\(\int \frac {1}{x^4 (a+b x^2)^2 (c+d x^2)^2} \, dx\) [309]

Optimal result

Integrand size = 22, antiderivative size = 271 \[ \int \frac {1}{x^4 \left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=-\frac {5 b^2 c^2-4 a b c d+5 a^2 d^2}{6 a^2 c^2 (b c-a d)^2 x^3}+\frac {(b c+a d) \left (5 b^2 c^2-9 a b c d+5 a^2 d^2\right )}{2 a^3 c^3 (b c-a d)^2 x}+\frac {d (b c+a d)}{2 a c (b c-a d)^2 x^3 \left (c+d x^2\right )}+\frac {b}{2 a (b c-a d) x^3 \left (a+b x^2\right ) \left (c+d x^2\right )}+\frac {b^{7/2} (5 b c-9 a d) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{7/2} (b c-a d)^3}+\frac {d^{7/2} (9 b c-5 a d) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 c^{7/2} (b c-a d)^3} \]

[Out]

1/6*(-5*a^2*d^2+4*a*b*c*d-5*b^2*c^2)/a^2/c^2/(-a*d+b*c)^2/x^3+1/2*(a*d+b*c)*(5*a^2*d^2-9*a*b*c*d+5*b^2*c^2)/a^
3/c^3/(-a*d+b*c)^2/x+1/2*d*(a*d+b*c)/a/c/(-a*d+b*c)^2/x^3/(d*x^2+c)+1/2*b/a/(-a*d+b*c)/x^3/(b*x^2+a)/(d*x^2+c)
+1/2*b^(7/2)*(-9*a*d+5*b*c)*arctan(x*b^(1/2)/a^(1/2))/a^(7/2)/(-a*d+b*c)^3+1/2*d^(7/2)*(-5*a*d+9*b*c)*arctan(x
*d^(1/2)/c^(1/2))/c^(7/2)/(-a*d+b*c)^3

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {483, 593, 597, 536, 211} \[ \int \frac {1}{x^4 \left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=\frac {b^{7/2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) (5 b c-9 a d)}{2 a^{7/2} (b c-a d)^3}-\frac {5 a^2 d^2-4 a b c d+5 b^2 c^2}{6 a^2 c^2 x^3 (b c-a d)^2}+\frac {(a d+b c) \left (5 a^2 d^2-9 a b c d+5 b^2 c^2\right )}{2 a^3 c^3 x (b c-a d)^2}+\frac {d^{7/2} (9 b c-5 a d) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 c^{7/2} (b c-a d)^3}+\frac {b}{2 a x^3 \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}+\frac {d (a d+b c)}{2 a c x^3 \left (c+d x^2\right ) (b c-a d)^2} \]

[In]

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

[Out]

