\(\int \frac {1}{x (a+b x^2) (c+d x^2)^3} \, dx\) [257]

Optimal result

Integrand size = 22, antiderivative size = 149 \[ \int \frac {1}{x \left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx=-\frac {d}{4 c (b c-a d) \left (c+d x^2\right )^2}-\frac {d (2 b c-a d)}{2 c^2 (b c-a d)^2 \left (c+d x^2\right )}+\frac {\log (x)}{a c^3}-\frac {b^3 \log \left (a+b x^2\right )}{2 a (b c-a d)^3}+\frac {d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) \log \left (c+d x^2\right )}{2 c^3 (b c-a d)^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {457, 84} \[ \int \frac {1}{x \left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx=\frac {d \left (a^2 d^2-3 a b c d+3 b^2 c^2\right ) \log \left (c+d x^2\right )}{2 c^3 (b c-a d)^3}-\frac {b^3 \log \left (a+b x^2\right )}{2 a (b c-a d)^3}-\frac {d (2 b c-a d)}{2 c^2 \left (c+d x^2\right ) (b c-a d)^2}-\frac {d}{4 c \left (c+d x^2\right )^2 (b c-a d)}+\frac {\log (x)}{a c^3} \]

[In]

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

[Out]

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

Rule 84

Int[((e_.) + (f_.)*(x_))^(p_.)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Int[ExpandIntegrand[(
e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[p]

Rule 457

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(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] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {1}{x (a+b x) (c+d x)^3} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (\frac {1}{a c^3 x}+\frac {b^4}{a (-b c+a d)^3 (a+b x)}+\frac {d^2}{c (b c-a d) (c+d x)^3}+\frac {d^2 (2 b c-a d)}{c^2 (b c-a d)^2 (c+d x)^2}+\frac {d^2 \left (3 b^2 c^2-3 a b c d+a^2 d^2\right )}{c^3 (b c-a d)^3 (c+d x)}\right ) \, dx,x,x^2\right ) \\ & = -\frac {d}{4 c (b c-a d) \left (c+d x^2\right )^2}-\frac {d (2 b c-a d)}{2 c^2 (b c-a d)^2 \left (c+d x^2\right )}+\frac {\log (x)}{a c^3}-\frac {b^3 \log \left (a+b x^2\right )}{2 a (b c-a d)^3}+\frac {d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) \log \left (c+d x^2\right )}{2 c^3 (b c-a d)^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.85 \[ \int \frac {1}{x \left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx=\frac {\log (x)}{a c^3}+\frac {\frac {2 b^3 \log \left (a+b x^2\right )}{a}+\frac {d \left (\frac {c (b c-a d) \left (-a d \left (3 c+2 d x^2\right )+b c \left (5 c+4 d x^2\right )\right )}{\left (c+d x^2\right )^2}-2 \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) \log \left (c+d x^2\right )\right )}{c^3}}{4 (-b c+a d)^3} \]

[In]

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

[Out]

Log[x]/(a*c^3) + ((2*b^3*Log[a + b*x^2])/a + (d*((c*(b*c - a*d)*(-(a*d*(3*c + 2*d*x^2)) + b*c*(5*c + 4*d*x^2))
)/(c + d*x^2)^2 - 2*(3*b^2*c^2 - 3*a*b*c*d + a^2*d^2)*Log[c + d*x^2]))/c^3)/(4*(-(b*c) + a*d)^3)

Maple [A] (verified)

