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

Optimal. Leaf size=218 \[ -\frac{3 a^2 d^2-4 a b c d+3 b^2 c^2}{2 a^2 c^2 x (b c-a d)^2}-\frac{b^{5/2} (3 b c-7 a d) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{5/2} (b c-a d)^3}-\frac{d^{5/2} (7 b c-3 a d) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{2 c^{5/2} (b c-a d)^3}+\frac{b}{2 a x \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}+\frac{d (a d+b c)}{2 a c x \left (c+d x^2\right ) (b c-a d)^2} \]

[Out]

-(3*b^2*c^2 - 4*a*b*c*d + 3*a^2*d^2)/(2*a^2*c^2*(b*c - a*d)^2*x) + (d*(b*c + a*d))/(2*a*c*(b*c - a*d)^2*x*(c +
d*x^2)) + b/(2*a*(b*c - a*d)*x*(a + b*x^2)*(c + d*x^2)) - (b^(5/2)*(3*b*c - 7*a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]
])/(2*a^(5/2)*(b*c - a*d)^3) - (d^(5/2)*(7*b*c - 3*a*d)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(2*c^(5/2)*(b*c - a*d)^3)

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Rubi [A]  time = 0.311306, antiderivative size = 218, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {472, 579, 583, 522, 205} \[ -\frac{3 a^2 d^2-4 a b c d+3 b^2 c^2}{2 a^2 c^2 x (b c-a d)^2}-\frac{b^{5/2} (3 b c-7 a d) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{5/2} (b c-a d)^3}-\frac{d^{5/2} (7 b c-3 a d) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{2 c^{5/2} (b c-a d)^3}+\frac{b}{2 a x \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}+\frac{d (a d+b c)}{2 a c x \left (c+d x^2\right ) (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(3*b^2*c^2 - 4*a*b*c*d + 3*a^2*d^2)/(2*a^2*c^2*(b*c - a*d)^2*x) + (d*(b*c + a*d))/(2*a*c*(b*c - a*d)^2*x*(c +
d*x^2)) + b/(2*a*(b*c - a*d)*x*(a + b*x^2)*(c + d*x^2)) - (b^(5/2)*(3*b*c - 7*a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]
])/(2*a^(5/2)*(b*c - a*d)^3) - (d^(5/2)*(7*b*c - 3*a*d)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(2*c^(5/2)*(b*c - a*d)^3)

Rule 472

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Simp[(b*(e*x
)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*e*n*(b*c - a*d)*(p + 1)), x] + Dist[1/(a*n*(b*c - a*d)*(
p + 1)), Int[(e*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*b*(m + 1) + n*(b*c - a*d)*(p + 1) + d*b*(m + n*(
p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, m, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p
, -1] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]

Rule 579

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_)*((e_) + (f_.)*(x_)^(n_)), x
_Symbol] :> -Simp[((b*e - a*f)*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*g*n*(b*c - a*d)*(p +
1)), x] + Dist[1/(a*n*(b*c - a*d)*(p + 1)), Int[(g*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f)*(
m + 1) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*f)*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d,
e, f, g, m, q}, x] && IGtQ[n, 0] && LtQ[p, -1]

Rule 583

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

Rule 522

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> Dist[(b*e - a*f
)/(b*c - a*d), Int[1/(a + b*x^n), x], x] - Dist[(d*e - c*f)/(b*c - a*d), Int[1/(c + d*x^n), x], x] /; FreeQ[{a
, b, c, d, e, f, n}, x]

