\(\int \frac {1}{x (a+b x^2)^2 (c+d x^2)^{5/2}} \, dx\) [781]

Optimal result

Integrand size = 24, antiderivative size = 225 \[ \int \frac {1}{x \left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}} \, dx=\frac {d (3 b c+2 a d)}{6 a c (b c-a d)^2 \left (c+d x^2\right )^{3/2}}+\frac {b}{2 a (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}+\frac {d \left (b^2 c^2+6 a b c d-2 a^2 d^2\right )}{2 a c^2 (b c-a d)^3 \sqrt {c+d x^2}}-\frac {\text {arctanh}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{a^2 c^{5/2}}+\frac {b^{5/2} (2 b c-7 a d) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{2 a^2 (b c-a d)^{7/2}} \]

[Out]

1/6*d*(2*a*d+3*b*c)/a/c/(-a*d+b*c)^2/(d*x^2+c)^(3/2)+1/2*b/a/(-a*d+b*c)/(b*x^2+a)/(d*x^2+c)^(3/2)-arctanh((d*x
^2+c)^(1/2)/c^(1/2))/a^2/c^(5/2)+1/2*b^(5/2)*(-7*a*d+2*b*c)*arctanh(b^(1/2)*(d*x^2+c)^(1/2)/(-a*d+b*c)^(1/2))/
a^2/(-a*d+b*c)^(7/2)+1/2*d*(-2*a^2*d^2+6*a*b*c*d+b^2*c^2)/a/c^2/(-a*d+b*c)^3/(d*x^2+c)^(1/2)

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {457, 105, 157, 162, 65, 214} \[ \int \frac {1}{x \left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}} \, dx=\frac {b^{5/2} (2 b c-7 a d) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{2 a^2 (b c-a d)^{7/2}}-\frac {\text {arctanh}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{a^2 c^{5/2}}+\frac {d \left (-2 a^2 d^2+6 a b c d+b^2 c^2\right )}{2 a c^2 \sqrt {c+d x^2} (b c-a d)^3}+\frac {b}{2 a \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2} (b c-a d)}+\frac {d (2 a d+3 b c)}{6 a c \left (c+d x^2\right )^{3/2} (b c-a d)^2} \]

[In]

Int[1/(x*(a + b*x^2)^2*(c + d*x^2)^(5/2)),x]

[Out]

(d*(3*b*c + 2*a*d))/(6*a*c*(b*c - a*d)^2*(c + d*x^2)^(3/2)) + b/(2*a*(b*c - a*d)*(a + b*x^2)*(c + d*x^2)^(3/2)
) + (d*(b^2*c^2 + 6*a*b*c*d - 2*a^2*d^2))/(2*a*c^2*(b*c - a*d)^3*Sqrt[c + d*x^2]) - ArcTanh[Sqrt[c + d*x^2]/Sq
rt[c]]/(a^2*c^(5/2)) + (b^(5/2)*(2*b*c - 7*a*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^2])/Sqrt[b*c - a*d]])/(2*a^2*(b*
c - a*d)^(7/2))

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 105

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Dist[1/((m + 1)*(b*
c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] &
& (IntegerQ[n] || IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])

Rule 157

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f
))), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegersQ[2*m, 2*n, 2*p]

Rule 162

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :>
Dist[(b*g - a*h)/(b*c - a*d), Int[(e + f*x)^p/(a + b*x), x], x] - Dist[(d*g - c*h)/(b*c - a*d), Int[(e + f*x)
^p/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, 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 457

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {1}{x (a+b x)^2 (c+d x)^{5/2}} \, dx,x,x^2\right ) \\ & = \frac {b}{2 a (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}+\frac {\text {Subst}\left (\int \frac {b c-a d+\frac {5 b d x}{2}}{x (a+b x) (c+d x)^{5/2}} \, dx,x,x^2\right )}{2 a (b c-a d)} \\ & = \frac {d (3 b c+2 a d)}{6 a c (b c-a d)^2 \left (c+d x^2\right )^{3/2}}+\frac {b}{2 a (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}-\frac {\text {Subst}\left (\int \frac {-\frac {3}{2} (b c-a d)^2-\frac {3}{4} b d (3 b c+2 a d) x}{x (a+b x) (c+d x)^{3/2}} \, dx,x,x^2\right )}{3 a c (b c-a d)^2} \\ & = \frac {d (3 b c+2 a d)}{6 a c (b c-a d)^2 \left (c+d x^2\right )^{3/2}}+\frac {b}{2 a (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}+\frac {d \left (b^2 c^2+6 a b c d-2 a^2 d^2\right )}{2 a c^2 (b c-a d)^3 \sqrt {c+d x^2}}+\frac {2 \text {Subst}\left (\int \frac {\frac {3}{4} (b c-a d)^3+\frac {3}{8} b d \left (b^2 c^2+6 a b c d-2 a^2 d^2\right ) x}{x (a+b x) \sqrt {c+d x}} \, dx,x,x^2\right )}{3 a c^2 (b c-a d)^3} \\ & = \frac {d (3 b c+2 a d)}{6 a c (b c-a d)^2 \left (c+d x^2\right )^{3/2}}+\frac {b}{2 a (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}+\frac {d \left (b^2 c^2+6 a b c d-2 a^2 d^2\right )}{2 a c^2 (b c-a d)^3 \sqrt {c+d x^2}}+\frac {\text {Subst}\left (\int \frac {1}{x \sqrt {c+d x}} \, dx,x,x^2\right )}{2 a^2 c^2}-\frac {\left (b^3 (2 b c-7 a d)\right ) \text {Subst}\left (\int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx,x,x^2\right )}{4 a^2 (b c-a d)^3} \\ & = \frac {d (3 b c+2 a d)}{6 a c (b c-a d)^2 \left (c+d x^2\right )^{3/2}}+\frac {b}{2 a (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}+\frac {d \left (b^2 c^2+6 a b c d-2 a^2 d^2\right )}{2 a c^2 (b c-a d)^3 \sqrt {c+d x^2}}+\frac {\text {Subst}\left (\int \frac {1}{-\frac {c}{d}+\frac {x^2}{d}} \, dx,x,\sqrt {c+d x^2}\right )}{a^2 c^2 d}-\frac {\left (b^3 (2 b c-7 a d)\right ) \text {Subst}\left (\int \frac {1}{a-\frac {b c}{d}+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d x^2}\right )}{2 a^2 d (b c-a d)^3} \\ & = \frac {d (3 b c+2 a d)}{6 a c (b c-a d)^2 \left (c+d x^2\right )^{3/2}}+\frac {b}{2 a (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}+\frac {d \left (b^2 c^2+6 a b c d-2 a^2 d^2\right )}{2 a c^2 (b c-a d)^3 \sqrt {c+d x^2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{a^2 c^{5/2}}+\frac {b^{5/2} (2 b c-7 a d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{2 a^2 (b c-a d)^{7/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.81 (sec) , antiderivative size = 215, normalized size of antiderivative = 0.96 \[ \int \frac {1}{x \left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}} \, dx=\frac {\frac {a \left (3 b^3 c^2 \left (c+d x^2\right )^2-2 a^3 d^3 \left (4 c+3 d x^2\right )+2 a b^2 c d^2 x^2 \left (10 c+9 d x^2\right )+2 a^2 b d^2 \left (10 c^2+5 c d x^2-3 d^2 x^4\right )\right )}{c^2 (b c-a d)^3 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}+\frac {3 b^{5/2} (2 b c-7 a d) \arctan \left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {-b c+a d}}\right )}{(-b c+a d)^{7/2}}-\frac {6 \text {arctanh}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{c^{5/2}}}{6 a^2} \]

