\(\int \frac {1}{x^2 (c+d x)^3 (a+b x^2)^2} \, dx\) [217]

Optimal result

Integrand size = 20, antiderivative size = 336 \[ \int \frac {1}{x^2 (c+d x)^3 \left (a+b x^2\right )^2} \, dx=-\frac {1}{a^2 c^3 x}-\frac {d^5}{2 c^2 \left (b c^2+a d^2\right )^2 (c+d x)^2}-\frac {2 d^5 \left (3 b c^2+a d^2\right )}{c^3 \left (b c^2+a d^2\right )^3 (c+d x)}-\frac {b^2 \left (a d \left (3 b c^2-a d^2\right )+b c \left (b c^2-3 a d^2\right ) x\right )}{2 a^2 \left (b c^2+a d^2\right )^3 \left (a+b x^2\right )}-\frac {3 b^{5/2} c \left (b^2 c^4+2 a b c^2 d^2-7 a^2 d^4\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{5/2} \left (b c^2+a d^2\right )^4}-\frac {3 d \log (x)}{a^2 c^4}+\frac {3 d^5 \left (7 b^2 c^4+4 a b c^2 d^2+a^2 d^4\right ) \log (c+d x)}{c^4 \left (b c^2+a d^2\right )^4}+\frac {3 b^2 d \left (b^2 c^4+4 a b c^2 d^2-a^2 d^4\right ) \log \left (a+b x^2\right )}{2 a^2 \left (b c^2+a d^2\right )^4} \] Output:

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

Mathematica [A] (verified)

Time = 0.47 (sec) , antiderivative size = 323, normalized size of antiderivative = 0.96 \[ \int \frac {1}{x^2 (c+d x)^3 \left (a+b x^2\right )^2} \, dx=\frac {1}{2} \left (-\frac {2}{a^2 c^3 x}-\frac {d^5}{c^2 \left (b c^2+a d^2\right )^2 (c+d x)^2}-\frac {4 d^5 \left (3 b c^2+a d^2\right )}{\left (b c^3+a c d^2\right )^3 (c+d x)}+\frac {b^2 \left (a^2 d^3-b^2 c^3 x-3 a b c d (c-d x)\right )}{a^2 \left (b c^2+a d^2\right )^3 \left (a+b x^2\right )}-\frac {3 b^{5/2} c \left (b^2 c^4+2 a b c^2 d^2-7 a^2 d^4\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{5/2} \left (b c^2+a d^2\right )^4}-\frac {6 d \log (x)}{a^2 c^4}+\frac {6 \left (7 b^2 c^4 d^5+4 a b c^2 d^7+a^2 d^9\right ) \log (c+d x)}{\left (b c^3+a c d^2\right )^4}+\frac {3 b^2 \left (b^2 c^4 d+4 a b c^2 d^3-a^2 d^5\right ) \log \left (a+b x^2\right )}{a^2 \left (b c^2+a d^2\right )^4}\right ) \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.22 (sec) , antiderivative size = 387, normalized size of antiderivative = 1.15, 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.