Rule 205

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

Rubi steps

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

Mathematica [A]  time = 0.294245, size = 158, normalized size = 0.72 \[ \frac{1}{2} \left (-\frac{b^3 x}{a^2 \left (a+b x^2\right ) (b c-a d)^2}+\frac{b^{5/2} (3 b c-7 a d) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{5/2} (a d-b c)^3}-\frac{2}{a^2 c^2 x}-\frac{d^3 x}{c^2 \left (c+d x^2\right ) (b c-a d)^2}+\frac{d^{5/2} (3 a d-7 b c) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{c^{5/2} (b c-a d)^3}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2/(a^2*c^2*x) - (b^3*x)/(a^2*(b*c - a*d)^2*(a + b*x^2)) - (d^3*x)/(c^2*(b*c - a*d)^2*(c + d*x^2)) + (b^(5/2)
*(3*b*c - 7*a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(a^(5/2)*(-(b*c) + a*d)^3) + (d^(5/2)*(-7*b*c + 3*a*d)*ArcTan[(S
qrt[d]*x)/Sqrt[c]])/(c^(5/2)*(b*c - a*d)^3))/2

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Maple [A]  time = 0.017, size = 261, normalized size = 1.2 \begin{align*} -{\frac{{d}^{4}xa}{2\,{c}^{2} \left ( ad-bc \right ) ^{3} \left ( d{x}^{2}+c \right ) }}+{\frac{{d}^{3}xb}{2\,c \left ( ad-bc \right ) ^{3} \left ( d{x}^{2}+c \right ) }}-{\frac{3\,{d}^{4}a}{2\,{c}^{2} \left ( ad-bc \right ) ^{3}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}+{\frac{7\,{d}^{3}b}{2\,c \left ( ad-bc \right ) ^{3}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-{\frac{1}{{a}^{2}{c}^{2}x}}-{\frac{{b}^{3}xd}{2\,a \left ( ad-bc \right ) ^{3} \left ( b{x}^{2}+a \right ) }}+{\frac{{b}^{4}xc}{2\,{a}^{2} \left ( ad-bc \right ) ^{3} \left ( b{x}^{2}+a \right ) }}-{\frac{7\,{b}^{3}d}{2\,a \left ( ad-bc \right ) ^{3}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{3\,{b}^{4}c}{2\,{a}^{2} \left ( ad-bc \right ) ^{3}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 12.5628, size = 4182, normalized size = 19.18 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/4*(4*a*b^3*c^4 - 12*a^2*b^2*c^3*d + 12*a^3*b*c^2*d^2 - 4*a^4*c*d^3 + 2*(3*b^4*c^3*d - 7*a*b^3*c^2*d^2 + 7*
