3.4.15 \(\int \frac {1}{x (a+b x^2)^2 (c+d x^2)^3} \, dx\) [315]

Optimal. Leaf size=192 \[ \frac {b^3}{2 a (b c-a d)^3 \left (a+b x^2\right )}+\frac {d^2}{4 c (b c-a d)^2 \left (c+d x^2\right )^2}+\frac {d^2 (3 b c-a d)}{2 c^2 (b c-a d)^3 \left (c+d x^2\right )}+\frac {\log (x)}{a^2 c^3}-\frac {b^3 (b c-4 a d) \log \left (a+b x^2\right )}{2 a^2 (b c-a d)^4}-\frac {d^2 \left (6 b^2 c^2-4 a b c d+a^2 d^2\right ) \log \left (c+d x^2\right )}{2 c^3 (b c-a d)^4} \]

[Out]

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

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Rubi [A]
time = 0.17, antiderivative size = 192, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {457, 90} \begin {gather*} -\frac {b^3 (b c-4 a d) \log \left (a+b x^2\right )}{2 a^2 (b c-a d)^4}-\frac {d^2 \left (a^2 d^2-4 a b c d+6 b^2 c^2\right ) \log \left (c+d x^2\right )}{2 c^3 (b c-a d)^4}+\frac {\log (x)}{a^2 c^3}+\frac {b^3}{2 a \left (a+b x^2\right ) (b c-a d)^3}+\frac {d^2 (3 b c-a d)}{2 c^2 \left (c+d x^2\right ) (b c-a d)^3}+\frac {d^2}{4 c \left (c+d x^2\right )^2 (b c-a d)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

b^3/(2*a*(b*c - a*d)^3*(a + b*x^2)) + d^2/(4*c*(b*c - a*d)^2*(c + d*x^2)^2) + (d^2*(3*b*c - a*d))/(2*c^2*(b*c
- a*d)^3*(c + d*x^2)) + Log[x]/(a^2*c^3) - (b^3*(b*c - 4*a*d)*Log[a + b*x^2])/(2*a^2*(b*c - a*d)^4) - (d^2*(6*
b^2*c^2 - 4*a*b*c*d + a^2*d^2)*Log[c + d*x^2])/(2*c^3*(b*c - a*d)^4)

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*} \int \frac {1}{x \left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{x (a+b x)^2 (c+d x)^3} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (\frac {1}{a^2 c^3 x}+\frac {b^4}{a (-b c+a d)^3 (a+b x)^2}+\frac {b^4 (-b c+4 a d)}{a^2 (-b c+a d)^4 (a+b x)}-\frac {d^3}{c (b c-a d)^2 (c+d x)^3}-\frac {d^3 (3 b c-a d)}{c^2 (b c-a d)^3 (c+d x)^2}-\frac {d^3 \left (6 b^2 c^2-4 a b c d+a^2 d^2\right )}{c^3 (b c-a d)^4 (c+d x)}\right ) \, dx,x,x^2\right )\\ &=\frac {b^3}{2 a (b c-a d)^3 \left (a+b x^2\right )}+\frac {d^2}{4 c (b c-a d)^2 \left (c+d x^2\right )^2}+\frac {d^2 (3 b c-a d)}{2 c^2 (b c-a d)^3 \left (c+d x^2\right )}+\frac {\log (x)}{a^2 c^3}-\frac {b^3 (b c-4 a d) \log \left (a+b x^2\right )}{2 a^2 (b c-a d)^4}-\frac {d^2 \left (6 b^2 c^2-4 a b c d+a^2 d^2\right ) \log \left (c+d x^2\right )}{2 c^3 (b c-a d)^4}\\ \end {align*}

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Mathematica [A]
time = 0.20, size = 187, normalized size = 0.97 \begin {gather*} \frac {1}{4} \left (-\frac {2 b^3}{a (-b c+a d)^3 \left (a+b x^2\right )}+\frac {d^2}{c (b c-a d)^2 \left (c+d x^2\right )^2}+\frac {2 d^2 (3 b c-a d)}{c^2 (b c-a d)^3 \left (c+d x^2\right )}+\frac {4 \log (x)}{a^2 c^3}+\frac {2 b^3 (-b c+4 a d) \log \left (a+b x^2\right )}{a^2 (b c-a d)^4}-\frac {2 d^2 \left (6 b^2 c^2-4 a b c d+a^2 d^2\right ) \log \left (c+d x^2\right )}{c^3 (b c-a d)^4}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-2*b^3)/(a*(-(b*c) + a*d)^3*(a + b*x^2)) + d^2/(c*(b*c - a*d)^2*(c + d*x^2)^2) + (2*d^2*(3*b*c - a*d))/(c^2*
