3.1.6 \(\int \frac {1}{\sqrt {a+b x+c x^2} (d+b x+c x^2)^4} \, dx\) [6]

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

[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 = 0.60, 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 = {988, 1074, 12, 996, 214} \begin {gather*} -\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} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*((b + 2*c*x)*Sqrt[a + b*x + c*x^2])/((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))*A
rcTanh[(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 214

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

Rule 988

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 996

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 1074

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)/((b^2 - 4*a*c)*((c*d - a*f)^
2 - (b*d - a*e)*(c*e - b*f))*(p + 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), 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 ) \text {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 [A]
time = 11.82, size = 604, normalized size = 1.84 \begin {gather*} \frac {-2 \sqrt {a-d} \sqrt {b^2-4 c d} (b+2 c x) \sqrt {a+x (b+c x)} \left (8 (a-d)^2 \left (b^2-4 c d\right )^2-10 \left (b^2+4 c (a-2 d)\right ) (a-d) \left (b^2-4 c d\right ) (d+x (b+c x))+\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 ) (d+x (b+c x))^2\right )-3 \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 ) (d+x (b+c x))^3 \log \left (b-\sqrt {b^2-4 c d}+2 c x\right )+3 \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 ) (d+x (b+c x))^3 \log \left (b+\sqrt {b^2-4 c d}+2 c x\right )-3 \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 ) (d+x (b+c x))^3 \log \left (b^2+b \sqrt {b^2-4 c d}+2 c \left (-2 a+\sqrt {b^2-4 c d} x-2 \sqrt {a-d} \sqrt {a+x (b+c x)}\right )\right )+3 \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 ) (d+x (b+c x))^3 \log \left (-b^2+b \sqrt {b^2-4 c d}+2 c \left (2 a+\sqrt {b^2-4 c d} x+2 \sqrt {a-d} \sqrt {a+x (b+c x)}\right )\right )}{48 (a-d)^{7/2} \left (b^2-4 c d\right )^{7/2} (d+x (b+c x))^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[a - d]*Sqrt[b^2 - 4*c*d]*(b + 2*c*x)*Sqrt[a + x*(b + c*x)]*(8*(a - d)^2*(b^2 - 4*c*d)^2 - 10*(b^2 + 4
*c*(a - 2*d))*(a - d)*(b^2 - 4*c*d)*(d + x*(b + c*x)) + (15*b^4 + 8*b^2*c*(7*a - 22*d) + 16*c^2*(15*a^2 - 44*a
*d + 44*d^2))*(d + x*(b + c*x))^2) - 3*(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))*(d + x*(b + c*x))^3*Log[b - Sqrt[b^2 - 4*c*d] + 2*c*x] + 3*(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))*(d + x*(b + c*x))^3*Log[b + Sqrt[b^2 - 4*c*d] + 2*c*x] - 3*(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))*(d + x*(b + c*x))^3*Log[b^2 + b*Sq
rt[b^2 - 4*c*d] + 2*c*(-2*a + Sqrt[b^2 - 4*c*d]*x - 2*Sqrt[a - d]*Sqrt[a + x*(b + c*x)])] + 3*(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))*(d + x*(b + c*x))^3*Log[-b^2 + b*Sqrt[b^2 -
4*c*d] + 2*c*(2*a + Sqrt[b^2 - 4*c*d]*x + 2*Sqrt[a - d]*Sqrt[a + x*(b + c*x)])])/(48*(a - d)^(7/2)*(b^2 - 4*c
*d)^(7/2)*(d + x*(b + c*x))^3)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(3648\) vs. \(2(304)=608\).
time = 0.17, size = 3649, normalized size = 11.12

method	result	size
default	\(\text {Expression too large to display}\)	\(3649\)

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,method=_RETURNVERBOSE)

[Out]