a^2*b^2*c*d^3 - 3*a^3*b*d^4)*x^4 + 2*(3*b^4*c^4 - 5*a*b^3*c^3*d + 5*a^3*b*c*d^3 - 3*a^4*d^4)*x^2 - ((3*b^4*c^3
*d - 7*a*b^3*c^2*d^2)*x^5 + (3*b^4*c^4 - 4*a*b^3*c^3*d - 7*a^2*b^2*c^2*d^2)*x^3 + (3*a*b^3*c^4 - 7*a^2*b^2*c^3
*d)*x)*sqrt(-b/a)*log((b*x^2 - 2*a*x*sqrt(-b/a) - a)/(b*x^2 + a)) - ((7*a^2*b^2*c*d^3 - 3*a^3*b*d^4)*x^5 + (7*
a^2*b^2*c^2*d^2 + 4*a^3*b*c*d^3 - 3*a^4*d^4)*x^3 + (7*a^3*b*c^2*d^2 - 3*a^4*c*d^3)*x)*sqrt(-d/c)*log((d*x^2 -
2*c*x*sqrt(-d/c) - c)/(d*x^2 + c)))/((a^2*b^4*c^5*d - 3*a^3*b^3*c^4*d^2 + 3*a^4*b^2*c^3*d^3 - a^5*b*c^2*d^4)*x
^5 + (a^2*b^4*c^6 - 2*a^3*b^3*c^5*d + 2*a^5*b*c^3*d^3 - a^6*c^2*d^4)*x^3 + (a^3*b^3*c^6 - 3*a^4*b^2*c^5*d + 3*
a^5*b*c^4*d^2 - a^6*c^3*d^3)*x), -1/4*(4*a*b^3*c^4 - 12*a^2*b^2*c^3*d + 12*a^3*b*c^2*d^2 - 4*a^4*c*d^3 + 2*(3*
b^4*c^3*d - 7*a*b^3*c^2*d^2 + 7*a^2*b^2*c*d^3 - 3*a^3*b*d^4)*x^4 + 2*(3*b^4*c^4 - 5*a*b^3*c^3*d + 5*a^3*b*c*d^
3 - 3*a^4*d^4)*x^2 + 2*((7*a^2*b^2*c*d^3 - 3*a^3*b*d^4)*x^5 + (7*a^2*b^2*c^2*d^2 + 4*a^3*b*c*d^3 - 3*a^4*d^4)*
x^3 + (7*a^3*b*c^2*d^2 - 3*a^4*c*d^3)*x)*sqrt(d/c)*arctan(x*sqrt(d/c)) - ((3*b^4*c^3*d - 7*a*b^3*c^2*d^2)*x^5
+ (3*b^4*c^4 - 4*a*b^3*c^3*d - 7*a^2*b^2*c^2*d^2)*x^3 + (3*a*b^3*c^4 - 7*a^2*b^2*c^3*d)*x)*sqrt(-b/a)*log((b*x
^2 - 2*a*x*sqrt(-b/a) - a)/(b*x^2 + a)))/((a^2*b^4*c^5*d - 3*a^3*b^3*c^4*d^2 + 3*a^4*b^2*c^3*d^3 - a^5*b*c^2*d
^4)*x^5 + (a^2*b^4*c^6 - 2*a^3*b^3*c^5*d + 2*a^5*b*c^3*d^3 - a^6*c^2*d^4)*x^3 + (a^3*b^3*c^6 - 3*a^4*b^2*c^5*d
+ 3*a^5*b*c^4*d^2 - a^6*c^3*d^3)*x), -1/4*(4*a*b^3*c^4 - 12*a^2*b^2*c^3*d + 12*a^3*b*c^2*d^2 - 4*a^4*c*d^3 +
2*(3*b^4*c^3*d - 7*a*b^3*c^2*d^2 + 7*a^2*b^2*c*d^3 - 3*a^3*b*d^4)*x^4 + 2*(3*b^4*c^4 - 5*a*b^3*c^3*d + 5*a^3*b
*c*d^3 - 3*a^4*d^4)*x^2 + 2*((3*b^4*c^3*d - 7*a*b^3*c^2*d^2)*x^5 + (3*b^4*c^4 - 4*a*b^3*c^3*d - 7*a^2*b^2*c^2*
d^2)*x^3 + (3*a*b^3*c^4 - 7*a^2*b^2*c^3*d)*x)*sqrt(b/a)*arctan(x*sqrt(b/a)) - ((7*a^2*b^2*c*d^3 - 3*a^3*b*d^4)
