\(\int \frac {x}{(c+d x)^2 (a+b x^2)^2} \, dx\) [203]

Optimal result

Integrand size = 18, antiderivative size = 200 \[ \int \frac {x}{(c+d x)^2 \left (a+b x^2\right )^2} \, dx=\frac {c d^2}{\left (b c^2+a d^2\right )^2 (c+d x)}-\frac {b c^2-a d^2-2 b c d x}{2 \left (b c^2+a d^2\right )^2 \left (a+b x^2\right )}-\frac {\sqrt {b} c d \left (b c^2-3 a d^2\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} \left (b c^2+a d^2\right )^3}-\frac {d^2 \left (3 b c^2-a d^2\right ) \log (c+d x)}{\left (b c^2+a d^2\right )^3}+\frac {d^2 \left (3 b c^2-a d^2\right ) \log \left (a+b x^2\right )}{2 \left (b c^2+a d^2\right )^3} \] Output:

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

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.84 \[ \int \frac {x}{(c+d x)^2 \left (a+b x^2\right )^2} \, dx=\frac {\frac {2 c d^2 \left (b c^2+a d^2\right )}{c+d x}+\frac {\left (b c^2+a d^2\right ) \left (a d^2-b c (c-2 d x)\right )}{a+b x^2}+\frac {2 \sqrt {b} c d \left (-b c^2+3 a d^2\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a}}+2 \left (-3 b c^2 d^2+a d^4\right ) \log (c+d x)+\left (3 b c^2 d^2-a d^4\right ) \log \left (a+b x^2\right )}{2 \left (b c^2+a d^2\right )^3} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.65 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.02, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {593, 27, 657, 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.

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

\(\Big \downarrow \) 593

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 657

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

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 593

Int[(x_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> 
 Simp[(c + d*x)^(n + 1)*(c - d*x)*((a + b*x^2)^(p + 1)/(2*(p + 1)*(b*c^2 + 
a*d^2))), x] - Simp[d/(2*(p + 1)*(b*c^2 + a*d^2))   Int[(c + d*x)^n*(a + b* 
x^2)^(p + 1)*(c*n - d*(n + 2*p + 4)*x), x], x] /; FreeQ[{a, b, c, d, n}, x] 
 && LtQ[p, -1] && NeQ[b*c^2 + a*d^2, 0]
 

rule 657

Int[(((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.))/((a_) + (c_.)*( 
x_)^2), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*((f + g*x)^n/(a + c*x^ 
2)), x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && IntegersQ[n]
 

rule 2009

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

Maple [A] (verified)

Time = 0.33 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.92

method	result	size
default	\(\frac {b \left (\frac {\left (a c \,d^{3}+b \,c^{3} d \right ) x +\frac {a^{2} d^{4}-b^{2} c^{4}}{2 b}}{b \,x^{2}+a}+d \left (\frac {\left (-a \,d^{3}+3 b \,c^{2} d \right ) \ln \left (b \,x^{2}+a \right )}{2 b}+\frac {\left (3 a \,d^{2} c -b \,c^{3}\right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}}\right )\right )}{\left (a \,d^{2}+b \,c^{2}\right )^{3}}+\frac {d^{2} \left (a \,d^{2}-3 b \,c^{2}\right ) \ln \left (d x +c \right )}{\left (a \,d^{2}+b \,c^{2}\right )^{3}}+\frac {c \,d^{2}}{\left (a \,d^{2}+b \,c^{2}\right )^{2} \left (d x +c \right )}\)	\(184\)
risch	\(\text {Expression too large to display}\)	\(2302\)

Input:

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

Output:

b/(a*d^2+b*c^2)^3*(((a*c*d^3+b*c^3*d)*x+1/2*(a^2*d^4-b^2*c^4)/b)/(b*x^2+a) 
+d*(1/2*(-a*d^3+3*b*c^2*d)/b*ln(b*x^2+a)+(3*a*c*d^2-b*c^3)/(a*b)^(1/2)*arc 
tan(b*x/(a*b)^(1/2))))+d^2*(a*d^2-3*b*c^2)/(a*d^2+b*c^2)^3*ln(d*x+c)+c*d^2 
/(a*d^2+b*c^2)^2/(d*x+c)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 564 vs. \(2 (188) = 376\).

