\(\int \frac {1}{x^2 (c+d x)^3 (a+b x^2)^3} \, dx\) [275]

Optimal result

Integrand size = 20, antiderivative size = 459 \[ \int \frac {1}{x^2 (c+d x)^3 \left (a+b x^2\right )^3} \, dx=-\frac {1}{a^3 c^3 x}-\frac {d^7}{2 c^2 \left (b c^2+a d^2\right )^3 (c+d x)^2}-\frac {2 d^7 \left (4 b c^2+a d^2\right )}{c^3 \left (b c^2+a d^2\right )^4 (c+d x)}-\frac {b^2 \left (a d \left (3 b c^2-a d^2\right )+b c \left (b c^2-3 a d^2\right ) x\right )}{4 a^2 \left (b c^2+a d^2\right )^3 \left (a+b x^2\right )^2}-\frac {b^2 \left (12 a d \left (b^2 c^4+4 a b c^2 d^2-a^2 d^4\right )+b c \left (7 b^2 c^4+10 a b c^2 d^2-45 a^2 d^4\right ) x\right )}{8 a^3 \left (b c^2+a d^2\right )^4 \left (a+b x^2\right )}-\frac {3 b^{5/2} c \left (5 b^3 c^6+19 a b^2 c^4 d^2+15 a^2 b c^2 d^4-63 a^3 d^6\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{7/2} \left (b c^2+a d^2\right )^5}-\frac {3 d \log (x)}{a^3 c^4}+\frac {3 d^7 \left (12 b^2 c^4+5 a b c^2 d^2+a^2 d^4\right ) \log (c+d x)}{c^4 \left (b c^2+a d^2\right )^5}+\frac {3 b^2 d \left (b^3 c^6+5 a b^2 c^4 d^2+10 a^2 b c^2 d^4-2 a^3 d^6\right ) \log \left (a+b x^2\right )}{2 a^3 \left (b c^2+a d^2\right )^5} \] Output:

-1/a^3/c^3/x-1/2*d^7/c^2/(a*d^2+b*c^2)^3/(d*x+c)^2-2*d^7*(a*d^2+4*b*c^2)/c 
^3/(a*d^2+b*c^2)^4/(d*x+c)-1/4*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)^2-1/8*b^2*(12*a*d*(-a^2*d^4+4*a*b*c^2 
*d^2+b^2*c^4)+b*c*(-45*a^2*d^4+10*a*b*c^2*d^2+7*b^2*c^4)*x)/a^3/(a*d^2+b*c 
^2)^4/(b*x^2+a)-3/8*b^(5/2)*c*(-63*a^3*d^6+15*a^2*b*c^2*d^4+19*a*b^2*c^4*d 
^2+5*b^3*c^6)*arctan(b^(1/2)*x/a^(1/2))/a^(7/2)/(a*d^2+b*c^2)^5-3*d*ln(x)/ 
a^3/c^4+3*d^7*(a^2*d^4+5*a*b*c^2*d^2+12*b^2*c^4)*ln(d*x+c)/c^4/(a*d^2+b*c^ 
2)^5+3/2*b^2*d*(-2*a^3*d^6+10*a^2*b*c^2*d^4+5*a*b^2*c^4*d^2+b^3*c^6)*ln(b* 
x^2+a)/a^3/(a*d^2+b*c^2)^5
 

Mathematica [A] (verified)