(b*c - a*d)^3*(c + d*x^2)) + (4*Log[x])/(a^2*c^3) + (2*b^3*(-(b*c) + 4*a*d)*Log[a + b*x^2])/(a^2*(b*c - a*d)^4
) - (2*d^2*(6*b^2*c^2 - 4*a*b*c*d + a^2*d^2)*Log[c + d*x^2])/(c^3*(b*c - a*d)^4))/4

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Maple [A]
time = 0.22, size = 202, normalized size = 1.05

method	result	size
default	\(\frac {b^{4} \left (\frac {\left (4 a d -b c \right ) \ln \left (b \,x^{2}+a \right )}{b}-\frac {a \left (a d -b c \right )}{b \left (b \,x^{2}+a \right )}\right )}{2 a^{2} \left (a d -b c \right )^{4}}-\frac {d^{3} \left (-\frac {c \left (a^{2} d^{2}-4 a b c d +3 b^{2} c^{2}\right )}{d \left (d \,x^{2}+c \right )}+\frac {\left (a^{2} d^{2}-4 a b c d +6 b^{2} c^{2}\right ) \ln \left (d \,x^{2}+c \right )}{d}-\frac {c^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{2 d \left (d \,x^{2}+c \right )^{2}}\right )}{2 c^{3} \left (a d -b c \right )^{4}}+\frac {\ln \left (x \right )}{a^{2} c^{3}}\)	\(202\)
norman	\(\frac {\frac {\left (-2 a^{4} d^{4}+4 a^{3} b c \,d^{3}+b^{4} c^{4}\right ) x^{2}}{2 c^{2} a^{2} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {d \left (-3 a^{4} d^{4}+3 a^{3} b c \,d^{3}+8 a^{2} b^{2} c^{2} d^{2}+4 b^{4} c^{4}\right ) x^{4}}{4 c^{3} a^{2} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {d^{2} b \left (-3 a^{3} d^{3}+7 a^{2} b c \,d^{2}+2 b^{3} c^{3}\right ) x^{6}}{4 c^{3} a^{2} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}}{\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )^{2}}+\frac {\ln \left (x \right )}{a^{2} c^{3}}+\frac {b^{3} \left (4 a d -b c \right ) \ln \left (b \,x^{2}+a \right )}{2 a^{2} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}-\frac {d^{2} \left (a^{2} d^{2}-4 a b c d +6 b^{2} c^{2}\right ) \ln \left (d \,x^{2}+c \right )}{2 c^{3} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}\)	\(442\)
risch	\(\frac {\frac {b \,d^{2} \left (a^{2} d^{2}-3 a b c d -b^{2} c^{2}\right ) x^{4}}{2 c^{2} a \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {d \left (2 a^{3} d^{3}-3 a^{2} b c \,d^{2}-7 a \,b^{2} c^{2} d -4 b^{3} c^{3}\right ) x^{2}}{4 a \,c^{2} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {3 a^{3} d^{3}-7 a^{2} b c \,d^{2}-2 b^{3} c^{3}}{4 a c \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}}{\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )^{2}}+\frac {\ln \left (x \right )}{a^{2} c^{3}}+\frac {2 b^{3} \ln \left (b \,x^{2}+a \right ) d}{a \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}-\frac {b^{4} \ln \left (b \,x^{2}+a \right ) c}{2 a^{2} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}-\frac {d^{4} \ln \left (-d \,x^{2}-c \right ) a^{2}}{2 c^{3} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}+\frac {2 d^{3} \ln \left (-d \,x^{2}-c \right ) a b}{c^{2} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}-\frac {3 d^{2} \ln \left (-d \,x^{2}-c \right ) b^{2}}{c \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}\)	\(620\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 527 vs. \(2 (182) = 364\).
