\(\int \frac {1}{x^5 (a+b x^2)^2 (c+d x^2)} \, dx\) [298]

Optimal result

Integrand size = 22, antiderivative size = 160 \[ \int \frac {1}{x^5 \left (a+b x^2\right )^2 \left (c+d x^2\right )} \, dx=-\frac {1}{4 a^2 c x^4}+\frac {2 b c+a d}{2 a^3 c^2 x^2}+\frac {b^3}{2 a^3 (b c-a d) \left (a+b x^2\right )}+\frac {\left (3 b^2 c^2+2 a b c d+a^2 d^2\right ) \log (x)}{a^4 c^3}-\frac {b^3 (3 b c-4 a d) \log \left (a+b x^2\right )}{2 a^4 (b c-a d)^2}-\frac {d^4 \log \left (c+d x^2\right )}{2 c^3 (b c-a d)^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 160, 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, 90} \[ \int \frac {1}{x^5 \left (a+b x^2\right )^2 \left (c+d x^2\right )} \, dx=-\frac {b^3 (3 b c-4 a d) \log \left (a+b x^2\right )}{2 a^4 (b c-a d)^2}+\frac {b^3}{2 a^3 \left (a+b x^2\right ) (b c-a d)}+\frac {a d+2 b c}{2 a^3 c^2 x^2}-\frac {1}{4 a^2 c x^4}+\frac {\log (x) \left (a^2 d^2+2 a b c d+3 b^2 c^2\right )}{a^4 c^3}-\frac {d^4 \log \left (c+d x^2\right )}{2 c^3 (b c-a d)^2} \]

[In]

Int[1/(x^5*(a + b*x^2)^2*(c + d*x^2)),x]

[Out]

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

Rule 90

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

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^3 (a+b x)^2 (c+d x)} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (\frac {1}{a^2 c x^3}+\frac {-2 b c-a d}{a^3 c^2 x^2}+\frac {3 b^2 c^2+2 a b c d+a^2 d^2}{a^4 c^3 x}+\frac {b^4}{a^3 (-b c+a d) (a+b x)^2}+\frac {b^4 (-3 b c+4 a d)}{a^4 (-b c+a d)^2 (a+b x)}-\frac {d^5}{c^3 (b c-a d)^2 (c+d x)}\right ) \, dx,x,x^2\right ) \\ & = -\frac {1}{4 a^2 c x^4}+\frac {2 b c+a d}{2 a^3 c^2 x^2}+\frac {b^3}{2 a^3 (b c-a d) \left (a+b x^2\right )}+\frac {\left (3 b^2 c^2+2 a b c d+a^2 d^2\right ) \log (x)}{a^4 c^3}-\frac {b^3 (3 b c-4 a d) \log \left (a+b x^2\right )}{2 a^4 (b c-a d)^2}-\frac {d^4 \log \left (c+d x^2\right )}{2 c^3 (b c-a d)^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.97 \[ \int \frac {1}{x^5 \left (a+b x^2\right )^2 \left (c+d x^2\right )} \, dx=\frac {1}{4} \left (-\frac {1}{a^2 c x^4}+\frac {4 b c+2 a d}{a^3 c^2 x^2}-\frac {2 b^3}{a^3 (-b c+a d) \left (a+b x^2\right )}+\frac {4 \left (3 b^2 c^2+2 a b c d+a^2 d^2\right ) \log (x)}{a^4 c^3}+\frac {2 b^3 (-3 b c+4 a d) \log \left (a+b x^2\right )}{a^4 (b c-a d)^2}-\frac {2 d^4 \log \left (c+d x^2\right )}{c^3 (b c-a d)^2}\right ) \]

[In]

Integrate[1/(x^5*(a + b*x^2)^2*(c + d*x^2)),x]

[Out]

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

Maple [A] (verified)

