\(\int \frac {1}{x^4 (a+b x^2) (c+d x^2)^3} \, dx\) [654]

Optimal result

Integrand size = 22, antiderivative size = 270 \[ \int \frac {1}{x^4 \left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx=-\frac {8 b^2 c^2-55 a b c d+35 a^2 d^2}{24 a c^3 (b c-a d)^2 x^3}+\frac {8 b^3 c^3+8 a b^2 c^2 d-55 a^2 b c d^2+35 a^3 d^3}{8 a^2 c^4 (b c-a d)^2 x}-\frac {d}{4 c (b c-a d) x^3 \left (c+d x^2\right )^2}-\frac {d (11 b c-7 a d)}{8 c^2 (b c-a d)^2 x^3 \left (c+d x^2\right )}+\frac {b^{9/2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{5/2} (b c-a d)^3}-\frac {d^{5/2} \left (63 b^2 c^2-90 a b c d+35 a^2 d^2\right ) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{8 c^{9/2} (b c-a d)^3} \] Output:

-1/24*(35*a^2*d^2-55*a*b*c*d+8*b^2*c^2)/a/c^3/(-a*d+b*c)^2/x^3+1/8*(35*a^3 
*d^3-55*a^2*b*c*d^2+8*a*b^2*c^2*d+8*b^3*c^3)/a^2/c^4/(-a*d+b*c)^2/x-1/4*d/ 
c/(-a*d+b*c)/x^3/(d*x^2+c)^2-1/8*d*(-7*a*d+11*b*c)/c^2/(-a*d+b*c)^2/x^3/(d 
*x^2+c)+b^(9/2)*arctan(b^(1/2)*x/a^(1/2))/a^(5/2)/(-a*d+b*c)^3-1/8*d^(5/2) 
*(35*a^2*d^2-90*a*b*c*d+63*b^2*c^2)*arctan(d^(1/2)*x/c^(1/2))/c^(9/2)/(-a* 
d+b*c)^3
 

Mathematica [A] (verified)

Time = 0.28 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.73 \[ \int \frac {1}{x^4 \left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx=-\frac {1}{3 a c^3 x^3}+\frac {b c+3 a d}{a^2 c^4 x}-\frac {d^3 x}{4 c^3 (b c-a d) \left (c+d x^2\right )^2}-\frac {d^3 (15 b c-11 a d) x}{8 c^4 (b c-a d)^2 \left (c+d x^2\right )}-\frac {b^{9/2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{5/2} (-b c+a d)^3}-\frac {d^{5/2} \left (63 b^2 c^2-90 a b c d+35 a^2 d^2\right ) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{8 c^{9/2} (b c-a d)^3} \] Input:

Integrate[1/(x^4*(a + b*x^2)*(c + d*x^2)^3),x]
 

Output:

-1/3*1/(a*c^3*x^3) + (b*c + 3*a*d)/(a^2*c^4*x) - (d^3*x)/(4*c^3*(b*c - a*d 
)*(c + d*x^2)^2) - (d^3*(15*b*c - 11*a*d)*x)/(8*c^4*(b*c - a*d)^2*(c + d*x 
^2)) - (b^(9/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(a^(5/2)*(-(b*c) + a*d)^3) - 
(d^(5/2)*(63*b^2*c^2 - 90*a*b*c*d + 35*a^2*d^2)*ArcTan[(Sqrt[d]*x)/Sqrt[c] 
])/(8*c^(9/2)*(b*c - a*d)^3)
 

Rubi [A] (verified)

Time = 0.57 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.11, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {374, 441, 445, 27, 445, 397, 218}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {1}{x^4 \left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx\)

\(\Big \downarrow \) 374

\(\displaystyle \frac {\int \frac {-7 b d x^2+4 b c-7 a d}{x^4 \left (b x^2+a\right ) \left (d x^2+c\right )^2}dx}{4 c (b c-a d)}-\frac {d}{4 c x^3 \left (c+d x^2\right )^2 (b c-a d)}\)