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

\(\Big \downarrow \) 615

\(\displaystyle \int \left (\frac {3 d^6 \left (a^2 d^4+4 a b c^2 d^2+7 b^2 c^4\right )}{c^4 (c+d x) \left (a d^2+b c^2\right )^4}+\frac {b^3 \left (3 d x \left (-a^2 d^4+4 a b c^2 d^2+b^2 c^4\right )-c \left (-9 a^2 d^4+4 a b c^2 d^2+b^2 c^4\right )\right )}{a^2 \left (a+b x^2\right ) \left (a d^2+b c^2\right )^4}-\frac {3 d}{a^2 c^4 x}+\frac {1}{a^2 c^3 x^2}+\frac {b^3 \left (d x \left (3 b c^2-a d^2\right )-c \left (b c^2-3 a d^2\right )\right )}{a \left (a+b x^2\right )^2 \left (a d^2+b c^2\right )^3}+\frac {d^6}{c^2 (c+d x)^3 \left (a d^2+b c^2\right )^2}+\frac {2 \left (a d^8+3 b c^2 d^6\right )}{c^3 (c+d x)^2 \left (a d^2+b c^2\right )^3}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {b^{5/2} c \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (b c^2-3 a d^2\right )}{2 a^{5/2} \left (a d^2+b c^2\right )^3}-\frac {b^2 \left (b c x \left (b c^2-3 a d^2\right )+a d \left (3 b c^2-a d^2\right )\right )}{2 a^2 \left (a+b x^2\right ) \left (a d^2+b c^2\right )^3}+\frac {3 b^2 d \left (-a^2 d^4+4 a b c^2 d^2+b^2 c^4\right ) \log \left (a+b x^2\right )}{2 a^2 \left (a d^2+b c^2\right )^4}+\frac {3 d^5 \left (a^2 d^4+4 a b c^2 d^2+7 b^2 c^4\right ) \log (c+d x)}{c^4 \left (a d^2+b c^2\right )^4}-\frac {3 d \log (x)}{a^2 c^4}-\frac {1}{a^2 c^3 x}-\frac {b^{5/2} c \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (-9 a^2 d^4+4 a b c^2 d^2+b^2 c^4\right )}{a^{5/2} \left (a d^2+b c^2\right )^4}-\frac {d^5}{2 c^2 (c+d x)^2 \left (a d^2+b c^2\right )^2}-\frac {2 d^5 \left (a d^2+3 b c^2\right )}{c^3 (c+d x) \left (a d^2+b c^2\right )^3}\)

Input:

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

Output:

-(1/(a^2*c^3*x)) - d^5/(2*c^2*(b*c^2 + a*d^2)^2*(c + d*x)^2) - (2*d^5*(3*b 
*c^2 + a*d^2))/(c^3*(b*c^2 + a*d^2)^3*(c + d*x)) - (b^2*(a*d*(3*b*c^2 - a* 
d^2) + b*c*(b*c^2 - 3*a*d^2)*x))/(2*a^2*(b*c^2 + a*d^2)^3*(a + b*x^2)) - ( 
b^(5/2)*c*(b*c^2 - 3*a*d^2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(5/2)*(b*c^2 
 + a*d^2)^3) - (b^(5/2)*c*(b^2*c^4 + 4*a*b*c^2*d^2 - 9*a^2*d^4)*ArcTan[(Sq 
rt[b]*x)/Sqrt[a]])/(a^(5/2)*(b*c^2 + a*d^2)^4) - (3*d*Log[x])/(a^2*c^4) + 
(3*d^5*(7*b^2*c^4 + 4*a*b*c^2*d^2 + a^2*d^4)*Log[c + d*x])/(c^4*(b*c^2 + a 
*d^2)^4) + (3*b^2*d*(b^2*c^4 + 4*a*b*c^2*d^2 - a^2*d^4)*Log[a + b*x^2])/(2 
*a^2*(b*c^2 + a*d^2)^4)
 

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 = 0.39 (sec) , antiderivative size = 324, normalized size of antiderivative = 0.96

method	result	size
default	\(-\frac {1}{a^{2} c^{3} x}-\frac {3 d \ln \left (x \right )}{a^{2} c^{4}}+\frac {b^{3} \left (\frac {\left (\frac {3}{2} a^{2} c \,d^{4}+a \,c^{3} d^{2} b -\frac {1}{2} c^{5} b^{2}\right ) x +\frac {a d \left (a^{2} d^{4}-2 b \,c^{2} d^{2} a -3 b^{2} c^{4}\right )}{2 b}}{b \,x^{2}+a}+\frac {3 \left (-2 a^{2} d^{5}+8 d^{3} a \,c^{2} b +2 b^{2} c^{4} d \right ) \ln \left (b \,x^{2}+a \right )}{4 b}+\frac {3 \left (7 a^{2} c \,d^{4}-2 a \,c^{3} d^{2} b -c^{5} b^{2}\right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{\left (a \,d^{2}+b \,c^{2}\right )^{4} a^{2}}-\frac {d^{5}}{2 c^{2} \left (a \,d^{2}+b \,c^{2}\right )^{2} \left (d x +c \right )^{2}}-\frac {2 d^{5} \left (a \,d^{2}+3 b \,c^{2}\right )}{c^{3} \left (a \,d^{2}+b \,c^{2}\right )^{3} \left (d x +c \right )}+\frac {3 d^{5} \left (a^{2} d^{4}+4 b \,c^{2} d^{2} a +7 b^{2} c^{4}\right ) \ln \left (d x +c \right )}{c^{4} \left (a \,d^{2}+b \,c^{2}\right )^{4}}\)	\(324\)
risch	\(\text {Expression too large to display}\)	\(1600\)