Time = 2.73 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.96

method	result	size
default	\(-\frac {1}{4 a^{2} c \,x^{4}}-\frac {-a d -2 b c}{2 x^{2} a^{3} c^{2}}+\frac {\left (a^{2} d^{2}+2 a b c d +3 b^{2} c^{2}\right ) \ln \left (x \right )}{a^{4} c^{3}}+\frac {b^{4} \left (\frac {\left (4 a d -3 b c \right ) \ln \left (b \,x^{2}+a \right )}{b}-\frac {\left (a d -b c \right ) a}{b \left (b \,x^{2}+a \right )}\right )}{2 a^{4} \left (a d -b c \right )^{2}}-\frac {d^{4} \ln \left (d \,x^{2}+c \right )}{2 c^{3} \left (a d -b c \right )^{2}}\)	\(153\)
norman	\(\frac {-\frac {1}{4 a c}+\frac {\left (2 a d +3 b c \right ) x^{2}}{4 a^{2} c^{2}}+\frac {\left (-a^{2} b \,d^{2}-a \,b^{2} c d +3 c^{2} b^{3}\right ) b \,x^{6}}{2 c^{2} a^{4} \left (a d -b c \right )}}{x^{4} \left (b \,x^{2}+a \right )}+\frac {\left (a^{2} d^{2}+2 a b c d +3 b^{2} c^{2}\right ) \ln \left (x \right )}{a^{4} c^{3}}-\frac {d^{4} \ln \left (d \,x^{2}+c \right )}{2 c^{3} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {b^{3} \left (4 a d -3 b c \right ) \ln \left (b \,x^{2}+a \right )}{2 a^{4} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}\)	\(210\)
risch	\(\frac {\frac {b \left (a^{2} d^{2}+a b c d -3 b^{2} c^{2}\right ) x^{4}}{2 a^{3} c^{2} \left (a d -b c \right )}+\frac {\left (2 a d +3 b c \right ) x^{2}}{4 a^{2} c^{2}}-\frac {1}{4 a c}}{x^{4} \left (b \,x^{2}+a \right )}+\frac {\ln \left (x \right ) d^{2}}{a^{2} c^{3}}+\frac {2 \ln \left (x \right ) b d}{a^{3} c^{2}}+\frac {3 \ln \left (x \right ) b^{2}}{a^{4} c}-\frac {d^{4} \ln \left (-d \,x^{2}-c \right )}{2 c^{3} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {2 b^{3} \ln \left (b \,x^{2}+a \right ) d}{a^{3} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}-\frac {3 b^{4} \ln \left (b \,x^{2}+a \right ) c}{2 a^{4} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}\)	\(246\)
parallelrisch	\(\frac {3 x^{2} a^{2} b^{3} c^{4}+12 \ln \left (x \right ) x^{6} b^{5} c^{4}+2 a^{4} b \,c^{3} d -a^{3} b^{2} c^{4}+8 \ln \left (b \,x^{2}+a \right ) x^{6} a \,b^{4} c^{3} d -16 \ln \left (x \right ) x^{4} a^{2} b^{3} c^{3} d -d^{2} c^{2} a^{5}-6 \ln \left (b \,x^{2}+a \right ) x^{6} b^{5} c^{4}+4 \ln \left (x \right ) x^{4} a^{5} d^{4}-2 \ln \left (d \,x^{2}+c \right ) x^{4} a^{5} d^{4}+8 \ln \left (b \,x^{2}+a \right ) x^{4} a^{2} b^{3} c^{3} d -2 x^{6} a^{3} b^{2} c \,d^{3}+8 x^{6} a \,b^{4} c^{3} d -6 \ln \left (b \,x^{2}+a \right ) x^{4} a \,b^{4} c^{4}+4 \ln \left (x \right ) x^{6} a^{4} b \,d^{4}-2 \ln \left (d \,x^{2}+c \right ) x^{6} a^{4} b \,d^{4}+12 \ln \left (x \right ) x^{4} a \,b^{4} c^{4}-x^{2} a^{4} b \,c^{2} d^{2}-4 x^{2} a^{3} b^{2} c^{3} d -6 x^{6} b^{5} c^{4}-16 \ln \left (x \right ) x^{6} a \,b^{4} c^{3} d +2 x^{2} a^{5} c \,d^{3}}{4 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (b \,x^{2}+a \right ) x^{4} a^{4} c^{3}}\)	\(380\)

[In]

int(1/x^5/(b*x^2+a)^2/(d*x^2+c),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 356 vs. \(2 (150) = 300\).

Time = 3.29 (sec) , antiderivative size = 356, normalized size of antiderivative = 2.22 \[ \int \frac {1}{x^5 \left (a+b x^2\right )^2 \left (c+d x^2\right )} \, dx=-\frac {a^{3} b^{2} c^{4} - 2 \, a^{4} b c^{3} d + a^{5} c^{2} d^{2} - 2 \, {\left (3 \, a b^{4} c^{4} - 4 \, a^{2} b^{3} c^{3} d + a^{4} b c d^{3}\right )} x^{4} - {\left (3 \, a^{2} b^{3} c^{4} - 4 \, a^{3} b^{2} c^{3} d - a^{4} b c^{2} d^{2} + 2 \, a^{5} c d^{3}\right )} x^{2} + 2 \, {\left ({\left (3 \, b^{5} c^{4} - 4 \, a b^{4} c^{3} d\right )} x^{6} + {\left (3 \, a b^{4} c^{4} - 4 \, a^{2} b^{3} c^{3} d\right )} x^{4}\right )} \log \left (b x^{2} + a\right ) + 2 \, {\left (a^{4} b d^{4} x^{6} + a^{5} d^{4} x^{4}\right )} \log \left (d x^{2} + c\right ) - 4 \, {\left ({\left (3 \, b^{5} c^{4} - 4 \, a b^{4} c^{3} d + a^{4} b d^{4}\right )} x^{6} + {\left (3 \, a b^{4} c^{4} - 4 \, a^{2} b^{3} c^{3} d + a^{5} d^{4}\right )} x^{4}\right )} \log \left (x\right )}{4 \, {\left ({\left (a^{4} b^{3} c^{5} - 2 \, a^{5} b^{2} c^{4} d + a^{6} b c^{3} d^{2}\right )} x^{6} + {\left (a^{5} b^{2} c^{5} - 2 \, a^{6} b c^{4} d + a^{7} c^{3} d^{2}\right )} x^{4}\right )}} \]