Time = 0.71 (sec) , antiderivative size = 1151, normalized size of antiderivative = 5.76 \[ \int \frac {x}{(c+d x)^2 \left (a+b x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

[-1/2*(b^2*c^5 - 2*a*b*c^3*d^2 - 3*a^2*c*d^4 - 4*(b^2*c^3*d^2 + a*b*c*d^4) 
*x^2 + (a*b*c^4*d - 3*a^2*c^2*d^3 + (b^2*c^3*d^2 - 3*a*b*c*d^4)*x^3 + (b^2 
*c^4*d - 3*a*b*c^2*d^3)*x^2 + (a*b*c^3*d^2 - 3*a^2*c*d^4)*x)*sqrt(-b/a)*lo 
g((b*x^2 + 2*a*x*sqrt(-b/a) - a)/(b*x^2 + a)) - (b^2*c^4*d + 2*a*b*c^2*d^3 
 + a^2*d^5)*x - (3*a*b*c^3*d^2 - a^2*c*d^4 + (3*b^2*c^2*d^3 - a*b*d^5)*x^3 
 + (3*b^2*c^3*d^2 - a*b*c*d^4)*x^2 + (3*a*b*c^2*d^3 - a^2*d^5)*x)*log(b*x^ 
2 + a) + 2*(3*a*b*c^3*d^2 - a^2*c*d^4 + (3*b^2*c^2*d^3 - a*b*d^5)*x^3 + (3 
*b^2*c^3*d^2 - a*b*c*d^4)*x^2 + (3*a*b*c^2*d^3 - a^2*d^5)*x)*log(d*x + c)) 
/(a*b^3*c^7 + 3*a^2*b^2*c^5*d^2 + 3*a^3*b*c^3*d^4 + a^4*c*d^6 + (b^4*c^6*d 
 + 3*a*b^3*c^4*d^3 + 3*a^2*b^2*c^2*d^5 + a^3*b*d^7)*x^3 + (b^4*c^7 + 3*a*b 
^3*c^5*d^2 + 3*a^2*b^2*c^3*d^4 + a^3*b*c*d^6)*x^2 + (a*b^3*c^6*d + 3*a^2*b 
^2*c^4*d^3 + 3*a^3*b*c^2*d^5 + a^4*d^7)*x), -1/2*(b^2*c^5 - 2*a*b*c^3*d^2 
- 3*a^2*c*d^4 - 4*(b^2*c^3*d^2 + a*b*c*d^4)*x^2 + 2*(a*b*c^4*d - 3*a^2*c^2 
*d^3 + (b^2*c^3*d^2 - 3*a*b*c*d^4)*x^3 + (b^2*c^4*d - 3*a*b*c^2*d^3)*x^2 + 
 (a*b*c^3*d^2 - 3*a^2*c*d^4)*x)*sqrt(b/a)*arctan(x*sqrt(b/a)) - (b^2*c^4*d 
 + 2*a*b*c^2*d^3 + a^2*d^5)*x - (3*a*b*c^3*d^2 - a^2*c*d^4 + (3*b^2*c^2*d^ 
3 - a*b*d^5)*x^3 + (3*b^2*c^3*d^2 - a*b*c*d^4)*x^2 + (3*a*b*c^2*d^3 - a^2* 
d^5)*x)*log(b*x^2 + a) + 2*(3*a*b*c^3*d^2 - a^2*c*d^4 + (3*b^2*c^2*d^3 - a 
*b*d^5)*x^3 + (3*b^2*c^3*d^2 - a*b*c*d^4)*x^2 + (3*a*b*c^2*d^3 - a^2*d^5)* 
x)*log(d*x + c))/(a*b^3*c^7 + 3*a^2*b^2*c^5*d^2 + 3*a^3*b*c^3*d^4 + a^4...
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 375, normalized size of antiderivative = 1.88 \[ \int \frac {x}{(c+d x)^2 \left (a+b x^2\right )^2} \, dx=\frac {{\left (3 \, b c^{2} d^{2} - a d^{4}\right )} \log \left (b x^{2} + a\right )}{2 \, {\left (b^{3} c^{6} + 3 \, a b^{2} c^{4} d^{2} + 3 \, a^{2} b c^{2} d^{4} + a^{3} d^{6}\right )}} - \frac {{\left (3 \, b c^{2} d^{2} - a d^{4}\right )} \log \left (d x + c\right )}{b^{3} c^{6} + 3 \, a b^{2} c^{4} d^{2} + 3 \, a^{2} b c^{2} d^{4} + a^{3} d^{6}} - \frac {{\left (b^{2} c^{3} d - 3 \, a b c d^{3}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{{\left (b^{3} c^{6} + 3 \, a b^{2} c^{4} d^{2} + 3 \, a^{2} b c^{2} d^{4} + a^{3} d^{6}\right )} \sqrt {a b}} + \frac {4 \, b c d^{2} x^{2} - b c^{3} + 3 \, a c d^{2} + {\left (b c^{2} d + a d^{3}\right )} x}{2 \, {\left (a b^{2} c^{5} + 2 \, a^{2} b c^{3} d^{2} + a^{3} c d^{4} + {\left (b^{3} c^{4} d + 2 \, a b^{2} c^{2} d^{3} + a^{2} b d^{5}\right )} x^{3} + {\left (b^{3} c^{5} + 2 \, a b^{2} c^{3} d^{2} + a^{2} b c d^{4}\right )} x^{2} + {\left (a b^{2} c^{4} d + 2 \, a^{2} b c^{2} d^{3} + a^{3} d^{5}\right )} x\right )}} \] Input:

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

Output:

1/2*(3*b*c^2*d^2 - a*d^4)*log(b*x^2 + a)/(b^3*c^6 + 3*a*b^2*c^4*d^2 + 3*a^ 
2*b*c^2*d^4 + a^3*d^6) - (3*b*c^2*d^2 - a*d^4)*log(d*x + c)/(b^3*c^6 + 3*a 
*b^2*c^4*d^2 + 3*a^2*b*c^2*d^4 + a^3*d^6) - (b^2*c^3*d - 3*a*b*c*d^3)*arct 
an(b*x/sqrt(a*b))/((b^3*c^6 + 3*a*b^2*c^4*d^2 + 3*a^2*b*c^2*d^4 + a^3*d^6) 
*sqrt(a*b)) + 1/2*(4*b*c*d^2*x^2 - b*c^3 + 3*a*c*d^2 + (b*c^2*d + a*d^3)*x 
)/(a*b^2*c^5 + 2*a^2*b*c^3*d^2 + a^3*c*d^4 + (b^3*c^4*d + 2*a*b^2*c^2*d^3 
+ a^2*b*d^5)*x^3 + (b^3*c^5 + 2*a*b^2*c^3*d^2 + a^2*b*c*d^4)*x^2 + (a*b^2* 
c^4*d + 2*a^2*b*c^2*d^3 + a^3*d^5)*x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 386 vs. \(2 (188) = 376\).