Input:

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

Output:

-1/a^2/c^3/x-3*d*ln(x)/a^2/c^4+b^3/(a*d^2+b*c^2)^4/a^2*(((3/2*a^2*c*d^4+a* 
c^3*d^2*b-1/2*c^5*b^2)*x+1/2*a*d*(a^2*d^4-2*a*b*c^2*d^2-3*b^2*c^4)/b)/(b*x 
^2+a)+3/4*(-2*a^2*d^5+8*a*b*c^2*d^3+2*b^2*c^4*d)/b*ln(b*x^2+a)+3/2*(7*a^2* 
c*d^4-2*a*b*c^3*d^2-b^2*c^5)/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2)))-1/2*d^5/ 
c^2/(a*d^2+b*c^2)^2/(d*x+c)^2-2*d^5*(a*d^2+3*b*c^2)/c^3/(a*d^2+b*c^2)^3/(d 
*x+c)+3*d^5*(a^2*d^4+4*a*b*c^2*d^2+7*b^2*c^4)*ln(d*x+c)/c^4/(a*d^2+b*c^2)^ 
4
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 857 vs. \(2 (320) = 640\).

Time = 0.14 (sec) , antiderivative size = 857, normalized size of antiderivative = 2.55 \[ \int \frac {1}{x^2 (c+d x)^3 \left (a+b x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

3/2*(b^4*c^4*d + 4*a*b^3*c^2*d^3 - a^2*b^2*d^5)*log(b*x^2 + a)/(a^2*b^4*c^ 
8 + 4*a^3*b^3*c^6*d^2 + 6*a^4*b^2*c^4*d^4 + 4*a^5*b*c^2*d^6 + a^6*d^8) + 3 
*(7*b^2*c^4*d^5 + 4*a*b*c^2*d^7 + a^2*d^9)*log(d*x + c)/(b^4*c^12 + 4*a*b^ 
3*c^10*d^2 + 6*a^2*b^2*c^8*d^4 + 4*a^3*b*c^6*d^6 + a^4*c^4*d^8) - 3/2*(b^5 
*c^5 + 2*a*b^4*c^3*d^2 - 7*a^2*b^3*c*d^4)*arctan(b*x/sqrt(a*b))/((a^2*b^4* 
c^8 + 4*a^3*b^3*c^6*d^2 + 6*a^4*b^2*c^4*d^4 + 4*a^5*b*c^2*d^6 + a^6*d^8)*s 
qrt(a*b)) - 1/2*(2*a*b^3*c^8 + 6*a^2*b^2*c^6*d^2 + 6*a^3*b*c^4*d^4 + 2*a^4 
*c^2*d^6 + 3*(b^4*c^6*d^2 + a*b^3*c^4*d^4 + 6*a^2*b^2*c^2*d^6 + 2*a^3*b*d^ 
8)*x^4 + 3*(2*b^4*c^7*d + 3*a*b^3*c^5*d^3 + 8*a^2*b^2*c^3*d^5 + 3*a^3*b*c* 
d^7)*x^3 + (3*b^4*c^8 + 11*a*b^3*c^6*d^2 + 10*a^2*b^2*c^4*d^4 + 20*a^3*b*c 
^2*d^6 + 6*a^4*d^8)*x^2 + (7*a*b^3*c^7*d + 11*a^2*b^2*c^5*d^3 + 25*a^3*b*c 
^3*d^5 + 9*a^4*c*d^7)*x)/((a^2*b^4*c^9*d^2 + 3*a^3*b^3*c^7*d^4 + 3*a^4*b^2 
*c^5*d^6 + a^5*b*c^3*d^8)*x^5 + 2*(a^2*b^4*c^10*d + 3*a^3*b^3*c^8*d^3 + 3* 
a^4*b^2*c^6*d^5 + a^5*b*c^4*d^7)*x^4 + (a^2*b^4*c^11 + 4*a^3*b^3*c^9*d^2 + 
 6*a^4*b^2*c^7*d^4 + 4*a^5*b*c^5*d^6 + a^6*c^3*d^8)*x^3 + 2*(a^3*b^3*c^10* 
d + 3*a^4*b^2*c^8*d^3 + 3*a^5*b*c^6*d^5 + a^6*c^4*d^7)*x^2 + (a^3*b^3*c^11 
 + 3*a^4*b^2*c^9*d^2 + 3*a^5*b*c^7*d^4 + a^6*c^5*d^6)*x) - 3*d*log(x)/(a^2 
*c^4)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 692 vs. \(2 (320) = 640\).