Time = 0.74 (sec) , antiderivative size = 441, normalized size of antiderivative = 0.96 \[ \int \frac {1}{x^2 (c+d x)^3 \left (a+b x^2\right )^3} \, dx=\frac {1}{8} \left (-\frac {8}{a^3 c^3 x}-\frac {4 d^7}{c^2 \left (b c^2+a d^2\right )^3 (c+d x)^2}-\frac {16 d^7 \left (4 b c^2+a d^2\right )}{c^3 \left (b c^2+a d^2\right )^4 (c+d x)}+\frac {2 b^2 \left (a^2 d^3-b^2 c^3 x-3 a b c d (c-d x)\right )}{a^2 \left (b c^2+a d^2\right )^3 \left (a+b x^2\right )^2}-\frac {b^2 \left (-12 a^3 d^5+7 b^3 c^5 x+3 a^2 b c d^3 (16 c-15 d x)+2 a b^2 c^3 d (6 c+5 d x)\right )}{a^3 \left (b c^2+a d^2\right )^4 \left (a+b x^2\right )}-\frac {3 b^{5/2} c \left (5 b^3 c^6+19 a b^2 c^4 d^2+15 a^2 b c^2 d^4-63 a^3 d^6\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{7/2} \left (b c^2+a d^2\right )^5}-\frac {24 d \log (x)}{a^3 c^4}+\frac {24 d^7 \left (12 b^2 c^4+5 a b c^2 d^2+a^2 d^4\right ) \log (c+d x)}{c^4 \left (b c^2+a d^2\right )^5}+\frac {12 b^2 \left (b^3 c^6 d+5 a b^2 c^4 d^3+10 a^2 b c^2 d^5-2 a^3 d^7\right ) \log \left (a+b x^2\right )}{a^3 \left (b c^2+a d^2\right )^5}\right ) \] Input:

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

Output:

(-8/(a^3*c^3*x) - (4*d^7)/(c^2*(b*c^2 + a*d^2)^3*(c + d*x)^2) - (16*d^7*(4 
*b*c^2 + a*d^2))/(c^3*(b*c^2 + a*d^2)^4*(c + d*x)) + (2*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)^2) - (b^ 
2*(-12*a^3*d^5 + 7*b^3*c^5*x + 3*a^2*b*c*d^3*(16*c - 15*d*x) + 2*a*b^2*c^3 
*d*(6*c + 5*d*x)))/(a^3*(b*c^2 + a*d^2)^4*(a + b*x^2)) - (3*b^(5/2)*c*(5*b 
^3*c^6 + 19*a*b^2*c^4*d^2 + 15*a^2*b*c^2*d^4 - 63*a^3*d^6)*ArcTan[(Sqrt[b] 
*x)/Sqrt[a]])/(a^(7/2)*(b*c^2 + a*d^2)^5) - (24*d*Log[x])/(a^3*c^4) + (24* 
d^7*(12*b^2*c^4 + 5*a*b*c^2*d^2 + a^2*d^4)*Log[c + d*x])/(c^4*(b*c^2 + a*d 
^2)^5) + (12*b^2*(b^3*c^6*d + 5*a*b^2*c^4*d^3 + 10*a^2*b*c^2*d^5 - 2*a^3*d 
^7)*Log[a + b*x^2])/(a^3*(b*c^2 + a*d^2)^5))/8
 

Rubi [A] (verified)

Time = 1.83 (sec) , antiderivative size = 621, normalized size of antiderivative = 1.35, 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.