-1/6*(5*b^2*c^2 - 4*a*b*c*d + 5*a^2*d^2)/(a^2*c^2*(b*c - a*d)^2*x^3) + ((b*c + a*d)*(5*b^2*c^2 - 9*a*b*c*d + 5
*a^2*d^2))/(2*a^3*c^3*(b*c - a*d)^2*x) + (d*(b*c + a*d))/(2*a*c*(b*c - a*d)^2*x^3*(c + d*x^2)) + b/(2*a*(b*c -
a*d)*x^3*(a + b*x^2)*(c + d*x^2)) + (b^(7/2)*(5*b*c - 9*a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(7/2)*(b*c - a
*d)^3) + (d^(7/2)*(9*b*c - 5*a*d)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(2*c^(7/2)*(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^3 \left (a+b x^2\right ) \left (c+d x^2\right )}-\frac {\int \frac {-5 b c+2 a d-7 b d x^2}{x^4 \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^3 \left (c+d x^2\right )}+\frac {b}{2 a (b c-a d) x^3 \left (a+b x^2\right ) \left (c+d x^2\right )}-\frac {\int \frac {-2 \left (5 b^2 c^2-4 a b c d+5 a^2 d^2\right )-10 b d (b c+a d) x^2}{x^4 \left (a+b x^2\right ) \left (c+d x^2\right )} \, dx}{4 a c (b c-a d)^2} \\ & = -\frac {5 b^2 c^2-4 a b c d+5 a^2 d^2}{6 a^2 c^2 (b c-a d)^2 x^3}+\frac {d (b c+a d)}{2 a c (b c-a d)^2 x^3 \left (c+d x^2\right )}+\frac {b}{2 a (b c-a d) x^3 \left (a+b x^2\right ) \left (c+d x^2\right )}+\frac {\int \frac {-6 (b c+a d) \left (5 b^2 c^2-9 a b c d+5 a^2 d^2\right )-6 b d \left (5 b^2 c^2-4 a b c d+5 a^2 d^2\right ) x^2}{x^2 \left (a+b x^2\right ) \left (c+d x^2\right )} \, dx}{12 a^2 c^2 (b c-a d)^2} \\ & = -\frac {5 b^2 c^2-4 a b c d+5 a^2 d^2}{6 a^2 c^2 (b c-a d)^2 x^3}+\frac {(b c+a d) \left (5 b^2 c^2-9 a b c d+5 a^2 d^2\right )}{2 a^3 c^3 (b c-a d)^2 x}+\frac {d (b c+a d)}{2 a c (b c-a d)^2 x^3 \left (c+d x^2\right )}+\frac {b}{2 a (b c-a d) x^3 \left (a+b x^2\right ) \left (c+d x^2\right )}-\frac {\int \frac {-6 \left (5 b^4 c^4-4 a b^3 c^3 d-4 a^2 b^2 c^2 d^2-4 a^3 b c d^3+5 a^4 d^4\right )-6 b d (b c+a d) \left (5 b^2 c^2-9 a b c d+5 a^2 d^2\right ) x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx}{12 a^3 c^3 (b c-a d)^2} \\ & = -\frac {5 b^2 c^2-4 a b c d+5 a^2 d^2}{6 a^2 c^2 (b c-a d)^2 x^3}+\frac {(b c+a d) \left (5 b^2 c^2-9 a b c d+5 a^2 d^2\right )}{2 a^3 c^3 (b c-a d)^2 x}+\frac {d (b c+a d)}{2 a c (b c-a d)^2 x^3 \left (c+d x^2\right )}+\frac {b}{2 a (b c-a d) x^3 \left (a+b x^2\right ) \left (c+d x^2\right )}+\frac {\left (b^4 (5 b c-9 a d)\right ) \int \frac {1}{a+b x^2} \, dx}{2 a^3 (b c-a d)^3}+\frac {\left (d^4 (9 b c-5 a d)\right ) \int \frac {1}{c+d x^2} \, dx}{2 c^3 (b c-a d)^3} \\ & = -\frac {5 b^2 c^2-4 a b c d+5 a^2 d^2}{6 a^2 c^2 (b c-a d)^2 x^3}+\frac {(b c+a d) \left (5 b^2 c^2-9 a b c d+5 a^2 d^2\right )}{2 a^3 c^3 (b c-a d)^2 x}+\frac {d (b c+a d)}{2 a c (b c-a d)^2 x^3 \left (c+d x^2\right )}+\frac {b}{2 a (b c-a d) x^3 \left (a+b x^2\right ) \left (c+d x^2\right )}+\frac {b^{7/2} (5 b c-9 a d) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{7/2} (b c-a d)^3}+\frac {d^{7/2} (9 b c-5 a d) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 c^{7/2} (b c-a d)^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.66 \[ \int \frac {1}{x^4 \left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=\frac {1}{6} \left (-\frac {2}{a^2 c^2 x^3}+\frac {12 (b c+a d)}{a^3 c^3 x}+\frac {3 b^4 x}{a^3 (b c-a d)^2 \left (a+b x^2\right )}+\frac {3 d^4 x}{c^3 (b c-a d)^2 \left (c+d x^2\right )}+\frac {3 b^{7/2} (-5 b c+9 a d) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{7/2} (-b c+a d)^3}+\frac {3 d^{7/2} (9 b c-5 a d) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{c^{7/2} (b c-a d)^3}\right ) \]

[In]

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

[Out]

(-2/(a^2*c^2*x^3) + (12*(b*c + a*d))/(a^3*c^3*x) + (3*b^4*x)/(a^3*(b*c - a*d)^2*(a + b*x^2)) + (3*d^4*x)/(c^3*
(b*c - a*d)^2*(c + d*x^2)) + (3*b^(7/2)*(-5*b*c + 9*a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(a^(7/2)*(-(b*c) + a*d)^
3) + (3*d^(7/2)*(9*b*c - 5*a*d)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(c^(7/2)*(b*c - a*d)^3))/6

Maple [A] (verified)

Time = 2.87 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.59

method	result	size
default	\(-\frac {1}{3 a^{2} c^{2} x^{3}}-\frac {-2 a d -2 b c}{x \,c^{3} a^{3}}+\frac {b^{4} \left (\frac {\left (\frac {a d}{2}-\frac {b c}{2}\right ) x}{b \,x^{2}+a}+\frac {\left (9 a d -5 b c \right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{a^{3} \left (a d -b c \right )^{3}}+\frac {d^{4} \left (\frac {\left (\frac {a d}{2}-\frac {b c}{2}\right ) x}{d \,x^{2}+c}+\frac {\left (5 a d -9 b c \right ) \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{2 \sqrt {c d}}\right )}{c^{3} \left (a d -b c \right )^{3}}\)	\(159\)
risch	\(\text {Expression too large to display}\)	\(2929\)

[In]

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

[Out]

-1/3/a^2/c^2/x^3-(-2*a*d-2*b*c)/x/c^3/a^3+b^4/a^3/(a*d-b*c)^3*((1/2*a*d-1/2*b*c)*x/(b*x^2+a)+1/2*(9*a*d-5*b*c)
/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2)))+d^4/c^3/(a*d-b*c)^3*((1/2*a*d-1/2*b*c)*x/(d*x^2+c)+1/2*(5*a*d-9*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. 591 vs. \(2 (243) = 486\).