[In]

Integrate[1/(x*(a + b*x^2)^2*(c + d*x^2)^(5/2)),x]

[Out]

((a*(3*b^3*c^2*(c + d*x^2)^2 - 2*a^3*d^3*(4*c + 3*d*x^2) + 2*a*b^2*c*d^2*x^2*(10*c + 9*d*x^2) + 2*a^2*b*d^2*(1
0*c^2 + 5*c*d*x^2 - 3*d^2*x^4)))/(c^2*(b*c - a*d)^3*(a + b*x^2)*(c + d*x^2)^(3/2)) + (3*b^(5/2)*(2*b*c - 7*a*d
)*ArcTan[(Sqrt[b]*Sqrt[c + d*x^2])/Sqrt[-(b*c) + a*d]])/(-(b*c) + a*d)^(7/2) - (6*ArcTanh[Sqrt[c + d*x^2]/Sqrt
[c]])/c^(5/2))/(6*a^2)

Maple [A] (verified)

Time = 3.51 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.14

method	result	size
pseudoelliptic	\(-\frac {-\left (b \,x^{2}+a \right ) b^{3} c^{\frac {9}{2}} \left (-\frac {7 a d}{2}+b c \right ) \left (d \,x^{2}+c \right )^{\frac {3}{2}} \arctan \left (\frac {b \sqrt {d \,x^{2}+c}}{\sqrt {\left (a d -b c \right ) b}}\right )+\left (\left (b \,x^{2}+a \right ) c^{2} \left (d \,x^{2}+c \right )^{\frac {3}{2}} \left (a d -b c \right )^{3} \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}}{\sqrt {c}}\right )+\frac {\left (c^{4} b^{3}+2 b^{3} c^{3} d \,x^{2}+\frac {20 \left (\frac {3}{20} b^{2} x^{4}+a b \,x^{2}+a^{2}\right ) b \,c^{2} d^{2}}{3}-\frac {8 \left (b \,x^{2}+a \right ) c a \left (-\frac {9 b \,x^{2}}{4}+a \right ) d^{3}}{3}-2 a^{2} d^{4} x^{2} \left (b \,x^{2}+a \right )\right ) c^{\frac {5}{2}} a}{2}\right ) \sqrt {\left (a d -b c \right ) b}}{\sqrt {\left (a d -b c \right ) b}\, \left (d \,x^{2}+c \right )^{\frac {3}{2}} a^{2} \left (b \,x^{2}+a \right ) \left (a d -b c \right )^{3} c^{\frac {9}{2}}}\)	\(257\)
default	\(\text {Expression too large to display}\)	\(3548\)

[In]

int(1/x/(b*x^2+a)^2/(d*x^2+c)^(5/2),x,method=_RETURNVERBOSE)

[Out]

-1/((a*d-b*c)*b)^(1/2)*(-(b*x^2+a)*b^3*c^(9/2)*(-7/2*a*d+b*c)*(d*x^2+c)^(3/2)*arctan(b*(d*x^2+c)^(1/2)/((a*d-b
*c)*b)^(1/2))+((b*x^2+a)*c^2*(d*x^2+c)^(3/2)*(a*d-b*c)^3*arctanh((d*x^2+c)^(1/2)/c^(1/2))+1/2*(c^4*b^3+2*b^3*c
^3*d*x^2+20/3*(3/20*b^2*x^4+a*b*x^2+a^2)*b*c^2*d^2-8/3*(b*x^2+a)*c*a*(-9/4*b*x^2+a)*d^3-2*a^2*d^4*x^2*(b*x^2+a
))*c^(5/2)*a)*((a*d-b*c)*b)^(1/2))/(d*x^2+c)^(3/2)/a^2/(b*x^2+a)/(a*d-b*c)^3/c^(9/2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 803 vs. \(2 (195) = 390\).

Time = 4.28 (sec) , antiderivative size = 3403, normalized size of antiderivative = 15.12 \[ \int \frac {1}{x \left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}} \, dx=\text {Too large to display} \]

[In]

integrate(1/x/(b*x^2+a)^2/(d*x^2+c)^(5/2),x, algorithm="fricas")

[Out]

