\(\int \frac {1}{x^2 (a+b x^2)^2 (c+d x^2)^2} \, dx\) [307]

Optimal result

Integrand size = 22, antiderivative size = 218 \[ \int \frac {1}{x^2 \left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=-\frac {3 b^2 c^2-4 a b c d+3 a^2 d^2}{2 a^2 c^2 (b c-a d)^2 x}+\frac {d (b c+a d)}{2 a c (b c-a d)^2 x \left (c+d x^2\right )}+\frac {b}{2 a (b c-a d) x \left (a+b x^2\right ) \left (c+d x^2\right )}-\frac {b^{5/2} (3 b c-7 a d) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{5/2} (b c-a d)^3}-\frac {d^{5/2} (7 b c-3 a d) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 c^{5/2} (b c-a d)^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {483, 593, 597, 536, 211} \[ \int \frac {1}{x^2 \left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=-\frac {b^{5/2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) (3 b c-7 a d)}{2 a^{5/2} (b c-a d)^3}-\frac {3 a^2 d^2-4 a b c d+3 b^2 c^2}{2 a^2 c^2 x (b c-a d)^2}-\frac {d^{5/2} (7 b c-3 a d) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 c^{5/2} (b c-a d)^3}+\frac {b}{2 a x \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}+\frac {d (a d+b c)}{2 a c x \left (c+d x^2\right ) (b c-a d)^2} \]

[In]

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

[Out]

-1/2*(3*b^2*c^2 - 4*a*b*c*d + 3*a^2*d^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 211

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

Rule 483

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 536

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 593

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 597

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]

Rubi steps \begin{align*} \text {integral}& = \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] (verified)

Time = 0.18 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.72 \[ \int \frac {1}{x^2 \left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=\frac {1}{2} \left (-\frac {2}{a^2 c^2 x}-\frac {b^3 x}{a^2 (b c-a d)^2 \left (a+b x^2\right )}-\frac {d^3 x}{c^2 (b c-a d)^2 \left (c+d x^2\right )}+\frac {b^{5/2} (3 b c-7 a d) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{5/2} (-b c+a d)^3}+\frac {d^{5/2} (-7 b c+3 a d) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{c^{5/2} (b c-a d)^3}\right ) \]

[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

Maple [A] (verified)

Time = 2.90 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.65

method	result	size
default	\(-\frac {1}{a^{2} c^{2} x}-\frac {b^{3} \left (\frac {\left (\frac {a d}{2}-\frac {b c}{2}\right ) x}{b \,x^{2}+a}+\frac {\left (7 a d -3 b c \right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{a^{2} \left (a d -b c \right )^{3}}-\frac {d^{3} \left (\frac {\left (\frac {a d}{2}-\frac {b c}{2}\right ) x}{d \,x^{2}+c}+\frac {\left (3 a d -7 b c \right ) \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{2 \sqrt {c d}}\right )}{c^{2} \left (a d -b c \right )^{3}}\)	\(141\)
risch	\(\text {Expression too large to display}\)	\(2299\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 502 vs. \(2 (192) = 384\).

Time = 1.31 (sec) , antiderivative size = 2113, normalized size of antiderivative = 9.69 \[ \int \frac {1}{x^2 \left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=\text {Too large to display} \]

[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)]