Time = 3.38 (sec) , antiderivative size = 2457, normalized size of antiderivative = 9.07 \[ \int \frac {1}{x^4 \left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=\text {Too large to display} \]

[In]

integrate(1/x^4/(b*x^2+a)^2/(d*x^2+c)^2,x, algorithm="fricas")

[Out]

[-1/12*(4*a^2*b^3*c^5 - 12*a^3*b^2*c^4*d + 12*a^4*b*c^3*d^2 - 4*a^5*c^2*d^3 - 6*(5*b^5*c^4*d - 9*a*b^4*c^3*d^2
+ 9*a^3*b^2*c*d^4 - 5*a^4*b*d^5)*x^6 - 2*(15*b^5*c^5 - 17*a*b^4*c^4*d - 18*a^2*b^3*c^3*d^2 + 18*a^3*b^2*c^2*d
^3 + 17*a^4*b*c*d^4 - 15*a^5*d^5)*x^4 - 20*(a*b^4*c^5 - 2*a^2*b^3*c^4*d + 2*a^4*b*c^2*d^3 - a^5*c*d^4)*x^2 - 3
*((5*b^5*c^4*d - 9*a*b^4*c^3*d^2)*x^7 + (5*b^5*c^5 - 4*a*b^4*c^4*d - 9*a^2*b^3*c^3*d^2)*x^5 + (5*a*b^4*c^5 - 9
*a^2*b^3*c^4*d)*x^3)*sqrt(-b/a)*log((b*x^2 + 2*a*x*sqrt(-b/a) - a)/(b*x^2 + a)) - 3*((9*a^3*b^2*c*d^4 - 5*a^4*
b*d^5)*x^7 + (9*a^3*b^2*c^2*d^3 + 4*a^4*b*c*d^4 - 5*a^5*d^5)*x^5 + (9*a^4*b*c^2*d^3 - 5*a^5*c*d^4)*x^3)*sqrt(-
d/c)*log((d*x^2 + 2*c*x*sqrt(-d/c) - c)/(d*x^2 + c)))/((a^3*b^4*c^6*d - 3*a^4*b^3*c^5*d^2 + 3*a^5*b^2*c^4*d^3
- a^6*b*c^3*d^4)*x^7 + (a^3*b^4*c^7 - 2*a^4*b^3*c^6*d + 2*a^6*b*c^4*d^3 - a^7*c^3*d^4)*x^5 + (a^4*b^3*c^7 - 3*
a^5*b^2*c^6*d + 3*a^6*b*c^5*d^2 - a^7*c^4*d^3)*x^3), -1/12*(4*a^2*b^3*c^5 - 12*a^3*b^2*c^4*d + 12*a^4*b*c^3*d^
2 - 4*a^5*c^2*d^3 - 6*(5*b^5*c^4*d - 9*a*b^4*c^3*d^2 + 9*a^3*b^2*c*d^4 - 5*a^4*b*d^5)*x^6 - 2*(15*b^5*c^5 - 17
*a*b^4*c^4*d - 18*a^2*b^3*c^3*d^2 + 18*a^3*b^2*c^2*d^3 + 17*a^4*b*c*d^4 - 15*a^5*d^5)*x^4 - 20*(a*b^4*c^5 - 2*
a^2*b^3*c^4*d + 2*a^4*b*c^2*d^3 - a^5*c*d^4)*x^2 - 6*((9*a^3*b^2*c*d^4 - 5*a^4*b*d^5)*x^7 + (9*a^3*b^2*c^2*d^3
+ 4*a^4*b*c*d^4 - 5*a^5*d^5)*x^5 + (9*a^4*b*c^2*d^3 - 5*a^5*c*d^4)*x^3)*sqrt(d/c)*arctan(x*sqrt(d/c)) - 3*((5
*b^5*c^4*d - 9*a*b^4*c^3*d^2)*x^7 + (5*b^5*c^5 - 4*a*b^4*c^4*d - 9*a^2*b^3*c^3*d^2)*x^5 + (5*a*b^4*c^5 - 9*a^2
*b^3*c^4*d)*x^3)*sqrt(-b/a)*log((b*x^2 + 2*a*x*sqrt(-b/a) - a)/(b*x^2 + a)))/((a^3*b^4*c^6*d - 3*a^4*b^3*c^5*d
