48*a^2*b*c^3)*d^3 - (5*b^7 + 12*a*b^5*c + 48*a^2*b^3*c^2 + 320*a^3*b*
c^3)*d^2)*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 + 1
28*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*(8*a^3*b^7 + 4608*b*c^3*d^6 - 2592*(b^3*c^2 + 4*a*b*c^3)*d^5 + 2*
(15*a*b^6*c^3 + 56*a^2*b^4*c^4 + 240*a^3*b^2*c^5 + 2816*c^6*d^4 - 1408*(b^2*c^5 + 4*a*c^6)*d^3 + 4*(59*b^4*c^4
+ 584*a*b^2*c^5 + 944*a^2*c^6)*d^2 - (15*b^6*c^3 + 292*a*b^4*c^4 + 1168*a^2*b^2*c^5 + 960*a^3*c^6)*d)*x^5 + 4
*(123*b^5*c + 1352*a*b^3*c^2 + 1968*a^2*b*c^3)*d^4 + 5*(15*a*b^7*c^2 + 56*a^2*b^5*c^3 + 240*a^3*b^3*c^4 + 2816
*b*c^5*d^4 - 1408*(b^3*c^4 + 4*a*b*c^5)*d^3 + 4*(59*b^5*c^3 + 584*a*b^3*c^4 + 944*a^2*b*c^5)*d^2 - (15*b^7*c^2
+ 292*a*b^5*c^3 + 1168*a^2*b^3*c^4 + 960*a^3*b*c^5)*d)*x^4 - (33*b^7 + 940*a*b^5*c + 3760*a^2*b^3*c^2 + 2112*
a^3*b*c^3)*d^3 + 4*(15*a*b^8*c + 51*a^2*b^6*c^2 + 220*a^3*b^4*c^3 + 3456*c^5*d^5 + 16*(63*b^2*c^4 - 452*a*c^5)
*d^4 - 4*(273*b^4*c^3 + 584*a*b^2*c^4 - 1264*a^2*c^5)*d^3 + 8*(27*b^6*c^2 + 233*a*b^4*c^3 + 236*a^2*b^2*c^4 -
160*a^3*c^5)*d^2 - (15*b^8*c + 267*a*b^6*c^2 + 992*a^2*b^4*c^3 + 560*a^3*b^2*c^4)*d)*x^3 + (59*a*b^7 + 584*a^2
*b^5*c + 944*a^3*b^3*c^2)*d^2 + (15*a*b^9 + 26*a^2*b^7*c + 120*a^3*b^5*c^2 + 20736*b*c^4*d^5 - 32*(251*b^3*c^3
+ 1356*a*b*c^4)*d^4 + 8*(61*b^5*c^2 + 1768*a*b^3*c^3 + 3792*a^2*b*c^4)*d^3 + 4*(29*b^7*c - 124*a*b^5*c^2 - 18
88*a^2*b^3*c^3 - 1920*a^3*b*c^4)*d^2 - (15*b^9 + 142*a*b^7*c + 112*a^2*b^5*c^2 - 1440*a^3*b^3*c^3)*d)*x^2 - 34
*(a^2*b^7 + 4*a^3*b^5*c)*d - 2*(5*a^2*b^8 + 12*a^3*b^6*c - 4608*c^4*d^6 - 864*(b^2*c^3 - 12*a*c^4)*d^5 + 4*(32
9*b^4*c^2 + 456*a*b^2*c^3 - 1968*a^2*c^4)*d^4 - (283*b^6*c + 2356*a*b^4*c^2 + 1296*a^2*b^2*c^3 - 2112*a^3*c^4)
*d^3 + (20*b^8 + 413*a*b^6*c + 1304*a^2*b^4*c^2 + 336*a^3*b^2*c^3)*d^2 - (25*a*b^8 + 142*a^2*b^6*c + 264*a^3*b
^4*c^2)*d)*x)*sqrt(c*x^2 + b*x + a))/(a^4*b^8*d^3 + 256*c^4*d^11 - 256*(b^2*c^3 + 4*a*c^4)*d^10 + 32*(3*b^4*c^
2 + 32*a*b^2*c^3 + 48*a^2*c^4)*d^9 - 16*(b^6*c + 24*a*b^4*c^2 + 96*a^2*b^2*c^3 + 64*a^3*c^4)*d^8 + (b^8 + 64*a