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{x^2 \left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 378, normalized size of antiderivative = 1.73 \[ \int \frac {1}{x^2 \left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=-\frac {{\left (3 \, b^{4} c - 7 \, a b^{3} d\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, {\left (a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - a^{5} d^{3}\right )} \sqrt {a b}} - \frac {{\left (7 \, b c d^{3} - 3 \, a d^{4}\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{2 \, {\left (b^{3} c^{5} - 3 \, a b^{2} c^{4} d + 3 \, a^{2} b c^{3} d^{2} - a^{3} c^{2} d^{3}\right )} \sqrt {c d}} - \frac {2 \, a b^{2} c^{3} - 4 \, a^{2} b c^{2} d + 2 \, a^{3} c d^{2} + {\left (3 \, b^{3} c^{2} d - 4 \, a b^{2} c d^{2} + 3 \, a^{2} b d^{3}\right )} x^{4} + {\left (3 \, b^{3} c^{3} - 2 \, a b^{2} c^{2} d - 2 \, a^{2} b c d^{2} + 3 \, a^{3} d^{3}\right )} x^{2}}{2 \, {\left ({\left (a^{2} b^{3} c^{4} d - 2 \, a^{3} b^{2} c^{3} d^{2} + a^{4} b c^{2} d^{3}\right )} x^{5} + {\left (a^{2} b^{3} c^{5} - a^{3} b^{2} c^{4} d - a^{4} b c^{3} d^{2} + a^{5} c^{2} d^{3}\right )} x^{3} + {\left (a^{3} b^{2} c^{5} - 2 \, a^{4} b c^{4} d + a^{5} c^{3} d^{2}\right )} x\right )}} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 321, normalized size of antiderivative = 1.47 \[ \int \frac {1}{x^2 \left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=-\frac {{\left (3 \, b^{4} c - 7 \, a b^{3} d\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, {\left (a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - a^{5} d^{3}\right )} \sqrt {a b}} - \frac {{\left (7 \, b c d^{3} - 3 \, a d^{4}\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{2 \, {\left (b^{3} c^{5} - 3 \, a b^{2} c^{4} d + 3 \, a^{2} b c^{3} d^{2} - a^{3} c^{2} d^{3}\right )} \sqrt {c d}} - \frac {3 \, b^{3} c^{2} d x^{4} - 4 \, a b^{2} c d^{2} x^{4} + 3 \, a^{2} b d^{3} x^{4} + 3 \, b^{3} c^{3} x^{2} - 2 \, a b^{2} c^{2} d x^{2} - 2 \, a^{2} b c d^{2} x^{2} + 3 \, a^{3} d^{3} x^{2} + 2 \, a b^{2} c^{3} - 4 \, a^{2} b c^{2} d + 2 \, a^{3} c d^{2}}{2 \, {\left (a^{2} b^{2} c^{4} - 2 \, a^{3} b c^{3} d + a^{4} c^{2} d^{2}\right )} {\left (b d x^{5} + b c x^{3} + a d x^{3} + a c x\right )}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 6.71 (sec) , antiderivative size = 3747, normalized size of antiderivative = 17.19 \[ \int \frac {1}{x^2 \left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