^2 + 3*a^5*b^2*c^4*d^3 - a^6*b*c^3*d^4)*x^7 + (a^3*b^4*c^7 - 2*a^4*b^3*c^6*d + 2*a^6*b*c^4*d^3 - a^7*c^3*d^4)*
x^5 + (a^4*b^3*c^7 - 3*a^5*b^2*c^6*d + 3*a^6*b*c^5*d^2 - a^7*c^4*d^3)*x^3), -1/12*(4*a^2*b^3*c^5 - 12*a^3*b^2*
c^4*d + 12*a^4*b*c^3*d^2 - 4*a^5*c^2*d^3 - 6*(5*b^5*c^4*d - 9*a*b^4*c^3*d^2 + 9*a^3*b^2*c*d^4 - 5*a^4*b*d^5)*x
^6 - 2*(15*b^5*c^5 - 17*a*b^4*c^4*d - 18*a^2*b^3*c^3*d^2 + 18*a^3*b^2*c^2*d^3 + 17*a^4*b*c*d^4 - 15*a^5*d^5)*x
^4 - 20*(a*b^4*c^5 - 2*a^2*b^3*c^4*d + 2*a^4*b*c^2*d^3 - a^5*c*d^4)*x^2 - 6*((5*b^5*c^4*d - 9*a*b^4*c^3*d^2)*x
^7 + (5*b^5*c^5 - 4*a*b^4*c^4*d - 9*a^2*b^3*c^3*d^2)*x^5 + (5*a*b^4*c^5 - 9*a^2*b^3*c^4*d)*x^3)*sqrt(b/a)*arct
an(x*sqrt(b/a)) - 3*((9*a^3*b^2*c*d^4 - 5*a^4*b*d^5)*x^7 + (9*a^3*b^2*c^2*d^3 + 4*a^4*b*c*d^4 - 5*a^5*d^5)*x^5
+ (9*a^4*b*c^2*d^3 - 5*a^5*c*d^4)*x^3)*sqrt(-d/c)*log((d*x^2 + 2*c*x*sqrt(-d/c) - c)/(d*x^2 + c)))/((a^3*b^4*
c^6*d - 3*a^4*b^3*c^5*d^2 + 3*a^5*b^2*c^4*d^3 - a^6*b*c^3*d^4)*x^7 + (a^3*b^4*c^7 - 2*a^4*b^3*c^6*d + 2*a^6*b*
c^4*d^3 - a^7*c^3*d^4)*x^5 + (a^4*b^3*c^7 - 3*a^5*b^2*c^6*d + 3*a^6*b*c^5*d^2 - a^7*c^4*d^3)*x^3), -1/6*(2*a^2
*b^3*c^5 - 6*a^3*b^2*c^4*d + 6*a^4*b*c^3*d^2 - 2*a^5*c^2*d^3 - 3*(5*b^5*c^4*d - 9*a*b^4*c^3*d^2 + 9*a^3*b^2*c*
d^4 - 5*a^4*b*d^5)*x^6 - (15*b^5*c^5 - 17*a*b^4*c^4*d - 18*a^2*b^3*c^3*d^2 + 18*a^3*b^2*c^2*d^3 + 17*a^4*b*c*d
^4 - 15*a^5*d^5)*x^4 - 10*(a*b^4*c^5 - 2*a^2*b^3*c^4*d + 2*a^4*b*c^2*d^3 - a^5*c*d^4)*x^2 - 3*((5*b^5*c^4*d -
9*a*b^4*c^3*d^2)*x^7 + (5*b^5*c^5 - 4*a*b^4*c^4*d - 9*a^2*b^3*c^3*d^2)*x^5 + (5*a*b^4*c^5 - 9*a^2*b^3*c^4*d)*x
^3)*sqrt(b/a)*arctan(x*sqrt(b/a)) - 3*((9*a^3*b^2*c*d^4 - 5*a^4*b*d^5)*x^7 + (9*a^3*b^2*c^2*d^3 + 4*a^4*b*c*d^
4 - 5*a^5*d^5)*x^5 + (9*a^4*b*c^2*d^3 - 5*a^5*c*d^4)*x^3)*sqrt(d/c)*arctan(x*sqrt(d/c)))/((a^3*b^4*c^6*d - 3*a
^4*b^3*c^5*d^2 + 3*a^5*b^2*c^4*d^3 - a^6*b*c^3*d^4)*x^7 + (a^3*b^4*c^7 - 2*a^4*b^3*c^6*d + 2*a^6*b*c^4*d^3 - a
^7*c^3*d^4)*x^5 + (a^4*b^3*c^7 - 3*a^5*b^2*c^6*d + 3*a^6*b*c^5*d^2 - a^7*c^4*d^3)*x^3)]

