\(\int \frac {x}{(a+b x^2)^2 (c+d x^2)^2} \, dx\) [304]

Optimal result

Integrand size = 20, antiderivative size = 92 \[ \int \frac {x}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=-\frac {b}{2 (b c-a d)^2 \left (a+b x^2\right )}-\frac {d}{2 (b c-a d)^2 \left (c+d x^2\right )}-\frac {b d \log \left (a+b x^2\right )}{(b c-a d)^3}+\frac {b d \log \left (c+d x^2\right )}{(b c-a d)^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {455, 46} \[ \int \frac {x}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=-\frac {b}{2 \left (a+b x^2\right ) (b c-a d)^2}-\frac {d}{2 \left (c+d x^2\right ) (b c-a d)^2}-\frac {b d \log \left (a+b x^2\right )}{(b c-a d)^3}+\frac {b d \log \left (c+d x^2\right )}{(b c-a d)^3} \]

[In]

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

[Out]

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

Rule 46

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x
)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && Lt
Q[m + n + 2, 0])

Rule 455

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[m
- n + 1, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {1}{(a+b x)^2 (c+d x)^2} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (\frac {b^2}{(b c-a d)^2 (a+b x)^2}-\frac {2 b^2 d}{(b c-a d)^3 (a+b x)}+\frac {d^2}{(b c-a d)^2 (c+d x)^2}+\frac {2 b d^2}{(b c-a d)^3 (c+d x)}\right ) \, dx,x,x^2\right ) \\ & = -\frac {b}{2 (b c-a d)^2 \left (a+b x^2\right )}-\frac {d}{2 (b c-a d)^2 \left (c+d x^2\right )}-\frac {b d \log \left (a+b x^2\right )}{(b c-a d)^3}+\frac {b d \log \left (c+d x^2\right )}{(b c-a d)^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.84 \[ \int \frac {x}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=\frac {\frac {b (-b c+a d)}{a+b x^2}+\frac {d (-b c+a d)}{c+d x^2}-2 b d \log \left (a+b x^2\right )+2 b d \log \left (c+d x^2\right )}{2 (b c-a d)^3} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 2.71 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.15

method	result	size
default	\(\frac {b^{2} \left (\frac {2 d \ln \left (b \,x^{2}+a \right )}{b}-\frac {a d -b c}{b \left (b \,x^{2}+a \right )}\right )}{2 \left (a d -b c \right )^{3}}+\frac {d^{2} \left (-\frac {2 b \ln \left (d \,x^{2}+c \right )}{d}-\frac {a d -b c}{d \left (d \,x^{2}+c \right )}\right )}{2 \left (a d -b c \right )^{3}}\)	\(106\)
risch	\(\frac {-\frac {b d \,x^{2}}{a^{2} d^{2}-2 a b c d +b^{2} c^{2}}-\frac {a d +b c}{2 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}}{\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}+\frac {b d \ln \left (b \,x^{2}+a \right )}{a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}}-\frac {b d \ln \left (-d \,x^{2}-c \right )}{a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}}\)	\(186\)
norman	\(\frac {-\frac {b d \,x^{2}}{a^{2} d^{2}-2 a b c d +b^{2} c^{2}}+\frac {-a b \,d^{2}-b^{2} c d}{2 d b \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}}{\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}+\frac {b d \ln \left (b \,x^{2}+a \right )}{a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}}-\frac {b d \ln \left (d \,x^{2}+c \right )}{a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}}\)	\(197\)
parallelrisch	\(\frac {2 \ln \left (b \,x^{2}+a \right ) x^{4} b^{3} d^{3}-2 \ln \left (d \,x^{2}+c \right ) x^{4} b^{3} d^{3}+2 \ln \left (b \,x^{2}+a \right ) x^{2} a \,b^{2} d^{3}+2 \ln \left (b \,x^{2}+a \right ) x^{2} b^{3} c \,d^{2}-2 \ln \left (d \,x^{2}+c \right ) x^{2} a \,b^{2} d^{3}-2 \ln \left (d \,x^{2}+c \right ) x^{2} b^{3} c \,d^{2}-2 x^{2} a \,b^{2} d^{3}+2 x^{2} b^{3} c \,d^{2}+2 \ln \left (b \,x^{2}+a \right ) a \,b^{2} c \,d^{2}-2 \ln \left (d \,x^{2}+c \right ) a \,b^{2} c \,d^{2}-a^{2} b \,d^{3}+b^{3} c^{2} d}{2 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right ) b d}\)	\(261\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 253 vs. \(2 (88) = 176\).

