\(\int \frac {1}{x^2 (c+d x) (a+b x^2)^3} \, dx\) [250]

Optimal result

Integrand size = 20, antiderivative size = 275 \[ \int \frac {1}{x^2 (c+d x) \left (a+b x^2\right )^3} \, dx=-\frac {1}{a^3 c x}-\frac {b (a d+b c x)}{4 a^2 \left (b c^2+a d^2\right ) \left (a+b x^2\right )^2}-\frac {b \left (4 a d \left (b c^2+2 a d^2\right )+b c \left (7 b c^2+11 a d^2\right ) x\right )}{8 a^3 \left (b c^2+a d^2\right )^2 \left (a+b x^2\right )}-\frac {b^{3/2} c \left (15 b^2 c^4+42 a b c^2 d^2+35 a^2 d^4\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{7/2} \left (b c^2+a d^2\right )^3}-\frac {d \log (x)}{a^3 c^2}+\frac {d^7 \log (c+d x)}{c^2 \left (b c^2+a d^2\right )^3}+\frac {b d \left (b^2 c^4+3 a b c^2 d^2+3 a^2 d^4\right ) \log \left (a+b x^2\right )}{2 a^3 \left (b c^2+a d^2\right )^3} \] Output:

-1/a^3/c/x-1/4*b*(b*c*x+a*d)/a^2/(a*d^2+b*c^2)/(b*x^2+a)^2-1/8*b*(4*a*d*(2 
*a*d^2+b*c^2)+b*c*(11*a*d^2+7*b*c^2)*x)/a^3/(a*d^2+b*c^2)^2/(b*x^2+a)-1/8* 
b^(3/2)*c*(35*a^2*d^4+42*a*b*c^2*d^2+15*b^2*c^4)*arctan(b^(1/2)*x/a^(1/2)) 
/a^(7/2)/(a*d^2+b*c^2)^3-d*ln(x)/a^3/c^2+d^7*ln(d*x+c)/c^2/(a*d^2+b*c^2)^3 
+1/2*b*d*(3*a^2*d^4+3*a*b*c^2*d^2+b^2*c^4)*ln(b*x^2+a)/a^3/(a*d^2+b*c^2)^3
 

Mathematica [A] (verified)

Time = 0.34 (sec) , antiderivative size = 269, normalized size of antiderivative = 0.98 \[ \int \frac {1}{x^2 (c+d x) \left (a+b x^2\right )^3} \, dx=\frac {1}{8} \left (-\frac {8}{a^3 c x}-\frac {2 b (a d+b c x)}{a^2 \left (b c^2+a d^2\right ) \left (a+b x^2\right )^2}-\frac {b \left (8 a^2 d^3+7 b^2 c^3 x+a b c d (4 c+11 d x)\right )}{a^3 \left (b c^2+a d^2\right )^2 \left (a+b x^2\right )}-\frac {b^{3/2} c \left (15 b^2 c^4+42 a b c^2 d^2+35 a^2 d^4\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{7/2} \left (b c^2+a d^2\right )^3}-\frac {8 d \log (x)}{a^3 c^2}+\frac {8 d^7 \log (c+d x)}{c^2 \left (b c^2+a d^2\right )^3}+\frac {4 b \left (b^2 c^4 d+3 a b c^2 d^3+3 a^2 d^5\right ) \log \left (a+b x^2\right )}{a^3 \left (b c^2+a d^2\right )^3}\right ) \] Input:

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

Output:

(-8/(a^3*c*x) - (2*b*(a*d + b*c*x))/(a^2*(b*c^2 + a*d^2)*(a + b*x^2)^2) - 
(b*(8*a^2*d^3 + 7*b^2*c^3*x + a*b*c*d*(4*c + 11*d*x)))/(a^3*(b*c^2 + a*d^2 
)^2*(a + b*x^2)) - (b^(3/2)*c*(15*b^2*c^4 + 42*a*b*c^2*d^2 + 35*a^2*d^4)*A 
rcTan[(Sqrt[b]*x)/Sqrt[a]])/(a^(7/2)*(b*c^2 + a*d^2)^3) - (8*d*Log[x])/(a^ 
3*c^2) + (8*d^7*Log[c + d*x])/(c^2*(b*c^2 + a*d^2)^3) + (4*b*(b^2*c^4*d + 
3*a*b*c^2*d^3 + 3*a^2*d^5)*Log[a + b*x^2])/(a^3*(b*c^2 + a*d^2)^3))/8
 

Rubi [A] (verified)

Time = 0.99 (sec) , antiderivative size = 386, normalized size of antiderivative = 1.40, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {615, 2009}

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.