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

\(\Big \downarrow \) 615

\(\displaystyle \int \left (-\frac {3 d}{a^3 c^4 x}+\frac {1}{a^3 c^3 x^2}+\frac {3 d^8 \left (a^2 d^4+5 a b c^2 d^2+12 b^2 c^4\right )}{c^4 (c+d x) \left (a d^2+b c^2\right )^5}+\frac {b^3 \left (3 d x \left (-a^2 d^4+4 a b c^2 d^2+b^2 c^4\right )-c \left (-9 a^2 d^4+4 a b c^2 d^2+b^2 c^4\right )\right )}{a^2 \left (a+b x^2\right )^2 \left (a d^2+b c^2\right )^4}+\frac {b^3 \left (3 d x \left (-2 a^3 d^6+10 a^2 b c^2 d^4+5 a b^2 c^4 d^2+b^3 c^6\right )-c \left (-18 a^3 d^6+10 a^2 b c^2 d^4+5 a b^2 c^4 d^2+b^3 c^6\right )\right )}{a^3 \left (a+b x^2\right ) \left (a d^2+b c^2\right )^5}+\frac {b^3 \left (d x \left (3 b c^2-a d^2\right )-c \left (b c^2-3 a d^2\right )\right )}{a \left (a+b x^2\right )^3 \left (a d^2+b c^2\right )^3}+\frac {d^8}{c^2 (c+d x)^3 \left (a d^2+b c^2\right )^3}+\frac {2 d^8 \left (a d^2+4 b c^2\right )}{c^3 (c+d x)^2 \left (a d^2+b c^2\right )^4}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {3 b^{5/2} c \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (b c^2-3 a d^2\right )}{8 a^{7/2} \left (a d^2+b c^2\right )^3}-\frac {3 b^3 c x \left (b c^2-3 a d^2\right )}{8 a^3 \left (a+b x^2\right ) \left (a d^2+b c^2\right )^3}-\frac {3 d \log (x)}{a^3 c^4}-\frac {1}{a^3 c^3 x}-\frac {b^2 \left (b c x \left (b c^2-3 a d^2\right )+a d \left (3 b c^2-a d^2\right )\right )}{4 a^2 \left (a+b x^2\right )^2 \left (a d^2+b c^2\right )^3}+\frac {3 d^7 \left (a^2 d^4+5 a b c^2 d^2+12 b^2 c^4\right ) \log (c+d x)}{c^4 \left (a d^2+b c^2\right )^5}-\frac {b^{5/2} c \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (-9 a^2 d^4+4 a b c^2 d^2+b^2 c^4\right )}{2 a^{7/2} \left (a d^2+b c^2\right )^4}-\frac {b^2 \left (b c x \left (-9 a^2 d^4+4 a b c^2 d^2+b^2 c^4\right )+3 a d \left (-a^2 d^4+4 a b c^2 d^2+b^2 c^4\right )\right )}{2 a^3 \left (a+b x^2\right ) \left (a d^2+b c^2\right )^4}+\frac {3 b^2 d \left (-2 a^3 d^6+10 a^2 b c^2 d^4+5 a b^2 c^4 d^2+b^3 c^6\right ) \log \left (a+b x^2\right )}{2 a^3 \left (a d^2+b c^2\right )^5}-\frac {b^{5/2} c \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (-18 a^3 d^6+10 a^2 b c^2 d^4+5 a b^2 c^4 d^2+b^3 c^6\right )}{a^{7/2} \left (a d^2+b c^2\right )^5}-\frac {d^7}{2 c^2 (c+d x)^2 \left (a d^2+b c^2\right )^3}-\frac {2 d^7 \left (a d^2+4 b c^2\right )}{c^3 (c+d x) \left (a d^2+b c^2\right )^4}\)

Input:

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

Output:

-(1/(a^3*c^3*x)) - d^7/(2*c^2*(b*c^2 + a*d^2)^3*(c + d*x)^2) - (2*d^7*(4*b 
*c^2 + a*d^2))/(c^3*(b*c^2 + a*d^2)^4*(c + d*x)) - (b^2*(a*d*(3*b*c^2 - a* 
d^2) + b*c*(b*c^2 - 3*a*d^2)*x))/(4*a^2*(b*c^2 + a*d^2)^3*(a + b*x^2)^2) - 
 (3*b^3*c*(b*c^2 - 3*a*d^2)*x)/(8*a^3*(b*c^2 + a*d^2)^3*(a + b*x^2)) - (b^ 
2*(3*a*d*(b^2*c^4 + 4*a*b*c^2*d^2 - a^2*d^4) + b*c*(b^2*c^4 + 4*a*b*c^2*d^ 
2 - 9*a^2*d^4)*x))/(2*a^3*(b*c^2 + a*d^2)^4*(a + b*x^2)) - (3*b^(5/2)*c*(b 
*c^2 - 3*a*d^2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(7/2)*(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[(Sqrt[b]*x)/Sqr 
t[a]])/(2*a^(7/2)*(b*c^2 + a*d^2)^4) - (b^(5/2)*c*(b^3*c^6 + 5*a*b^2*c^4*d 
^2 + 10*a^2*b*c^2*d^4 - 18*a^3*d^6)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(a^(7/2)* 
(b*c^2 + a*d^2)^5) - (3*d*Log[x])/(a^3*c^4) + (3*d^7*(12*b^2*c^4 + 5*a*b*c 
^2*d^2 + a^2*d^4)*Log[c + d*x])/(c^4*(b*c^2 + a*d^2)^5) + (3*b^2*d*(b^3*c^ 
6 + 5*a*b^2*c^4*d^2 + 10*a^2*b*c^2*d^4 - 2*a^3*d^6)*Log[a + b*x^2])/(2*a^3 
*(b*c^2 + a*d^2)^5)
 