*b^6*c + 576*a^2*b^4*c^2 + 1024*a^3*b^2*c^3 + 256*a^4*c^4)*d^7 - 4*(a*b^8 + 24*a^2*b^6*c + 96*a^3*b^4*c^2 + 64
*a^4*b^2*c^3)*d^6 + (a^4*b^8*c^3 + 256*c^7*d^8 - 256*(b^2*c^6 + 4*a*c^7)*d^7 + 32*(3*b^4*c^5 + 32*a*b^2*c^6 +
48*a^2*c^7)*d^6 - 16*(b^6*c^4 + 24*a*b^4*c^5 + 96*a^2*b^2*c^6 + 64*a^3*c^7)*d^5 + (b^8*c^3 + 64*a*b^6*c^4 + 57
6*a^2*b^4*c^5 + 1024*a^3*b^2*c^6 + 256*a^4*c^7)*d^4 - 4*(a*b^8*c^3 + 24*a^2*b^6*c^4 + 96*a^3*b^4*c^5 + 64*a^4*
b^2*c^6)*d^3 + 2*(3*a^2*b^8*c^3 + 32*a^3*b^6*c^4 + 48*a^4*b^4*c^5)*d^2 - 4*(a^3*b^8*c^3 + 4*a^4*b^6*c^4)*d)*x^
6 + 2*(3*a^2*b^8 + 32*a^3*b^6*c + 48*a^4*b^4*c^2)*d^5 + 3*(a^4*b^9*c^2 + 256*b*c^6*d^8 - 256*(b^3*c^5 + 4*a*b*
c^6)*d^7 + 32*(3*b^5*c^4 + 32*a*b^3*c^5 + 48*a^2*b*c^6)*d^6 - 16*(b^7*c^3 + 24*a*b^5*c^4 + 96*a^2*b^3*c^5 + 64
*a^3*b*c^6)*d^5 + (b^9*c^2 + 64*a*b^7*c^3 + 576*a^2*b^5*c^4 + 1024*a^3*b^3*c^5 + 256*a^4*b*c^6)*d^4 - 4*(a*b^9
*c^2 + 24*a^2*b^7*c^3 + 96*a^3*b^5*c^4 + 64*a^4*b^3*c^5)*d^3 + 2*(3*a^2*b^9*c^2 + 32*a^3*b^7*c^3 + 48*a^4*b^5*
c^4)*d^2 - 4*(a^3*b^9*c^2 + 4*a^4*b^7*c^3)*d)*x^5 - 4*(a^3*b^8 + 4*a^4*b^6*c)*d^4 + 3*(a^4*b^10*c - 1024*a*c^6
*d^8 + 256*c^6*d^9 - 32*(5*b^4*c^4 - 48*a^2*c^6)*d^7 + 16*(5*b^6*c^3 + 40*a*b^4*c^4 - 64*a^3*c^6)*d^6 - (15*b^
8*c^2 + 320*a*b^6*c^3 + 960*a^2*b^4*c^4 - 256*a^4*c^6)*d^5 + (b^10*c + 60*a*b^8*c^2 + 480*a^2*b^6*c^3 + 640*a^
3*b^4*c^4)*d^4 - 2*(2*a*b^10*c + 45*a^2*b^8*c^2 + 160*a^3*b^6*c^3 + 80*a^4*b^4*c^4)*d^3 + 2*(3*a^2*b^10*c + 30
*a^3*b^8*c^2 + 40*a^4*b^6*c^3)*d^2 - (4*a^3*b^10*c + 15*a^4*b^8*c^2)*d)*x^4 + (a^4*b^11 + 1536*b*c^5*d^9 - 256
*(5*b^3*c^4 + 24*a*b*c^5)*d^8 + 64*(5*b^5*c^3 + 80*a*b^3*c^4 + 144*a^2*b*c^5)*d^7 - 256*(5*a*b^5*c^3 + 30*a^2*
b^3*c^4 + 24*a^3*b*c^5)*d^6 - 2*(5*b^9*c - 960*a^2*b^5*c^3 - 2560*a^3*b^3*c^4 - 768*a^4*b*c^5)*d^5 + (b^11 + 4
0*a*b^9*c - 1280*a^3*b^5*c^3 - 1280*a^4*b^3*c^4)*d^4 - 4*(a*b^11 + 15*a^2*b^9*c - 80*a^4*b^5*c^3)*d^3 + 2*(3*a
^2*b^11 + 20*a^3*b^9*c)*d^2 - 2*(2*a^3*b^11 + 5*a^4*b^9*c)*d)*x^3 + 3*(a^4*b^10*d - 1024*a*c^5*d^9 + 256*c^5*d