- (1/(a*c) + (x^2*(3*a^3*d^3 + 3*b^3*c^3 - 2*a*b^2*c^2*d - 2*a^2*b*c*d^2))/(2*a^2*c^2*(a^2*d^2 + b^2*c^2 - 2*a
*b*c*d)) + (b*d*x^4*(3*a^2*d^2 + 3*b^2*c^2 - 4*a*b*c*d))/(2*a^2*c^2*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)))/(x^3*(a*
d + b*c) + a*c*x + b*d*x^5) - (atan((a^7*d^3*x*(-c^5*d^5)^(3/2)*9i + b^7*c^12*d*x*(-c^5*d^5)^(1/2)*9i + a^2*b^
5*c^10*d^3*x*(-c^5*d^5)^(1/2)*49i - a^6*b*c*d^2*x*(-c^5*d^5)^(3/2)*42i + a^5*b^2*c^2*d*x*(-c^5*d^5)^(3/2)*49i
- a*b^6*c^11*d^2*x*(-c^5*d^5)^(1/2)*42i)/(9*a^7*c^8*d^10 - 9*b^7*c^15*d^3 + 42*a*b^6*c^14*d^4 - 42*a^6*b*c^9*d
^9 - 49*a^2*b^5*c^13*d^5 + 49*a^5*b^2*c^10*d^8))*(3*a*d - 7*b*c)*(-c^5*d^5)^(1/2)*1i)/(2*(b^3*c^8 - a^3*c^5*d^
3 + 3*a^2*b*c^6*d^2 - 3*a*b^2*c^7*d)) - (atan((((7*a*d - 3*b*c)*(x*(144*a^6*b^15*c^18*d^3 - 1536*a^7*b^14*c^17
*d^4 + 6976*a^8*b^13*c^16*d^5 - 17664*a^9*b^12*c^15*d^6 + 28144*a^10*b^11*c^14*d^7 - 32000*a^11*b^10*c^13*d^8
+ 31872*a^12*b^9*c^12*d^9 - 32000*a^13*b^8*c^11*d^10 + 28144*a^14*b^7*c^10*d^11 - 17664*a^15*b^6*c^9*d^12 + 69
76*a^16*b^5*c^8*d^13 - 1536*a^17*b^4*c^7*d^14 + 144*a^18*b^3*c^6*d^15) - ((7*a*d - 3*b*c)*(-a^5*b^5)^(1/2)*(19
2*a^8*b^15*c^21*d^2 - 2176*a^9*b^14*c^20*d^3 + 10944*a^10*b^13*c^19*d^4 - 31808*a^11*b^12*c^18*d^5 + 57600*a^1
2*b^11*c^17*d^6 - 62784*a^13*b^10*c^16*d^7 + 28032*a^14*b^9*c^15*d^8 + 28032*a^15*b^8*c^14*d^9 - 62784*a^16*b^
7*c^13*d^10 + 57600*a^17*b^6*c^12*d^11 - 31808*a^18*b^5*c^11*d^12 + 10944*a^19*b^4*c^10*d^13 - 2176*a^20*b^3*c
^9*d^14 + 192*a^21*b^2*c^8*d^15 - (x*(7*a*d - 3*b*c)*(-a^5*b^5)^(1/2)*(256*a^10*b^15*c^23*d^2 - 2816*a^11*b^14
*c^22*d^3 + 13824*a^12*b^13*c^21*d^4 - 39424*a^13*b^12*c^20*d^5 + 70400*a^14*b^11*c^19*d^6 - 76032*a^15*b^10*c
^18*d^7 + 33792*a^16*b^9*c^17*d^8 + 33792*a^17*b^8*c^16*d^9 - 76032*a^18*b^7*c^15*d^10 + 70400*a^19*b^6*c^14*d
^11 - 39424*a^20*b^5*c^13*d^12 + 13824*a^21*b^4*c^12*d^13 - 2816*a^22*b^3*c^11*d^14 + 256*a^23*b^2*c^10*d^15))
/(4*(a^8*d^3 - a^5*b^3*c^3 + 3*a^6*b^2*c^2*d - 3*a^7*b*c*d^2))))/(4*(a^8*d^3 - a^5*b^3*c^3 + 3*a^6*b^2*c^2*d -
3*a^7*b*c*d^2)))*(-a^5*b^5)^(1/2)*1i)/(4*(a^8*d^3 - a^5*b^3*c^3 + 3*a^6*b^2*c^2*d - 3*a^7*b*c*d^2)) + ((7*a*d
- 3*b*c)*(x*(144*a^6*b^15*c^18*d^3 - 1536*a^7*b^14*c^17*d^4 + 6976*a^8*b^13*c^16*d^5 - 17664*a^9*b^12*c^15*d^