Time = 2.77 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.11

method	result	size
default	\(\frac {\ln \left (x \right )}{a \,c^{3}}+\frac {b^{3} \ln \left (b \,x^{2}+a \right )}{2 a \left (a d -b c \right )^{3}}-\frac {d^{2} \left (-\frac {c^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{2 d \left (d \,x^{2}+c \right )^{2}}+\frac {\left (a^{2} d^{2}-3 a b c d +3 b^{2} c^{2}\right ) \ln \left (d \,x^{2}+c \right )}{d}-\frac {c \left (a^{2} d^{2}-3 a b c d +2 b^{2} c^{2}\right )}{d \left (d \,x^{2}+c \right )}\right )}{2 \left (a d -b c \right )^{3} c^{3}}\)	\(165\)
norman	\(\frac {\frac {\left (-3 a \,d^{2}+5 b c d \right ) d^{2} x^{4}}{4 c^{3} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {\left (-2 a \,d^{2}+3 b c d \right ) d \,x^{2}}{2 c^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}}{\left (d \,x^{2}+c \right )^{2}}+\frac {\ln \left (x \right )}{a \,c^{3}}+\frac {b^{3} \ln \left (b \,x^{2}+a \right )}{2 a \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}-\frac {d \left (a^{2} d^{2}-3 a b c d +3 b^{2} c^{2}\right ) \ln \left (d \,x^{2}+c \right )}{2 c^{3} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}\)	\(240\)
risch	\(\frac {\frac {d^{2} \left (a d -2 b c \right ) x^{2}}{2 c^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {d \left (3 a d -5 b c \right )}{4 c \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}}{\left (d \,x^{2}+c \right )^{2}}+\frac {\ln \left (x \right )}{a \,c^{3}}+\frac {b^{3} \ln \left (b \,x^{2}+a \right )}{2 a \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}-\frac {d^{3} \ln \left (-d \,x^{2}-c \right ) a^{2}}{2 c^{3} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {3 d^{2} \ln \left (-d \,x^{2}-c \right ) a b}{2 c^{2} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}-\frac {3 d \ln \left (-d \,x^{2}-c \right ) b^{2}}{2 c \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}\)	\(333\)
parallelrisch	\(\frac {4 \ln \left (x \right ) x^{4} a^{3} d^{5}-2 \ln \left (d \,x^{2}+c \right ) x^{4} a^{3} d^{5}-4 x^{2} a^{3} c \,d^{4}-12 \ln \left (x \right ) x^{4} a^{2} b c \,d^{4}+6 \ln \left (d \,x^{2}+c \right ) x^{4} a^{2} b c \,d^{4}-24 \ln \left (x \right ) x^{2} a^{2} b \,c^{2} d^{3}+12 \ln \left (d \,x^{2}+c \right ) x^{2} a^{2} b \,c^{2} d^{3}+4 \ln \left (x \right ) a^{3} c^{2} d^{3}-2 \ln \left (d \,x^{2}+c \right ) a^{3} c^{2} d^{3}-12 \ln \left (d \,x^{2}+c \right ) x^{2} a \,b^{2} c^{3} d^{2}+24 \ln \left (x \right ) x^{2} a \,b^{2} c^{3} d^{2}+12 \ln \left (x \right ) x^{4} a \,b^{2} c^{2} d^{3}-4 \ln \left (x \right ) x^{4} b^{3} c^{3} d^{2}-4 \ln \left (x \right ) b^{3} c^{5}-12 \ln \left (x \right ) a^{2} b \,c^{3} d^{2}+12 \ln \left (x \right ) a \,b^{2} c^{4} d +6 \ln \left (d \,x^{2}+c \right ) a^{2} b \,c^{3} d^{2}-6 \ln \left (d \,x^{2}+c \right ) a \,b^{2} c^{4} d +8 x^{4} a^{2} b c \,d^{4}+2 \ln \left (b \,x^{2}+a \right ) x^{4} b^{3} c^{3} d^{2}+8 \ln \left (x \right ) x^{2} a^{3} c \,d^{4}-8 \ln \left (x \right ) x^{2} b^{3} c^{4} d +4 \ln \left (b \,x^{2}+a \right ) x^{2} b^{3} c^{4} d -4 \ln \left (d \,x^{2}+c \right ) x^{2} a^{3} c \,d^{4}+10 x^{2} a^{2} b \,c^{2} d^{3}-6 x^{2} a \,b^{2} c^{3} d^{2}-3 x^{4} a^{3} d^{5}+2 \ln \left (b \,x^{2}+a \right ) b^{3} c^{5}-6 \ln \left (d \,x^{2}+c \right ) x^{4} a \,b^{2} c^{2} d^{3}-5 x^{4} a \,b^{2} c^{2} d^{3}}{4 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) a \left (d \,x^{2}+c \right )^{2} c^{3}}\)	\(554\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 520 vs. \(2 (141) = 282\).