^10 - 32*(5*b^4*c^3 - 48*a^2*c^5)*d^8 + 16*(5*b^6*c^2 + 40*a*b^4*c^3 - 64*a^3*c^5)*d^7 - (15*b^8*c + 320*a*b^6
*c^2 + 960*a^2*b^4*c^3 - 256*a^4*c^5)*d^6 + (b^10 + 60*a*b^8*c + 480*a^2*b^6*c^2 + 640*a^3*b^4*c^3)*d^5 - 2*(2
*a*b^10 + 45*a^2*b^8*c + 160*a^3*b^6*c^2 + 80*a^4*b^4*c^3)*d^4 + 2*(3*a^2*b^10 + 30*a^3*b^8*c + 40*a^4*b^6*c^2
)*d^3 - (4*a^3*b^10 + 15*a^4*b^8*c)*d^2)*x^2 + 3*(a^4*b^9*d^2 + 256*b*c^4*d^10 - 256*(b^3*c^3 + 4*a*b*c^4)*d^9
+ 32*(3*b^5*c^2 + 32*a*b^3*c^3 + 48*a^2*b*c^4)*d^8 - 16*(b^7*c + 24*a*b^5*c^2 + 96*a^2*b^3*c^3 + 64*a^3*b*c^4
)*d^7 + (b^9 + 64*a*b^7*c + 576*a^2*b^5*c^2 + 1024*a^3*b^3*c^3 + 256*a^4*b*c^4)*d^6 - 4*(a*b^9 + 24*a^2*b^7*c
+ 96*a^3*b^5*c^2 + 64*a^4*b^3*c^3)*d^5 + 2*(3*a^2*b^9 + 32*a^3*b^7*c + 48*a^4*b^5*c^2)*d^4 - 4*(a^3*b^9 + 4*a^
4*b^7*c)*d^3)*x), -1/48*(3*(1024*c^3*d^6 - (5*b^6*c^3 + 12*a*b^4*c^4 + 48*a^2*b^2*c^5 + 320*a^3*c^6 - 1024*c^6
*d^3 + 384*(b^2*c^5 + 4*a*c^6)*d^2 - 24*(3*b^4*c^4 + 8*a*b^2*c^5 + 48*a^2*c^6)*d)*x^6 - 384*(b^2*c^2 + 4*a*c^3
)*d^5 - 3*(5*b^7*c^2 + 12*a*b^5*c^3 + 48*a^2*b^3*c^4 + 320*a^3*b*c^5 - 1024*b*c^5*d^3 + 384*(b^3*c^4 + 4*a*b*c
^5)*d^2 - 24*(3*b^5*c^3 + 8*a*b^3*c^4 + 48*a^2*b*c^5)*d)*x^5 + 24*(3*b^4*c + 8*a*b^2*c^2 + 48*a^2*c^3)*d^4 - 3
*(5*b^8*c + 12*a*b^6*c^2 + 48*a^2*b^4*c^3 + 320*a^3*b^2*c^4 - 1024*c^5*d^4 - 128*(5*b^2*c^4 - 12*a*c^5)*d^3 +
24*(13*b^4*c^3 + 56*a*b^2*c^4 - 48*a^2*c^5)*d^2 - (67*b^6*c^2 + 180*a*b^4*c^3 + 1104*a^2*b^2*c^4 - 320*a^3*c^5
)*d)*x^4 - (5*b^6 + 12*a*b^4*c + 48*a^2*b^2*c^2 + 320*a^3*c^3)*d^3 - (5*b^9 + 12*a*b^7*c + 48*a^2*b^5*c^2 + 32
0*a^3*b^3*c^3 - 6144*b*c^4*d^4 + 256*(5*b^3*c^3 + 36*a*b*c^4)*d^3 - 48*(b^5*c^2 - 8*a*b^3*c^3 + 144*a^2*b*c^4)
*d^2 - 6*(7*b^7*c + 20*a*b^5*c^2 + 144*a^2*b^3*c^3 - 320*a^3*b*c^4)*d)*x^3 + 3*(1024*c^4*d^5 + 128*(5*b^2*c^3
- 12*a*c^4)*d^4 - 24*(13*b^4*c^2 + 56*a*b^2*c^3 - 48*a^2*c^4)*d^3 + (67*b^6*c + 180*a*b^4*c^2 + 1104*a^2*b^2*c
^3 - 320*a^3*c^4)*d^2 - (5*b^8 + 12*a*b^6*c + 48*a^2*b^4*c^2 + 320*a^3*b^2*c^3)*d)*x^2 + 3*(1024*b*c^3*d^5 - 3
84*(b^3*c^2 + 4*a*b*c^3)*d^4 + 24*(3*b^5*c + 8*a*b^3*c^2 + 48*a^2*b*c^3)*d^3 - (5*b^7 + 12*a*b^5*c + 48*a^2*b^