6 + 28144*a^10*b^11*c^14*d^7 - 32000*a^11*b^10*c^13*d^8 + 31872*a^12*b^9*c^12*d^9 - 32000*a^13*b^8*c^11*d^10 +
28144*a^14*b^7*c^10*d^11 - 17664*a^15*b^6*c^9*d^12 + 6976*a^16*b^5*c^8*d^13 - 1536*a^17*b^4*c^7*d^14 + 144*a^
18*b^3*c^6*d^15) + ((7*a*d - 3*b*c)*(-a^5*b^5)^(1/2)*(192*a^8*b^15*c^21*d^2 - 2176*a^9*b^14*c^20*d^3 + 10944*a
^10*b^13*c^19*d^4 - 31808*a^11*b^12*c^18*d^5 + 57600*a^12*b^11*c^17*d^6 - 62784*a^13*b^10*c^16*d^7 + 28032*a^1
4*b^9*c^15*d^8 + 28032*a^15*b^8*c^14*d^9 - 62784*a^16*b^7*c^13*d^10 + 57600*a^17*b^6*c^12*d^11 - 31808*a^18*b^
5*c^11*d^12 + 10944*a^19*b^4*c^10*d^13 - 2176*a^20*b^3*c^9*d^14 + 192*a^21*b^2*c^8*d^15 + (x*(7*a*d - 3*b*c)*(
-a^5*b^5)^(1/2)*(256*a^10*b^15*c^23*d^2 - 2816*a^11*b^14*c^22*d^3 + 13824*a^12*b^13*c^21*d^4 - 39424*a^13*b^12
*c^20*d^5 + 70400*a^14*b^11*c^19*d^6 - 76032*a^15*b^10*c^18*d^7 + 33792*a^16*b^9*c^17*d^8 + 33792*a^17*b^8*c^1
6*d^9 - 76032*a^18*b^7*c^15*d^10 + 70400*a^19*b^6*c^14*d^11 - 39424*a^20*b^5*c^13*d^12 + 13824*a^21*b^4*c^12*d
^13 - 2816*a^22*b^3*c^11*d^14 + 256*a^23*b^2*c^10*d^15))/(4*(a^8*d^3 - a^5*b^3*c^3 + 3*a^6*b^2*c^2*d - 3*a^7*b
*c*d^2))))/(4*(a^8*d^3 - a^5*b^3*c^3 + 3*a^6*b^2*c^2*d - 3*a^7*b*c*d^2)))*(-a^5*b^5)^(1/2)*1i)/(4*(a^8*d^3 - a
^5*b^3*c^3 + 3*a^6*b^2*c^2*d - 3*a^7*b*c*d^2)))/(((7*a*d - 3*b*c)*(x*(144*a^6*b^15*c^18*d^3 - 1536*a^7*b^14*c^
17*d^4 + 6976*a^8*b^13*c^16*d^5 - 17664*a^9*b^12*c^15*d^6 + 28144*a^10*b^11*c^14*d^7 - 32000*a^11*b^10*c^13*d^
8 + 31872*a^12*b^9*c^12*d^9 - 32000*a^13*b^8*c^11*d^10 + 28144*a^14*b^7*c^10*d^11 - 17664*a^15*b^6*c^9*d^12 +
6976*a^16*b^5*c^8*d^13 - 1536*a^17*b^4*c^7*d^14 + 144*a^18*b^3*c^6*d^15) + ((7*a*d - 3*b*c)*(-a^5*b^5)^(1/2)*(
192*a^8*b^15*c^21*d^2 - 2176*a^9*b^14*c^20*d^3 + 10944*a^10*b^13*c^19*d^4 - 31808*a^11*b^12*c^18*d^5 + 57600*a
^12*b^11*c^17*d^6 - 62784*a^13*b^10*c^16*d^7 + 28032*a^14*b^9*c^15*d^8 + 28032*a^15*b^8*c^14*d^9 - 62784*a^16*
b^7*c^13*d^10 + 57600*a^17*b^6*c^12*d^11 - 31808*a^18*b^5*c^11*d^12 + 10944*a^19*b^4*c^10*d^13 - 2176*a^20*b^3
*c^9*d^14 + 192*a^21*b^2*c^8*d^15 + (x*(7*a*d - 3*b*c)*(-a^5*b^5)^(1/2)*(256*a^10*b^15*c^23*d^2 - 2816*a^11*b^
14*c^22*d^3 + 13824*a^12*b^13*c^21*d^4 - 39424*a^13*b^12*c^20*d^5 + 70400*a^14*b^11*c^19*d^6 - 76032*a^15*b^10