time = 0.35, size = 527, normalized size = 2.74 \begin {gather*} -\frac {{\left (b^{4} c - 4 \, a b^{3} d\right )} \log \left (b x^{2} + a\right )}{2 \, {\left (a^{2} b^{4} c^{4} - 4 \, a^{3} b^{3} c^{3} d + 6 \, a^{4} b^{2} c^{2} d^{2} - 4 \, a^{5} b c d^{3} + a^{6} d^{4}\right )}} - \frac {{\left (6 \, b^{2} c^{2} d^{2} - 4 \, a b c d^{3} + a^{2} d^{4}\right )} \log \left (d x^{2} + c\right )}{2 \, {\left (b^{4} c^{7} - 4 \, a b^{3} c^{6} d + 6 \, a^{2} b^{2} c^{5} d^{2} - 4 \, a^{3} b c^{4} d^{3} + a^{4} c^{3} d^{4}\right )}} + \frac {2 \, b^{3} c^{4} + 7 \, a^{2} b c^{2} d^{2} - 3 \, a^{3} c d^{3} + 2 \, {\left (b^{3} c^{2} d^{2} + 3 \, a b^{2} c d^{3} - a^{2} b d^{4}\right )} x^{4} + {\left (4 \, b^{3} c^{3} d + 7 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3} - 2 \, a^{3} d^{4}\right )} x^{2}}{4 \, {\left (a^{2} b^{3} c^{7} - 3 \, a^{3} b^{2} c^{6} d + 3 \, a^{4} b c^{5} d^{2} - a^{5} c^{4} d^{3} + {\left (a b^{4} c^{5} d^{2} - 3 \, a^{2} b^{3} c^{4} d^{3} + 3 \, a^{3} b^{2} c^{3} d^{4} - a^{4} b c^{2} d^{5}\right )} x^{6} + {\left (2 \, a b^{4} c^{6} d - 5 \, a^{2} b^{3} c^{5} d^{2} + 3 \, a^{3} b^{2} c^{4} d^{3} + a^{4} b c^{3} d^{4} - a^{5} c^{2} d^{5}\right )} x^{4} + {\left (a b^{4} c^{7} - a^{2} b^{3} c^{6} d - 3 \, a^{3} b^{2} c^{5} d^{2} + 5 \, a^{4} b c^{4} d^{3} - 2 \, a^{5} c^{3} d^{4}\right )} x^{2}\right )}} + \frac {\log \left (x^{2}\right )}{2 \, a^{2} c^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1058 vs. \(2 (182) = 364\).