4/(b^2-4*c*d)^(5/2)*c*(-1/2/(a-d)/(x+1/2*(b+(b^2-4*c*d)^(1/2))/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)^(1/2)/(a-d)*(-1/(a-d)/(x+1/2*(b+(b^2-4
*c*d)^(1/2))/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)^(1/2)/(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)))+1/2*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)))+1/(b^2-4*c*d)^2*(-1/3/(a-d)/(x-1/2*(-b+(b^2-4*c*d)^(1/2))/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)-5/6*(b^2-4*c*d)^(1/2)/(
a-d)*(-1/2/(a-d)/(x-1/2*(-b+(b^2-4*c*d)^(1/2))/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)^(1/2)/(a-d)*(-1/(a-d)/(x-1/2*(-b+(b^2-4*c*d)^(1/2))/
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)^(1/2)/(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)))+1/2*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)))-2/3*c/(a-d)*(-1/(a-d)/(x-1/2*(-b+(b^2-4*c*d)^(1/2))/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)^(1/2)/(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))))-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)*((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))+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)*((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))+10*c^2/(b^2-4*c*d)^3*(-1/(a-d)/(x+1/2*(b+(b^2-4*c*d)^(1/2))/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)^(1/2)/(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)))+10*c^2/(b^2-4*c*d)^3*(-1/(a-d)/(x-1/2*(-b+(b^2-4*c*d)^(1/2))/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)^(1/2)/(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)))+1/(b^2-4*c*d)^2*(
-1/3/(a-d)/(x+1/2*(b+(b^2-4*c*d)^(1/2))/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)+5/6*(b^2-4*c*d)^(1/2)/(a-d)*(-1/2/(a-d)/(x+1/2*(b+(b^2-4*c*d)^(1/2))/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)
^(1/2)/(a-d)*(-1/(a-d)/(x+1/2*(b+(b^2-4*c*d)^(1/2))/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)^(1/2)/(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)))+1/2*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)))-2/3*c/(a-d)*(-1/(a-d)/(x+1/2*(b+(b^2-
4*c*d)^(1/2))/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)^(1/2)/(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))))-4/(b^2-4*c*d)^(5/2)*c*(-1/2/(a-d)/(x-1/2*(-b+(b^2-4*c*d)^(1/2))/c)^2*((x-1/2*(-b+
(b^2-4*c*d)^(1/2))/c)^2*c+(b^2-4*c*d)^(1/2)*(x-...

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(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)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 3962 vs. \(2 (304) = 608\).
time = 28.66, size = 8134, normalized size = 24.80 \begin {gather*} \text {Too large to display} \end {gather*}

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...

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 30280 vs. \(2 (304) = 608\).
time = 10.03, size = 30280, normalized size = 92.32 \begin {gather*} \text {Too large to display} \end {gather*}

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]