3*c^2 + 320*a^3*b*c^3)*d^2)*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)*sqr
t(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*(8*a^3*b^7 + 4608*b*c^3*d^6 - 2592*(b^3*c^2 + 4*a*b*c^3)*d^5
+ 2*(15*a*b^6*c^3 + 56*a^2*b^4*c^4 + 240*a^3*b^2*c^5 + 2816*c^6*d^4 - 1408*(b^2*c^5 + 4*a*c^6)*d^3 + 4*(59*b^
4*c^4 + 584*a*b^2*c^5 + 944*a^2*c^6)*d^2 - (15*b^6*c^3 + 292*a*b^4*c^4 + 1168*a^2*b^2*c^5 + 960*a^3*c^6)*d)*x^
5 + 4*(123*b^5*c + 1352*a*b^3*c^2 + 1968*a^2*b*c^3)*d^4 + 5*(15*a*b^7*c^2 + 56*a^2*b^5*c^3 + 240*a^3*b^3*c^4 +
2816*b*c^5*d^4 - 1408*(b^3*c^4 + 4*a*b*c^5)*d^3 + 4*(59*b^5*c^3 + 584*a*b^3*c^4 + 944*a^2*b*c^5)*d^2 - (15*b^
7*c^2 + 292*a*b^5*c^3 + 1168*a^2*b^3*c^4 + 960*a^3*b*c^5)*d)*x^4 - (33*b^7 + 940*a*b^5*c + 3760*a^2*b^3*c^2 +
2112*a^3*b*c^3)*d^3 + 4*(15*a*b^8*c + 51*a^2*b^6*c^2 + 220*a^3*b^4*c^3 + 3456*c^5*d^5 + 16*(63*b^2*c^4 - 452*a
*c^5)*d^4 - 4*(273*b^4*c^3 + 584*a*b^2*c^4 - 1264*a^2*c^5)*d^3 + 8*(27*b^6*c^2 + 233*a*b^4*c^3 + 236*a^2*b^2*c
^4 - 160*a^3*c^5)*d^2 - (15*b^8*c + 267*a*b^6*c^2 + 992*a^2*b^4*c^3 + 560*a^3*b^2*c^4)*d)*x^3 + (59*a*b^7 + 58
4*a^2*b^5*c + 944*a^3*b^3*c^2)*d^2 + (15*a*b^9 + 26*a^2*b^7*c + 120*a^3*b^5*c^2 + 20736*b*c^4*d^5 - 32*(251*b^
3*c^3 + 1356*a*b*c^4)*d^4 + 8*(61*b^5*c^2 + 1768*a*b^3*c^3 + 3792*a^2*b*c^4)*d^3 + 4*(29*b^7*c - 124*a*b^5*c^2
- 1888*a^2*b^3*c^3 - 1920*a^3*b*c^4)*d^2 - (15*b^9 + 142*a*b^7*c + 112*a^2*b^5*c^2 - 1440*a^3*b^3*c^3)*d)*x^2
- 34*(a^2*b^7 + 4*a^3*b^5*c)*d - 2*(5*a^2*b^8 + 12*a^3*b^6*c - 4608*c^4*d^6 - 864*(b^2*c^3 - 12*a*c^4)*d^5 +
4*(329*b^4*c^2 + 456*a*b^2*c^3 - 1968*a^2*c^4)*d^4 - (283*b^6*c + 2356*a*b^4*c^2 + 1296*a^2*b^2*c^3 - 2112*a^3
*c^4)*d^3 + (20*b^8 + 413*a*b^6*c + 1304*a^2*b^4*c^2 + 336*a^3*b^2*c^3)*d^2 - (25*a*b^8 + 142*a^2*b^6*c + 264*
a^3*b^4*c^2)*d)*x)*sqrt(c*x^2 + b*x + a))/(a^4*b^8*d^3 + 256*c^4*d^11 - 256*(b^2*c^3 + 4*a*c^4)*d^10 + 32*(3*b
^4*c^2 + 32*a*b^2*c^3 + 48*a^2*c^4)*d^9 - 16*(b^6*c + 24*a*b^4*c^2 + 96*a^2*b^2*c^3 + 64*a^3*c^4)*d^8 + (b^8 +