Time = 2.39 (sec) , antiderivative size = 520, normalized size of antiderivative = 3.49 \[ \int \frac {1}{x \left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx=-\frac {5 \, a b^{2} c^{4} d - 8 \, a^{2} b c^{3} d^{2} + 3 \, a^{3} c^{2} d^{3} + 2 \, {\left (2 \, a b^{2} c^{3} d^{2} - 3 \, a^{2} b c^{2} d^{3} + a^{3} c d^{4}\right )} x^{2} + 2 \, {\left (b^{3} c^{3} d^{2} x^{4} + 2 \, b^{3} c^{4} d x^{2} + b^{3} c^{5}\right )} \log \left (b x^{2} + a\right ) - 2 \, {\left (3 \, a b^{2} c^{4} d - 3 \, a^{2} b c^{3} d^{2} + a^{3} c^{2} d^{3} + {\left (3 \, a b^{2} c^{2} d^{3} - 3 \, a^{2} b c d^{4} + a^{3} d^{5}\right )} x^{4} + 2 \, {\left (3 \, a b^{2} c^{3} d^{2} - 3 \, a^{2} b c^{2} d^{3} + a^{3} c d^{4}\right )} x^{2}\right )} \log \left (d x^{2} + c\right ) - 4 \, {\left (b^{3} c^{5} - 3 \, a b^{2} c^{4} d + 3 \, a^{2} b c^{3} d^{2} - a^{3} c^{2} d^{3} + {\left (b^{3} c^{3} d^{2} - 3 \, a b^{2} c^{2} d^{3} + 3 \, a^{2} b c d^{4} - a^{3} d^{5}\right )} x^{4} + 2 \, {\left (b^{3} c^{4} d - 3 \, a b^{2} c^{3} d^{2} + 3 \, a^{2} b c^{2} d^{3} - a^{3} c d^{4}\right )} x^{2}\right )} \log \left (x\right )}{4 \, {\left (a b^{3} c^{8} - 3 \, a^{2} b^{2} c^{7} d + 3 \, a^{3} b c^{6} d^{2} - a^{4} c^{5} d^{3} + {\left (a b^{3} c^{6} d^{2} - 3 \, a^{2} b^{2} c^{5} d^{3} + 3 \, a^{3} b c^{4} d^{4} - a^{4} c^{3} d^{5}\right )} x^{4} + 2 \, {\left (a b^{3} c^{7} d - 3 \, a^{2} b^{2} c^{6} d^{2} + 3 \, a^{3} b c^{5} d^{3} - a^{4} c^{4} d^{4}\right )} x^{2}\right )}} \]

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.87 \[ \int \frac {1}{x \left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx=-\frac {b^{3} \log \left (b x^{2} + a\right )}{2 \, {\left (a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}\right )}} + \frac {{\left (3 \, b^{2} c^{2} d - 3 \, a b c d^{2} + a^{2} d^{3}\right )} \log \left (d x^{2} + c\right )}{2 \, {\left (b^{3} c^{6} - 3 \, a b^{2} c^{5} d + 3 \, a^{2} b c^{4} d^{2} - a^{3} c^{3} d^{3}\right )}} - \frac {5 \, b c^{2} d - 3 \, a c d^{2} + 2 \, {\left (2 \, b c d^{2} - a d^{3}\right )} x^{2}}{4 \, {\left (b^{2} c^{6} - 2 \, a b c^{5} d + a^{2} c^{4} d^{2} + {\left (b^{2} c^{4} d^{2} - 2 \, a b c^{3} d^{3} + a^{2} c^{2} d^{4}\right )} x^{4} + 2 \, {\left (b^{2} c^{5} d - 2 \, a b c^{4} d^{2} + a^{2} c^{3} d^{3}\right )} x^{2}\right )}} + \frac {\log \left (x^{2}\right )}{2 \, a c^{3}} \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 315 vs. \(2 (141) = 282\).