Time = 0.13 (sec) , antiderivative size = 386, normalized size of antiderivative = 1.93 \[ \int \frac {x}{(c+d x)^2 \left (a+b x^2\right )^2} \, dx=\frac {\frac {2 \, c d^{7}}{{\left (b^{2} c^{4} d^{4} + 2 \, a b c^{2} d^{6} + a^{2} d^{8}\right )} {\left (d x + c\right )}} + \frac {{\left (3 \, b c^{2} d^{3} - a d^{5}\right )} \log \left (b - \frac {2 \, b c}{d x + c} + \frac {b c^{2}}{{\left (d x + c\right )}^{2}} + \frac {a d^{2}}{{\left (d x + c\right )}^{2}}\right )}{b^{3} c^{6} + 3 \, a b^{2} c^{4} d^{2} + 3 \, a^{2} b c^{2} d^{4} + a^{3} d^{6}} - \frac {2 \, {\left (b^{2} c^{3} d^{4} - 3 \, a b c d^{6}\right )} \arctan \left (\frac {b c - \frac {b c^{2}}{d x + c} - \frac {a d^{2}}{d x + c}}{\sqrt {a b} d}\right )}{{\left (b^{3} c^{6} + 3 \, a b^{2} c^{4} d^{2} + 3 \, a^{2} b c^{2} d^{4} + a^{3} d^{6}\right )} \sqrt {a b} d^{2}} + \frac {\frac {3 \, b^{2} c^{2} d^{3} - a b d^{5}}{b c^{2} + a d^{2}} - \frac {4 \, {\left (b^{2} c^{3} d^{4} - a b c d^{6}\right )}}{{\left (b c^{2} + a d^{2}\right )} {\left (d x + c\right )} d}}{{\left (b c^{2} + a d^{2}\right )}^{2} {\left (b - \frac {2 \, b c}{d x + c} + \frac {b c^{2}}{{\left (d x + c\right )}^{2}} + \frac {a d^{2}}{{\left (d x + c\right )}^{2}}\right )}}}{2 \, d} \] Input:

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

Output:

1/2*(2*c*d^7/((b^2*c^4*d^4 + 2*a*b*c^2*d^6 + a^2*d^8)*(d*x + c)) + (3*b*c^ 
2*d^3 - a*d^5)*log(b - 2*b*c/(d*x + c) + b*c^2/(d*x + c)^2 + a*d^2/(d*x + 
c)^2)/(b^3*c^6 + 3*a*b^2*c^4*d^2 + 3*a^2*b*c^2*d^4 + a^3*d^6) - 2*(b^2*c^3 
*d^4 - 3*a*b*c*d^6)*arctan((b*c - b*c^2/(d*x + c) - a*d^2/(d*x + c))/(sqrt 
(a*b)*d))/((b^3*c^6 + 3*a*b^2*c^4*d^2 + 3*a^2*b*c^2*d^4 + a^3*d^6)*sqrt(a* 
b)*d^2) + ((3*b^2*c^2*d^3 - a*b*d^5)/(b*c^2 + a*d^2) - 4*(b^2*c^3*d^4 - a* 
b*c*d^6)/((b*c^2 + a*d^2)*(d*x + c)*d))/((b*c^2 + a*d^2)^2*(b - 2*b*c/(d*x 
 + c) + b*c^2/(d*x + c)^2 + a*d^2/(d*x + c)^2)))/d
 

Mupad [B] (verification not implemented)