64*a*b^6*c + 576*a^2*b^4*c^2 + 1024*a^3*b^2*c^3 + 256*a^4*c^4)*d^7 - 4*(a*b^8 + 24*a^2*b^6*c + 96*a^3*b^4*c^2
+ 64*a^4*b^2*c^3)*d^6 + (a^4*b^8*c^3 + 256*c^7*d^8 - 256*(b^2*c^6 + 4*a*c^7)*d^7 + 32*(3*b^4*c^5 + 32*a*b^2*c
^6 + 48*a^2*c^7)*d^6 - 16*(b^6*c^4 + 24*a*b^4*c^5 + 96*a^2*b^2*c^6 + 64*a^3*c^7)*d^5 + (b^8*c^3 + 64*a*b^6*c^4
+ 576*a^2*b^4*c^5 + 1024*a^3*b^2*c^6 + 256*a^4*c^7)*d^4 - 4*(a*b^8*c^3 + 24*a^2*b^6*c^4 + 96*a^3*b^4*c^5 + 64
*a^4*b^2*c^6)*d^3 + 2*(3*a^2*b^8*c^3 + 32*a^3*b^6*c^4 + 48*a^4*b^4*c^5)*d^2 - 4*(a^3*b^8*c^3 + 4*a^4*b^6*c^4)*
d)*x^6 + 2*(3*a^2*b^8 + 32*a^3*b^6*c + 48*a^4*b^4*c^2)*d^5 + 3*(a^4*b^9*c^2 + 256*b*c^6*d^8 - 256*(b^3*c^5 + 4
*a*b*c^6)*d^7 + 32*(3*b^5*c^4 + 32*a*b^3*c^5 + 48*a^2*b*c^6)*d^6 - 16*(b^7*c^3 + 24*a*b^5*c^4 + 96*a^2*b^3*c^5
+ 64*a^3*b*c^6)*d^5 + (b^9*c^2 + 64*a*b^7*c^3 + 576*a^2*b^5*c^4 + 1024*a^3*b^3*c^5 + 256*a^4*b*c^6)*d^4 - 4*(
a*b^9*c^2 + 24*a^2*b^7*c^3 + 96*a^3*b^5*c^4 + 64*a^4*b^3*c^5)*d^3 + 2*(3*a^2*b^9*c^2 + 32*a^3*b^7*c^3 + 48*a^4
*b^5*c^4)*d^2 - 4*(a^3*b^9*c^2 + 4*a^4*b^7*c^3)*d)*x^5 - 4*(a^3*b^8 + 4*a^4*b^6*c)*d^4 + 3*(a^4*b^10*c - 1024*
a*c^6*d^8 + 256*c^6*d^9 - 32*(5*b^4*c^4 - 48*a^2*c^6)*d^7 + 16*(5*b^6*c^3 + 40*a*b^4*c^4 - 64*a^3*c^6)*d^6 - (
15*b^8*c^2 + 320*a*b^6*c^3 + 960*a^2*b^4*c^4 - 256*a^4*c^6)*d^5 + (b^10*c + 60*a*b^8*c^2 + 480*a^2*b^6*c^3 + 6
40*a^3*b^4*c^4)*d^4 - 2*(2*a*b^10*c + 45*a^2*b^8*c^2 + 160*a^3*b^6*c^3 + 80*a^4*b^4*c^4)*d^3 + 2*(3*a^2*b^10*c
+ 30*a^3*b^8*c^2 + 40*a^4*b^6*c^3)*d^2 - (4*a^3*b^10*c + 15*a^4*b^8*c^2)*d)*x^4 + (a^4*b^11 + 1536*b*c^5*d^9
- 256*(5*b^3*c^4 + 24*a*b*c^5)*d^8 + 64*(5*b^5*c^3 + 80*a*b^3*c^4 + 144*a^2*b*c^5)*d^7 - 256*(5*a*b^5*c^3 + 30
*a^2*b^3*c^4 + 24*a^3*b*c^5)*d^6 - 2*(5*b^9*c - 960*a^2*b^5*c^3 - 2560*a^3*b^3*c^4 - 768*a^4*b*c^5)*d^5 + (b^1
1 + 40*a*b^9*c - 1280*a^3*b^5*c^3 - 1280*a^4*b^3*c^4)*d^4 - 4*(a*b^11 + 15*a^2*b^9*c - 80*a^4*b^5*c^3)*d^3 + 2
*(3*a^2*b^11 + 20*a^3*b^9*c)*d^2 - 2*(2*a^3*b^11 + 5*a^4*b^9*c)*d)*x^3 + 3*(a^4*b^10*d - 1024*a*c^5*d^9 + 256*
c^5*d^10 - 32*(5*b^4*c^3 - 48*a^2*c^5)*d^8 + 16*(5*b^6*c^2 + 40*a*b^4*c^3 - 64*a^3*c^5)*d^7 - (15*b^8*c + 320*