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

\(\Big \downarrow \) 615

\(\displaystyle \int \left (-\frac {d}{a^3 c^2 x}+\frac {1}{a^3 c x^2}-\frac {b^2 (c-d x) \left (2 a d^2+b c^2\right )}{a^2 \left (a+b x^2\right )^2 \left (a d^2+b c^2\right )^2}-\frac {b^2 (c-d x) \left (3 a^2 d^4+3 a b c^2 d^2+b^2 c^4\right )}{a^3 \left (a+b x^2\right ) \left (a d^2+b c^2\right )^3}-\frac {b^2 (c-d x)}{a \left (a+b x^2\right )^3 \left (a d^2+b c^2\right )}+\frac {d^8}{c^2 (c+d x) \left (a d^2+b c^2\right )^3}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {b^{3/2} c \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (2 a d^2+b c^2\right )}{2 a^{7/2} \left (a d^2+b c^2\right )^2}-\frac {3 b^{3/2} c \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{7/2} \left (a d^2+b c^2\right )}-\frac {3 b^2 c x}{8 a^3 \left (a+b x^2\right ) \left (a d^2+b c^2\right )}-\frac {b \left (2 a d^2+b c^2\right ) (a d+b c x)}{2 a^3 \left (a+b x^2\right ) \left (a d^2+b c^2\right )^2}-\frac {d \log (x)}{a^3 c^2}-\frac {1}{a^3 c x}-\frac {b (a d+b c x)}{4 a^2 \left (a+b x^2\right )^2 \left (a d^2+b c^2\right )}-\frac {b^{3/2} c \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (3 a^2 d^4+3 a b c^2 d^2+b^2 c^4\right )}{a^{7/2} \left (a d^2+b c^2\right )^3}+\frac {b d \left (3 a^2 d^4+3 a b c^2 d^2+b^2 c^4\right ) \log \left (a+b x^2\right )}{2 a^3 \left (a d^2+b c^2\right )^3}+\frac {d^7 \log (c+d x)}{c^2 \left (a d^2+b c^2\right )^3}\)

Input:

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

Output:

-(1/(a^3*c*x)) - (b*(a*d + b*c*x))/(4*a^2*(b*c^2 + a*d^2)*(a + b*x^2)^2) - 
 (3*b^2*c*x)/(8*a^3*(b*c^2 + a*d^2)*(a + b*x^2)) - (b*(b*c^2 + 2*a*d^2)*(a 
*d + b*c*x))/(2*a^3*(b*c^2 + a*d^2)^2*(a + b*x^2)) - (3*b^(3/2)*c*ArcTan[( 
Sqrt[b]*x)/Sqrt[a]])/(8*a^(7/2)*(b*c^2 + a*d^2)) - (b^(3/2)*c*(b*c^2 + 2*a 
*d^2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(7/2)*(b*c^2 + a*d^2)^2) - (b^(3/2 
)*c*(b^2*c^4 + 3*a*b*c^2*d^2 + 3*a^2*d^4)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(a^ 
(7/2)*(b*c^2 + a*d^2)^3) - (d*Log[x])/(a^3*c^2) + (d^7*Log[c + d*x])/(c^2* 
(b*c^2 + a*d^2)^3) + (b*d*(b^2*c^4 + 3*a*b*c^2*d^2 + 3*a^2*d^4)*Log[a + b* 
x^2])/(2*a^3*(b*c^2 + a*d^2)^3)
 

Defintions of rubi rules used

rule 615

Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), 
 x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*(c + d*x)^n*(a + b*x^2)^p, x], x] 
 /; FreeQ[{a, b, c, d, e, m, n}, x] && ILtQ[p, 0]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [A] (verified)