*x^5 + (7*a^2*b^2*c^2*d^2 + 4*a^3*b*c*d^3 - 3*a^4*d^4)*x^3 + (7*a^3*b*c^2*d^2 - 3*a^4*c*d^3)*x)*sqrt(-d/c)*log
((d*x^2 - 2*c*x*sqrt(-d/c) - c)/(d*x^2 + c)))/((a^2*b^4*c^5*d - 3*a^3*b^3*c^4*d^2 + 3*a^4*b^2*c^3*d^3 - a^5*b*
c^2*d^4)*x^5 + (a^2*b^4*c^6 - 2*a^3*b^3*c^5*d + 2*a^5*b*c^3*d^3 - a^6*c^2*d^4)*x^3 + (a^3*b^3*c^6 - 3*a^4*b^2*
c^5*d + 3*a^5*b*c^4*d^2 - a^6*c^3*d^3)*x), -1/2*(2*a*b^3*c^4 - 6*a^2*b^2*c^3*d + 6*a^3*b*c^2*d^2 - 2*a^4*c*d^3
+ (3*b^4*c^3*d - 7*a*b^3*c^2*d^2 + 7*a^2*b^2*c*d^3 - 3*a^3*b*d^4)*x^4 + (3*b^4*c^4 - 5*a*b^3*c^3*d + 5*a^3*b*
c*d^3 - 3*a^4*d^4)*x^2 + ((3*b^4*c^3*d - 7*a*b^3*c^2*d^2)*x^5 + (3*b^4*c^4 - 4*a*b^3*c^3*d - 7*a^2*b^2*c^2*d^2
)*x^3 + (3*a*b^3*c^4 - 7*a^2*b^2*c^3*d)*x)*sqrt(b/a)*arctan(x*sqrt(b/a)) + ((7*a^2*b^2*c*d^3 - 3*a^3*b*d^4)*x^
5 + (7*a^2*b^2*c^2*d^2 + 4*a^3*b*c*d^3 - 3*a^4*d^4)*x^3 + (7*a^3*b*c^2*d^2 - 3*a^4*c*d^3)*x)*sqrt(d/c)*arctan(
x*sqrt(d/c)))/((a^2*b^4*c^5*d - 3*a^3*b^3*c^4*d^2 + 3*a^4*b^2*c^3*d^3 - a^5*b*c^2*d^4)*x^5 + (a^2*b^4*c^6 - 2*
a^3*b^3*c^5*d + 2*a^5*b*c^3*d^3 - a^6*c^2*d^4)*x^3 + (a^3*b^3*c^6 - 3*a^4*b^2*c^5*d + 3*a^5*b*c^4*d^2 - a^6*c^
3*d^3)*x)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [B]  time = 1.53009, size = 2430, normalized size = 11.15 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(3*sqrt(c*d)*a^2*b^6*c^7*abs(d) - 13*sqrt(c*d)*a^3*b^5*c^6*d*abs(d) + 10*sqrt(c*d)*a^4*b^4*c^5*d^2*abs(d)
+ 10*sqrt(c*d)*a^5*b^3*c^4*d^3*abs(d) - 13*sqrt(c*d)*a^6*b^2*c^3*d^4*abs(d) + 3*sqrt(c*d)*a^7*b*c^2*d^5*abs(d)
- 3*sqrt(c*d)*b^3*c^2*abs(a^2*b^3*c^5 - 3*a^3*b^2*c^4*d + 3*a^4*b*c^3*d^2 - a^5*c^2*d^3)*abs(d) + 4*sqrt(c*d)
*a*b^2*c*d*abs(a^2*b^3*c^5 - 3*a^3*b^2*c^4*d + 3*a^4*b*c^3*d^2 - a^5*c^2*d^3)*abs(d) - 3*sqrt(c*d)*a^2*b*d^2*a
bs(a^2*b^3*c^5 - 3*a^3*b^2*c^4*d + 3*a^4*b*c^3*d^2 - a^5*c^2*d^3)*abs(d))*arctan(2*sqrt(1/2)*x/sqrt((a^2*b^3*c
^5 - a^3*b^2*c^4*d - a^4*b*c^3*d^2 + a^5*c^2*d^3 + sqrt((a^2*b^3*c^5 - a^3*b^2*c^4*d - a^4*b*c^3*d^2 + a^5*c^2
*d^3)^2 - 4*(a^3*b^2*c^5 - 2*a^4*b*c^4*d + a^5*c^3*d^2)*(a^2*b^3*c^4*d - 2*a^3*b^2*c^3*d^2 + a^4*b*c^2*d^3)))/