[1/24*(3*(2*a*b^3*c^6 - 7*a^2*b^2*c^5*d + (2*b^4*c^4*d^2 - 7*a*b^3*c^3*d^3)*x^6 + (4*b^4*c^5*d - 12*a*b^3*c^4*
d^2 - 7*a^2*b^2*c^3*d^3)*x^4 + (2*b^4*c^6 - 3*a*b^3*c^5*d - 14*a^2*b^2*c^4*d^2)*x^2)*sqrt(b/(b*c - a*d))*log((
b^2*d^2*x^4 + 8*b^2*c^2 - 8*a*b*c*d + a^2*d^2 + 2*(4*b^2*c*d - 3*a*b*d^2)*x^2 + 4*(2*b^2*c^2 - 3*a*b*c*d + a^2
*d^2 + (b^2*c*d - a*b*d^2)*x^2)*sqrt(d*x^2 + c)*sqrt(b/(b*c - a*d)))/(b^2*x^4 + 2*a*b*x^2 + a^2)) + 12*(a*b^3*
c^5 - 3*a^2*b^2*c^4*d + 3*a^3*b*c^3*d^2 - a^4*c^2*d^3 + (b^4*c^3*d^2 - 3*a*b^3*c^2*d^3 + 3*a^2*b^2*c*d^4 - a^3
*b*d^5)*x^6 + (2*b^4*c^4*d - 5*a*b^3*c^3*d^2 + 3*a^2*b^2*c^2*d^3 + a^3*b*c*d^4 - a^4*d^5)*x^4 + (b^4*c^5 - a*b
^3*c^4*d - 3*a^2*b^2*c^3*d^2 + 5*a^3*b*c^2*d^3 - 2*a^4*c*d^4)*x^2)*sqrt(c)*log(-(d*x^2 - 2*sqrt(d*x^2 + c)*sqr
t(c) + 2*c)/x^2) + 4*(3*a*b^3*c^5 + 20*a^3*b*c^3*d^2 - 8*a^4*c^2*d^3 + 3*(a*b^3*c^3*d^2 + 6*a^2*b^2*c^2*d^3 -
2*a^3*b*c*d^4)*x^4 + 2*(3*a*b^3*c^4*d + 10*a^2*b^2*c^3*d^2 + 5*a^3*b*c^2*d^3 - 3*a^4*c*d^4)*x^2)*sqrt(d*x^2 +
c))/(a^3*b^3*c^8 - 3*a^4*b^2*c^7*d + 3*a^5*b*c^6*d^2 - a^6*c^5*d^3 + (a^2*b^4*c^6*d^2 - 3*a^3*b^3*c^5*d^3 + 3*
a^4*b^2*c^4*d^4 - a^5*b*c^3*d^5)*x^6 + (2*a^2*b^4*c^7*d - 5*a^3*b^3*c^6*d^2 + 3*a^4*b^2*c^5*d^3 + a^5*b*c^4*d^
4 - a^6*c^3*d^5)*x^4 + (a^2*b^4*c^8 - a^3*b^3*c^7*d - 3*a^4*b^2*c^6*d^2 + 5*a^5*b*c^5*d^3 - 2*a^6*c^4*d^4)*x^2
), 1/24*(24*(a*b^3*c^5 - 3*a^2*b^2*c^4*d + 3*a^3*b*c^3*d^2 - a^4*c^2*d^3 + (b^4*c^3*d^2 - 3*a*b^3*c^2*d^3 + 3*
a^2*b^2*c*d^4 - a^3*b*d^5)*x^6 + (2*b^4*c^4*d - 5*a*b^3*c^3*d^2 + 3*a^2*b^2*c^2*d^3 + a^3*b*c*d^4 - a^4*d^5)*x
^4 + (b^4*c^5 - a*b^3*c^4*d - 3*a^2*b^2*c^3*d^2 + 5*a^3*b*c^2*d^3 - 2*a^4*c*d^4)*x^2)*sqrt(-c)*arctan(sqrt(-c)
/sqrt(d*x^2 + c)) + 3*(2*a*b^3*c^6 - 7*a^2*b^2*c^5*d + (2*b^4*c^4*d^2 - 7*a*b^3*c^3*d^3)*x^6 + (4*b^4*c^5*d -
12*a*b^3*c^4*d^2 - 7*a^2*b^2*c^3*d^3)*x^4 + (2*b^4*c^6 - 3*a*b^3*c^5*d - 14*a^2*b^2*c^4*d^2)*x^2)*sqrt(b/(b*c
- a*d))*log((b^2*d^2*x^4 + 8*b^2*c^2 - 8*a*b*c*d + a^2*d^2 + 2*(4*b^2*c*d - 3*a*b*d^2)*x^2 + 4*(2*b^2*c^2 - 3*
a*b*c*d + a^2*d^2 + (b^2*c*d - a*b*d^2)*x^2)*sqrt(d*x^2 + c)*sqrt(b/(b*c - a*d)))/(b^2*x^4 + 2*a*b*x^2 + a^2))
+ 4*(3*a*b^3*c^5 + 20*a^3*b*c^3*d^2 - 8*a^4*c^2*d^3 + 3*(a*b^3*c^3*d^2 + 6*a^2*b^2*c^2*d^3 - 2*a^3*b*c*d^4)*x
^4 + 2*(3*a*b^3*c^4*d + 10*a^2*b^2*c^3*d^2 + 5*a^3*b*c^2*d^3 - 3*a^4*c*d^4)*x^2)*sqrt(d*x^2 + c))/(a^3*b^3*c^8
- 3*a^4*b^2*c^7*d + 3*a^5*b*c^6*d^2 - a^6*c^5*d^3 + (a^2*b^4*c^6*d^2 - 3*a^3*b^3*c^5*d^3 + 3*a^4*b^2*c^4*d^4
- a^5*b*c^3*d^5)*x^6 + (2*a^2*b^4*c^7*d - 5*a^3*b^3*c^6*d^2 + 3*a^4*b^2*c^5*d^3 + a^5*b*c^4*d^4 - a^6*c^3*d^5)
*x^4 + (a^2*b^4*c^8 - a^3*b^3*c^7*d - 3*a^4*b^2*c^6*d^2 + 5*a^5*b*c^5*d^3 - 2*a^6*c^4*d^4)*x^2), -1/12*(3*(2*a
*b^3*c^6 - 7*a^2*b^2*c^5*d + (2*b^4*c^4*d^2 - 7*a*b^3*c^3*d^3)*x^6 + (4*b^4*c^5*d - 12*a*b^3*c^4*d^2 - 7*a^2*b
^2*c^3*d^3)*x^4 + (2*b^4*c^6 - 3*a*b^3*c^5*d - 14*a^2*b^2*c^4*d^2)*x^2)*sqrt(-b/(b*c - a*d))*arctan(1/2*(b*d*x
^2 + 2*b*c - a*d)*sqrt(d*x^2 + c)*sqrt(-b/(b*c - a*d))/(b*d*x^2 + b*c)) - 6*(a*b^3*c^5 - 3*a^2*b^2*c^4*d + 3*a
^3*b*c^3*d^2 - a^4*c^2*d^3 + (b^4*c^3*d^2 - 3*a*b^3*c^2*d^3 + 3*a^2*b^2*c*d^4 - a^3*b*d^5)*x^6 + (2*b^4*c^4*d
- 5*a*b^3*c^3*d^2 + 3*a^2*b^2*c^2*d^3 + a^3*b*c*d^4 - a^4*d^5)*x^4 + (b^4*c^5 - a*b^3*c^4*d - 3*a^2*b^2*c^3*d^
2 + 5*a^3*b*c^2*d^3 - 2*a^4*c*d^4)*x^2)*sqrt(c)*log(-(d*x^2 - 2*sqrt(d*x^2 + c)*sqrt(c) + 2*c)/x^2) - 2*(3*a*b
^3*c^5 + 20*a^3*b*c^3*d^2 - 8*a^4*c^2*d^3 + 3*(a*b^3*c^3*d^2 + 6*a^2*b^2*c^2*d^3 - 2*a^3*b*c*d^4)*x^4 + 2*(3*a
*b^3*c^4*d + 10*a^2*b^2*c^3*d^2 + 5*a^3*b*c^2*d^3 - 3*a^4*c*d^4)*x^2)*sqrt(d*x^2 + c))/(a^3*b^3*c^8 - 3*a^4*b^
2*c^7*d + 3*a^5*b*c^6*d^2 - a^6*c^5*d^3 + (a^2*b^4*c^6*d^2 - 3*a^3*b^3*c^5*d^3 + 3*a^4*b^2*c^4*d^4 - a^5*b*c^3
*d^5)*x^6 + (2*a^2*b^4*c^7*d - 5*a^3*b^3*c^6*d^2 + 3*a^4*b^2*c^5*d^3 + a^5*b*c^4*d^4 - a^6*c^3*d^5)*x^4 + (a^2
*b^4*c^8 - a^3*b^3*c^7*d - 3*a^4*b^2*c^6*d^2 + 5*a^5*b*c^5*d^3 - 2*a^6*c^4*d^4)*x^2), -1/12*(3*(2*a*b^3*c^6 -
7*a^2*b^2*c^5*d + (2*b^4*c^4*d^2 - 7*a*b^3*c^3*d^3)*x^6 + (4*b^4*c^5*d - 12*a*b^3*c^4*d^2 - 7*a^2*b^2*c^3*d^3)
*x^4 + (2*b^4*c^6 - 3*a*b^3*c^5*d - 14*a^2*b^2*c^4*d^2)*x^2)*sqrt(-b/(b*c - a*d))*arctan(1/2*(b*d*x^2 + 2*b*c
- a*d)*sqrt(d*x^2 + c)*sqrt(-b/(b*c - a*d))/(b*d*x^2 + b*c)) - 12*(a*b^3*c^5 - 3*a^2*b^2*c^4*d + 3*a^3*b*c^3*d
^2 - a^4*c^2*d^3 + (b^4*c^3*d^2 - 3*a*b^3*c^2*d^3 + 3*a^2*b^2*c*d^4 - a^3*b*d^5)*x^6 + (2*b^4*c^4*d - 5*a*b^3*
c^3*d^2 + 3*a^2*b^2*c^2*d^3 + a^3*b*c*d^4 - a^4*d^5)*x^4 + (b^4*c^5 - a*b^3*c^4*d - 3*a^2*b^2*c^3*d^2 + 5*a^3*
b*c^2*d^3 - 2*a^4*c*d^4)*x^2)*sqrt(-c)*arctan(sqrt(-c)/sqrt(d*x^2 + c)) - 2*(3*a*b^3*c^5 + 20*a^3*b*c^3*d^2 -
8*a^4*c^2*d^3 + 3*(a*b^3*c^3*d^2 + 6*a^2*b^2*c^2*d^3 - 2*a^3*b*c*d^4)*x^4 + 2*(3*a*b^3*c^4*d + 10*a^2*b^2*c^3*
d^2 + 5*a^3*b*c^2*d^3 - 3*a^4*c*d^4)*x^2)*sqrt(d*x^2 + c))/(a^3*b^3*c^8 - 3*a^4*b^2*c^7*d + 3*a^5*b*c^6*d^2 -
a^6*c^5*d^3 + (a^2*b^4*c^6*d^2 - 3*a^3*b^3*c^5*d^3 + 3*a^4*b^2*c^4*d^4 - a^5*b*c^3*d^5)*x^6 + (2*a^2*b^4*c^7*d
- 5*a^3*b^3*c^6*d^2 + 3*a^4*b^2*c^5*d^3 + a^5*b*c^4*d^4 - a^6*c^3*d^5)*x^4 + (a^2*b^4*c^8 - a^3*b^3*c^7*d - 3
*a^4*b^2*c^6*d^2 + 5*a^5*b*c^5*d^3 - 2*a^6*c^4*d^4)*x^2)]