Time = 0.24 (sec) , antiderivative size = 253, normalized size of antiderivative = 2.75 \[ \int \frac {x}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=-\frac {b^{2} c^{2} - a^{2} d^{2} + 2 \, {\left (b^{2} c d - a b d^{2}\right )} x^{2} + 2 \, {\left (b^{2} d^{2} x^{4} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x^{2}\right )} \log \left (b x^{2} + a\right ) - 2 \, {\left (b^{2} d^{2} x^{4} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x^{2}\right )} \log \left (d x^{2} + c\right )}{2 \, {\left (a b^{3} c^{4} - 3 \, a^{2} b^{2} c^{3} d + 3 \, a^{3} b c^{2} d^{2} - a^{4} c d^{3} + {\left (b^{4} c^{3} d - 3 \, a b^{3} c^{2} d^{2} + 3 \, a^{2} b^{2} c d^{3} - a^{3} b d^{4}\right )} x^{4} + {\left (b^{4} c^{4} - 2 \, a b^{3} c^{3} d + 2 \, a^{3} b c d^{3} - a^{4} d^{4}\right )} x^{2}\right )}} \]

[In]

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

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 410 vs. \(2 (76) = 152\).

Time = 1.96 (sec) , antiderivative size = 410, normalized size of antiderivative = 4.46 \[ \int \frac {x}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=- \frac {b d \log {\left (x^{2} + \frac {- \frac {a^{4} b d^{5}}{\left (a d - b c\right )^{3}} + \frac {4 a^{3} b^{2} c d^{4}}{\left (a d - b c\right )^{3}} - \frac {6 a^{2} b^{3} c^{2} d^{3}}{\left (a d - b c\right )^{3}} + \frac {4 a b^{4} c^{3} d^{2}}{\left (a d - b c\right )^{3}} + a b d^{2} - \frac {b^{5} c^{4} d}{\left (a d - b c\right )^{3}} + b^{2} c d}{2 b^{2} d^{2}} \right )}}{\left (a d - b c\right )^{3}} + \frac {b d \log {\left (x^{2} + \frac {\frac {a^{4} b d^{5}}{\left (a d - b c\right )^{3}} - \frac {4 a^{3} b^{2} c d^{4}}{\left (a d - b c\right )^{3}} + \frac {6 a^{2} b^{3} c^{2} d^{3}}{\left (a d - b c\right )^{3}} - \frac {4 a b^{4} c^{3} d^{2}}{\left (a d - b c\right )^{3}} + a b d^{2} + \frac {b^{5} c^{4} d}{\left (a d - b c\right )^{3}} + b^{2} c d}{2 b^{2} d^{2}} \right )}}{\left (a d - b c\right )^{3}} + \frac {- a d - b c - 2 b d x^{2}}{2 a^{3} c d^{2} - 4 a^{2} b c^{2} d + 2 a b^{2} c^{3} + x^{4} \cdot \left (2 a^{2} b d^{3} - 4 a b^{2} c d^{2} + 2 b^{3} c^{2} d\right ) + x^{2} \cdot \left (2 a^{3} d^{3} - 2 a^{2} b c d^{2} - 2 a b^{2} c^{2} d + 2 b^{3} c^{3}\right )} \]

[In]

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

[Out]

-b*d*log(x**2 + (-a**4*b*d**5/(a*d - b*c)**3 + 4*a**3*b**2*c*d**4/(a*d - b*c)**3 - 6*a**2*b**3*c**2*d**3/(a*d
- b*c)**3 + 4*a*b**4*c**3*d**2/(a*d - b*c)**3 + a*b*d**2 - b**5*c**4*d/(a*d - b*c)**3 + b**2*c*d)/(2*b**2*d**2
))/(a*d - b*c)**3 + b*d*log(x**2 + (a**4*b*d**5/(a*d - b*c)**3 - 4*a**3*b**2*c*d**4/(a*d - b*c)**3 + 6*a**2*b*
*3*c**2*d**3/(a*d - b*c)**3 - 4*a*b**4*c**3*d**2/(a*d - b*c)**3 + a*b*d**2 + b**5*c**4*d/(a*d - b*c)**3 + b**2
*c*d)/(2*b**2*d**2))/(a*d - b*c)**3 + (-a*d - b*c - 2*b*d*x**2)/(2*a**3*c*d**2 - 4*a**2*b*c**2*d + 2*a*b**2*c*
*3 + x**4*(2*a**2*b*d**3 - 4*a*b**2*c*d**2 + 2*b**3*c**2*d) + x**2*(2*a**3*d**3 - 2*a**2*b*c*d**2 - 2*a*b**2*c
**2*d + 2*b**3*c**3))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 215 vs. \(2 (88) = 176\).