(a^2*b^3*c^4*d - 2*a^3*b^2*c^3*d^2 + a^4*b*c^2*d^3)))/(a^2*b^3*c^5*d*abs(a^2*b^3*c^5 - 3*a^3*b^2*c^4*d + 3*a^4
*b*c^3*d^2 - a^5*c^2*d^3) - a^3*b^2*c^4*d^2*abs(a^2*b^3*c^5 - 3*a^3*b^2*c^4*d + 3*a^4*b*c^3*d^2 - a^5*c^2*d^3)
- a^4*b*c^3*d^3*abs(a^2*b^3*c^5 - 3*a^3*b^2*c^4*d + 3*a^4*b*c^3*d^2 - a^5*c^2*d^3) + a^5*c^2*d^4*abs(a^2*b^3*
c^5 - 3*a^3*b^2*c^4*d + 3*a^4*b*c^3*d^2 - a^5*c^2*d^3) + (a^2*b^3*c^5 - 3*a^3*b^2*c^4*d + 3*a^4*b*c^3*d^2 - a^
5*c^2*d^3)^2*d) - 1/2*(3*sqrt(a*b)*a^2*b^5*c^7*d*abs(b) - 13*sqrt(a*b)*a^3*b^4*c^6*d^2*abs(b) + 10*sqrt(a*b)*a
^4*b^3*c^5*d^3*abs(b) + 10*sqrt(a*b)*a^5*b^2*c^4*d^4*abs(b) - 13*sqrt(a*b)*a^6*b*c^3*d^5*abs(b) + 3*sqrt(a*b)*
a^7*c^2*d^6*abs(b) + 3*sqrt(a*b)*b^2*c^2*d*abs(a^2*b^3*c^5 - 3*a^3*b^2*c^4*d + 3*a^4*b*c^3*d^2 - a^5*c^2*d^3)*
abs(b) - 4*sqrt(a*b)*a*b*c*d^2*abs(a^2*b^3*c^5 - 3*a^3*b^2*c^4*d + 3*a^4*b*c^3*d^2 - a^5*c^2*d^3)*abs(b) + 3*s
qrt(a*b)*a^2*d^3*abs(a^2*b^3*c^5 - 3*a^3*b^2*c^4*d + 3*a^4*b*c^3*d^2 - a^5*c^2*d^3)*abs(b))*arctan(2*sqrt(1/2)
*x/sqrt((a^2*b^3*c^5 - a^3*b^2*c^4*d - a^4*b*c^3*d^2 + a^5*c^2*d^3 - sqrt((a^2*b^3*c^5 - a^3*b^2*c^4*d - a^4*b
*c^3*d^2 + a^5*c^2*d^3)^2 - 4*(a^3*b^2*c^5 - 2*a^4*b*c^4*d + a^5*c^3*d^2)*(a^2*b^3*c^4*d - 2*a^3*b^2*c^3*d^2 +
a^4*b*c^2*d^3)))/(a^2*b^3*c^4*d - 2*a^3*b^2*c^3*d^2 + a^4*b*c^2*d^3)))/(a^2*b^4*c^5*abs(a^2*b^3*c^5 - 3*a^3*b
^2*c^4*d + 3*a^4*b*c^3*d^2 - a^5*c^2*d^3) - a^3*b^3*c^4*d*abs(a^2*b^3*c^5 - 3*a^3*b^2*c^4*d + 3*a^4*b*c^3*d^2
- a^5*c^2*d^3) - a^4*b^2*c^3*d^2*abs(a^2*b^3*c^5 - 3*a^3*b^2*c^4*d + 3*a^4*b*c^3*d^2 - a^5*c^2*d^3) + a^5*b*c^
2*d^3*abs(a^2*b^3*c^5 - 3*a^3*b^2*c^4*d + 3*a^4*b*c^3*d^2 - a^5*c^2*d^3) - (a^2*b^3*c^5 - 3*a^3*b^2*c^4*d + 3*
a^4*b*c^3*d^2 - a^5*c^2*d^3)^2*b) - 1/2*(3*b^3*c^2*d*x^4 - 4*a*b^2*c*d^2*x^4 + 3*a^2*b*d^3*x^4 + 3*b^3*c^3*x^2
- 2*a*b^2*c^2*d*x^2 - 2*a^2*b*c*d^2*x^2 + 3*a^3*d^3*x^2 + 2*a*b^2*c^3 - 4*a^2*b*c^2*d + 2*a^3*c*d^2)/((a^2*b^
2*c^4 - 2*a^3*b*c^3*d + a^4*c^2*d^2)*(b*d*x^5 + b*c*x^3 + a*d*x^3 + a*c*x))