Sympy [F]

\[ \int \frac {1}{x \left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}} \, dx=\int \frac {1}{x \left (a + b x^{2}\right )^{2} \left (c + d x^{2}\right )^{\frac {5}{2}}}\, dx \]

[In]

integrate(1/x/(b*x**2+a)**2/(d*x**2+c)**(5/2),x)

[Out]

Integral(1/(x*(a + b*x**2)**2*(c + d*x**2)**(5/2)), x)

Maxima [F]

\[ \int \frac {1}{x \left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{2} {\left (d x^{2} + c\right )}^{\frac {5}{2}} x} \,d x } \]

[In]

integrate(1/x/(b*x^2+a)^2/(d*x^2+c)^(5/2),x, algorithm="maxima")

[Out]

integrate(1/((b*x^2 + a)^2*(d*x^2 + c)^(5/2)*x), x)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.32 \[ \int \frac {1}{x \left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}} \, dx=\frac {\sqrt {d x^{2} + c} b^{3} d}{2 \, {\left (a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}\right )} {\left ({\left (d x^{2} + c\right )} b - b c + a d\right )}} - \frac {{\left (2 \, b^{4} c - 7 \, a b^{3} d\right )} \arctan \left (\frac {\sqrt {d x^{2} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{2 \, {\left (a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - a^{5} d^{3}\right )} \sqrt {-b^{2} c + a b d}} + \frac {9 \, {\left (d x^{2} + c\right )} b c d^{2} + b c^{2} d^{2} - 3 \, {\left (d x^{2} + c\right )} a d^{3} - a c d^{3}}{3 \, {\left (b^{3} c^{5} - 3 \, a b^{2} c^{4} d + 3 \, a^{2} b c^{3} d^{2} - a^{3} c^{2} d^{3}\right )} {\left (d x^{2} + c\right )}^{\frac {3}{2}}} + \frac {\arctan \left (\frac {\sqrt {d x^{2} + c}}{\sqrt {-c}}\right )}{a^{2} \sqrt {-c} c^{2}} \]

[In]

integrate(1/x/(b*x^2+a)^2/(d*x^2+c)^(5/2),x, algorithm="giac")

[Out]

1/2*sqrt(d*x^2 + c)*b^3*d/((a*b^3*c^3 - 3*a^2*b^2*c^2*d + 3*a^3*b*c*d^2 - a^4*d^3)*((d*x^2 + c)*b - b*c + a*d)
) - 1/2*(2*b^4*c - 7*a*b^3*d)*arctan(sqrt(d*x^2 + c)*b/sqrt(-b^2*c + a*b*d))/((a^2*b^3*c^3 - 3*a^3*b^2*c^2*d +
3*a^4*b*c*d^2 - a^5*d^3)*sqrt(-b^2*c + a*b*d)) + 1/3*(9*(d*x^2 + c)*b*c*d^2 + b*c^2*d^2 - 3*(d*x^2 + c)*a*d^3
- a*c*d^3)/((b^3*c^5 - 3*a*b^2*c^4*d + 3*a^2*b*c^3*d^2 - a^3*c^2*d^3)*(d*x^2 + c)^(3/2)) + arctan(sqrt(d*x^2
+ c)/sqrt(-c))/(a^2*sqrt(-c)*c^2)

Mupad [B] (verification not implemented)

Time = 9.96 (sec) , antiderivative size = 8467, normalized size of antiderivative = 37.63 \[ \int \frac {1}{x \left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}} \, dx=\text {Too large to display} \]

[In]

int(1/(x*(a + b*x^2)^2*(c + d*x^2)^(5/2)),x)

[Out]