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 = 0.41 (sec) , antiderivative size = 484, normalized size of antiderivative = 1.05

method	result	size
default	\(-\frac {1}{a^{3} c^{3} x}-\frac {3 d \ln \left (x \right )}{a^{3} c^{4}}+\frac {b^{3} \left (\frac {\left (\frac {45}{8} d^{6} c \,a^{3} b +\frac {35}{8} a^{2} c^{3} d^{4} b^{2}-\frac {17}{8} a \,c^{5} d^{2} b^{3}-\frac {7}{8} c^{7} b^{4}\right ) x^{3}+\left (\frac {3}{2} a^{4} d^{7}-\frac {9}{2} a^{3} b \,c^{2} d^{5}-\frac {15}{2} a^{2} b^{2} c^{4} d^{3}-\frac {3}{2} a \,b^{3} c^{6} d \right ) x^{2}+\frac {3 a c \left (17 a^{3} d^{6}+15 a^{2} b \,c^{2} d^{4}-5 a \,b^{2} c^{4} d^{2}-3 b^{3} c^{6}\right ) x}{8}+\frac {a^{2} d \left (7 a^{3} d^{6}-19 a^{2} b \,c^{2} d^{4}-35 a \,b^{2} c^{4} d^{2}-9 b^{3} c^{6}\right )}{4 b}}{\left (b \,x^{2}+a \right )^{2}}+\frac {3 \left (-16 a^{3} d^{7}+80 a^{2} b \,c^{2} d^{5}+40 a \,b^{2} c^{4} d^{3}+8 b^{3} c^{6} d \right ) \ln \left (b \,x^{2}+a \right )}{16 b}+\frac {3 \left (63 d^{6} c \,a^{3}-15 a^{2} b \,c^{3} d^{4}-19 a \,b^{2} c^{5} d^{2}-5 b^{3} c^{7}\right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \sqrt {a b}}\right )}{\left (a \,d^{2}+b \,c^{2}\right )^{5} a^{3}}-\frac {d^{7}}{2 c^{2} \left (a \,d^{2}+b \,c^{2}\right )^{3} \left (d x +c \right )^{2}}-\frac {2 d^{7} \left (a \,d^{2}+4 b \,c^{2}\right )}{c^{3} \left (a \,d^{2}+b \,c^{2}\right )^{4} \left (d x +c \right )}+\frac {3 d^{7} \left (a^{2} d^{4}+5 b \,c^{2} d^{2} a +12 b^{2} c^{4}\right ) \ln \left (d x +c \right )}{c^{4} \left (a \,d^{2}+b \,c^{2}\right )^{5}}\)	\(484\)
risch	\(\text {Expression too large to display}\)	\(2291\)

Input:

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

Output:

-1/a^3/c^3/x-3*d*ln(x)/a^3/c^4+b^3/(a*d^2+b*c^2)^5/a^3*(((45/8*d^6*c*a^3*b 
+35/8*a^2*c^3*d^4*b^2-17/8*a*c^5*d^2*b^3-7/8*c^7*b^4)*x^3+(3/2*a^4*d^7-9/2 
*a^3*b*c^2*d^5-15/2*a^2*b^2*c^4*d^3-3/2*a*b^3*c^6*d)*x^2+3/8*a*c*(17*a^3*d 
^6+15*a^2*b*c^2*d^4-5*a*b^2*c^4*d^2-3*b^3*c^6)*x+1/4*a^2*d*(7*a^3*d^6-19*a 
^2*b*c^2*d^4-35*a*b^2*c^4*d^2-9*b^3*c^6)/b)/(b*x^2+a)^2+3/16*(-16*a^3*d^7+ 
80*a^2*b*c^2*d^5+40*a*b^2*c^4*d^3+8*b^3*c^6*d)/b*ln(b*x^2+a)+3/8*(63*a^3*c 
*d^6-15*a^2*b*c^3*d^4-19*a*b^2*c^5*d^2-5*b^3*c^7)/(a*b)^(1/2)*arctan(b*x/( 
a*b)^(1/2)))-1/2*d^7/c^2/(a*d^2+b*c^2)^3/(d*x+c)^2-2*d^7*(a*d^2+4*b*c^2)/c 
^3/(a*d^2+b*c^2)^4/(d*x+c)+3*d^7*(a^2*d^4+5*a*b*c^2*d^2+12*b^2*c^4)*ln(d*x 
+c)/c^4/(a*d^2+b*c^2)^5
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F(-1)]

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

integrate(1/x**2/(d*x+c)**3/(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. 1383 vs. \(2 (441) = 882\).