Time = 0.38 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.11

method	result	size
default	\(-\frac {1}{a^{3} c x}-\frac {d \ln \left (x \right )}{a^{3} c^{2}}-\frac {b^{2} \left (\frac {\left (\frac {11}{8} a^{2} c \,d^{4} b +\frac {9}{4} a \,b^{2} c^{3} d^{2}+\frac {7}{8} b^{3} c^{5}\right ) x^{3}+\left (a^{3} d^{5}+\frac {3}{2} a^{2} b \,c^{2} d^{3}+\frac {1}{2} a \,b^{2} c^{4} d \right ) x^{2}+\frac {a c \left (13 a^{2} d^{4}+22 b \,c^{2} d^{2} a +9 b^{2} c^{4}\right ) x}{8}+\frac {a^{2} d \left (5 a^{2} d^{4}+8 b \,c^{2} d^{2} a +3 b^{2} c^{4}\right )}{4 b}}{\left (b \,x^{2}+a \right )^{2}}+\frac {\left (-24 a^{2} d^{5}-24 d^{3} a \,c^{2} b -8 b^{2} c^{4} d \right ) \ln \left (b \,x^{2}+a \right )}{16 b}+\frac {\left (35 a^{2} c \,d^{4}+42 a \,c^{3} d^{2} b +15 c^{5} b^{2}\right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \sqrt {a b}}\right )}{\left (a \,d^{2}+b \,c^{2}\right )^{3} a^{3}}+\frac {d^{7} \ln \left (d x +c \right )}{c^{2} \left (a \,d^{2}+b \,c^{2}\right )^{3}}\)	\(305\)
risch	\(\text {Expression too large to display}\)	\(1080\)

Input:

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

Output:

-1/a^3/c/x-d*ln(x)/a^3/c^2-b^2/(a*d^2+b*c^2)^3/a^3*(((11/8*a^2*c*d^4*b+9/4 
*a*b^2*c^3*d^2+7/8*b^3*c^5)*x^3+(a^3*d^5+3/2*a^2*b*c^2*d^3+1/2*a*b^2*c^4*d 
)*x^2+1/8*a*c*(13*a^2*d^4+22*a*b*c^2*d^2+9*b^2*c^4)*x+1/4*a^2*d*(5*a^2*d^4 
+8*a*b*c^2*d^2+3*b^2*c^4)/b)/(b*x^2+a)^2+1/16*(-24*a^2*d^5-24*a*b*c^2*d^3- 
8*b^2*c^4*d)/b*ln(b*x^2+a)+1/8*(35*a^2*c*d^4+42*a*b*c^3*d^2+15*b^2*c^5)/(a 
*b)^(1/2)*arctan(b*x/(a*b)^(1/2)))+d^7*ln(d*x+c)/c^2/(a*d^2+b*c^2)^3
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 522 vs. \(2 (259) = 518\).

Time = 0.14 (sec) , antiderivative size = 522, normalized size of antiderivative = 1.90 \[ \int \frac {1}{x^2 (c+d x) \left (a+b x^2\right )^3} \, dx=\frac {d^{7} \log \left (d x + c\right )}{b^{3} c^{8} + 3 \, a b^{2} c^{6} d^{2} + 3 \, a^{2} b c^{4} d^{4} + a^{3} c^{2} d^{6}} + \frac {{\left (b^{3} c^{4} d + 3 \, a b^{2} c^{2} d^{3} + 3 \, a^{2} b d^{5}\right )} \log \left (b x^{2} + a\right )}{2 \, {\left (a^{3} b^{3} c^{6} + 3 \, a^{4} b^{2} c^{4} d^{2} + 3 \, a^{5} b c^{2} d^{4} + a^{6} d^{6}\right )}} - \frac {{\left (15 \, b^{4} c^{5} + 42 \, a b^{3} c^{3} d^{2} + 35 \, a^{2} b^{2} c d^{4}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, {\left (a^{3} b^{3} c^{6} + 3 \, a^{4} b^{2} c^{4} d^{2} + 3 \, a^{5} b c^{2} d^{4} + a^{6} d^{6}\right )} \sqrt {a b}} - \frac {8 \, a^{2} b^{2} c^{4} + 16 \, a^{3} b c^{2} d^{2} + 8 \, a^{4} d^{4} + {\left (15 \, b^{4} c^{4} + 27 \, a b^{3} c^{2} d^{2} + 8 \, a^{2} b^{2} d^{4}\right )} x^{4} + 4 \, {\left (a b^{3} c^{3} d + 2 \, a^{2} b^{2} c d^{3}\right )} x^{3} + {\left (25 \, a b^{3} c^{4} + 45 \, a^{2} b^{2} c^{2} d^{2} + 16 \, a^{3} b d^{4}\right )} x^{2} + 2 \, {\left (3 \, a^{2} b^{2} c^{3} d + 5 \, a^{3} b c d^{3}\right )} x}{8 \, {\left ({\left (a^{3} b^{4} c^{5} + 2 \, a^{4} b^{3} c^{3} d^{2} + a^{5} b^{2} c d^{4}\right )} x^{5} + 2 \, {\left (a^{4} b^{3} c^{5} + 2 \, a^{5} b^{2} c^{3} d^{2} + a^{6} b c d^{4}\right )} x^{3} + {\left (a^{5} b^{2} c^{5} + 2 \, a^{6} b c^{3} d^{2} + a^{7} c d^{4}\right )} x\right )}} - \frac {d \log \left (x\right )}{a^{3} c^{2}} \] Input:

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

Output:

d^7*log(d*x + c)/(b^3*c^8 + 3*a*b^2*c^6*d^2 + 3*a^2*b*c^4*d^4 + a^3*c^2*d^ 
6) + 1/2*(b^3*c^4*d + 3*a*b^2*c^2*d^3 + 3*a^2*b*d^5)*log(b*x^2 + a)/(a^3*b 
^3*c^6 + 3*a^4*b^2*c^4*d^2 + 3*a^5*b*c^2*d^4 + a^6*d^6) - 1/8*(15*b^4*c^5 
+ 42*a*b^3*c^3*d^2 + 35*a^2*b^2*c*d^4)*arctan(b*x/sqrt(a*b))/((a^3*b^3*c^6 
 + 3*a^4*b^2*c^4*d^2 + 3*a^5*b*c^2*d^4 + a^6*d^6)*sqrt(a*b)) - 1/8*(8*a^2* 
b^2*c^4 + 16*a^3*b*c^2*d^2 + 8*a^4*d^4 + (15*b^4*c^4 + 27*a*b^3*c^2*d^2 + 
8*a^2*b^2*d^4)*x^4 + 4*(a*b^3*c^3*d + 2*a^2*b^2*c*d^3)*x^3 + (25*a*b^3*c^4 
 + 45*a^2*b^2*c^2*d^2 + 16*a^3*b*d^4)*x^2 + 2*(3*a^2*b^2*c^3*d + 5*a^3*b*c 
*d^3)*x)/((a^3*b^4*c^5 + 2*a^4*b^3*c^3*d^2 + a^5*b^2*c*d^4)*x^5 + 2*(a^4*b 
^3*c^5 + 2*a^5*b^2*c^3*d^2 + a^6*b*c*d^4)*x^3 + (a^5*b^2*c^5 + 2*a^6*b*c^3 
*d^2 + a^7*c*d^4)*x) - d*log(x)/(a^3*c^2)
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 518, normalized size of antiderivative = 1.88 \[ \int \frac {1}{x^2 (c+d x) \left (a+b x^2\right )^3} \, dx=\frac {d^{8} \log \left ({\left | d x + c \right |}\right )}{b^{3} c^{8} d + 3 \, a b^{2} c^{6} d^{3} + 3 \, a^{2} b c^{4} d^{5} + a^{3} c^{2} d^{7}} + \frac {{\left (b^{3} c^{4} d + 3 \, a b^{2} c^{2} d^{3} + 3 \, a^{2} b d^{5}\right )} \log \left (b x^{2} + a\right )}{2 \, {\left (a^{3} b^{3} c^{6} + 3 \, a^{4} b^{2} c^{4} d^{2} + 3 \, a^{5} b c^{2} d^{4} + a^{6} d^{6}\right )}} - \frac {{\left (15 \, b^{4} c^{5} + 42 \, a b^{3} c^{3} d^{2} + 35 \, a^{2} b^{2} c d^{4}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, {\left (a^{3} b^{3} c^{6} + 3 \, a^{4} b^{2} c^{4} d^{2} + 3 \, a^{5} b c^{2} d^{4} + a^{6} d^{6}\right )} \sqrt {a b}} - \frac {d \log \left ({\left | x \right |}\right )}{a^{3} c^{2}} - \frac {8 \, a^{2} b^{3} c^{7} + 24 \, a^{3} b^{2} c^{5} d^{2} + 24 \, a^{4} b c^{3} d^{4} + 8 \, a^{5} c d^{6} + {\left (15 \, b^{5} c^{7} + 42 \, a b^{4} c^{5} d^{2} + 35 \, a^{2} b^{3} c^{3} d^{4} + 8 \, a^{3} b^{2} c d^{6}\right )} x^{4} + 4 \, {\left (a b^{4} c^{6} d + 3 \, a^{2} b^{3} c^{4} d^{3} + 2 \, a^{3} b^{2} c^{2} d^{5}\right )} x^{3} + {\left (25 \, a b^{4} c^{7} + 70 \, a^{2} b^{3} c^{5} d^{2} + 61 \, a^{3} b^{2} c^{3} d^{4} + 16 \, a^{4} b c d^{6}\right )} x^{2} + 2 \, {\left (3 \, a^{2} b^{3} c^{6} d + 8 \, a^{3} b^{2} c^{4} d^{3} + 5 \, a^{4} b c^{2} d^{5}\right )} x}{8 \, {\left (b c^{2} + a d^{2}\right )}^{3} {\left (b x^{2} + a\right )}^{2} a^{3} c^{2} x} \] Input:

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

Output:

d^8*log(abs(d*x + c))/(b^3*c^8*d + 3*a*b^2*c^6*d^3 + 3*a^2*b*c^4*d^5 + a^3 
*c^2*d^7) + 1/2*(b^3*c^4*d + 3*a*b^2*c^2*d^3 + 3*a^2*b*d^5)*log(b*x^2 + a) 
/(a^3*b^3*c^6 + 3*a^4*b^2*c^4*d^2 + 3*a^5*b*c^2*d^4 + a^6*d^6) - 1/8*(15*b 
^4*c^5 + 42*a*b^3*c^3*d^2 + 35*a^2*b^2*c*d^4)*arctan(b*x/sqrt(a*b))/((a^3* 
b^3*c^6 + 3*a^4*b^2*c^4*d^2 + 3*a^5*b*c^2*d^4 + a^6*d^6)*sqrt(a*b)) - d*lo 
g(abs(x))/(a^3*c^2) - 1/8*(8*a^2*b^3*c^7 + 24*a^3*b^2*c^5*d^2 + 24*a^4*b*c 
^3*d^4 + 8*a^5*c*d^6 + (15*b^5*c^7 + 42*a*b^4*c^5*d^2 + 35*a^2*b^3*c^3*d^4 
 + 8*a^3*b^2*c*d^6)*x^4 + 4*(a*b^4*c^6*d + 3*a^2*b^3*c^4*d^3 + 2*a^3*b^2*c 
^2*d^5)*x^3 + (25*a*b^4*c^7 + 70*a^2*b^3*c^5*d^2 + 61*a^3*b^2*c^3*d^4 + 16 
*a^4*b*c*d^6)*x^2 + 2*(3*a^2*b^3*c^6*d + 8*a^3*b^2*c^4*d^3 + 5*a^4*b*c^2*d 
^5)*x)/((b*c^2 + a*d^2)^3*(b*x^2 + a)^2*a^3*c^2*x)
 

Mupad [B] (verification not implemented)

Time = 9.29 (sec) , antiderivative size = 1870, normalized size of antiderivative = 6.80 \[ \int \frac {1}{x^2 (c+d x) \left (a+b x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