((d^2*(c + d*x^2)*(3*a*d - 8*b*c))/(3*(b*c^2 - a*c*d)^2) - d^2/(3*(b*c^2 - a*c*d)) + (d*(c + d*x^2)^2*(b^3*c^2
- 2*a^2*b*d^2 + 6*a*b^2*c*d))/(2*a*c*(b*c^2 - a*c*d)*(a*d - b*c)^2))/(b*(c + d*x^2)^(5/2) + (c + d*x^2)^(3/2)
*(a*d - b*c)) - atanh((560*a^3*b^16*c^19*d^4*(c + d*x^2)^(1/2))/((c^5)^(1/2)*(560*a^3*b^16*c^17*d^4 - 7280*a^4
*b^15*c^16*d^5 + 42560*a^5*b^14*c^15*d^6 - 149184*a^6*b^13*c^14*d^7 + 351904*a^7*b^12*c^13*d^8 - 593440*a^8*b^
11*c^12*d^9 + 741120*a^9*b^10*c^11*d^10 - 699840*a^10*b^9*c^10*d^11 + 505008*a^11*b^8*c^9*d^12 - 278768*a^12*b
^7*c^8*d^13 + 116480*a^13*b^6*c^7*d^14 - 35840*a^14*b^5*c^6*d^15 + 7680*a^15*b^4*c^5*d^16 - 1024*a^16*b^3*c^4*
d^17 + 64*a^17*b^2*c^3*d^18)) - (7280*a^4*b^15*c^18*d^5*(c + d*x^2)^(1/2))/((c^5)^(1/2)*(560*a^3*b^16*c^17*d^4
- 7280*a^4*b^15*c^16*d^5 + 42560*a^5*b^14*c^15*d^6 - 149184*a^6*b^13*c^14*d^7 + 351904*a^7*b^12*c^13*d^8 - 59
3440*a^8*b^11*c^12*d^9 + 741120*a^9*b^10*c^11*d^10 - 699840*a^10*b^9*c^10*d^11 + 505008*a^11*b^8*c^9*d^12 - 27
8768*a^12*b^7*c^8*d^13 + 116480*a^13*b^6*c^7*d^14 - 35840*a^14*b^5*c^6*d^15 + 7680*a^15*b^4*c^5*d^16 - 1024*a^
16*b^3*c^4*d^17 + 64*a^17*b^2*c^3*d^18)) + (42560*a^5*b^14*c^17*d^6*(c + d*x^2)^(1/2))/((c^5)^(1/2)*(560*a^3*b
^16*c^17*d^4 - 7280*a^4*b^15*c^16*d^5 + 42560*a^5*b^14*c^15*d^6 - 149184*a^6*b^13*c^14*d^7 + 351904*a^7*b^12*c
^13*d^8 - 593440*a^8*b^11*c^12*d^9 + 741120*a^9*b^10*c^11*d^10 - 699840*a^10*b^9*c^10*d^11 + 505008*a^11*b^8*c
^9*d^12 - 278768*a^12*b^7*c^8*d^13 + 116480*a^13*b^6*c^7*d^14 - 35840*a^14*b^5*c^6*d^15 + 7680*a^15*b^4*c^5*d^
16 - 1024*a^16*b^3*c^4*d^17 + 64*a^17*b^2*c^3*d^18)) - (149184*a^6*b^13*c^16*d^7*(c + d*x^2)^(1/2))/((c^5)^(1/
2)*(560*a^3*b^16*c^17*d^4 - 7280*a^4*b^15*c^16*d^5 + 42560*a^5*b^14*c^15*d^6 - 149184*a^6*b^13*c^14*d^7 + 3519
04*a^7*b^12*c^13*d^8 - 593440*a^8*b^11*c^12*d^9 + 741120*a^9*b^10*c^11*d^10 - 699840*a^10*b^9*c^10*d^11 + 5050
08*a^11*b^8*c^9*d^12 - 278768*a^12*b^7*c^8*d^13 + 116480*a^13*b^6*c^7*d^14 - 35840*a^14*b^5*c^6*d^15 + 7680*a^
15*b^4*c^5*d^16 - 1024*a^16*b^3*c^4*d^17 + 64*a^17*b^2*c^3*d^18)) + (351904*a^7*b^12*c^15*d^8*(c + d*x^2)^(1/2
))/((c^5)^(1/2)*(560*a^3*b^16*c^17*d^4 - 7280*a^4*b^15*c^16*d^5 + 42560*a^5*b^14*c^15*d^6 - 149184*a^6*b^13*c^
14*d^7 + 351904*a^7*b^12*c^13*d^8 - 593440*a^8*b^11*c^12*d^9 + 741120*a^9*b^10*c^11*d^10 - 699840*a^10*b^9*c^1
0*d^11 + 505008*a^11*b^8*c^9*d^12 - 278768*a^12*b^7*c^8*d^13 + 116480*a^13*b^6*c^7*d^14 - 35840*a^14*b^5*c^6*d
^15 + 7680*a^15*b^4*c^5*d^16 - 1024*a^16*b^3*c^4*d^17 + 64*a^17*b^2*c^3*d^18)) - (593440*a^8*b^11*c^14*d^9*(c
+ d*x^2)^(1/2))/((c^5)^(1/2)*(560*a^3*b^16*c^17*d^4 - 7280*a^4*b^15*c^16*d^5 + 42560*a^5*b^14*c^15*d^6 - 14918
4*a^6*b^13*c^14*d^7 + 351904*a^7*b^12*c^13*d^8 - 593440*a^8*b^11*c^12*d^9 + 741120*a^9*b^10*c^11*d^10 - 699840
*a^10*b^9*c^10*d^11 + 505008*a^11*b^8*c^9*d^12 - 278768*a^12*b^7*c^8*d^13 + 116480*a^13*b^6*c^7*d^14 - 35840*a
^14*b^5*c^6*d^15 + 7680*a^15*b^4*c^5*d^16 - 1024*a^16*b^3*c^4*d^17 + 64*a^17*b^2*c^3*d^18)) + (741120*a^9*b^10
*c^13*d^10*(c + d*x^2)^(1/2))/((c^5)^(1/2)*(560*a^3*b^16*c^17*d^4 - 7280*a^4*b^15*c^16*d^5 + 42560*a^5*b^14*c^
15*d^6 - 149184*a^6*b^13*c^14*d^7 + 351904*a^7*b^12*c^13*d^8 - 593440*a^8*b^11*c^12*d^9 + 741120*a^9*b^10*c^11
*d^10 - 699840*a^10*b^9*c^10*d^11 + 505008*a^11*b^8*c^9*d^12 - 278768*a^12*b^7*c^8*d^13 + 116480*a^13*b^6*c^7*
d^14 - 35840*a^14*b^5*c^6*d^15 + 7680*a^15*b^4*c^5*d^16 - 1024*a^16*b^3*c^4*d^17 + 64*a^17*b^2*c^3*d^18)) - (6
99840*a^10*b^9*c^12*d^11*(c + d*x^2)^(1/2))/((c^5)^(1/2)*(560*a^3*b^16*c^17*d^4 - 7280*a^4*b^15*c^16*d^5 + 425
60*a^5*b^14*c^15*d^6 - 149184*a^6*b^13*c^14*d^7 + 351904*a^7*b^12*c^13*d^8 - 593440*a^8*b^11*c^12*d^9 + 741120
*a^9*b^10*c^11*d^10 - 699840*a^10*b^9*c^10*d^11 + 505008*a^11*b^8*c^9*d^12 - 278768*a^12*b^7*c^8*d^13 + 116480
*a^13*b^6*c^7*d^14 - 35840*a^14*b^5*c^6*d^15 + 7680*a^15*b^4*c^5*d^16 - 1024*a^16*b^3*c^4*d^17 + 64*a^17*b^2*c