\(\Big \downarrow \) 441

\(\displaystyle \frac {\frac {\int \frac {8 b^2 c^2-55 a b d c+35 a^2 d^2-5 b d (11 b c-7 a d) x^2}{x^4 \left (b x^2+a\right ) \left (d x^2+c\right )}dx}{2 c (b c-a d)}-\frac {d (11 b c-7 a d)}{2 c x^3 \left (c+d x^2\right ) (b c-a d)}}{4 c (b c-a d)}-\frac {d}{4 c x^3 \left (c+d x^2\right )^2 (b c-a d)}\)

\(\Big \downarrow \) 445

\(\displaystyle \frac {\frac {-\frac {\int \frac {3 \left (8 b^3 c^3+8 a b^2 d c^2-55 a^2 b d^2 c+35 a^3 d^3+b d \left (8 b^2 c^2-55 a b d c+35 a^2 d^2\right ) x^2\right )}{x^2 \left (b x^2+a\right ) \left (d x^2+c\right )}dx}{3 a c}-\frac {\frac {8 b^2 c}{a}+\frac {35 a d^2}{c}-55 b d}{3 x^3}}{2 c (b c-a d)}-\frac {d (11 b c-7 a d)}{2 c x^3 \left (c+d x^2\right ) (b c-a d)}}{4 c (b c-a d)}-\frac {d}{4 c x^3 \left (c+d x^2\right )^2 (b c-a d)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {-\frac {\int \frac {8 b^3 c^3+8 a b^2 d c^2-55 a^2 b d^2 c+35 a^3 d^3+b d \left (8 b^2 c^2-55 a b d c+35 a^2 d^2\right ) x^2}{x^2 \left (b x^2+a\right ) \left (d x^2+c\right )}dx}{a c}-\frac {\frac {8 b^2 c}{a}+\frac {35 a d^2}{c}-55 b d}{3 x^3}}{2 c (b c-a d)}-\frac {d (11 b c-7 a d)}{2 c x^3 \left (c+d x^2\right ) (b c-a d)}}{4 c (b c-a d)}-\frac {d}{4 c x^3 \left (c+d x^2\right )^2 (b c-a d)}\)

\(\Big \downarrow \) 445

\(\displaystyle \frac {\frac {-\frac {-\frac {\int \frac {8 b^4 c^4+8 a b^3 d c^3+8 a^2 b^2 d^2 c^2-55 a^3 b d^3 c+35 a^4 d^4+b d \left (8 b^3 c^3+8 a b^2 d c^2-55 a^2 b d^2 c+35 a^3 d^3\right ) x^2}{\left (b x^2+a\right ) \left (d x^2+c\right )}dx}{a c}-\frac {35 a^3 d^3-55 a^2 b c d^2+8 a b^2 c^2 d+8 b^3 c^3}{a c x}}{a c}-\frac {\frac {8 b^2 c}{a}+\frac {35 a d^2}{c}-55 b d}{3 x^3}}{2 c (b c-a d)}-\frac {d (11 b c-7 a d)}{2 c x^3 \left (c+d x^2\right ) (b c-a d)}}{4 c (b c-a d)}-\frac {d}{4 c x^3 \left (c+d x^2\right )^2 (b c-a d)}\)