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{x^4 \left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=\text {Timed out} \]

[In]

integrate(1/x**4/(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 = 460, normalized size of antiderivative = 1.70 \[ \int \frac {1}{x^4 \left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=\frac {{\left (5 \, b^{5} c - 9 \, a b^{4} d\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, {\left (a^{3} b^{3} c^{3} - 3 \, a^{4} b^{2} c^{2} d + 3 \, a^{5} b c d^{2} - a^{6} d^{3}\right )} \sqrt {a b}} + \frac {{\left (9 \, b c d^{4} - 5 \, a d^{5}\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{2 \, {\left (b^{3} c^{6} - 3 \, a b^{2} c^{5} d + 3 \, a^{2} b c^{4} d^{2} - a^{3} c^{3} d^{3}\right )} \sqrt {c d}} - \frac {2 \, a^{2} b^{2} c^{4} - 4 \, a^{3} b c^{3} d + 2 \, a^{4} c^{2} d^{2} - 3 \, {\left (5 \, b^{4} c^{3} d - 4 \, a b^{3} c^{2} d^{2} - 4 \, a^{2} b^{2} c d^{3} + 5 \, a^{3} b d^{4}\right )} x^{6} - {\left (15 \, b^{4} c^{4} - 2 \, a b^{3} c^{3} d - 20 \, a^{2} b^{2} c^{2} d^{2} - 2 \, a^{3} b c d^{3} + 15 \, a^{4} d^{4}\right )} x^{4} - 10 \, {\left (a b^{3} c^{4} - a^{2} b^{2} c^{3} d - a^{3} b c^{2} d^{2} + a^{4} c d^{3}\right )} x^{2}}{6 \, {\left ({\left (a^{3} b^{3} c^{5} d - 2 \, a^{4} b^{2} c^{4} d^{2} + a^{5} b c^{3} d^{3}\right )} x^{7} + {\left (a^{3} b^{3} c^{6} - a^{4} b^{2} c^{5} d - a^{5} b c^{4} d^{2} + a^{6} c^{3} d^{3}\right )} x^{5} + {\left (a^{4} b^{2} c^{6} - 2 \, a^{5} b c^{5} d + a^{6} c^{4} d^{2}\right )} x^{3}\right )}} \]

[In]

integrate(1/x^4/(b*x^2+a)^2/(d*x^2+c)^2,x, algorithm="maxima")

[Out]

1/2*(5*b^5*c - 9*a*b^4*d)*arctan(b*x/sqrt(a*b))/((a^3*b^3*c^3 - 3*a^4*b^2*c^2*d + 3*a^5*b*c*d^2 - a^6*d^3)*sqr
t(a*b)) + 1/2*(9*b*c*d^4 - 5*a*d^5)*arctan(d*x/sqrt(c*d))/((b^3*c^6 - 3*a*b^2*c^5*d + 3*a^2*b*c^4*d^2 - a^3*c^
3*d^3)*sqrt(c*d)) - 1/6*(2*a^2*b^2*c^4 - 4*a^3*b*c^3*d + 2*a^4*c^2*d^2 - 3*(5*b^4*c^3*d - 4*a*b^3*c^2*d^2 - 4*
a^2*b^2*c*d^3 + 5*a^3*b*d^4)*x^6 - (15*b^4*c^4 - 2*a*b^3*c^3*d - 20*a^2*b^2*c^2*d^2 - 2*a^3*b*c*d^3 + 15*a^4*d
^4)*x^4 - 10*(a*b^3*c^4 - a^2*b^2*c^3*d - a^3*b*c^2*d^2 + a^4*c*d^3)*x^2)/((a^3*b^3*c^5*d - 2*a^4*b^2*c^4*d^2
+ a^5*b*c^3*d^3)*x^7 + (a^3*b^3*c^6 - a^4*b^2*c^5*d - a^5*b*c^4*d^2 + a^6*c^3*d^3)*x^5 + (a^4*b^2*c^6 - 2*a^5*
b*c^5*d + a^6*c^4*d^2)*x^3)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.01 \[ \int \frac {1}{x^4 \left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=\frac {{\left (5 \, b^{5} c - 9 \, a b^{4} d\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, {\left (a^{3} b^{3} c^{3} - 3 \, a^{4} b^{2} c^{2} d + 3 \, a^{5} b c d^{2} - a^{6} d^{3}\right )} \sqrt {a b}} + \frac {{\left (9 \, b c d^{4} - 5 \, a d^{5}\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{2 \, {\left (b^{3} c^{6} - 3 \, a b^{2} c^{5} d + 3 \, a^{2} b c^{4} d^{2} - a^{3} c^{3} d^{3}\right )} \sqrt {c d}} + \frac {b^{4} c^{3} d x^{3} + a^{3} b d^{4} x^{3} + b^{4} c^{4} x + a^{4} d^{4} x}{2 \, {\left (a^{3} b^{2} c^{5} - 2 \, a^{4} b c^{4} d + a^{5} c^{3} d^{2}\right )} {\left (b d x^{4} + b c x^{2} + a d x^{2} + a c\right )}} + \frac {6 \, b c x^{2} + 6 \, a d x^{2} - a c}{3 \, a^{3} c^{3} x^{3}} \]

[In]

integrate(1/x^4/(b*x^2+a)^2/(d*x^2+c)^2,x, algorithm="giac")

[Out]

1/2*(5*b^5*c - 9*a*b^4*d)*arctan(b*x/sqrt(a*b))/((a^3*b^3*c^3 - 3*a^4*b^2*c^2*d + 3*a^5*b*c*d^2 - a^6*d^3)*sqr
t(a*b)) + 1/2*(9*b*c*d^4 - 5*a*d^5)*arctan(d*x/sqrt(c*d))/((b^3*c^6 - 3*a*b^2*c^5*d + 3*a^2*b*c^4*d^2 - a^3*c^
3*d^3)*sqrt(c*d)) + 1/2*(b^4*c^3*d*x^3 + a^3*b*d^4*x^3 + b^4*c^4*x + a^4*d^4*x)/((a^3*b^2*c^5 - 2*a^4*b*c^4*d
+ a^5*c^3*d^2)*(b*d*x^4 + b*c*x^2 + a*d*x^2 + a*c)) + 1/3*(6*b*c*x^2 + 6*a*d*x^2 - a*c)/(a^3*c^3*x^3)