Time = 0.12 (sec) , antiderivative size = 692, normalized size of antiderivative = 2.06 \[ \int \frac {1}{x^2 (c+d x)^3 \left (a+b x^2\right )^2} \, dx=\frac {3 \, {\left (b^{4} c^{4} d + 4 \, a b^{3} c^{2} d^{3} - a^{2} b^{2} d^{5}\right )} \log \left (b x^{2} + a\right )}{2 \, {\left (a^{2} b^{4} c^{8} + 4 \, a^{3} b^{3} c^{6} d^{2} + 6 \, a^{4} b^{2} c^{4} d^{4} + 4 \, a^{5} b c^{2} d^{6} + a^{6} d^{8}\right )}} + \frac {3 \, {\left (7 \, b^{2} c^{4} d^{6} + 4 \, a b c^{2} d^{8} + a^{2} d^{10}\right )} \log \left ({\left | d x + c \right |}\right )}{b^{4} c^{12} d + 4 \, a b^{3} c^{10} d^{3} + 6 \, a^{2} b^{2} c^{8} d^{5} + 4 \, a^{3} b c^{6} d^{7} + a^{4} c^{4} d^{9}} - \frac {3 \, {\left (b^{5} c^{5} + 2 \, a b^{4} c^{3} d^{2} - 7 \, a^{2} b^{3} c d^{4}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, {\left (a^{2} b^{4} c^{8} + 4 \, a^{3} b^{3} c^{6} d^{2} + 6 \, a^{4} b^{2} c^{4} d^{4} + 4 \, a^{5} b c^{2} d^{6} + a^{6} d^{8}\right )} \sqrt {a b}} - \frac {3 \, d \log \left ({\left | x \right |}\right )}{a^{2} c^{4}} - \frac {2 \, a b^{4} c^{11} + 8 \, a^{2} b^{3} c^{9} d^{2} + 12 \, a^{3} b^{2} c^{7} d^{4} + 8 \, a^{4} b c^{5} d^{6} + 2 \, a^{5} c^{3} d^{8} + 3 \, {\left (b^{5} c^{9} d^{2} + 2 \, a b^{4} c^{7} d^{4} + 7 \, a^{2} b^{3} c^{5} d^{6} + 8 \, a^{3} b^{2} c^{3} d^{8} + 2 \, a^{4} b c d^{10}\right )} x^{4} + 3 \, {\left (2 \, b^{5} c^{10} d + 5 \, a b^{4} c^{8} d^{3} + 11 \, a^{2} b^{3} c^{6} d^{5} + 11 \, a^{3} b^{2} c^{4} d^{7} + 3 \, a^{4} b c^{2} d^{9}\right )} x^{3} + {\left (3 \, b^{5} c^{11} + 14 \, a b^{4} c^{9} d^{2} + 21 \, a^{2} b^{3} c^{7} d^{4} + 30 \, a^{3} b^{2} c^{5} d^{6} + 26 \, a^{4} b c^{3} d^{8} + 6 \, a^{5} c d^{10}\right )} x^{2} + {\left (7 \, a b^{4} c^{10} d + 18 \, a^{2} b^{3} c^{8} d^{3} + 36 \, a^{3} b^{2} c^{6} d^{5} + 34 \, a^{4} b c^{4} d^{7} + 9 \, a^{5} c^{2} d^{9}\right )} x}{2 \, {\left (b c^{2} + a d^{2}\right )}^{4} {\left (b x^{2} + a\right )} {\left (d x + c\right )}^{2} a^{2} c^{4} x} \] Input:

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

Output:

3/2*(b^4*c^4*d + 4*a*b^3*c^2*d^3 - a^2*b^2*d^5)*log(b*x^2 + a)/(a^2*b^4*c^ 
8 + 4*a^3*b^3*c^6*d^2 + 6*a^4*b^2*c^4*d^4 + 4*a^5*b*c^2*d^6 + a^6*d^8) + 3 
*(7*b^2*c^4*d^6 + 4*a*b*c^2*d^8 + a^2*d^10)*log(abs(d*x + c))/(b^4*c^12*d 
+ 4*a*b^3*c^10*d^3 + 6*a^2*b^2*c^8*d^5 + 4*a^3*b*c^6*d^7 + a^4*c^4*d^9) - 
3/2*(b^5*c^5 + 2*a*b^4*c^3*d^2 - 7*a^2*b^3*c*d^4)*arctan(b*x/sqrt(a*b))/(( 
a^2*b^4*c^8 + 4*a^3*b^3*c^6*d^2 + 6*a^4*b^2*c^4*d^4 + 4*a^5*b*c^2*d^6 + a^ 
6*d^8)*sqrt(a*b)) - 3*d*log(abs(x))/(a^2*c^4) - 1/2*(2*a*b^4*c^11 + 8*a^2* 
b^3*c^9*d^2 + 12*a^3*b^2*c^7*d^4 + 8*a^4*b*c^5*d^6 + 2*a^5*c^3*d^8 + 3*(b^ 
5*c^9*d^2 + 2*a*b^4*c^7*d^4 + 7*a^2*b^3*c^5*d^6 + 8*a^3*b^2*c^3*d^8 + 2*a^ 
4*b*c*d^10)*x^4 + 3*(2*b^5*c^10*d + 5*a*b^4*c^8*d^3 + 11*a^2*b^3*c^6*d^5 + 
 11*a^3*b^2*c^4*d^7 + 3*a^4*b*c^2*d^9)*x^3 + (3*b^5*c^11 + 14*a*b^4*c^9*d^ 
2 + 21*a^2*b^3*c^7*d^4 + 30*a^3*b^2*c^5*d^6 + 26*a^4*b*c^3*d^8 + 6*a^5*c*d 
^10)*x^2 + (7*a*b^4*c^10*d + 18*a^2*b^3*c^8*d^3 + 36*a^3*b^2*c^6*d^5 + 34* 
a^4*b*c^4*d^7 + 9*a^5*c^2*d^9)*x)/((b*c^2 + a*d^2)^4*(b*x^2 + a)*(d*x + c) 
^2*a^2*c^4*x)
 

Mupad [B] (verification not implemented)

Time = 9.79 (sec) , antiderivative size = 2394, normalized size of antiderivative = 7.12 \[ \int \frac {1}{x^2 (c+d x)^3 \left (a+b x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