a*b^6*c^2 + 960*a^2*b^4*c^3 - 256*a^4*c^5)*d^6 + (b^10 + 60*a*b^8*c + 480*a^2*b^6*c^2 + 640*a^3*b^4*c^3)*d^5 -
2*(2*a*b^10 + 45*a^2*b^8*c + 160*a^3*b^6*c^2 + 80*a^4*b^4*c^3)*d^4 + 2*(3*a^2*b^10 + 30*a^3*b^8*c + 40*a^4*b^
6*c^2)*d^3 - (4*a^3*b^10 + 15*a^4*b^8*c)*d^2)*x^2 + 3*(a^4*b^9*d^2 + 256*b*c^4*d^10 - 256*(b^3*c^3 + 4*a*b*c^4
)*d^9 + 32*(3*b^5*c^2 + 32*a*b^3*c^3 + 48*a^2*b*c^4)*d^8 - 16*(b^7*c + 24*a*b^5*c^2 + 96*a^2*b^3*c^3 + 64*a^3*
b*c^4)*d^7 + (b^9 + 64*a*b^7*c + 576*a^2*b^5*c^2 + 1024*a^3*b^3*c^3 + 256*a^4*b*c^4)*d^6 - 4*(a*b^9 + 24*a^2*b
^7*c + 96*a^3*b^5*c^2 + 64*a^4*b^3*c^3)*d^5 + 2*(3*a^2*b^9 + 32*a^3*b^7*c + 48*a^4*b^5*c^2)*d^4 - 4*(a^3*b^9 +
4*a^4*b^7*c)*d^3)*x)]

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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

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maple [B]  time = 0.04, size = 3695, normalized size = 11.27 \[ \text {output too large to display} \]

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]

5/2/(b^2-4*c*d)^2/(a-d)^3/(x+1/2*b/c+1/2*(b^2-4*c*d)^(1/2)/c)*(a+(x+1/2*(b+(b^2-4*c*d)^(1/2))/c)^2*c-d-(b^2-4*
c*d)^(1/2)*(x+1/2*(b+(b^2-4*c*d)^(1/2))/c))^(1/2)*c*d+5/2/(b^2-4*c*d)^2/(a-d)^3/(x+1/2*b/c-1/2*(b^2-4*c*d)^(1/
2)/c)*(a+(x-1/2*(-b+(b^2-4*c*d)^(1/2))/c)^2*c-d+(b^2-4*c*d)^(1/2)*(x-1/2*(-b+(b^2-4*c*d)^(1/2))/c))^(1/2)*c*d-
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)*(a+(x-1/2*(-b+(b^2-4*c*d)^(1/2))/c)^2*c-d+(b^2-4*c*d)^(1/2)*(x-1/2*(-b+(b^2-4*c*d)^(1/2))/c))^(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*b/c+1/2*(b^2-4*c*d)^(1/2)/c)^3*(a+(x+1/2*(b+(b^2-4*
c*d)^(1/2))/c)^2*c-d-(b^2-4*c*d)^(1/2)*(x+1/2*(b+(b^2-4*c*d)^(1/2))/c))^(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)*(a+(x+1/2*(b+(b^2-4*c*d)^(1/2)
)/c)^2*c-d-(b^2-4*c*d)^(1/2)*(x+1/2*(b+(b^2-4*c*d)^(1/2))/c))^(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*b/c-1/2*(b^2-4*c*d)^(1/2)/c)^3*(a+(x-1/2*(-b+(b^2-4*c*d)^(1/2))/c)^2*c-d+(b^2-4*c*d)^(1