Time = 0.16 (sec) , antiderivative size = 1383, normalized size of antiderivative = 3.01 \[ \int \frac {1}{x^2 (c+d x)^3 \left (a+b x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

3/2*(b^5*c^6*d + 5*a*b^4*c^4*d^3 + 10*a^2*b^3*c^2*d^5 - 2*a^3*b^2*d^7)*log 
(b*x^2 + a)/(a^3*b^5*c^10 + 5*a^4*b^4*c^8*d^2 + 10*a^5*b^3*c^6*d^4 + 10*a^ 
6*b^2*c^4*d^6 + 5*a^7*b*c^2*d^8 + a^8*d^10) + 3*(12*b^2*c^4*d^7 + 5*a*b*c^ 
2*d^9 + a^2*d^11)*log(d*x + c)/(b^5*c^14 + 5*a*b^4*c^12*d^2 + 10*a^2*b^3*c 
^10*d^4 + 10*a^3*b^2*c^8*d^6 + 5*a^4*b*c^6*d^8 + a^5*c^4*d^10) - 3/8*(5*b^ 
6*c^7 + 19*a*b^5*c^5*d^2 + 15*a^2*b^4*c^3*d^4 - 63*a^3*b^3*c*d^6)*arctan(b 
*x/sqrt(a*b))/((a^3*b^5*c^10 + 5*a^4*b^4*c^8*d^2 + 10*a^5*b^3*c^6*d^4 + 10 
*a^6*b^2*c^4*d^6 + 5*a^7*b*c^2*d^8 + a^8*d^10)*sqrt(a*b)) - 1/8*(8*a^2*b^4 
*c^10 + 32*a^3*b^3*c^8*d^2 + 48*a^4*b^2*c^6*d^4 + 32*a^5*b*c^4*d^6 + 8*a^6 
*c^2*d^8 + 3*(5*b^6*c^8*d^2 + 14*a*b^5*c^6*d^4 + a^2*b^4*c^4*d^6 + 32*a^3* 
b^3*c^2*d^8 + 8*a^4*b^2*d^10)*x^6 + 6*(5*b^6*c^9*d + 16*a*b^5*c^7*d^3 + 9* 
a^2*b^4*c^5*d^5 + 20*a^3*b^3*c^3*d^7 + 6*a^4*b^2*c*d^9)*x^5 + (15*b^6*c^10 
 + 91*a*b^5*c^8*d^2 + 169*a^2*b^4*c^6*d^4 + 53*a^3*b^3*c^4*d^6 + 200*a^4*b 
^2*c^2*d^8 + 48*a^5*b*d^10)*x^4 + 2*(31*a*b^5*c^9*d + 103*a^2*b^4*c^7*d^3 
+ 65*a^3*b^3*c^5*d^5 + 125*a^4*b^2*c^3*d^7 + 36*a^5*b*c*d^9)*x^3 + (25*a*b 
^5*c^10 + 114*a^2*b^4*c^8*d^2 + 181*a^3*b^3*c^6*d^4 + 84*a^4*b^2*c^4*d^6 + 
 112*a^5*b*c^2*d^8 + 24*a^6*d^10)*x^2 + 2*(17*a^2*b^4*c^9*d + 58*a^3*b^3*c 
^7*d^3 + 41*a^4*b^2*c^5*d^5 + 66*a^5*b*c^3*d^7 + 18*a^6*c*d^9)*x)/((a^3*b^ 
6*c^11*d^2 + 4*a^4*b^5*c^9*d^4 + 6*a^5*b^4*c^7*d^6 + 4*a^6*b^3*c^5*d^8 + a 
^7*b^2*c^3*d^10)*x^7 + 2*(a^3*b^6*c^12*d + 4*a^4*b^5*c^10*d^3 + 6*a^5*b...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 922 vs. \(2 (441) = 882\).