\(\Big \downarrow \) 397

\(\displaystyle \frac {\frac {-\frac {-\frac {\frac {8 b^5 c^4 \int \frac {1}{b x^2+a}dx}{b c-a d}-\frac {a^2 d^3 \left (35 a^2 d^2-90 a b c d+63 b^2 c^2\right ) \int \frac {1}{d x^2+c}dx}{b c-a d}}{a c}-\frac {35 a^3 d^3-55 a^2 b c d^2+8 a b^2 c^2 d+8 b^3 c^3}{a c x}}{a c}-\frac {\frac {8 b^2 c}{a}+\frac {35 a d^2}{c}-55 b d}{3 x^3}}{2 c (b c-a d)}-\frac {d (11 b c-7 a d)}{2 c x^3 \left (c+d x^2\right ) (b c-a d)}}{4 c (b c-a d)}-\frac {d}{4 c x^3 \left (c+d x^2\right )^2 (b c-a d)}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {\frac {-\frac {-\frac {\frac {8 b^{9/2} c^4 \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} (b c-a d)}-\frac {a^2 d^{5/2} \left (35 a^2 d^2-90 a b c d+63 b^2 c^2\right ) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {c} (b c-a d)}}{a c}-\frac {35 a^3 d^3-55 a^2 b c d^2+8 a b^2 c^2 d+8 b^3 c^3}{a c x}}{a c}-\frac {\frac {8 b^2 c}{a}+\frac {35 a d^2}{c}-55 b d}{3 x^3}}{2 c (b c-a d)}-\frac {d (11 b c-7 a d)}{2 c x^3 \left (c+d x^2\right ) (b c-a d)}}{4 c (b c-a d)}-\frac {d}{4 c x^3 \left (c+d x^2\right )^2 (b c-a d)}\)

Input:

Int[1/(x^4*(a + b*x^2)*(c + d*x^2)^3),x]
 

Output:

-1/4*d/(c*(b*c - a*d)*x^3*(c + d*x^2)^2) + (-1/2*(d*(11*b*c - 7*a*d))/(c*( 
b*c - a*d)*x^3*(c + d*x^2)) + (-1/3*((8*b^2*c)/a - 55*b*d + (35*a*d^2)/c)/ 
x^3 - (-((8*b^3*c^3 + 8*a*b^2*c^2*d - 55*a^2*b*c*d^2 + 35*a^3*d^3)/(a*c*x) 
) - ((8*b^(9/2)*c^4*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[a]*(b*c - a*d)) - ( 
a^2*d^(5/2)*(63*b^2*c^2 - 90*a*b*c*d + 35*a^2*d^2)*ArcTan[(Sqrt[d]*x)/Sqrt 
[c]])/(Sqrt[c]*(b*c - a*d)))/(a*c))/(a*c))/(2*c*(b*c - a*d)))/(4*c*(b*c - 
a*d))
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 218

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

rule 374

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

rule 397

Int[((e_) + (f_.)*(x_)^2)/(((a_) + (b_.)*(x_)^2)*((c_) + (d_.)*(x_)^2)), x_ 
Symbol] :> Simp[(b*e - a*f)/(b*c - a*d)   Int[1/(a + b*x^2), x], x] - Simp[ 
(d*e - c*f)/(b*c - a*d)   Int[1/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d, e 
, f}, x]
 

rule 441

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

rule 445

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

Maple [A] (verified)

Time = 0.76 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.70

method	result	size
default	\(-\frac {b^{5} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{a^{2} \left (a d -b c \right )^{3} \sqrt {a b}}-\frac {1}{3 a \,c^{3} x^{3}}-\frac {-3 a d -b c}{x \,a^{2} c^{4}}+\frac {d^{3} \left (\frac {\left (\frac {11}{8} a^{2} d^{3}-\frac {13}{4} a c \,d^{2} b +\frac {15}{8} b^{2} c^{2} d \right ) x^{3}+\frac {c \left (13 a^{2} d^{2}-30 a b c d +17 b^{2} c^{2}\right ) x}{8}}{\left (x^{2} d +c \right )^{2}}+\frac {\left (35 a^{2} d^{2}-90 a b c d +63 b^{2} c^{2}\right ) \arctan \left (\frac {x d}{\sqrt {c d}}\right )}{8 \sqrt {c d}}\right )}{c^{4} \left (a d -b c \right )^{3}}\)	\(190\)
risch	\(\text {Expression too large to display}\)	\(1765\)

Input:

int(1/x^4/(b*x^2+a)/(d*x^2+c)^3,x,method=_RETURNVERBOSE)
 

Output:

-1/a^2*b^5/(a*d-b*c)^3/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))-1/3/a/c^3/x^3-( 
-3*a*d-b*c)/x/a^2/c^4+d^3/c^4/(a*d-b*c)^3*(((11/8*a^2*d^3-13/4*a*c*d^2*b+1 
5/8*b^2*c^2*d)*x^3+1/8*c*(13*a^2*d^2-30*a*b*c*d+17*b^2*c^2)*x)/(d*x^2+c)^2 
+1/8*(35*a^2*d^2-90*a*b*c*d+63*b^2*c^2)/(c*d)^(1/2)*arctan(x*d/(c*d)^(1/2) 
))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 576 vs. \(2 (244) = 488\).