^3*d^18)) + (505008*a^11*b^8*c^11*d^12*(c + d*x^2)^(1/2))/((c^5)^(1/2)*(560*a^3*b^16*c^17*d^4 - 7280*a^4*b^15*
c^16*d^5 + 42560*a^5*b^14*c^15*d^6 - 149184*a^6*b^13*c^14*d^7 + 351904*a^7*b^12*c^13*d^8 - 593440*a^8*b^11*c^1
2*d^9 + 741120*a^9*b^10*c^11*d^10 - 699840*a^10*b^9*c^10*d^11 + 505008*a^11*b^8*c^9*d^12 - 278768*a^12*b^7*c^8
*d^13 + 116480*a^13*b^6*c^7*d^14 - 35840*a^14*b^5*c^6*d^15 + 7680*a^15*b^4*c^5*d^16 - 1024*a^16*b^3*c^4*d^17 +
64*a^17*b^2*c^3*d^18)) - (278768*a^12*b^7*c^10*d^13*(c + d*x^2)^(1/2))/((c^5)^(1/2)*(560*a^3*b^16*c^17*d^4 -
7280*a^4*b^15*c^16*d^5 + 42560*a^5*b^14*c^15*d^6 - 149184*a^6*b^13*c^14*d^7 + 351904*a^7*b^12*c^13*d^8 - 59344
0*a^8*b^11*c^12*d^9 + 741120*a^9*b^10*c^11*d^10 - 699840*a^10*b^9*c^10*d^11 + 505008*a^11*b^8*c^9*d^12 - 27876
8*a^12*b^7*c^8*d^13 + 116480*a^13*b^6*c^7*d^14 - 35840*a^14*b^5*c^6*d^15 + 7680*a^15*b^4*c^5*d^16 - 1024*a^16*
b^3*c^4*d^17 + 64*a^17*b^2*c^3*d^18)) + (116480*a^13*b^6*c^9*d^14*(c + d*x^2)^(1/2))/((c^5)^(1/2)*(560*a^3*b^1
6*c^17*d^4 - 7280*a^4*b^15*c^16*d^5 + 42560*a^5*b^14*c^15*d^6 - 149184*a^6*b^13*c^14*d^7 + 351904*a^7*b^12*c^1
3*d^8 - 593440*a^8*b^11*c^12*d^9 + 741120*a^9*b^10*c^11*d^10 - 699840*a^10*b^9*c^10*d^11 + 505008*a^11*b^8*c^9
*d^12 - 278768*a^12*b^7*c^8*d^13 + 116480*a^13*b^6*c^7*d^14 - 35840*a^14*b^5*c^6*d^15 + 7680*a^15*b^4*c^5*d^16
- 1024*a^16*b^3*c^4*d^17 + 64*a^17*b^2*c^3*d^18)) - (35840*a^14*b^5*c^8*d^15*(c + d*x^2)^(1/2))/((c^5)^(1/2)*
(560*a^3*b^16*c^17*d^4 - 7280*a^4*b^15*c^16*d^5 + 42560*a^5*b^14*c^15*d^6 - 149184*a^6*b^13*c^14*d^7 + 351904*
a^7*b^12*c^13*d^8 - 593440*a^8*b^11*c^12*d^9 + 741120*a^9*b^10*c^11*d^10 - 699840*a^10*b^9*c^10*d^11 + 505008*
a^11*b^8*c^9*d^12 - 278768*a^12*b^7*c^8*d^13 + 116480*a^13*b^6*c^7*d^14 - 35840*a^14*b^5*c^6*d^15 + 7680*a^15*
b^4*c^5*d^16 - 1024*a^16*b^3*c^4*d^17 + 64*a^17*b^2*c^3*d^18)) + (7680*a^15*b^4*c^7*d^16*(c + d*x^2)^(1/2))/((
c^5)^(1/2)*(560*a^3*b^16*c^17*d^4 - 7280*a^4*b^15*c^16*d^5 + 42560*a^5*b^14*c^15*d^6 - 149184*a^6*b^13*c^14*d^
7 + 351904*a^7*b^12*c^13*d^8 - 593440*a^8*b^11*c^12*d^9 + 741120*a^9*b^10*c^11*d^10 - 699840*a^10*b^9*c^10*d^1
1 + 505008*a^11*b^8*c^9*d^12 - 278768*a^12*b^7*c^8*d^13 + 116480*a^13*b^6*c^7*d^14 - 35840*a^14*b^5*c^6*d^15 +
7680*a^15*b^4*c^5*d^16 - 1024*a^16*b^3*c^4*d^17 + 64*a^17*b^2*c^3*d^18)) - (1024*a^16*b^3*c^6*d^17*(c + d*x^2
)^(1/2))/((c^5)^(1/2)*(560*a^3*b^16*c^17*d^4 - 7280*a^4*b^15*c^16*d^5 + 42560*a^5*b^14*c^15*d^6 - 149184*a^6*b
^13*c^14*d^7 + 351904*a^7*b^12*c^13*d^8 - 593440*a^8*b^11*c^12*d^9 + 741120*a^9*b^10*c^11*d^10 - 699840*a^10*b
^9*c^10*d^11 + 505008*a^11*b^8*c^9*d^12 - 278768*a^12*b^7*c^8*d^13 + 116480*a^13*b^6*c^7*d^14 - 35840*a^14*b^5
*c^6*d^15 + 7680*a^15*b^4*c^5*d^16 - 1024*a^16*b^3*c^4*d^17 + 64*a^17*b^2*c^3*d^18)) + (64*a^17*b^2*c^5*d^18*(
c + d*x^2)^(1/2))/((c^5)^(1/2)*(560*a^3*b^16*c^17*d^4 - 7280*a^4*b^15*c^16*d^5 + 42560*a^5*b^14*c^15*d^6 - 149
184*a^6*b^13*c^14*d^7 + 351904*a^7*b^12*c^13*d^8 - 593440*a^8*b^11*c^12*d^9 + 741120*a^9*b^10*c^11*d^10 - 6998
40*a^10*b^9*c^10*d^11 + 505008*a^11*b^8*c^9*d^12 - 278768*a^12*b^7*c^8*d^13 + 116480*a^13*b^6*c^7*d^14 - 35840
*a^14*b^5*c^6*d^15 + 7680*a^15*b^4*c^5*d^16 - 1024*a^16*b^3*c^4*d^17 + 64*a^17*b^2*c^3*d^18)))/(a^2*(c^5)^(1/2
)) + (atan((((-b^5*(a*d - b*c)^7)^(1/2)*((c + d*x^2)^(1/2)*(128*a^3*b^18*c^21*d^2 - 1984*a^4*b^17*c^20*d^3 + 1
3840*a^5*b^16*c^19*d^4 - 57680*a^6*b^15*c^18*d^5 + 161280*a^7*b^14*c^17*d^6 - 322560*a^8*b^13*c^16*d^7 + 48092
8*a^9*b^12*c^15*d^8 - 550560*a^10*b^11*c^14*d^9 + 494400*a^11*b^10*c^13*d^10 - 352640*a^12*b^9*c^12*d^11 + 199
696*a^13*b^8*c^11*d^12 - 88144*a^14*b^7*c^10*d^13 + 29120*a^15*b^6*c^9*d^14 - 6720*a^16*b^5*c^8*d^15 + 960*a^1
7*b^4*c^7*d^16 - 64*a^18*b^3*c^6*d^17) - ((-b^5*(a*d - b*c)^7)^(1/2)*(7*a*d - 2*b*c)*(1536*a^7*b^16*c^22*d^4 -
64*a^6*b^17*c^23*d^3 - 13952*a^8*b^15*c^21*d^5 + 71040*a^9*b^14*c^20*d^6 - 235968*a^10*b^13*c^19*d^7 + 551936