(3*log(a^10*b^26*c^40*x + 256*a^30*b^6*d^40*x - 616435*a*c^24*d^16*(-a^5*b 
^5)^(7/2) - 556696*b*c^26*d^14*(-a^5*b^5)^(7/2) + a^3*b^18*c^40*(-a^5*b^5) 
^(3/2) - 256*a^28*b^3*d^40*(-a^5*b^5)^(1/2) - 7012480*a^11*c^14*d^26*(-a^5 
*b^5)^(5/2) + 39680*a^21*c^4*d^36*(-a^5*b^5)^(3/2) - 717*b^11*c^36*d^4*(-a 
^5*b^5)^(5/2) - 5120*a*b^10*c^34*d^6*(-a^5*b^5)^(5/2) - 8283728*a^10*b*c^1 
6*d^24*(-a^5*b^5)^(5/2) + 214016*a^20*b*c^6*d^34*(-a^5*b^5)^(3/2) - 18338* 
a^2*b^9*c^32*d^8*(-a^5*b^5)^(5/2) - 10956*a^3*b^8*c^30*d^10*(-a^5*b^5)^(5/ 
2) + 155886*a^4*b^7*c^28*d^12*(-a^5*b^5)^(5/2) - 815250*a^7*b^4*c^22*d^18* 
(-a^5*b^5)^(5/2) - 3945441*a^8*b^3*c^20*d^20*(-a^5*b^5)^(5/2) - 7113064*a^ 
9*b^2*c^18*d^22*(-a^5*b^5)^(5/2) + 50*a^4*b^17*c^38*d^2*(-a^5*b^5)^(3/2) + 
 4489920*a^17*b^4*c^12*d^28*(-a^5*b^5)^(3/2) + 2187552*a^18*b^3*c^10*d^30* 
(-a^5*b^5)^(3/2) + 800640*a^19*b^2*c^8*d^32*(-a^5*b^5)^(3/2) - 4608*a^27*b 
^4*c^2*d^38*(-a^5*b^5)^(1/2) + 50*a^11*b^25*c^38*d^2*x + 717*a^12*b^24*c^3 
6*d^4*x + 5120*a^13*b^23*c^34*d^6*x + 18338*a^14*b^22*c^32*d^8*x + 10956*a 
^15*b^21*c^30*d^10*x - 155886*a^16*b^20*c^28*d^12*x - 556696*a^17*b^19*c^2 
6*d^14*x - 616435*a^18*b^18*c^24*d^16*x + 815250*a^19*b^17*c^22*d^18*x + 3 
945441*a^20*b^16*c^20*d^20*x + 7113064*a^21*b^15*c^18*d^22*x + 8283728*a^2 
2*b^14*c^16*d^24*x + 7012480*a^23*b^13*c^14*d^26*x + 4489920*a^24*b^12*c^1 
2*d^28*x + 2187552*a^25*b^11*c^10*d^30*x + 800640*a^26*b^10*c^8*d^32*x + 2 
14016*a^27*b^9*c^6*d^34*x + 39680*a^28*b^8*c^4*d^36*x + 4608*a^29*b^7*c...
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 2399, normalized size of antiderivative = 7.14 \[ \int \frac {1}{x^2 (c+d x)^3 \left (a+b x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

(42*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**3*b**2*c**7*d**4*x + 
84*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**3*b**2*c**6*d**5*x**2 
+ 42*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**3*b**2*c**5*d**6*x** 
3 - 12*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**2*b**3*c**9*d**2*x 
 - 24*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**2*b**3*c**8*d**3*x* 
*2 + 30*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**2*b**3*c**7*d**4* 
x**3 + 84*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**2*b**3*c**6*d** 
5*x**4 + 42*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**2*b**3*c**5*d 
**6*x**5 - 6*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a*b**4*c**11*x 
- 12*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a*b**4*c**10*d*x**2 - 1 
8*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a*b**4*c**9*d**2*x**3 - 24 
*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a*b**4*c**8*d**3*x**4 - 12* 
sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a*b**4*c**7*d**4*x**5 - 6*sq 
rt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*b**5*c**11*x**3 - 12*sqrt(b)*s 
qrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*b**5*c**10*d*x**4 - 6*sqrt(b)*sqrt(a) 
*atan((b*x)/(sqrt(b)*sqrt(a)))*b**5*c**9*d**2*x**5 - 6*log(a + b*x**2)*a** 
4*b**2*c**6*d**5*x - 12*log(a + b*x**2)*a**4*b**2*c**5*d**6*x**2 - 6*log(a 
 + b*x**2)*a**4*b**2*c**4*d**7*x**3 + 24*log(a + b*x**2)*a**3*b**3*c**8*d* 
*3*x + 48*log(a + b*x**2)*a**3*b**3*c**7*d**4*x**2 + 18*log(a + b*x**2)*a* 
*3*b**3*c**6*d**5*x**3 - 12*log(a + b*x**2)*a**3*b**3*c**5*d**6*x**4 - ...