Time = 0.14 (sec) , antiderivative size = 922, normalized size of antiderivative = 2.01 \[ \int \frac {1}{x^2 (c+d x)^3 \left (a+b x^2\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

3/2*(b^5*c^6*d + 5*a*b^4*c^4*d^3 + 10*a^2*b^3*c^2*d^5 - 2*a^3*b^2*d^7)*log 
(b*x^2 + a)/(a^3*b^5*c^10 + 5*a^4*b^4*c^8*d^2 + 10*a^5*b^3*c^6*d^4 + 10*a^ 
6*b^2*c^4*d^6 + 5*a^7*b*c^2*d^8 + a^8*d^10) + 3*(12*b^2*c^4*d^8 + 5*a*b*c^ 
2*d^10 + a^2*d^12)*log(abs(d*x + c))/(b^5*c^14*d + 5*a*b^4*c^12*d^3 + 10*a 
^2*b^3*c^10*d^5 + 10*a^3*b^2*c^8*d^7 + 5*a^4*b*c^6*d^9 + a^5*c^4*d^11) - 3 
/8*(5*b^6*c^7 + 19*a*b^5*c^5*d^2 + 15*a^2*b^4*c^3*d^4 - 63*a^3*b^3*c*d^6)* 
arctan(b*x/sqrt(a*b))/((a^3*b^5*c^10 + 5*a^4*b^4*c^8*d^2 + 10*a^5*b^3*c^6* 
d^4 + 10*a^6*b^2*c^4*d^6 + 5*a^7*b*c^2*d^8 + a^8*d^10)*sqrt(a*b)) - 3*d*lo 
g(abs(x))/(a^3*c^4) - 1/8*(8*a^2*b^4*c^11 + 32*a^3*b^3*c^9*d^2 + 48*a^4*b^ 
2*c^7*d^4 + 32*a^5*b*c^5*d^6 + 8*a^6*c^3*d^8 + 3*(5*b^6*c^9*d^2 + 14*a*b^5 
*c^7*d^4 + a^2*b^4*c^5*d^6 + 32*a^3*b^3*c^3*d^8 + 8*a^4*b^2*c*d^10)*x^6 + 
6*(5*b^6*c^10*d + 16*a*b^5*c^8*d^3 + 9*a^2*b^4*c^6*d^5 + 20*a^3*b^3*c^4*d^ 
7 + 6*a^4*b^2*c^2*d^9)*x^5 + (15*b^6*c^11 + 91*a*b^5*c^9*d^2 + 169*a^2*b^4 
*c^7*d^4 + 53*a^3*b^3*c^5*d^6 + 200*a^4*b^2*c^3*d^8 + 48*a^5*b*c*d^10)*x^4 
 + 2*(31*a*b^5*c^10*d + 103*a^2*b^4*c^8*d^3 + 65*a^3*b^3*c^6*d^5 + 125*a^4 
*b^2*c^4*d^7 + 36*a^5*b*c^2*d^9)*x^3 + (25*a*b^5*c^11 + 114*a^2*b^4*c^9*d^ 
2 + 181*a^3*b^3*c^7*d^4 + 84*a^4*b^2*c^5*d^6 + 112*a^5*b*c^3*d^8 + 24*a^6* 
c*d^10)*x^2 + 2*(17*a^2*b^4*c^10*d + 58*a^3*b^3*c^8*d^3 + 41*a^4*b^2*c^6*d 
^5 + 66*a^5*b*c^4*d^7 + 18*a^6*c^2*d^9)*x)/((b*c^2 + a*d^2)^4*(b*x^2 + a)^ 
2*(d*x + c)^2*a^3*c^4*x)
 

Mupad [B] (verification not implemented)

Time = 12.13 (sec) , antiderivative size = 3238, normalized size of antiderivative = 7.05 \[ \int \frac {1}{x^2 (c+d x)^3 \left (a+b x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