/2)*(x-1/2*(-b+(b^2-4*c*d)^(1/2))/c))^(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)*(a+(x-1/2*(-b+(b^2-4*c*d)^(1/2))/c)^2*c-d+(b^2-4*c*d)^(1/2)*(x-
1/2*(-b+(b^2-4*c*d)^(1/2))/c))^(1/2))/(x-1/2*(-b+(b^2-4*c*d)^(1/2))/c))+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)*(a+(x-1/2*(-b+(b^2-4*c*d)^(1/2))/c
)^2*c-d+(b^2-4*c*d)^(1/2)*(x-1/2*(-b+(b^2-4*c*d)^(1/2))/c))^(1/2))/(x-1/2*(-b+(b^2-4*c*d)^(1/2))/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+(b^2-4*c*d)^(1/2))/c)+2*(a-d)^(1/2)*(a+(x+
1/2*(b+(b^2-4*c*d)^(1/2))/c)^2*c-d-(b^2-4*c*d)^(1/2)*(x+1/2*(b+(b^2-4*c*d)^(1/2))/c))^(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*b/c+1/2*(b^2-4*c*d)^(1/2)/c)^2*(a+(x+1/2*(b+(b^2-4*c*d)^
(1/2))/c)^2*c-d-(b^2-4*c*d)^(1/2)*(x+1/2*(b+(b^2-4*c*d)^(1/2))/c))^(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)*(a+(x+1/2*(b+(b^2-4*c*d)^(1/2))/c)^2
*c-d-(b^2-4*c*d)^(1/2)*(x+1/2*(b+(b^2-4*c*d)^(1/2))/c))^(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)*(a+(x+1/
2*(b+(b^2-4*c*d)^(1/2))/c)^2*c-d-(b^2-4*c*d)^(1/2)*(x+1/2*(b+(b^2-4*c*d)^(1/2))/c))^(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*b/c-1/2*(b^2-4*c*d)^(1/2)/c)^2*(a+(x-1/2*(-b+(b^2-4*c*d)^(
1/2))/c)^2*c-d+(b^2-4*c*d)^(1/2)*(x-1/2*(-b+(b^2-4*c*d)^(1/2))/c))^(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)*(a+(x-1/2*(-b+(b^2-4*c*d)^(1/2))/c)
^2*c-d+(b^2-4*c*d)^(1/2)*(x-1/2*(-b+(b^2-4*c*d)^(1/2))/c))^(1/2))/(x-1/2*(-b+(b^2-4*c*d)^(1/2))/c))*b^2-5/8/(b
^2-4*c*d)^2/(a-d)^3/(x+1/2*b/c+1/2*(b^2-4*c*d)^(1/2)/c)*(a+(x+1/2*(b+(b^2-4*c*d)^(1/2))/c)^2*c-d-(b^2-4*c*d)^(
1/2)*(x+1/2*(b+(b^2-4*c*d)^(1/2))/c))^(1/2)*b^2-10/(b^2-4*c*d)^3*c^2/(a-d)/(x+1/2*b/c+1/2*(b^2-4*c*d)^(1/2)/c)
*(a+(x+1/2*(b+(b^2-4*c*d)^(1/2))/c)^2*c-d-(b^2-4*c*d)^(1/2)*(x+1/2*(b+(b^2-4*c*d)^(1/2))/c))^(1/2)+2/(b^2-4*c*
d)^(5/2)*c/(a-d)/(x+1/2*b/c-1/2*(b^2-4*c*d)^(1/2)/c)^2*(a+(x-1/2*(-b+(b^2-4*c*d)^(1/2))/c)^2*c-d+(b^2-4*c*d)^(
1/2)*(x-1/2*(-b+(b^2-4*c*d)^(1/2))/c))^(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)*(a+(x-1/2*(-b+(b^2-4*c*d)^(1/2))/c)^2*c-d+(b^2-4*c*d)^(1/2)*(x