Time = 7.29 (sec) , antiderivative size = 704, normalized size of antiderivative = 3.52 \[ \int \frac {x}{(c+d x)^2 \left (a+b x^2\right )^2} \, dx=\frac {\frac {d\,x}{2\,\left (b\,c^2+a\,d^2\right )}+\frac {c\,\left (3\,a\,d^2-b\,c^2\right )}{2\,{\left (b\,c^2+a\,d^2\right )}^2}+\frac {2\,b\,c\,d^2\,x^2}{a^2\,d^4+2\,a\,b\,c^2\,d^2+b^2\,c^4}}{b\,d\,x^3+b\,c\,x^2+a\,d\,x+a\,c}-\frac {\ln \left (b^4\,c^{10}\,{\left (-a\,b\right )}^{3/2}-9\,a^6\,d^{10}\,\sqrt {-a\,b}+106\,c^6\,d^4\,{\left (-a\,b\right )}^{7/2}+6\,a^2\,c^4\,d^6\,{\left (-a\,b\right )}^{5/2}-27\,a^4\,c^2\,d^8\,{\left (-a\,b\right )}^{3/2}-77\,b^2\,c^8\,d^2\,{\left (-a\,b\right )}^{5/2}+a\,b^6\,c^{10}\,x+9\,a^6\,b\,d^{10}\,x+77\,a^2\,b^5\,c^8\,d^2\,x+106\,a^3\,b^4\,c^6\,d^4\,x-6\,a^4\,b^3\,c^4\,d^6\,x-27\,a^5\,b^2\,c^2\,d^8\,x\right )\,\left (\frac {a^2\,d^4}{2}-b\,\left (\frac {c^3\,d\,\sqrt {-a\,b}}{2}+\frac {3\,a\,c^2\,d^2}{2}\right )+\frac {3\,a\,c\,d^3\,\sqrt {-a\,b}}{2}\right )}{a^4\,d^6+3\,a^3\,b\,c^2\,d^4+3\,a^2\,b^2\,c^4\,d^2+a\,b^3\,c^6}-\frac {\ln \left (9\,a^6\,d^{10}\,\sqrt {-a\,b}-b^4\,c^{10}\,{\left (-a\,b\right )}^{3/2}-106\,c^6\,d^4\,{\left (-a\,b\right )}^{7/2}-6\,a^2\,c^4\,d^6\,{\left (-a\,b\right )}^{5/2}+27\,a^4\,c^2\,d^8\,{\left (-a\,b\right )}^{3/2}+77\,b^2\,c^8\,d^2\,{\left (-a\,b\right )}^{5/2}+a\,b^6\,c^{10}\,x+9\,a^6\,b\,d^{10}\,x+77\,a^2\,b^5\,c^8\,d^2\,x+106\,a^3\,b^4\,c^6\,d^4\,x-6\,a^4\,b^3\,c^4\,d^6\,x-27\,a^5\,b^2\,c^2\,d^8\,x\right )\,\left (b\,\left (\frac {c^3\,d\,\sqrt {-a\,b}}{2}-\frac {3\,a\,c^2\,d^2}{2}\right )+\frac {a^2\,d^4}{2}-\frac {3\,a\,c\,d^3\,\sqrt {-a\,b}}{2}\right )}{a^4\,d^6+3\,a^3\,b\,c^2\,d^4+3\,a^2\,b^2\,c^4\,d^2+a\,b^3\,c^6}+\frac {\ln \left (c+d\,x\right )\,\left (a\,d^4-3\,b\,c^2\,d^2\right )}{a^3\,d^6+3\,a^2\,b\,c^2\,d^4+3\,a\,b^2\,c^4\,d^2+b^3\,c^6} \] Input:

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

Output:

((d*x)/(2*(a*d^2 + b*c^2)) + (c*(3*a*d^2 - b*c^2))/(2*(a*d^2 + b*c^2)^2) + 
 (2*b*c*d^2*x^2)/(a^2*d^4 + b^2*c^4 + 2*a*b*c^2*d^2))/(a*c + a*d*x + b*c*x 
^2 + b*d*x^3) - (log(b^4*c^10*(-a*b)^(3/2) - 9*a^6*d^10*(-a*b)^(1/2) + 106 
*c^6*d^4*(-a*b)^(7/2) + 6*a^2*c^4*d^6*(-a*b)^(5/2) - 27*a^4*c^2*d^8*(-a*b) 
^(3/2) - 77*b^2*c^8*d^2*(-a*b)^(5/2) + a*b^6*c^10*x + 9*a^6*b*d^10*x + 77* 
a^2*b^5*c^8*d^2*x + 106*a^3*b^4*c^6*d^4*x - 6*a^4*b^3*c^4*d^6*x - 27*a^5*b 
^2*c^2*d^8*x)*((a^2*d^4)/2 - b*((c^3*d*(-a*b)^(1/2))/2 + (3*a*c^2*d^2)/2) 
+ (3*a*c*d^3*(-a*b)^(1/2))/2))/(a^4*d^6 + a*b^3*c^6 + 3*a^3*b*c^2*d^4 + 3* 
a^2*b^2*c^4*d^2) - (log(9*a^6*d^10*(-a*b)^(1/2) - b^4*c^10*(-a*b)^(3/2) - 
106*c^6*d^4*(-a*b)^(7/2) - 6*a^2*c^4*d^6*(-a*b)^(5/2) + 27*a^4*c^2*d^8*(-a 
*b)^(3/2) + 77*b^2*c^8*d^2*(-a*b)^(5/2) + a*b^6*c^10*x + 9*a^6*b*d^10*x + 
77*a^2*b^5*c^8*d^2*x + 106*a^3*b^4*c^6*d^4*x - 6*a^4*b^3*c^4*d^6*x - 27*a^ 
5*b^2*c^2*d^8*x)*(b*((c^3*d*(-a*b)^(1/2))/2 - (3*a*c^2*d^2)/2) + (a^2*d^4) 
/2 - (3*a*c*d^3*(-a*b)^(1/2))/2))/(a^4*d^6 + a*b^3*c^6 + 3*a^3*b*c^2*d^4 + 
 3*a^2*b^2*c^4*d^2) + (log(c + d*x)*(a*d^4 - 3*b*c^2*d^2))/(a^3*d^6 + b^3* 
c^6 + 3*a*b^2*c^4*d^2 + 3*a^2*b*c^2*d^4)
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 820, normalized size of antiderivative = 4.10 \[ \int \frac {x}{(c+d x)^2 \left (a+b x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