Time = 2.96 (sec) , antiderivative size = 2397, normalized size of antiderivative = 8.88 \[ \int \frac {1}{x^4 \left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx=\text {Too large to display} \] Input:

integrate(1/x^4/(b*x^2+a)/(d*x^2+c)^3,x, algorithm="fricas")
 

Output:

[-1/48*(16*a*b^3*c^6 - 48*a^2*b^2*c^5*d + 48*a^3*b*c^4*d^2 - 16*a^4*c^3*d^ 
3 - 6*(8*b^4*c^4*d^2 - 63*a^2*b^2*c^2*d^4 + 90*a^3*b*c*d^5 - 35*a^4*d^6)*x 
^6 - 2*(48*b^4*c^5*d - 8*a*b^3*c^4*d^2 - 315*a^2*b^2*c^3*d^3 + 450*a^3*b*c 
^2*d^4 - 175*a^4*c*d^5)*x^4 - 16*(3*b^4*c^6 - 2*a*b^3*c^5*d - 12*a^2*b^2*c 
^4*d^2 + 18*a^3*b*c^3*d^3 - 7*a^4*c^2*d^4)*x^2 + 24*(b^4*c^4*d^2*x^7 + 2*b 
^4*c^5*d*x^5 + b^4*c^6*x^3)*sqrt(-b/a)*log((b*x^2 - 2*a*x*sqrt(-b/a) - a)/ 
(b*x^2 + a)) + 3*((63*a^2*b^2*c^2*d^4 - 90*a^3*b*c*d^5 + 35*a^4*d^6)*x^7 + 
 2*(63*a^2*b^2*c^3*d^3 - 90*a^3*b*c^2*d^4 + 35*a^4*c*d^5)*x^5 + (63*a^2*b^ 
2*c^4*d^2 - 90*a^3*b*c^3*d^3 + 35*a^4*c^2*d^4)*x^3)*sqrt(-d/c)*log((d*x^2 
+ 2*c*x*sqrt(-d/c) - c)/(d*x^2 + c)))/((a^2*b^3*c^7*d^2 - 3*a^3*b^2*c^6*d^ 
3 + 3*a^4*b*c^5*d^4 - a^5*c^4*d^5)*x^7 + 2*(a^2*b^3*c^8*d - 3*a^3*b^2*c^7* 
d^2 + 3*a^4*b*c^6*d^3 - a^5*c^5*d^4)*x^5 + (a^2*b^3*c^9 - 3*a^3*b^2*c^8*d 
+ 3*a^4*b*c^7*d^2 - a^5*c^6*d^3)*x^3), -1/24*(8*a*b^3*c^6 - 24*a^2*b^2*c^5 
*d + 24*a^3*b*c^4*d^2 - 8*a^4*c^3*d^3 - 3*(8*b^4*c^4*d^2 - 63*a^2*b^2*c^2* 
d^4 + 90*a^3*b*c*d^5 - 35*a^4*d^6)*x^6 - (48*b^4*c^5*d - 8*a*b^3*c^4*d^2 - 
 315*a^2*b^2*c^3*d^3 + 450*a^3*b*c^2*d^4 - 175*a^4*c*d^5)*x^4 - 8*(3*b^4*c 
^6 - 2*a*b^3*c^5*d - 12*a^2*b^2*c^4*d^2 + 18*a^3*b*c^3*d^3 - 7*a^4*c^2*d^4 
)*x^2 + 3*((63*a^2*b^2*c^2*d^4 - 90*a^3*b*c*d^5 + 35*a^4*d^6)*x^7 + 2*(63* 
a^2*b^2*c^3*d^3 - 90*a^3*b*c^2*d^4 + 35*a^4*c*d^5)*x^5 + (63*a^2*b^2*c^4*d 
^2 - 90*a^3*b*c^3*d^3 + 35*a^4*c^2*d^4)*x^3)*sqrt(d/c)*arctan(x*sqrt(d/...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{x^4 \left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx=\text {Timed out} \] Input:

integrate(1/x**4/(b*x**2+a)/(d*x**2+c)**3,x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 440, normalized size of antiderivative = 1.63 \[ \int \frac {1}{x^4 \left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx=\frac {b^{5} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{{\left (a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - a^{5} d^{3}\right )} \sqrt {a b}} - \frac {{\left (63 \, b^{2} c^{2} d^{3} - 90 \, a b c d^{4} + 35 \, a^{2} d^{5}\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{8 \, {\left (b^{3} c^{7} - 3 \, a b^{2} c^{6} d + 3 \, a^{2} b c^{5} d^{2} - a^{3} c^{4} d^{3}\right )} \sqrt {c d}} - \frac {8 \, a b^{2} c^{5} - 16 \, a^{2} b c^{4} d + 8 \, a^{3} c^{3} d^{2} - 3 \, {\left (8 \, b^{3} c^{3} d^{2} + 8 \, a b^{2} c^{2} d^{3} - 55 \, a^{2} b c d^{4} + 35 \, a^{3} d^{5}\right )} x^{6} - {\left (48 \, b^{3} c^{4} d + 40 \, a b^{2} c^{3} d^{2} - 275 \, a^{2} b c^{2} d^{3} + 175 \, a^{3} c d^{4}\right )} x^{4} - 8 \, {\left (3 \, b^{3} c^{5} + a b^{2} c^{4} d - 11 \, a^{2} b c^{3} d^{2} + 7 \, a^{3} c^{2} d^{3}\right )} x^{2}}{24 \, {\left ({\left (a^{2} b^{2} c^{6} d^{2} - 2 \, a^{3} b c^{5} d^{3} + a^{4} c^{4} d^{4}\right )} x^{7} + 2 \, {\left (a^{2} b^{2} c^{7} d - 2 \, a^{3} b c^{6} d^{2} + a^{4} c^{5} d^{3}\right )} x^{5} + {\left (a^{2} b^{2} c^{8} - 2 \, a^{3} b c^{7} d + a^{4} c^{6} d^{2}\right )} x^{3}\right )}} \] Input:

integrate(1/x^4/(b*x^2+a)/(d*x^2+c)^3,x, algorithm="maxima")
 

Output:

b^5*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)*sqrt(a*b)) - 1/8*(63*b^2*c^2*d^3 - 90*a*b*c*d^4 + 35*a^2*d^5)*a 
rctan(d*x/sqrt(c*d))/((b^3*c^7 - 3*a*b^2*c^6*d + 3*a^2*b*c^5*d^2 - a^3*c^4 
*d^3)*sqrt(c*d)) - 1/24*(8*a*b^2*c^5 - 16*a^2*b*c^4*d + 8*a^3*c^3*d^2 - 3* 
(8*b^3*c^3*d^2 + 8*a*b^2*c^2*d^3 - 55*a^2*b*c*d^4 + 35*a^3*d^5)*x^6 - (48* 
b^3*c^4*d + 40*a*b^2*c^3*d^2 - 275*a^2*b*c^2*d^3 + 175*a^3*c*d^4)*x^4 - 8* 
(3*b^3*c^5 + a*b^2*c^4*d - 11*a^2*b*c^3*d^2 + 7*a^3*c^2*d^3)*x^2)/((a^2*b^ 
2*c^6*d^2 - 2*a^3*b*c^5*d^3 + a^4*c^4*d^4)*x^7 + 2*(a^2*b^2*c^7*d - 2*a^3* 
b*c^6*d^2 + a^4*c^5*d^3)*x^5 + (a^2*b^2*c^8 - 2*a^3*b*c^7*d + a^4*c^6*d^2) 
*x^3)
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 256, normalized size of antiderivative = 0.95 \[ \int \frac {1}{x^4 \left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx=\frac {b^{5} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{{\left (a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - a^{5} d^{3}\right )} \sqrt {a b}} - \frac {{\left (63 \, b^{2} c^{2} d^{3} - 90 \, a b c d^{4} + 35 \, a^{2} d^{5}\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{8 \, {\left (b^{3} c^{7} - 3 \, a b^{2} c^{6} d + 3 \, a^{2} b c^{5} d^{2} - a^{3} c^{4} d^{3}\right )} \sqrt {c d}} - \frac {15 \, b c d^{4} x^{3} - 11 \, a d^{5} x^{3} + 17 \, b c^{2} d^{3} x - 13 \, a c d^{4} x}{8 \, {\left (b^{2} c^{6} - 2 \, a b c^{5} d + a^{2} c^{4} d^{2}\right )} {\left (d x^{2} + c\right )}^{2}} + \frac {3 \, b c x^{2} + 9 \, a d x^{2} - a c}{3 \, a^{2} c^{4} x^{3}} \] Input:

integrate(1/x^4/(b*x^2+a)/(d*x^2+c)^3,x, algorithm="giac")
 

Output:

b^5*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)*sqrt(a*b)) - 1/8*(63*b^2*c^2*d^3 - 90*a*b*c*d^4 + 35*a^2*d^5)*a 
rctan(d*x/sqrt(c*d))/((b^3*c^7 - 3*a*b^2*c^6*d + 3*a^2*b*c^5*d^2 - a^3*c^4 
*d^3)*sqrt(c*d)) - 1/8*(15*b*c*d^4*x^3 - 11*a*d^5*x^3 + 17*b*c^2*d^3*x - 1 
3*a*c*d^4*x)/((b^2*c^6 - 2*a*b*c^5*d + a^2*c^4*d^2)*(d*x^2 + c)^2) + 1/3*( 
3*b*c*x^2 + 9*a*d*x^2 - a*c)/(a^2*c^4*x^3)
 

Mupad [B] (verification not implemented)

Time = 1.93 (sec) , antiderivative size = 785, normalized size of antiderivative = 2.91 \[ \int \frac {1}{x^4 \left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx=\frac {\frac {x^2\,\left (7\,a\,d+3\,b\,c\right )}{3\,a^2\,c^2}-\frac {1}{3\,a\,c}+\frac {x^4\,\left (175\,a^3\,d^4-275\,a^2\,b\,c\,d^3+40\,a\,b^2\,c^2\,d^2+48\,b^3\,c^3\,d\right )}{24\,a^2\,c^3\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {x^6\,\left (35\,a^3\,d^5-55\,a^2\,b\,c\,d^4+8\,a\,b^2\,c^2\,d^3+8\,b^3\,c^3\,d^2\right )}{8\,a^2\,c^4\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}}{c^2\,x^3+2\,c\,d\,x^5+d^2\,x^7}+\frac {\mathrm {atan}\left (\frac {b\,c^9\,x\,{\left (-a^5\,b^9\right )}^{3/2}\,64{}\mathrm {i}+a^{14}\,b\,d^9\,x\,\sqrt {-a^5\,b^9}\,1225{}\mathrm {i}+a^{10}\,b^5\,c^4\,d^5\,x\,\sqrt {-a^5\,b^9}\,3969{}\mathrm {i}-a^{11}\,b^4\,c^3\,d^6\,x\,\sqrt {-a^5\,b^9}\,11340{}\mathrm {i}+a^{12}\,b^3\,c^2\,d^7\,x\,\sqrt {-a^5\,b^9}\,12510{}\mathrm {i}-a^{13}\,b^2\,c\,d^8\,x\,\sqrt {-a^5\,b^9}\,6300{}\mathrm {i}}{-1225\,a^{17}\,b^5\,d^9+6300\,a^{16}\,b^6\,c\,d^8-12510\,a^{15}\,b^7\,c^2\,d^7+11340\,a^{14}\,b^8\,c^3\,d^6-3969\,a^{13}\,b^9\,c^4\,d^5+64\,a^8\,b^{14}\,c^9}\right )\,\sqrt {-a^5\,b^9}\,1{}\mathrm {i}}{a^8\,d^3-3\,a^7\,b\,c\,d^2+3\,a^6\,b^2\,c^2\,d-a^5\,b^3\,c^3}+\frac {\mathrm {atan}\left (\frac {a^9\,d^5\,x\,{\left (-c^9\,d^5\right )}^{3/2}\,1225{}\mathrm {i}+b^9\,c^{18}\,d\,x\,\sqrt {-c^9\,d^5}\,64{}\mathrm {i}-a^6\,b^3\,c^3\,d^2\,x\,{\left (-c^9\,d^5\right )}^{3/2}\,11340{}\mathrm {i}+a^7\,b^2\,c^2\,d^3\,x\,{\left (-c^9\,d^5\right )}^{3/2}\,12510{}\mathrm {i}-a^8\,b\,c\,d^4\,x\,{\left (-c^9\,d^5\right )}^{3/2}\,6300{}\mathrm {i}+a^5\,b^4\,c^4\,d\,x\,{\left (-c^9\,d^5\right )}^{3/2}\,3969{}\mathrm {i}}{1225\,a^9\,c^{14}\,d^{12}-6300\,a^8\,b\,c^{15}\,d^{11}+12510\,a^7\,b^2\,c^{16}\,d^{10}-11340\,a^6\,b^3\,c^{17}\,d^9+3969\,a^5\,b^4\,c^{18}\,d^8-64\,b^9\,c^{23}\,d^3}\right )\,\sqrt {-c^9\,d^5}\,\left (35\,a^2\,d^2-90\,a\,b\,c\,d+63\,b^2\,c^2\right )\,1{}\mathrm {i}}{8\,\left (-a^3\,c^9\,d^3+3\,a^2\,b\,c^{10}\,d^2-3\,a\,b^2\,c^{11}\,d+b^3\,c^{12}\right )} \] Input:

int(1/(x^4*(a + b*x^2)*(c + d*x^2)^3),x)
 

Output:

((x^2*(7*a*d + 3*b*c))/(3*a^2*c^2) - 1/(3*a*c) + (x^4*(175*a^3*d^4 + 48*b^ 
3*c^3*d + 40*a*b^2*c^2*d^2 - 275*a^2*b*c*d^3))/(24*a^2*c^3*(a^2*d^2 + b^2* 
c^2 - 2*a*b*c*d)) + (x^6*(35*a^3*d^5 + 8*b^3*c^3*d^2 + 8*a*b^2*c^2*d^3 - 5 
5*a^2*b*c*d^4))/(8*a^2*c^4*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)))/(c^2*x^3 + d^ 
2*x^7 + 2*c*d*x^5) + (atan((b*c^9*x*(-a^5*b^9)^(3/2)*64i + a^14*b*d^9*x*(- 
a^5*b^9)^(1/2)*1225i + a^10*b^5*c^4*d^5*x*(-a^5*b^9)^(1/2)*3969i - a^11*b^ 
4*c^3*d^6*x*(-a^5*b^9)^(1/2)*11340i + a^12*b^3*c^2*d^7*x*(-a^5*b^9)^(1/2)* 
12510i - a^13*b^2*c*d^8*x*(-a^5*b^9)^(1/2)*6300i)/(64*a^8*b^14*c^9 - 1225* 
a^17*b^5*d^9 + 6300*a^16*b^6*c*d^8 - 3969*a^13*b^9*c^4*d^5 + 11340*a^14*b^ 
8*c^3*d^6 - 12510*a^15*b^7*c^2*d^7))*(-a^5*b^9)^(1/2)*1i)/(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) + (atan((a^9*d^5*x*(-c^9*d^5)^(3 
/2)*1225i + b^9*c^18*d*x*(-c^9*d^5)^(1/2)*64i - a^6*b^3*c^3*d^2*x*(-c^9*d^ 
5)^(3/2)*11340i + a^7*b^2*c^2*d^3*x*(-c^9*d^5)^(3/2)*12510i - a^8*b*c*d^4* 
x*(-c^9*d^5)^(3/2)*6300i + a^5*b^4*c^4*d*x*(-c^9*d^5)^(3/2)*3969i)/(1225*a 
^9*c^14*d^12 - 64*b^9*c^23*d^3 - 6300*a^8*b*c^15*d^11 + 3969*a^5*b^4*c^18* 
d^8 - 11340*a^6*b^3*c^17*d^9 + 12510*a^7*b^2*c^16*d^10))*(-c^9*d^5)^(1/2)* 
(35*a^2*d^2 + 63*b^2*c^2 - 90*a*b*c*d)*1i)/(8*(b^3*c^12 - a^3*c^9*d^3 + 3* 
a^2*b*c^10*d^2 - 3*a*b^2*c^11*d))
 