*c^18*d^7 + 33792*a^16*b^9*c^17*d^8 + 33792*a^17*b^8*c^16*d^9 - 76032*a^18*b^7*c^15*d^10 + 70400*a^19*b^6*c^14
*d^11 - 39424*a^20*b^5*c^13*d^12 + 13824*a^21*b^4*c^12*d^13 - 2816*a^22*b^3*c^11*d^14 + 256*a^23*b^2*c^10*d^15
))/(4*(a^8*d^3 - a^5*b^3*c^3 + 3*a^6*b^2*c^2*d - 3*a^7*b*c*d^2))))/(4*(a^8*d^3 - a^5*b^3*c^3 + 3*a^6*b^2*c^2*d
- 3*a^7*b*c*d^2)))*(-a^5*b^5)^(1/2))/(4*(a^8*d^3 - a^5*b^3*c^3 + 3*a^6*b^2*c^2*d - 3*a^7*b*c*d^2)) - ((7*a*d
- 3*b*c)*(x*(144*a^6*b^15*c^18*d^3 - 1536*a^7*b^14*c^17*d^4 + 6976*a^8*b^13*c^16*d^5 - 17664*a^9*b^12*c^15*d^6
+ 28144*a^10*b^11*c^14*d^7 - 32000*a^11*b^10*c^13*d^8 + 31872*a^12*b^9*c^12*d^9 - 32000*a^13*b^8*c^11*d^10 +
28144*a^14*b^7*c^10*d^11 - 17664*a^15*b^6*c^9*d^12 + 6976*a^16*b^5*c^8*d^13 - 1536*a^17*b^4*c^7*d^14 + 144*a^1
8*b^3*c^6*d^15) - ((7*a*d - 3*b*c)*(-a^5*b^5)^(1/2)*(192*a^8*b^15*c^21*d^2 - 2176*a^9*b^14*c^20*d^3 + 10944*a^
10*b^13*c^19*d^4 - 31808*a^11*b^12*c^18*d^5 + 57600*a^12*b^11*c^17*d^6 - 62784*a^13*b^10*c^16*d^7 + 28032*a^14
*b^9*c^15*d^8 + 28032*a^15*b^8*c^14*d^9 - 62784*a^16*b^7*c^13*d^10 + 57600*a^17*b^6*c^12*d^11 - 31808*a^18*b^5
*c^11*d^12 + 10944*a^19*b^4*c^10*d^13 - 2176*a^20*b^3*c^9*d^14 + 192*a^21*b^2*c^8*d^15 - (x*(7*a*d - 3*b*c)*(-
a^5*b^5)^(1/2)*(256*a^10*b^15*c^23*d^2 - 2816*a^11*b^14*c^22*d^3 + 13824*a^12*b^13*c^21*d^4 - 39424*a^13*b^12*
c^20*d^5 + 70400*a^14*b^11*c^19*d^6 - 76032*a^15*b^10*c^18*d^7 + 33792*a^16*b^9*c^17*d^8 + 33792*a^17*b^8*c^16
*d^9 - 76032*a^18*b^7*c^15*d^10 + 70400*a^19*b^6*c^14*d^11 - 39424*a^20*b^5*c^13*d^12 + 13824*a^21*b^4*c^12*d^
13 - 2816*a^22*b^3*c^11*d^14 + 256*a^23*b^2*c^10*d^15))/(4*(a^8*d^3 - a^5*b^3*c^3 + 3*a^6*b^2*c^2*d - 3*a^7*b*
c*d^2))))/(4*(a^8*d^3 - a^5*b^3*c^3 + 3*a^6*b^2*c^2*d - 3*a^7*b*c*d^2)))*(-a^5*b^5)^(1/2))/(4*(a^8*d^3 - a^5*b
^3*c^3 + 3*a^6*b^2*c^2*d - 3*a^7*b*c*d^2)) + 504*a^6*b^13*c^14*d^5 - 4080*a^7*b^12*c^13*d^6 + 14144*a^8*b^11*c
^12*d^7 - 27920*a^9*b^10*c^11*d^8 + 34704*a^10*b^9*c^10*d^9 - 27920*a^11*b^8*c^9*d^10 + 14144*a^12*b^7*c^8*d^1
1 - 4080*a^13*b^6*c^7*d^12 + 504*a^14*b^5*c^6*d^13))*(7*a*d - 3*b*c)*(-a^5*b^5)^(1/2)*1i)/(2*(a^8*d^3 - a^5*b^
3*c^3 + 3*a^6*b^2*c^2*d - 3*a^7*b*c*d^2))