Time = 0.20 (sec) , antiderivative size = 215, normalized size of antiderivative = 2.34 \[ \int \frac {x}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=-\frac {b d \log \left (b x^{2} + a\right )}{b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}} + \frac {b d \log \left (d x^{2} + c\right )}{b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}} - \frac {2 \, b d x^{2} + b c + a d}{2 \, {\left (a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2} + {\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{4} + {\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} x^{2}\right )}} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.77 \[ \int \frac {x}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=\frac {b^{2} d \log \left ({\left | \frac {b c}{b x^{2} + a} - \frac {a d}{b x^{2} + a} + d \right |}\right )}{b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}} - \frac {b^{3}}{2 \, {\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} {\left (b x^{2} + a\right )}} + \frac {b d^{2}}{2 \, {\left (b c - a d\right )}^{3} {\left (\frac {b c}{b x^{2} + a} - \frac {a d}{b x^{2} + a} + d\right )}} \]

[In]

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

[Out]

b^2*d*log(abs(b*c/(b*x^2 + a) - a*d/(b*x^2 + a) + d))/(b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2*c*d^2 - a^3*b*d^3)
- 1/2*b^3/((b^4*c^2 - 2*a*b^3*c*d + a^2*b^2*d^2)*(b*x^2 + a)) + 1/2*b*d^2/((b*c - a*d)^3*(b*c/(b*x^2 + a) - a*
d/(b*x^2 + a) + d))

Mupad [B] (verification not implemented)

Time = 5.40 (sec) , antiderivative size = 378, normalized size of antiderivative = 4.11 \[ \int \frac {x}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=-\frac {b^2\,c^2-a^2\,d^2+b^2\,d^2\,x^4\,\mathrm {atan}\left (\frac {a\,d\,x^2\,1{}\mathrm {i}-b\,c\,x^2\,1{}\mathrm {i}}{2\,a\,c+a\,d\,x^2+b\,c\,x^2}\right )\,4{}\mathrm {i}-2\,a\,b\,d^2\,x^2+2\,b^2\,c\,d\,x^2+a\,b\,d^2\,x^2\,\mathrm {atan}\left (\frac {a\,d\,x^2\,1{}\mathrm {i}-b\,c\,x^2\,1{}\mathrm {i}}{2\,a\,c+a\,d\,x^2+b\,c\,x^2}\right )\,4{}\mathrm {i}+b^2\,c\,d\,x^2\,\mathrm {atan}\left (\frac {a\,d\,x^2\,1{}\mathrm {i}-b\,c\,x^2\,1{}\mathrm {i}}{2\,a\,c+a\,d\,x^2+b\,c\,x^2}\right )\,4{}\mathrm {i}+a\,b\,c\,d\,\mathrm {atan}\left (\frac {a\,d\,x^2\,1{}\mathrm {i}-b\,c\,x^2\,1{}\mathrm {i}}{2\,a\,c+a\,d\,x^2+b\,c\,x^2}\right )\,4{}\mathrm {i}}{-2\,a^4\,c\,d^3-2\,a^4\,d^4\,x^2+6\,a^3\,b\,c^2\,d^2+4\,a^3\,b\,c\,d^3\,x^2-2\,a^3\,b\,d^4\,x^4-6\,a^2\,b^2\,c^3\,d+6\,a^2\,b^2\,c\,d^3\,x^4+2\,a\,b^3\,c^4-4\,a\,b^3\,c^3\,d\,x^2-6\,a\,b^3\,c^2\,d^2\,x^4+2\,b^4\,c^4\,x^2+2\,b^4\,c^3\,d\,x^4} \]

[In]

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

[Out]

-(b^2*c^2 - a^2*d^2 + b^2*d^2*x^4*atan((a*d*x^2*1i - b*c*x^2*1i)/(2*a*c + a*d*x^2 + b*c*x^2))*4i - 2*a*b*d^2*x
^2 + 2*b^2*c*d*x^2 + a*b*d^2*x^2*atan((a*d*x^2*1i - b*c*x^2*1i)/(2*a*c + a*d*x^2 + b*c*x^2))*4i + b^2*c*d*x^2*
atan((a*d*x^2*1i - b*c*x^2*1i)/(2*a*c + a*d*x^2 + b*c*x^2))*4i + a*b*c*d*atan((a*d*x^2*1i - b*c*x^2*1i)/(2*a*c
+ a*d*x^2 + b*c*x^2))*4i)/(2*a*b^3*c^4 - 2*a^4*c*d^3 - 2*a^4*d^4*x^2 + 2*b^4*c^4*x^2 - 6*a^2*b^2*c^3*d + 6*a^
3*b*c^2*d^2 - 2*a^3*b*d^4*x^4 + 2*b^4*c^3*d*x^4 - 4*a*b^3*c^3*d*x^2 + 4*a^3*b*c*d^3*x^2 - 6*a*b^3*c^2*d^2*x^4
+ 6*a^2*b^2*c*d^3*x^4)