Time = 0.30 (sec) , antiderivative size = 315, normalized size of antiderivative = 2.11 \[ \int \frac {1}{x \left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx=-\frac {b^{4} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, {\left (a b^{4} c^{3} - 3 \, a^{2} b^{3} c^{2} d + 3 \, a^{3} b^{2} c d^{2} - a^{4} b d^{3}\right )}} + \frac {{\left (3 \, b^{2} c^{2} d^{2} - 3 \, a b c d^{3} + a^{2} d^{4}\right )} \log \left ({\left | d x^{2} + c \right |}\right )}{2 \, {\left (b^{3} c^{6} d - 3 \, a b^{2} c^{5} d^{2} + 3 \, a^{2} b c^{4} d^{3} - a^{3} c^{3} d^{4}\right )}} - \frac {9 \, b^{2} c^{2} d^{3} x^{4} - 9 \, a b c d^{4} x^{4} + 3 \, a^{2} d^{5} x^{4} + 22 \, b^{2} c^{3} d^{2} x^{2} - 24 \, a b c^{2} d^{3} x^{2} + 8 \, a^{2} c d^{4} x^{2} + 14 \, b^{2} c^{4} d - 17 \, a b c^{3} d^{2} + 6 \, a^{2} c^{2} d^{3}}{4 \, {\left (b^{3} c^{6} - 3 \, a b^{2} c^{5} d + 3 \, a^{2} b c^{4} d^{2} - a^{3} c^{3} d^{3}\right )} {\left (d x^{2} + c\right )}^{2}} + \frac {\log \left (x^{2}\right )}{2 \, a c^{3}} \]

[In]

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

[Out]

-1/2*b^4*log(abs(b*x^2 + a))/(a*b^4*c^3 - 3*a^2*b^3*c^2*d + 3*a^3*b^2*c*d^2 - a^4*b*d^3) + 1/2*(3*b^2*c^2*d^2
- 3*a*b*c*d^3 + a^2*d^4)*log(abs(d*x^2 + c))/(b^3*c^6*d - 3*a*b^2*c^5*d^2 + 3*a^2*b*c^4*d^3 - a^3*c^3*d^4) - 1
/4*(9*b^2*c^2*d^3*x^4 - 9*a*b*c*d^4*x^4 + 3*a^2*d^5*x^4 + 22*b^2*c^3*d^2*x^2 - 24*a*b*c^2*d^3*x^2 + 8*a^2*c*d^
4*x^2 + 14*b^2*c^4*d - 17*a*b*c^3*d^2 + 6*a^2*c^2*d^3)/((b^3*c^6 - 3*a*b^2*c^5*d + 3*a^2*b*c^4*d^2 - a^3*c^3*d
^3)*(d*x^2 + c)^2) + 1/2*log(x^2)/(a*c^3)

Mupad [B] (verification not implemented)

Time = 6.49 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.65 \[ \int \frac {1}{x \left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx=\frac {\frac {3\,a\,d^2-5\,b\,c\,d}{4\,c\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {d^2\,x^2\,\left (a\,d-2\,b\,c\right )}{2\,c^2\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}}{c^2+2\,c\,d\,x^2+d^2\,x^4}+\frac {b^3\,\ln \left (b\,x^2+a\right )}{2\,a^4\,d^3-6\,a^3\,b\,c\,d^2+6\,a^2\,b^2\,c^2\,d-2\,a\,b^3\,c^3}+\frac {\ln \left (x\right )}{a\,c^3}+\frac {\ln \left (d\,x^2+c\right )\,\left (a^2\,d^3-3\,a\,b\,c\,d^2+3\,b^2\,c^2\,d\right )}{-2\,a^3\,c^3\,d^3+6\,a^2\,b\,c^4\,d^2-6\,a\,b^2\,c^5\,d+2\,b^3\,c^6} \]

[In]

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

[Out]

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