(6*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**2*c**2*d**3 + 6*sqrt(b 
)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**2*c*d**4*x - 2*sqrt(b)*sqrt(a)* 
atan((b*x)/(sqrt(b)*sqrt(a)))*a*b*c**4*d - 2*sqrt(b)*sqrt(a)*atan((b*x)/(s 
qrt(b)*sqrt(a)))*a*b*c**3*d**2*x + 6*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*s 
qrt(a)))*a*b*c**2*d**3*x**2 + 6*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a 
)))*a*b*c*d**4*x**3 - 2*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*b**2 
*c**4*d*x**2 - 2*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*b**2*c**3*d 
**2*x**3 - log(a + b*x**2)*a**3*c*d**4 - log(a + b*x**2)*a**3*d**5*x + 3*l 
og(a + b*x**2)*a**2*b*c**3*d**2 + 3*log(a + b*x**2)*a**2*b*c**2*d**3*x - l 
og(a + b*x**2)*a**2*b*c*d**4*x**2 - log(a + b*x**2)*a**2*b*d**5*x**3 + 3*l 
og(a + b*x**2)*a*b**2*c**3*d**2*x**2 + 3*log(a + b*x**2)*a*b**2*c**2*d**3* 
x**3 + 2*log(c + d*x)*a**3*c*d**4 + 2*log(c + d*x)*a**3*d**5*x - 6*log(c + 
 d*x)*a**2*b*c**3*d**2 - 6*log(c + d*x)*a**2*b*c**2*d**3*x + 2*log(c + d*x 
)*a**2*b*c*d**4*x**2 + 2*log(c + d*x)*a**2*b*d**5*x**3 - 6*log(c + d*x)*a* 
b**2*c**3*d**2*x**2 - 6*log(c + d*x)*a*b**2*c**2*d**3*x**3 - a**3*c*d**4 - 
 3*a**3*d**5*x - 2*a**2*b*c**3*d**2 - 2*a**2*b*c**2*d**3*x - 4*a**2*b*d**5 
*x**3 - a*b**2*c**5 + a*b**2*c**4*d*x - 4*a*b**2*c**2*d**3*x**3)/(2*a*(a** 
4*c*d**6 + a**4*d**7*x + 3*a**3*b*c**3*d**4 + 3*a**3*b*c**2*d**5*x + a**3* 
b*c*d**6*x**2 + a**3*b*d**7*x**3 + 3*a**2*b**2*c**5*d**2 + 3*a**2*b**2*c** 
4*d**3*x + 3*a**2*b**2*c**3*d**4*x**2 + 3*a**2*b**2*c**2*d**5*x**3 + a*...