Mupad [B] (verification not implemented)

Time = 6.71 (sec) , antiderivative size = 3978, normalized size of antiderivative = 14.68 \[ \int \frac {1}{x^4 \left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

(atan((a^9*d^3*x*(-c^7*d^7)^(3/2)*25i + b^9*c^16*d*x*(-c^7*d^7)^(1/2)*25i + a^2*b^7*c^14*d^3*x*(-c^7*d^7)^(1/2
)*81i - a^8*b*c*d^2*x*(-c^7*d^7)^(3/2)*90i + a^7*b^2*c^2*d*x*(-c^7*d^7)^(3/2)*81i - a*b^8*c^15*d^2*x*(-c^7*d^7
)^(1/2)*90i)/(25*a^9*c^11*d^13 - 25*b^9*c^20*d^4 + 90*a*b^8*c^19*d^5 - 90*a^8*b*c^12*d^12 - 81*a^2*b^7*c^18*d^
6 + 81*a^7*b^2*c^13*d^11))*(5*a*d - 9*b*c)*(-c^7*d^7)^(1/2)*1i)/(2*(b^3*c^10 - a^3*c^7*d^3 + 3*a^2*b*c^8*d^2 -
3*a*b^2*c^9*d)) - (1/(3*a*c) - (5*x^2*(a*d + b*c))/(3*a^2*c^2) + (x^4*(20*a^2*b^2*c^2*d^2 - 15*b^4*c^4 - 15*a
^4*d^4 + 2*a*b^3*c^3*d + 2*a^3*b*c*d^3))/(6*a^3*c^3*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)) - (b*d*x^6*(5*a^3*d^3 + 5
*b^3*c^3 - 4*a*b^2*c^2*d - 4*a^2*b*c*d^2))/(2*a^3*c^3*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)))/(x^5*(a*d + b*c) + a*c
*x^3 + b*d*x^7) + (atan((((x*(400*a^9*b^17*c^23*d^3 - 3840*a^10*b^16*c^22*d^4 + 15936*a^11*b^15*c^21*d^5 - 373
76*a^12*b^14*c^20*d^6 + 54240*a^13*b^13*c^19*d^7 - 49920*a^14*b^12*c^18*d^8 + 29776*a^15*b^11*c^17*d^9 - 18432
*a^16*b^10*c^16*d^10 + 29776*a^17*b^9*c^15*d^11 - 49920*a^18*b^8*c^14*d^12 + 54240*a^19*b^7*c^13*d^13 - 37376*
a^20*b^6*c^12*d^14 + 15936*a^21*b^5*c^11*d^15 - 3840*a^22*b^4*c^10*d^16 + 400*a^23*b^3*c^9*d^17) - ((9*a*d - 5
*b*c)*(-a^7*b^7)^(1/2)*(320*a^12*b^16*c^26*d^2 - 3456*a^13*b^15*c^25*d^3 + 16704*a^14*b^14*c^24*d^4 - 47616*a^
15*b^13*c^23*d^5 + 89280*a^16*b^12*c^22*d^6 - 118400*a^17*b^11*c^21*d^7 + 123072*a^18*b^10*c^20*d^8 - 119808*a
^19*b^9*c^19*d^9 + 123072*a^20*b^8*c^18*d^10 - 118400*a^21*b^7*c^17*d^11 + 89280*a^22*b^6*c^16*d^12 - 47616*a^
23*b^5*c^15*d^13 + 16704*a^24*b^4*c^14*d^14 - 3456*a^25*b^3*c^13*d^15 + 320*a^26*b^2*c^12*d^16 - (x*(9*a*d - 5
*b*c)*(-a^7*b^7)^(1/2)*(256*a^15*b^15*c^28*d^2 - 2816*a^16*b^14*c^27*d^3 + 13824*a^17*b^13*c^26*d^4 - 39424*a^
18*b^12*c^25*d^5 + 70400*a^19*b^11*c^24*d^6 - 76032*a^20*b^10*c^23*d^7 + 33792*a^21*b^9*c^22*d^8 + 33792*a^22*
b^8*c^21*d^9 - 76032*a^23*b^7*c^20*d^10 + 70400*a^24*b^6*c^19*d^11 - 39424*a^25*b^5*c^18*d^12 + 13824*a^26*b^4
*c^17*d^13 - 2816*a^27*b^3*c^16*d^14 + 256*a^28*b^2*c^15*d^15))/(4*(a^10*d^3 - a^7*b^3*c^3 + 3*a^8*b^2*c^2*d -
3*a^9*b*c*d^2))))/(4*(a^10*d^3 - a^7*b^3*c^3 + 3*a^8*b^2*c^2*d - 3*a^9*b*c*d^2)))*(9*a*d - 5*b*c)*(-a^7*b^7)^