*a^11*b^12*c^18*d^8 - 948992*a^12*b^11*c^17*d^9 + 1229184*a^13*b^10*c^16*d^10 - 1214400*a^14*b^9*c^15*d^11 + 9
18016*a^15*b^8*c^14*d^12 - 528000*a^16*b^7*c^13*d^13 + 227456*a^17*b^6*c^12*d^14 - 71232*a^18*b^5*c^11*d^15 +
15360*a^19*b^4*c^10*d^16 - 2048*a^20*b^3*c^9*d^17 + 128*a^21*b^2*c^8*d^18 + ((-b^5*(a*d - b*c)^7)^(1/2)*(c + d
*x^2)^(1/2)*(7*a*d - 2*b*c)*(512*a^7*b^18*c^26*d^2 - 7936*a^8*b^17*c^25*d^3 + 57600*a^9*b^16*c^24*d^4 - 259840
*a^10*b^15*c^23*d^5 + 815360*a^11*b^14*c^22*d^6 - 1886976*a^12*b^13*c^21*d^7 + 3331328*a^13*b^12*c^20*d^8 - 45
76000*a^14*b^11*c^19*d^9 + 4942080*a^15*b^10*c^18*d^10 - 4209920*a^16*b^9*c^17*d^11 + 2818816*a^17*b^8*c^16*d^
12 - 1467648*a^18*b^7*c^15*d^13 + 582400*a^19*b^6*c^14*d^14 - 170240*a^20*b^5*c^13*d^15 + 34560*a^21*b^4*c^12*
d^16 - 4352*a^22*b^3*c^11*d^17 + 256*a^23*b^2*c^10*d^18))/(4*(a^9*d^7 - a^2*b^7*c^7 + 7*a^3*b^6*c^6*d - 21*a^4
*b^5*c^5*d^2 + 35*a^5*b^4*c^4*d^3 - 35*a^6*b^3*c^3*d^4 + 21*a^7*b^2*c^2*d^5 - 7*a^8*b*c*d^6))))/(4*(a^9*d^7 -
a^2*b^7*c^7 + 7*a^3*b^6*c^6*d - 21*a^4*b^5*c^5*d^2 + 35*a^5*b^4*c^4*d^3 - 35*a^6*b^3*c^3*d^4 + 21*a^7*b^2*c^2*
d^5 - 7*a^8*b*c*d^6)))*(7*a*d - 2*b*c)*1i)/(4*(a^9*d^7 - a^2*b^7*c^7 + 7*a^3*b^6*c^6*d - 21*a^4*b^5*c^5*d^2 +
35*a^5*b^4*c^4*d^3 - 35*a^6*b^3*c^3*d^4 + 21*a^7*b^2*c^2*d^5 - 7*a^8*b*c*d^6)) + ((-b^5*(a*d - b*c)^7)^(1/2)*(
(c + d*x^2)^(1/2)*(128*a^3*b^18*c^21*d^2 - 1984*a^4*b^17*c^20*d^3 + 13840*a^5*b^16*c^19*d^4 - 57680*a^6*b^15*c
^18*d^5 + 161280*a^7*b^14*c^17*d^6 - 322560*a^8*b^13*c^16*d^7 + 480928*a^9*b^12*c^15*d^8 - 550560*a^10*b^11*c^
14*d^9 + 494400*a^11*b^10*c^13*d^10 - 352640*a^12*b^9*c^12*d^11 + 199696*a^13*b^8*c^11*d^12 - 88144*a^14*b^7*c
^10*d^13 + 29120*a^15*b^6*c^9*d^14 - 6720*a^16*b^5*c^8*d^15 + 960*a^17*b^4*c^7*d^16 - 64*a^18*b^3*c^6*d^17) -
((-b^5*(a*d - b*c)^7)^(1/2)*(7*a*d - 2*b*c)*(64*a^6*b^17*c^23*d^3 - 1536*a^7*b^16*c^22*d^4 + 13952*a^8*b^15*c^
21*d^5 - 71040*a^9*b^14*c^20*d^6 + 235968*a^10*b^13*c^19*d^7 - 551936*a^11*b^12*c^18*d^8 + 948992*a^12*b^11*c^
17*d^9 - 1229184*a^13*b^10*c^16*d^10 + 1214400*a^14*b^9*c^15*d^11 - 918016*a^15*b^8*c^14*d^12 + 528000*a^16*b^
7*c^13*d^13 - 227456*a^17*b^6*c^12*d^14 + 71232*a^18*b^5*c^11*d^15 - 15360*a^19*b^4*c^10*d^16 + 2048*a^20*b^3*
c^9*d^17 - 128*a^21*b^2*c^8*d^18 + ((-b^5*(a*d - b*c)^7)^(1/2)*(c + d*x^2)^(1/2)*(7*a*d - 2*b*c)*(512*a^7*b^18
*c^26*d^2 - 7936*a^8*b^17*c^25*d^3 + 57600*a^9*b^16*c^24*d^4 - 259840*a^10*b^15*c^23*d^5 + 815360*a^11*b^14*c^
22*d^6 - 1886976*a^12*b^13*c^21*d^7 + 3331328*a^13*b^12*c^20*d^8 - 4576000*a^14*b^11*c^19*d^9 + 4942080*a^15*b
^10*c^18*d^10 - 4209920*a^16*b^9*c^17*d^11 + 2818816*a^17*b^8*c^16*d^12 - 1467648*a^18*b^7*c^15*d^13 + 582400*
a^19*b^6*c^14*d^14 - 170240*a^20*b^5*c^13*d^15 + 34560*a^21*b^4*c^12*d^16 - 4352*a^22*b^3*c^11*d^17 + 256*a^23
*b^2*c^10*d^18))/(4*(a^9*d^7 - a^2*b^7*c^7 + 7*a^3*b^6*c^6*d - 21*a^4*b^5*c^5*d^2 + 35*a^5*b^4*c^4*d^3 - 35*a^
6*b^3*c^3*d^4 + 21*a^7*b^2*c^2*d^5 - 7*a^8*b*c*d^6))))/(4*(a^9*d^7 - a^2*b^7*c^7 + 7*a^3*b^6*c^6*d - 21*a^4*b^
5*c^5*d^2 + 35*a^5*b^4*c^4*d^3 - 35*a^6*b^3*c^3*d^4 + 21*a^7*b^2*c^2*d^5 - 7*a^8*b*c*d^6)))*(7*a*d - 2*b*c)*1i
)/(4*(a^9*d^7 - a^2*b^7*c^7 + 7*a^3*b^6*c^6*d - 21*a^4*b^5*c^5*d^2 + 35*a^5*b^4*c^4*d^3 - 35*a^6*b^3*c^3*d^4 +
21*a^7*b^2*c^2*d^5 - 7*a^8*b*c*d^6)))/(208*a^3*b^16*c^17*d^4 - 32*a^2*b^17*c^18*d^3 + 304*a^4*b^15*c^16*d^5 -
7040*a^5*b^14*c^15*d^6 + 31200*a^6*b^13*c^14*d^7 - 75936*a^7*b^12*c^13*d^8 + 118944*a^8*b^11*c^12*d^9 - 12652
8*a^9*b^10*c^11*d^10 + 92640*a^10*b^9*c^10*d^11 - 46000*a^11*b^8*c^9*d^12 + 14768*a^12*b^7*c^8*d^13 - 2752*a^1
3*b^6*c^7*d^14 + 224*a^14*b^5*c^6*d^15 + ((-b^5*(a*d - b*c)^7)^(1/2)*((c + d*x^2)^(1/2)*(128*a^3*b^18*c^21*d^2
- 1984*a^4*b^17*c^20*d^3 + 13840*a^5*b^16*c^19*d^4 - 57680*a^6*b^15*c^18*d^5 + 161280*a^7*b^14*c^17*d^6 - 322