-1/2*(-b+(b^2-4*c*d)^(1/2))/c))^(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)*(a+(x-1/2*(-b+(b^2-4*c*d)^(1/2
))/c)^2*c-d+(b^2-4*c*d)^(1/2)*(x-1/2*(-b+(b^2-4*c*d)^(1/2))/c))^(1/2))/(x-1/2*(-b+(b^2-4*c*d)^(1/2))/c))*d-2/(
b^2-4*c*d)^(5/2)*c/(a-d)/(x+1/2*b/c+1/2*(b^2-4*c*d)^(1/2)/c)^2*(a+(x+1/2*(b+(b^2-4*c*d)^(1/2))/c)^2*c-d-(b^2-4
*c*d)^(1/2)*(x+1/2*(b+(b^2-4*c*d)^(1/2))/c))^(1/2)-7/3/(b^2-4*c*d)^2*c/(a-d)^2/(x+1/2*b/c+1/2*(b^2-4*c*d)^(1/2
)/c)*(a+(x+1/2*(b+(b^2-4*c*d)^(1/2))/c)^2*c-d-(b^2-4*c*d)^(1/2)*(x+1/2*(b+(b^2-4*c*d)^(1/2))/c))^(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)*(a+(x
+1/2*(b+(b^2-4*c*d)^(1/2))/c)^2*c-d-(b^2-4*c*d)^(1/2)*(x+1/2*(b+(b^2-4*c*d)^(1/2))/c))^(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)*(a+(x+1/2*(b+(b^2-4*c*d)^(1/2))/c)^2*c-d-(b^2-4*c*d)^(1/2)*(x+1/2*(b+(b^2-4*c*d)^(1/2))
/c))^(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*b/c-1/2*(b^2-4*c*d)^(1/2)/c)*
(a+(x-1/2*(-b+(b^2-4*c*d)^(1/2))/c)^2*c-d+(b^2-4*c*d)^(1/2)*(x-1/2*(-b+(b^2-4*c*d)^(1/2))/c))^(1/2)-5/8/(b^2-4
*c*d)^2/(a-d)^3/(x+1/2*b/c-1/2*(b^2-4*c*d)^(1/2)/c)*(a+(x-1/2*(-b+(b^2-4*c*d)^(1/2))/c)^2*c-d+(b^2-4*c*d)^(1/2
)*(x-1/2*(-b+(b^2-4*c*d)^(1/2))/c))^(1/2)*b^2-7/3/(b^2-4*c*d)^2*c/(a-d)^2/(x+1/2*b/c-1/2*(b^2-4*c*d)^(1/2)/c)*
(a+(x-1/2*(-b+(b^2-4*c*d)^(1/2))/c)^2*c-d+(b^2-4*c*d)^(1/2)*(x-1/2*(-b+(b^2-4*c*d)^(1/2))/c))^(1/2)+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)*(a+(x+1/2*
(b+(b^2-4*c*d)^(1/2))/c)^2*c-d-(b^2-4*c*d)^(1/2)*(x+1/2*(b+(b^2-4*c*d)^(1/2))/c))^(1/2))/(x+1/2*(b+(b^2-4*c*d)
^(1/2))/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)*(a+(x-1/2*(-b+(b^2-4*c*d)^(1/2))/c)^2*c-d+(b^2-4*c*d)^(1/2)*(x-1/2*(-b+(b^2-4*c*d)^(1/2))/c))
^(1/2))/(x-1/2*(-b+(b^2-4*c*d)^(1/2))/c))*c*d

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {c x^{2} + b x + a} {\left (c x^{2} + b x + d\right )}^{4}}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{\sqrt {c\,x^2+b\,x+a}\,{\left (c\,x^2+b\,x+d\right )}^4} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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