(1/2)*1i)/(4*(a^10*d^3 - a^7*b^3*c^3 + 3*a^8*b^2*c^2*d - 3*a^9*b*c*d^2)) + ((x*(400*a^9*b^17*c^23*d^3 - 3840*a
^10*b^16*c^22*d^4 + 15936*a^11*b^15*c^21*d^5 - 37376*a^12*b^14*c^20*d^6 + 54240*a^13*b^13*c^19*d^7 - 49920*a^1
4*b^12*c^18*d^8 + 29776*a^15*b^11*c^17*d^9 - 18432*a^16*b^10*c^16*d^10 + 29776*a^17*b^9*c^15*d^11 - 49920*a^18
*b^8*c^14*d^12 + 54240*a^19*b^7*c^13*d^13 - 37376*a^20*b^6*c^12*d^14 + 15936*a^21*b^5*c^11*d^15 - 3840*a^22*b^
4*c^10*d^16 + 400*a^23*b^3*c^9*d^17) + ((9*a*d - 5*b*c)*(-a^7*b^7)^(1/2)*(320*a^12*b^16*c^26*d^2 - 3456*a^13*b
^15*c^25*d^3 + 16704*a^14*b^14*c^24*d^4 - 47616*a^15*b^13*c^23*d^5 + 89280*a^16*b^12*c^22*d^6 - 118400*a^17*b^
11*c^21*d^7 + 123072*a^18*b^10*c^20*d^8 - 119808*a^19*b^9*c^19*d^9 + 123072*a^20*b^8*c^18*d^10 - 118400*a^21*b
^7*c^17*d^11 + 89280*a^22*b^6*c^16*d^12 - 47616*a^23*b^5*c^15*d^13 + 16704*a^24*b^4*c^14*d^14 - 3456*a^25*b^3*
c^13*d^15 + 320*a^26*b^2*c^12*d^16 + (x*(9*a*d - 5*b*c)*(-a^7*b^7)^(1/2)*(256*a^15*b^15*c^28*d^2 - 2816*a^16*b
^14*c^27*d^3 + 13824*a^17*b^13*c^26*d^4 - 39424*a^18*b^12*c^25*d^5 + 70400*a^19*b^11*c^24*d^6 - 76032*a^20*b^1
0*c^23*d^7 + 33792*a^21*b^9*c^22*d^8 + 33792*a^22*b^8*c^21*d^9 - 76032*a^23*b^7*c^20*d^10 + 70400*a^24*b^6*c^1
9*d^11 - 39424*a^25*b^5*c^18*d^12 + 13824*a^26*b^4*c^17*d^13 - 2816*a^27*b^3*c^16*d^14 + 256*a^28*b^2*c^15*d^1
5))/(4*(a^10*d^3 - a^7*b^3*c^3 + 3*a^8*b^2*c^2*d - 3*a^9*b*c*d^2))))/(4*(a^10*d^3 - a^7*b^3*c^3 + 3*a^8*b^2*c^
2*d - 3*a^9*b*c*d^2)))*(9*a*d - 5*b*c)*(-a^7*b^7)^(1/2)*1i)/(4*(a^10*d^3 - a^7*b^3*c^3 + 3*a^8*b^2*c^2*d - 3*a
^9*b*c*d^2)))/(((x*(400*a^9*b^17*c^23*d^3 - 3840*a^10*b^16*c^22*d^4 + 15936*a^11*b^15*c^21*d^5 - 37376*a^12*b^
14*c^20*d^6 + 54240*a^13*b^13*c^19*d^7 - 49920*a^14*b^12*c^18*d^8 + 29776*a^15*b^11*c^17*d^9 - 18432*a^16*b^10
*c^16*d^10 + 29776*a^17*b^9*c^15*d^11 - 49920*a^18*b^8*c^14*d^12 + 54240*a^19*b^7*c^13*d^13 - 37376*a^20*b^6*c
^12*d^14 + 15936*a^21*b^5*c^11*d^15 - 3840*a^22*b^4*c^10*d^16 + 400*a^23*b^3*c^9*d^17) + ((9*a*d - 5*b*c)*(-a^
7*b^7)^(1/2)*(320*a^12*b^16*c^26*d^2 - 3456*a^13*b^15*c^25*d^3 + 16704*a^14*b^14*c^24*d^4 - 47616*a^15*b^13*c^
23*d^5 + 89280*a^16*b^12*c^22*d^6 - 118400*a^17*b^11*c^21*d^7 + 123072*a^18*b^10*c^20*d^8 - 119808*a^19*b^9*c^
19*d^9 + 123072*a^20*b^8*c^18*d^10 - 118400*a^21*b^7*c^17*d^11 + 89280*a^22*b^6*c^16*d^12 - 47616*a^23*b^5*c^1
5*d^13 + 16704*a^24*b^4*c^14*d^14 - 3456*a^25*b^3*c^13*d^15 + 320*a^26*b^2*c^12*d^16 + (x*(9*a*d - 5*b*c)*(-a^