time = 12.13, size = 1058, normalized size = 5.51 \begin {gather*} \frac {2 \, a b^{4} c^{6} - 2 \, a^{2} b^{3} c^{5} d + 7 \, a^{3} b^{2} c^{4} d^{2} - 10 \, a^{4} b c^{3} d^{3} + 3 \, a^{5} c^{2} d^{4} + 2 \, {\left (a b^{4} c^{4} d^{2} + 2 \, a^{2} b^{3} c^{3} d^{3} - 4 \, a^{3} b^{2} c^{2} d^{4} + a^{4} b c d^{5}\right )} x^{4} + {\left (4 \, a b^{4} c^{5} d + 3 \, a^{2} b^{3} c^{4} d^{2} - 4 \, a^{3} b^{2} c^{3} d^{3} - 5 \, a^{4} b c^{2} d^{4} + 2 \, a^{5} c d^{5}\right )} x^{2} - 2 \, {\left (a b^{4} c^{6} - 4 \, a^{2} b^{3} c^{5} d + {\left (b^{5} c^{4} d^{2} - 4 \, a b^{4} c^{3} d^{3}\right )} x^{6} + {\left (2 \, b^{5} c^{5} d - 7 \, a b^{4} c^{4} d^{2} - 4 \, a^{2} b^{3} c^{3} d^{3}\right )} x^{4} + {\left (b^{5} c^{6} - 2 \, a b^{4} c^{5} d - 8 \, a^{2} b^{3} c^{4} d^{2}\right )} x^{2}\right )} \log \left (b x^{2} + a\right ) - 2 \, {\left (6 \, a^{3} b^{2} c^{4} d^{2} - 4 \, a^{4} b c^{3} d^{3} + a^{5} c^{2} d^{4} + {\left (6 \, a^{2} b^{3} c^{2} d^{4} - 4 \, a^{3} b^{2} c d^{5} + a^{4} b d^{6}\right )} x^{6} + {\left (12 \, a^{2} b^{3} c^{3} d^{3} - 2 \, a^{3} b^{2} c^{2} d^{4} - 2 \, a^{4} b c d^{5} + a^{5} d^{6}\right )} x^{4} + {\left (6 \, a^{2} b^{3} c^{4} d^{2} + 8 \, a^{3} b^{2} c^{3} d^{3} - 7 \, a^{4} b c^{2} d^{4} + 2 \, a^{5} c d^{5}\right )} x^{2}\right )} \log \left (d x^{2} + c\right ) + 4 \, {\left (a b^{4} c^{6} - 4 \, a^{2} b^{3} c^{5} d + 6 \, a^{3} b^{2} c^{4} d^{2} - 4 \, a^{4} b c^{3} d^{3} + a^{5} c^{2} d^{4} + {\left (b^{5} c^{4} d^{2} - 4 \, a b^{4} c^{3} d^{3} + 6 \, a^{2} b^{3} c^{2} d^{4} - 4 \, a^{3} b^{2} c d^{5} + a^{4} b d^{6}\right )} x^{6} + {\left (2 \, b^{5} c^{5} d - 7 \, a b^{4} c^{4} d^{2} + 8 \, a^{2} b^{3} c^{3} d^{3} - 2 \, a^{3} b^{2} c^{2} d^{4} - 2 \, a^{4} b c d^{5} + a^{5} d^{6}\right )} x^{4} + {\left (b^{5} c^{6} - 2 \, a b^{4} c^{5} d - 2 \, a^{2} b^{3} c^{4} d^{2} + 8 \, a^{3} b^{2} c^{3} d^{3} - 7 \, a^{4} b c^{2} d^{4} + 2 \, a^{5} c d^{5}\right )} x^{2}\right )} \log \left (x\right )}{4 \, {\left (a^{3} b^{4} c^{9} - 4 \, a^{4} b^{3} c^{8} d + 6 \, a^{5} b^{2} c^{7} d^{2} - 4 \, a^{6} b c^{6} d^{3} + a^{7} c^{5} d^{4} + {\left (a^{2} b^{5} c^{7} d^{2} - 4 \, a^{3} b^{4} c^{6} d^{3} + 6 \, a^{4} b^{3} c^{5} d^{4} - 4 \, a^{5} b^{2} c^{4} d^{5} + a^{6} b c^{3} d^{6}\right )} x^{6} + {\left (2 \, a^{2} b^{5} c^{8} d - 7 \, a^{3} b^{4} c^{7} d^{2} + 8 \, a^{4} b^{3} c^{6} d^{3} - 2 \, a^{5} b^{2} c^{5} d^{4} - 2 \, a^{6} b c^{4} d^{5} + a^{7} c^{3} d^{6}\right )} x^{4} + {\left (a^{2} b^{5} c^{9} - 2 \, a^{3} b^{4} c^{8} d - 2 \, a^{4} b^{3} c^{7} d^{2} + 8 \, a^{5} b^{2} c^{6} d^{3} - 7 \, a^{6} b c^{5} d^{4} + 2 \, a^{7} c^{4} d^{5}\right )} x^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(2*a*b^4*c^6 - 2*a^2*b^3*c^5*d + 7*a^3*b^2*c^4*d^2 - 10*a^4*b*c^3*d^3 + 3*a^5*c^2*d^4 + 2*(a*b^4*c^4*d^2 +
2*a^2*b^3*c^3*d^3 - 4*a^3*b^2*c^2*d^4 + a^4*b*c*d^5)*x^4 + (4*a*b^4*c^5*d + 3*a^2*b^3*c^4*d^2 - 4*a^3*b^2*c^3
*d^3 - 5*a^4*b*c^2*d^4 + 2*a^5*c*d^5)*x^2 - 2*(a*b^4*c^6 - 4*a^2*b^3*c^5*d + (b^5*c^4*d^2 - 4*a*b^4*c^3*d^3)*x
^6 + (2*b^5*c^5*d - 7*a*b^4*c^4*d^2 - 4*a^2*b^3*c^3*d^3)*x^4 + (b^5*c^6 - 2*a*b^4*c^5*d - 8*a^2*b^3*c^4*d^2)*x
^2)*log(b*x^2 + a) - 2*(6*a^3*b^2*c^4*d^2 - 4*a^4*b*c^3*d^3 + a^5*c^2*d^4 + (6*a^2*b^3*c^2*d^4 - 4*a^3*b^2*c*d
^5 + a^4*b*d^6)*x^6 + (12*a^2*b^3*c^3*d^3 - 2*a^3*b^2*c^2*d^4 - 2*a^4*b*c*d^5 + a^5*d^6)*x^4 + (6*a^2*b^3*c^4*
d^2 + 8*a^3*b^2*c^3*d^3 - 7*a^4*b*c^2*d^4 + 2*a^5*c*d^5)*x^2)*log(d*x^2 + c) + 4*(a*b^4*c^6 - 4*a^2*b^3*c^5*d
+ 6*a^3*b^2*c^4*d^2 - 4*a^4*b*c^3*d^3 + a^5*c^2*d^4 + (b^5*c^4*d^2 - 4*a*b^4*c^3*d^3 + 6*a^2*b^3*c^2*d^4 - 4*a
^3*b^2*c*d^5 + a^4*b*d^6)*x^6 + (2*b^5*c^5*d - 7*a*b^4*c^4*d^2 + 8*a^2*b^3*c^3*d^3 - 2*a^3*b^2*c^2*d^4 - 2*a^4
*b*c*d^5 + a^5*d^6)*x^4 + (b^5*c^6 - 2*a*b^4*c^5*d - 2*a^2*b^3*c^4*d^2 + 8*a^3*b^2*c^3*d^3 - 7*a^4*b*c^2*d^4 +
2*a^5*c*d^5)*x^2)*log(x))/(a^3*b^4*c^9 - 4*a^4*b^3*c^8*d + 6*a^5*b^2*c^7*d^2 - 4*a^6*b*c^6*d^3 + a^7*c^5*d^4
+ (a^2*b^5*c^7*d^2 - 4*a^3*b^4*c^6*d^3 + 6*a^4*b^3*c^5*d^4 - 4*a^5*b^2*c^4*d^5 + a^6*b*c^3*d^6)*x^6 + (2*a^2*b
^5*c^8*d - 7*a^3*b^4*c^7*d^2 + 8*a^4*b^3*c^6*d^3 - 2*a^5*b^2*c^5*d^4 - 2*a^6*b*c^4*d^5 + a^7*c^3*d^6)*x^4 + (a
^2*b^5*c^9 - 2*a^3*b^4*c^8*d - 2*a^4*b^3*c^7*d^2 + 8*a^5*b^2*c^6*d^3 - 7*a^6*b*c^5*d^4 + 2*a^7*c^4*d^5)*x^2)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 470 vs. \(2 (182) = 364\).