-1/16*((5*b^6 + 12*a*b^4*c + 48*a^2*b^2*c^2 + 320*a^3*c^3 - 72*b^4*c*d - 192*a*b^2*c^2*d - 1152*a^2*c^3*d + 38
4*b^2*c^2*d^2 + 1536*a*c^3*d^2 - 1024*c^3*d^3)*log(abs((sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b^2*c + 4*(sqrt(c
)*x - sqrt(c*x^2 + b*x + a))^2*a*c^2 - 8*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*c^2*d + (sqrt(c)*x - sqrt(c*x^2
+ b*x + a))*b^3*sqrt(c) + 4*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b*c^(3/2) - 8*(sqrt(c)*x - sqrt(c*x^2 + b*x
+ a))*b*c^(3/2)*d + 3*a*b^2*c + 4*sqrt(a*b^2 - b^2*d - 4*a*c*d + 4*c*d^2)*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))
^2*c^(3/2) - 4*a^2*c^2 - 2*b^2*c*d + 4*sqrt(a*b^2 - b^2*d - 4*a*c*d + 4*c*d^2)*(sqrt(c)*x - sqrt(c*x^2 + b*x +
a))*b*c + sqrt(a*b^2 - b^2*d - 4*a*c*d + 4*c*d^2)*b^2*sqrt(c)))/sqrt(a*b^2 - b^2*d - 4*a*c*d + 4*c*d^2) - (5*
b^6 + 12*a*b^4*c + 48*a^2*b^2*c^2 + 320*a^3*c^3 - 72*b^4*c*d - 192*a*b^2*c^2*d - 1152*a^2*c^3*d + 384*b^2*c^2*
d^2 + 1536*a*c^3*d^2 - 1024*c^3*d^3)*log(abs((sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b^2*c + 4*(sqrt(c)*x - sqrt
(c*x^2 + b*x + a))^2*a*c^2 - 8*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*c^2*d + (sqrt(c)*x - sqrt(c*x^2 + b*x + a
))*b^3*sqrt(c) + 4*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b*c^(3/2) - 8*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b*c
^(3/2)*d + 3*a*b^2*c - 4*sqrt(a*b^2 - b^2*d - 4*a*c*d + 4*c*d^2)*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*c^(3/2)
- 4*a^2*c^2 - 2*b^2*c*d - 4*sqrt(a*b^2 - b^2*d - 4*a*c*d + 4*c*d^2)*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b*c -
sqrt(a*b^2 - b^2*d - 4*a*c*d + 4*c*d^2)*b^2*sqrt(c)))/sqrt(a*b^2 - b^2*d - 4*a*c*d + 4*c*d^2))/(a^3*b^6 - 3*a
^2*b^6*d - 12*a^3*b^4*c*d + 3*a*b^6*d^2 + 36*a^2*b^4*c*d^2 + 48*a^3*b^2*c^2*d^2 - b^6*d^3 - 36*a*b^4*c*d^3 - 1
44*a^2*b^2*c^2*d^3 - 64*a^3*c^3*d^3 + 12*b^4*c*d^4 + 144*a*b^2*c^2*d^4 + 192*a^2*c^3*d^4 - 48*b^2*c^2*d^5 - 19
2*a*c^3*d^5 + 64*c^3*d^6) + 1/24*(15*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^10*a^11*b^6*c^(5/2) + 36*(sqrt(c)*x -
sqrt(c*x^2 + b*x + a))^10*a^12*b^4*c^(7/2) + 144*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^10*a^13*b^2*c^(9/2) + 96
0*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^10*a^14*c^(11/2) - 165*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^10*a^10*b^6*c
^(5/2)*d - 612*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^10*a^11*b^4*c^(7/2)*d - 2160*(sqrt(c)*x - sqrt(c*x^2 + b*x
+ a))^10*a^12*b^2*c^(9/2)*d - 14016*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^10*a^13*c^(11/2)*d + 825*(sqrt(c)*x -
sqrt(c*x^2 + b*x + a))^10*a^9*b^6*c^(5/2)*d^2 + 4356*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^10*a^10*b^4*c^(7/2)*d
^2 + 15408*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^10*a^11*b^2*c^(9/2)*d^2 + 95424*(sqrt(c)*x - sqrt(c*x^2 + b*x +
a))^10*a^12*c^(11/2)*d^2 - 2475*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^10*a^8*b^6*c^(5/2)*d^3 - 17820*(sqrt(c)*x
- sqrt(c*x^2 + b*x + a))^10*a^9*b^4*c^(7/2)*d^3 - 68112*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^10*a^10*b^2*c^(9/
2)*d^3 - 402240*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^10*a^11*c^(11/2)*d^3 + 4950*(sqrt(c)*x - sqrt(c*x^2 + b*x
+ a))^10*a^7*b^6*c^(5/2)*d^4 + 47520*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^10*a^8*b^4*c^(7/2)*d^4 + 205920*(sqrt
(c)*x - sqrt(c*x^2 + b*x + a))^10*a^9*b^2*c^(9/2)*d^4 + 1174272*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^10*a^10*c^
(11/2)*d^4 - 6930*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^10*a^6*b^6*c^(5/2)*d^5 - 87912*(sqrt(c)*x - sqrt(c*x^2 +
b*x + a))^10*a^7*b^4*c^(7/2)*d^5 - 446688*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^10*a^8*b^2*c^(9/2)*d^5 - 251328
0*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^10*a^9*c^(11/2)*d^5 + 6930*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^10*a^5*b^
6*c^(5/2)*d^6 + 116424*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^10*a^6*b^4*c^(7/2)*d^6 + 712800*(sqrt(c)*x - sqrt(c
*x^2 + b*x + a))^10*a^7*b^2*c^(9/2)*d^6 + 4067712*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^10*a^8*c^(11/2)*d^6 - 49
50*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^10*a^4*b^6*c^(5/2)*d^7 - 111672*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^10*
a^5*b^4*c^(7/2)*d^7 - 845856*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^10*a^6*b^2*c^(9/2)*d^7 - 5056128*(sqrt(c)*x -
sqrt(c*x^2 + b*x + a))^10*a^7*c^(11/2)*d^7 + 2475*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^10*a^3*b^6*c^(5/2)*d^8
+ 77220*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^10*a^4*b^4*c^(7/2)*d^8 + 746064*(sqrt(c)*x - sqrt(c*x^2 + b*x + a)
)^10*a^5*b^2*c^(9/2)*d^8 + 4847040*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^10*a^6*c^(11/2)*d^8 - 825*(sqrt(c)*x -
sqrt(c*x^2 + b*x + a))^10*a^2*b^6*c^(5/2)*d^9 - 37620*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^10*a^3*b^4*c^(7/2)*d
^9 - 483120*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^10*a^4*b^2*c^(9/2)*d^9 - 3562944*(sqrt(c)*x - sqrt(c*x^2 + b*x
+ a))^10*a^5*c^(11/2)*d^9 + 165*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^10*a*b^6*c^(5/2)*d^10 + 12276*(sqrt(c)*x
- sqrt(c*x^2 + b*x + a))^10*a^2*b^4*c^(7/2)*d^10 + 223344*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^10*a^3*b^2*c^(9/
2)*d^10 + 1974720*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^10*a^4*c^(11/2)*d^10 - 15*(sqrt(c)*x - sqrt(c*x^2 + b*x
+ a))^10*b^6*c^(5/2)*d^11 - 2412*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^10*a*b^4*c^(7/2)*d^11 - 69840*(sqrt(c)*x
- sqrt(c*x^2 + b*x + a))^10*a^2*b^2*c^(9/2)*d^11 - 799296*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^10*a^3*c^(11/2)*
d^11 + 216*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^...

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

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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