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 775, normalized size of antiderivative = 2.87 \[ \int \frac {1}{x^4 \left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx =\text {Too large to display} \] Input:

int(1/x^4/(b*x^2+a)/(d*x^2+c)^3,x)
 

Output:

( - 24*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*b**4*c**7*x**3 - 48*s 
qrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*b**4*c**6*d*x**5 - 24*sqrt(b) 
*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*b**4*c**5*d**2*x**7 + 105*sqrt(d)*s 
qrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a**5*c**2*d**4*x**3 + 210*sqrt(d)*sqr 
t(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a**5*c*d**5*x**5 + 105*sqrt(d)*sqrt(c)* 
atan((d*x)/(sqrt(d)*sqrt(c)))*a**5*d**6*x**7 - 270*sqrt(d)*sqrt(c)*atan((d 
*x)/(sqrt(d)*sqrt(c)))*a**4*b*c**3*d**3*x**3 - 540*sqrt(d)*sqrt(c)*atan((d 
*x)/(sqrt(d)*sqrt(c)))*a**4*b*c**2*d**4*x**5 - 270*sqrt(d)*sqrt(c)*atan((d 
*x)/(sqrt(d)*sqrt(c)))*a**4*b*c*d**5*x**7 + 189*sqrt(d)*sqrt(c)*atan((d*x) 
/(sqrt(d)*sqrt(c)))*a**3*b**2*c**4*d**2*x**3 + 378*sqrt(d)*sqrt(c)*atan((d 
*x)/(sqrt(d)*sqrt(c)))*a**3*b**2*c**3*d**3*x**5 + 189*sqrt(d)*sqrt(c)*atan 
((d*x)/(sqrt(d)*sqrt(c)))*a**3*b**2*c**2*d**4*x**7 - 8*a**5*c**4*d**3 + 56 
*a**5*c**3*d**4*x**2 + 175*a**5*c**2*d**5*x**4 + 105*a**5*c*d**6*x**6 + 24 
*a**4*b*c**5*d**2 - 144*a**4*b*c**4*d**3*x**2 - 450*a**4*b*c**3*d**4*x**4 
- 270*a**4*b*c**2*d**5*x**6 - 24*a**3*b**2*c**6*d + 96*a**3*b**2*c**5*d**2 
*x**2 + 315*a**3*b**2*c**4*d**3*x**4 + 189*a**3*b**2*c**3*d**4*x**6 + 8*a* 
*2*b**3*c**7 + 16*a**2*b**3*c**6*d*x**2 + 8*a**2*b**3*c**5*d**2*x**4 - 24* 
a*b**4*c**7*x**2 - 48*a*b**4*c**6*d*x**4 - 24*a*b**4*c**5*d**2*x**6)/(24*a 
**3*c**5*x**3*(a**3*c**2*d**3 + 2*a**3*c*d**4*x**2 + a**3*d**5*x**4 - 3*a* 
*2*b*c**3*d**2 - 6*a**2*b*c**2*d**3*x**2 - 3*a**2*b*c*d**4*x**4 + 3*a*b...