7*b^7)^(1/2)*(256*a^15*b^15*c^28*d^2 - 2816*a^16*b^14*c^27*d^3 + 13824*a^17*b^13*c^26*d^4 - 39424*a^18*b^12*c^
25*d^5 + 70400*a^19*b^11*c^24*d^6 - 76032*a^20*b^10*c^23*d^7 + 33792*a^21*b^9*c^22*d^8 + 33792*a^22*b^8*c^21*d
^9 - 76032*a^23*b^7*c^20*d^10 + 70400*a^24*b^6*c^19*d^11 - 39424*a^25*b^5*c^18*d^12 + 13824*a^26*b^4*c^17*d^13
- 2816*a^27*b^3*c^16*d^14 + 256*a^28*b^2*c^15*d^15))/(4*(a^10*d^3 - a^7*b^3*c^3 + 3*a^8*b^2*c^2*d - 3*a^9*b*c
*d^2))))/(4*(a^10*d^3 - a^7*b^3*c^3 + 3*a^8*b^2*c^2*d - 3*a^9*b*c*d^2)))*(9*a*d - 5*b*c)*(-a^7*b^7)^(1/2))/(4*
(a^10*d^3 - a^7*b^3*c^3 + 3*a^8*b^2*c^2*d - 3*a^9*b*c*d^2)) - ((x*(400*a^9*b^17*c^23*d^3 - 3840*a^10*b^16*c^22
*d^4 + 15936*a^11*b^15*c^21*d^5 - 37376*a^12*b^14*c^20*d^6 + 54240*a^13*b^13*c^19*d^7 - 49920*a^14*b^12*c^18*d
^8 + 29776*a^15*b^11*c^17*d^9 - 18432*a^16*b^10*c^16*d^10 + 29776*a^17*b^9*c^15*d^11 - 49920*a^18*b^8*c^14*d^1
2 + 54240*a^19*b^7*c^13*d^13 - 37376*a^20*b^6*c^12*d^14 + 15936*a^21*b^5*c^11*d^15 - 3840*a^22*b^4*c^10*d^16 +
400*a^23*b^3*c^9*d^17) - ((9*a*d - 5*b*c)*(-a^7*b^7)^(1/2)*(320*a^12*b^16*c^26*d^2 - 3456*a^13*b^15*c^25*d^3
+ 16704*a^14*b^14*c^24*d^4 - 47616*a^15*b^13*c^23*d^5 + 89280*a^16*b^12*c^22*d^6 - 118400*a^17*b^11*c^21*d^7 +
123072*a^18*b^10*c^20*d^8 - 119808*a^19*b^9*c^19*d^9 + 123072*a^20*b^8*c^18*d^10 - 118400*a^21*b^7*c^17*d^11
+ 89280*a^22*b^6*c^16*d^12 - 47616*a^23*b^5*c^15*d^13 + 16704*a^24*b^4*c^14*d^14 - 3456*a^25*b^3*c^13*d^15 + 3
20*a^26*b^2*c^12*d^16 - (x*(9*a*d - 5*b*c)*(-a^7*b^7)^(1/2)*(256*a^15*b^15*c^28*d^2 - 2816*a^16*b^14*c^27*d^3
+ 13824*a^17*b^13*c^26*d^4 - 39424*a^18*b^12*c^25*d^5 + 70400*a^19*b^11*c^24*d^6 - 76032*a^20*b^10*c^23*d^7 +
33792*a^21*b^9*c^22*d^8 + 33792*a^22*b^8*c^21*d^9 - 76032*a^23*b^7*c^20*d^10 + 70400*a^24*b^6*c^19*d^11 - 3942
4*a^25*b^5*c^18*d^12 + 13824*a^26*b^4*c^17*d^13 - 2816*a^27*b^3*c^16*d^14 + 256*a^28*b^2*c^15*d^15))/(4*(a^10*
d^3 - a^7*b^3*c^3 + 3*a^8*b^2*c^2*d - 3*a^9*b*c*d^2))))/(4*(a^10*d^3 - a^7*b^3*c^3 + 3*a^8*b^2*c^2*d - 3*a^9*b
*c*d^2)))*(9*a*d - 5*b*c)*(-a^7*b^7)^(1/2))/(4*(a^10*d^3 - a^7*b^3*c^3 + 3*a^8*b^2*c^2*d - 3*a^9*b*c*d^2)) + 1
800*a^9*b^15*c^18*d^6 - 12880*a^10*b^14*c^17*d^7 + 37272*a^11*b^13*c^16*d^8 - 52536*a^12*b^12*c^15*d^9 + 26344
*a^13*b^11*c^14*d^10 + 26344*a^14*b^10*c^13*d^11 - 52536*a^15*b^9*c^12*d^12 + 37272*a^16*b^8*c^11*d^13 - 12880
*a^17*b^7*c^10*d^14 + 1800*a^18*b^6*c^9*d^15))*(9*a*d - 5*b*c)*(-a^7*b^7)^(1/2)*1i)/(2*(a^10*d^3 - a^7*b^3*c^3
+ 3*a^8*b^2*c^2*d - 3*a^9*b*c*d^2))