560*a^8*b^13*c^16*d^7 + 480928*a^9*b^12*c^15*d^8 - 550560*a^10*b^11*c^14*d^9 + 494400*a^11*b^10*c^13*d^10 - 35
2640*a^12*b^9*c^12*d^11 + 199696*a^13*b^8*c^11*d^12 - 88144*a^14*b^7*c^10*d^13 + 29120*a^15*b^6*c^9*d^14 - 672
0*a^16*b^5*c^8*d^15 + 960*a^17*b^4*c^7*d^16 - 64*a^18*b^3*c^6*d^17) - ((-b^5*(a*d - b*c)^7)^(1/2)*(7*a*d - 2*b
*c)*(1536*a^7*b^16*c^22*d^4 - 64*a^6*b^17*c^23*d^3 - 13952*a^8*b^15*c^21*d^5 + 71040*a^9*b^14*c^20*d^6 - 23596
8*a^10*b^13*c^19*d^7 + 551936*a^11*b^12*c^18*d^8 - 948992*a^12*b^11*c^17*d^9 + 1229184*a^13*b^10*c^16*d^10 - 1
214400*a^14*b^9*c^15*d^11 + 918016*a^15*b^8*c^14*d^12 - 528000*a^16*b^7*c^13*d^13 + 227456*a^17*b^6*c^12*d^14
- 71232*a^18*b^5*c^11*d^15 + 15360*a^19*b^4*c^10*d^16 - 2048*a^20*b^3*c^9*d^17 + 128*a^21*b^2*c^8*d^18 + ((-b^
5*(a*d - b*c)^7)^(1/2)*(c + d*x^2)^(1/2)*(7*a*d - 2*b*c)*(512*a^7*b^18*c^26*d^2 - 7936*a^8*b^17*c^25*d^3 + 576
00*a^9*b^16*c^24*d^4 - 259840*a^10*b^15*c^23*d^5 + 815360*a^11*b^14*c^22*d^6 - 1886976*a^12*b^13*c^21*d^7 + 33
31328*a^13*b^12*c^20*d^8 - 4576000*a^14*b^11*c^19*d^9 + 4942080*a^15*b^10*c^18*d^10 - 4209920*a^16*b^9*c^17*d^
11 + 2818816*a^17*b^8*c^16*d^12 - 1467648*a^18*b^7*c^15*d^13 + 582400*a^19*b^6*c^14*d^14 - 170240*a^20*b^5*c^1
3*d^15 + 34560*a^21*b^4*c^12*d^16 - 4352*a^22*b^3*c^11*d^17 + 256*a^23*b^2*c^10*d^18))/(4*(a^9*d^7 - a^2*b^7*c
^7 + 7*a^3*b^6*c^6*d - 21*a^4*b^5*c^5*d^2 + 35*a^5*b^4*c^4*d^3 - 35*a^6*b^3*c^3*d^4 + 21*a^7*b^2*c^2*d^5 - 7*a
^8*b*c*d^6))))/(4*(a^9*d^7 - a^2*b^7*c^7 + 7*a^3*b^6*c^6*d - 21*a^4*b^5*c^5*d^2 + 35*a^5*b^4*c^4*d^3 - 35*a^6*
b^3*c^3*d^4 + 21*a^7*b^2*c^2*d^5 - 7*a^8*b*c*d^6)))*(7*a*d - 2*b*c))/(4*(a^9*d^7 - a^2*b^7*c^7 + 7*a^3*b^6*c^6
*d - 21*a^4*b^5*c^5*d^2 + 35*a^5*b^4*c^4*d^3 - 35*a^6*b^3*c^3*d^4 + 21*a^7*b^2*c^2*d^5 - 7*a^8*b*c*d^6)) - ((-
b^5*(a*d - b*c)^7)^(1/2)*((c + d*x^2)^(1/2)*(128*a^3*b^18*c^21*d^2 - 1984*a^4*b^17*c^20*d^3 + 13840*a^5*b^16*c
^19*d^4 - 57680*a^6*b^15*c^18*d^5 + 161280*a^7*b^14*c^17*d^6 - 322560*a^8*b^13*c^16*d^7 + 480928*a^9*b^12*c^15
*d^8 - 550560*a^10*b^11*c^14*d^9 + 494400*a^11*b^10*c^13*d^10 - 352640*a^12*b^9*c^12*d^11 + 199696*a^13*b^8*c^
11*d^12 - 88144*a^14*b^7*c^10*d^13 + 29120*a^15*b^6*c^9*d^14 - 6720*a^16*b^5*c^8*d^15 + 960*a^17*b^4*c^7*d^16
- 64*a^18*b^3*c^6*d^17) - ((-b^5*(a*d - b*c)^7)^(1/2)*(7*a*d - 2*b*c)*(64*a^6*b^17*c^23*d^3 - 1536*a^7*b^16*c^
22*d^4 + 13952*a^8*b^15*c^21*d^5 - 71040*a^9*b^14*c^20*d^6 + 235968*a^10*b^13*c^19*d^7 - 551936*a^11*b^12*c^18
*d^8 + 948992*a^12*b^11*c^17*d^9 - 1229184*a^13*b^10*c^16*d^10 + 1214400*a^14*b^9*c^15*d^11 - 918016*a^15*b^8*
c^14*d^12 + 528000*a^16*b^7*c^13*d^13 - 227456*a^17*b^6*c^12*d^14 + 71232*a^18*b^5*c^11*d^15 - 15360*a^19*b^4*
c^10*d^16 + 2048*a^20*b^3*c^9*d^17 - 128*a^21*b^2*c^8*d^18 + ((-b^5*(a*d - b*c)^7)^(1/2)*(c + d*x^2)^(1/2)*(7*
a*d - 2*b*c)*(512*a^7*b^18*c^26*d^2 - 7936*a^8*b^17*c^25*d^3 + 57600*a^9*b^16*c^24*d^4 - 259840*a^10*b^15*c^23
*d^5 + 815360*a^11*b^14*c^22*d^6 - 1886976*a^12*b^13*c^21*d^7 + 3331328*a^13*b^12*c^20*d^8 - 4576000*a^14*b^11
*c^19*d^9 + 4942080*a^15*b^10*c^18*d^10 - 4209920*a^16*b^9*c^17*d^11 + 2818816*a^17*b^8*c^16*d^12 - 1467648*a^
18*b^7*c^15*d^13 + 582400*a^19*b^6*c^14*d^14 - 170240*a^20*b^5*c^13*d^15 + 34560*a^21*b^4*c^12*d^16 - 4352*a^2
2*b^3*c^11*d^17 + 256*a^23*b^2*c^10*d^18))/(4*(a^9*d^7 - a^2*b^7*c^7 + 7*a^3*b^6*c^6*d - 21*a^4*b^5*c^5*d^2 +
35*a^5*b^4*c^4*d^3 - 35*a^6*b^3*c^3*d^4 + 21*a^7*b^2*c^2*d^5 - 7*a^8*b*c*d^6))))/(4*(a^9*d^7 - a^2*b^7*c^7 + 7
*a^3*b^6*c^6*d - 21*a^4*b^5*c^5*d^2 + 35*a^5*b^4*c^4*d^3 - 35*a^6*b^3*c^3*d^4 + 21*a^7*b^2*c^2*d^5 - 7*a^8*b*c
*d^6)))*(7*a*d - 2*b*c))/(4*(a^9*d^7 - a^2*b^7*c^7 + 7*a^3*b^6*c^6*d - 21*a^4*b^5*c^5*d^2 + 35*a^5*b^4*c^4*d^3
- 35*a^6*b^3*c^3*d^4 + 21*a^7*b^2*c^2*d^5 - 7*a^8*b*c*d^6))))*(-b^5*(a*d - b*c)^7)^(1/2)*(7*a*d - 2*b*c)*1i)/
(2*(a^9*d^7 - a^2*b^7*c^7 + 7*a^3*b^6*c^6*d - 21*a^4*b^5*c^5*d^2 + 35*a^5*b^4*c^4*d^3 - 35*a^6*b^3*c^3*d^4 + 2
1*a^7*b^2*c^2*d^5 - 7*a^8*b*c*d^6))