[In]

integrate(1/x^5/(b*x^2+a)^2/(d*x^2+c),x, algorithm="fricas")

[Out]

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

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{x^5 \left (a+b x^2\right )^2 \left (c+d x^2\right )} \, dx=\text {Timed out} \]

[In]

integrate(1/x**5/(b*x**2+a)**2/(d*x**2+c),x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.61 \[ \int \frac {1}{x^5 \left (a+b x^2\right )^2 \left (c+d x^2\right )} \, dx=-\frac {d^{4} \log \left (d x^{2} + c\right )}{2 \, {\left (b^{2} c^{5} - 2 \, a b c^{4} d + a^{2} c^{3} d^{2}\right )}} - \frac {{\left (3 \, b^{4} c - 4 \, a b^{3} d\right )} \log \left (b x^{2} + a\right )}{2 \, {\left (a^{4} b^{2} c^{2} - 2 \, a^{5} b c d + a^{6} d^{2}\right )}} - \frac {a^{2} b c^{2} - a^{3} c d - 2 \, {\left (3 \, b^{3} c^{2} - a b^{2} c d - a^{2} b d^{2}\right )} x^{4} - {\left (3 \, a b^{2} c^{2} - a^{2} b c d - 2 \, a^{3} d^{2}\right )} x^{2}}{4 \, {\left ({\left (a^{3} b^{2} c^{3} - a^{4} b c^{2} d\right )} x^{6} + {\left (a^{4} b c^{3} - a^{5} c^{2} d\right )} x^{4}\right )}} + \frac {{\left (3 \, b^{2} c^{2} + 2 \, a b c d + a^{2} d^{2}\right )} \log \left (x^{2}\right )}{2 \, a^{4} c^{3}} \]

[In]

integrate(1/x^5/(b*x^2+a)^2/(d*x^2+c),x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.76 \[ \int \frac {1}{x^5 \left (a+b x^2\right )^2 \left (c+d x^2\right )} \, dx=-\frac {d^{5} \log \left ({\left | d x^{2} + c \right |}\right )}{2 \, {\left (b^{2} c^{5} d - 2 \, a b c^{4} d^{2} + a^{2} c^{3} d^{3}\right )}} - \frac {{\left (3 \, b^{5} c - 4 \, a b^{4} d\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, {\left (a^{4} b^{3} c^{2} - 2 \, a^{5} b^{2} c d + a^{6} b d^{2}\right )}} + \frac {3 \, b^{5} c x^{2} - 4 \, a b^{4} d x^{2} + 4 \, a b^{4} c - 5 \, a^{2} b^{3} d}{2 \, {\left (a^{4} b^{2} c^{2} - 2 \, a^{5} b c d + a^{6} d^{2}\right )} {\left (b x^{2} + a\right )}} + \frac {{\left (3 \, b^{2} c^{2} + 2 \, a b c d + a^{2} d^{2}\right )} \log \left (x^{2}\right )}{2 \, a^{4} c^{3}} - \frac {9 \, b^{2} c^{2} x^{4} + 6 \, a b c d x^{4} + 3 \, a^{2} d^{2} x^{4} - 4 \, a b c^{2} x^{2} - 2 \, a^{2} c d x^{2} + a^{2} c^{2}}{4 \, a^{4} c^{3} x^{4}} \]

[In]

integrate(1/x^5/(b*x^2+a)^2/(d*x^2+c),x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 6.02 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.36 \[ \int \frac {1}{x^5 \left (a+b x^2\right )^2 \left (c+d x^2\right )} \, dx=\frac {\frac {x^2\,\left (2\,a\,d+3\,b\,c\right )}{4\,a^2\,c^2}-\frac {1}{4\,a\,c}+\frac {x^4\,\left (a^2\,b\,d^2+a\,b^2\,c\,d-3\,b^3\,c^2\right )}{2\,a^3\,c^2\,\left (a\,d-b\,c\right )}}{b\,x^6+a\,x^4}-\frac {\ln \left (b\,x^2+a\right )\,\left (3\,b^4\,c-4\,a\,b^3\,d\right )}{2\,a^6\,d^2-4\,a^5\,b\,c\,d+2\,a^4\,b^2\,c^2}-\frac {d^4\,\ln \left (d\,x^2+c\right )}{2\,\left (a^2\,c^3\,d^2-2\,a\,b\,c^4\,d+b^2\,c^5\right )}+\frac {\ln \left (x\right )\,\left (a^2\,d^2+2\,a\,b\,c\,d+3\,b^2\,c^2\right )}{a^4\,c^3} \]

[In]

int(1/(x^5*(a + b*x^2)^2*(c + d*x^2)),x)

[Out]

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