log(50625*a^14*b^20*c^32*x + 65536*a^30*b^4*d^32*x + 50625*a^4*b^15*c^32*( 
-a^7*b^3)^(3/2) - 65536*a^27*b^2*d^32*(-a^7*b^3)^(1/2) + 6954496*a^9*c^8*d 
^24*(-a^7*b^3)^(5/2) + 524288*a^19*c^2*d^30*(-a^7*b^3)^(3/2) - 27489336*b^ 
9*c^26*d^6*(-a^7*b^3)^(5/2) - 85675410*a*b^8*c^24*d^8*(-a^7*b^3)^(5/2) - 9 
376384*a^8*b*c^10*d^22*(-a^7*b^3)^(5/2) + 950272*a^18*b*c^4*d^28*(-a^7*b^3 
)^(3/2) - 192648028*a^2*b^7*c^22*d^10*(-a^7*b^3)^(5/2) - 320475698*a^3*b^6 
*c^20*d^12*(-a^7*b^3)^(5/2) - 396451768*a^4*b^5*c^18*d^14*(-a^7*b^3)^(5/2) 
 - 358992685*a^5*b^4*c^16*d^16*(-a^7*b^3)^(5/2) - 226410842*a^6*b^3*c^14*d 
^18*(-a^7*b^3)^(5/2) - 86527441*a^7*b^2*c^12*d^20*(-a^7*b^3)^(5/2) + 81225 
0*a^5*b^14*c^30*d^2*(-a^7*b^3)^(3/2) + 6033789*a^6*b^13*c^28*d^4*(-a^7*b^3 
)^(3/2) - 1895936*a^17*b^2*c^6*d^26*(-a^7*b^3)^(3/2) + 812250*a^15*b^19*c^ 
30*d^2*x + 6033789*a^16*b^18*c^28*d^4*x + 27489336*a^17*b^17*c^26*d^6*x + 
85675410*a^18*b^16*c^24*d^8*x + 192648028*a^19*b^15*c^22*d^10*x + 32047569 
8*a^20*b^14*c^20*d^12*x + 396451768*a^21*b^13*c^18*d^14*x + 358992685*a^22 
*b^12*c^16*d^16*x + 226410842*a^23*b^11*c^14*d^18*x + 86527441*a^24*b^10*c 
^12*d^20*x + 9376384*a^25*b^9*c^10*d^22*x - 6954496*a^26*b^8*c^8*d^24*x - 
1895936*a^27*b^7*c^6*d^26*x + 950272*a^28*b^6*c^4*d^28*x + 524288*a^29*b^5 
*c^2*d^30*x)*(((15*b^2*c^5*(-a^7*b^3)^(1/2))/16 - (a^7*d^7)/(2*c^2) + (35* 
a^2*c*d^4*(-a^7*b^3)^(1/2))/16 + (21*a*b*c^3*d^2*(-a^7*b^3)^(1/2))/8)/(a^1 
0*d^6 + a^7*b^3*c^6 + 3*a^9*b*c^2*d^4 + 3*a^8*b^2*c^4*d^2) + d/(2*a^3*c...
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 1158, normalized size of antiderivative = 4.21 \[ \int \frac {1}{x^2 (c+d x) \left (a+b x^2\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 35*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**4*b*c**3*d**4*x - 
42*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**3*b**2*c**5*d**2*x - 7 
0*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**3*b**2*c**3*d**4*x**3 - 
 15*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**2*b**3*c**7*x - 84*sq 
rt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**2*b**3*c**5*d**2*x**3 - 35* 
sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**2*b**3*c**3*d**4*x**5 - 3 
0*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a*b**4*c**7*x**3 - 42*sqrt 
(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a*b**4*c**5*d**2*x**5 - 15*sqrt( 
b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*b**5*c**7*x**5 + 12*log(a + b*x** 
2)*a**5*b*c**2*d**5*x + 12*log(a + b*x**2)*a**4*b**2*c**4*d**3*x + 24*log( 
a + b*x**2)*a**4*b**2*c**2*d**5*x**3 + 4*log(a + b*x**2)*a**3*b**3*c**6*d* 
x + 24*log(a + b*x**2)*a**3*b**3*c**4*d**3*x**3 + 12*log(a + b*x**2)*a**3* 
b**3*c**2*d**5*x**5 + 8*log(a + b*x**2)*a**2*b**4*c**6*d*x**3 + 12*log(a + 
 b*x**2)*a**2*b**4*c**4*d**3*x**5 + 4*log(a + b*x**2)*a*b**5*c**6*d*x**5 + 
 8*log(c + d*x)*a**6*d**7*x + 16*log(c + d*x)*a**5*b*d**7*x**3 + 8*log(c + 
 d*x)*a**4*b**2*d**7*x**5 - 8*log(x)*a**6*d**7*x - 24*log(x)*a**5*b*c**2*d 
**5*x - 16*log(x)*a**5*b*d**7*x**3 - 24*log(x)*a**4*b**2*c**4*d**3*x - 48* 
log(x)*a**4*b**2*c**2*d**5*x**3 - 8*log(x)*a**4*b**2*d**7*x**5 - 8*log(x)* 
a**3*b**3*c**6*d*x - 48*log(x)*a**3*b**3*c**4*d**3*x**3 - 24*log(x)*a**3*b 
**3*c**2*d**5*x**5 - 16*log(x)*a**2*b**4*c**6*d*x**3 - 24*log(x)*a**2*b...