(3*log(625*a^14*b^30*c^48*x + 65536*a^38*b^6*d^48*x + 625*a^4*b^22*c^48*(- 
a^7*b^5)^(3/2) - 65536*a^35*b^3*d^48*(-a^7*b^5)^(1/2) - 4917680373*a^2*c^2 
4*d^24*(-a^7*b^5)^(7/2) - 9370756608*a^14*c^14*d^34*(-a^7*b^5)^(5/2) + 154 
00960*a^26*c^4*d^44*(-a^7*b^5)^(3/2) - 4613286135*b^2*c^28*d^20*(-a^7*b^5) 
^(7/2) - 3977228*b^14*c^42*d^6*(-a^7*b^5)^(5/2) - 24290745*a*b^13*c^40*d^8 
*(-a^7*b^5)^(5/2) - 14841962496*a^13*b*c^16*d^32*(-a^7*b^5)^(5/2) + 104071 
168*a^25*b*c^6*d^42*(-a^7*b^5)^(3/2) - 104240162*a^2*b^12*c^38*d^10*(-a^7* 
b^5)^(5/2) - 314918267*a^3*b^11*c^36*d^12*(-a^7*b^5)^(5/2) - 618576168*a^4 
*b^10*c^34*d^14*(-a^7*b^5)^(5/2) - 505303755*a^5*b^9*c^32*d^16*(-a^7*b^5)^ 
(5/2) + 1116815390*a^6*b^8*c^30*d^18*(-a^7*b^5)^(5/2) - 3822520126*a^10*b^ 
4*c^22*d^26*(-a^7*b^5)^(5/2) - 13385963265*a^11*b^3*c^20*d^28*(-a^7*b^5)^( 
5/2) - 17386060800*a^12*b^2*c^18*d^30*(-a^7*b^5)^(5/2) + 26750*a^5*b^21*c^ 
46*d^2*(-a^7*b^5)^(3/2) + 431739*a^6*b^20*c^44*d^4*(-a^7*b^5)^(3/2) + 4558 
181376*a^22*b^4*c^12*d^36*(-a^7*b^5)^(3/2) + 1716322816*a^23*b^3*c^10*d^38 
*(-a^7*b^5)^(3/2) + 491978752*a^24*b^2*c^8*d^40*(-a^7*b^5)^(3/2) - 1441792 
*a^34*b^4*c^2*d^46*(-a^7*b^5)^(1/2) + 26750*a^15*b^29*c^46*d^2*x + 431739* 
a^16*b^28*c^44*d^4*x + 3977228*a^17*b^27*c^42*d^6*x + 24290745*a^18*b^26*c 
^40*d^8*x + 104240162*a^19*b^25*c^38*d^10*x + 314918267*a^20*b^24*c^36*d^1 
2*x + 618576168*a^21*b^23*c^34*d^14*x + 505303755*a^22*b^22*c^32*d^16*x - 
1116815390*a^23*b^21*c^30*d^18*x - 4613286135*a^24*b^20*c^28*d^20*x - 7...
 

Reduce [B] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 4273, normalized size of antiderivative = 9.31 \[ \int \frac {1}{x^2 (c+d x)^3 \left (a+b x^2\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

(378*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**5*b**2*c**7*d**6*x + 
 756*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**5*b**2*c**6*d**7*x** 
2 + 378*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**5*b**2*c**5*d**8* 
x**3 - 90*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**4*b**3*c**9*d** 
4*x - 180*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**4*b**3*c**8*d** 
5*x**2 + 666*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**4*b**3*c**7* 
d**6*x**3 + 1512*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**4*b**3*c 
**6*d**7*x**4 + 756*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**4*b** 
3*c**5*d**8*x**5 - 114*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**3* 
b**4*c**11*d**2*x - 228*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**3 
*b**4*c**10*d**3*x**2 - 294*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))* 
a**3*b**4*c**9*d**4*x**3 - 360*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a) 
))*a**3*b**4*c**8*d**5*x**4 + 198*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt 
(a)))*a**3*b**4*c**7*d**6*x**5 + 756*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*s 
qrt(a)))*a**3*b**4*c**6*d**7*x**6 + 378*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b 
)*sqrt(a)))*a**3*b**4*c**5*d**8*x**7 - 30*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt 
(b)*sqrt(a)))*a**2*b**5*c**13*x - 60*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*s 
qrt(a)))*a**2*b**5*c**12*d*x**2 - 258*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)* 
sqrt(a)))*a**2*b**5*c**11*d**2*x**3 - 456*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt 
(b)*sqrt(a)))*a**2*b**5*c**10*d**3*x**4 - 318*sqrt(b)*sqrt(a)*atan((b*x...