time = 0.66, size = 470, normalized size = 2.45 \begin {gather*} -\frac {{\left (b^{5} c - 4 \, a b^{4} d\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, {\left (a^{2} b^{5} c^{4} - 4 \, a^{3} b^{4} c^{3} d + 6 \, a^{4} b^{3} c^{2} d^{2} - 4 \, a^{5} b^{2} c d^{3} + a^{6} b d^{4}\right )}} - \frac {{\left (6 \, b^{2} c^{2} d^{3} - 4 \, a b c d^{4} + a^{2} d^{5}\right )} \log \left ({\left | d x^{2} + c \right |}\right )}{2 \, {\left (b^{4} c^{7} d - 4 \, a b^{3} c^{6} d^{2} + 6 \, a^{2} b^{2} c^{5} d^{3} - 4 \, a^{3} b c^{4} d^{4} + a^{4} c^{3} d^{5}\right )}} + \frac {b^{5} c x^{2} - 4 \, a b^{4} d x^{2} + 2 \, a b^{4} c - 5 \, a^{2} b^{3} d}{2 \, {\left (a^{2} b^{4} c^{4} - 4 \, a^{3} b^{3} c^{3} d + 6 \, a^{4} b^{2} c^{2} d^{2} - 4 \, a^{5} b c d^{3} + a^{6} d^{4}\right )} {\left (b x^{2} + a\right )}} + \frac {18 \, b^{2} c^{2} d^{4} x^{4} - 12 \, a b c d^{5} x^{4} + 3 \, a^{2} d^{6} x^{4} + 42 \, b^{2} c^{3} d^{3} x^{2} - 32 \, a b c^{2} d^{4} x^{2} + 8 \, a^{2} c d^{5} x^{2} + 25 \, b^{2} c^{4} d^{2} - 22 \, a b c^{3} d^{3} + 6 \, a^{2} c^{2} d^{4}}{4 \, {\left (b^{4} c^{7} - 4 \, a b^{3} c^{6} d + 6 \, a^{2} b^{2} c^{5} d^{2} - 4 \, a^{3} b c^{4} d^{3} + a^{4} c^{3} d^{4}\right )} {\left (d x^{2} + c\right )}^{2}} + \frac {\log \left (x^{2}\right )}{2 \, a^{2} c^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(b^5*c - 4*a*b^4*d)*log(abs(b*x^2 + a))/(a^2*b^5*c^4 - 4*a^3*b^4*c^3*d + 6*a^4*b^3*c^2*d^2 - 4*a^5*b^2*c*
d^3 + a^6*b*d^4) - 1/2*(6*b^2*c^2*d^3 - 4*a*b*c*d^4 + a^2*d^5)*log(abs(d*x^2 + c))/(b^4*c^7*d - 4*a*b^3*c^6*d^
2 + 6*a^2*b^2*c^5*d^3 - 4*a^3*b*c^4*d^4 + a^4*c^3*d^5) + 1/2*(b^5*c*x^2 - 4*a*b^4*d*x^2 + 2*a*b^4*c - 5*a^2*b^
3*d)/((a^2*b^4*c^4 - 4*a^3*b^3*c^3*d + 6*a^4*b^2*c^2*d^2 - 4*a^5*b*c*d^3 + a^6*d^4)*(b*x^2 + a)) + 1/4*(18*b^2
*c^2*d^4*x^4 - 12*a*b*c*d^5*x^4 + 3*a^2*d^6*x^4 + 42*b^2*c^3*d^3*x^2 - 32*a*b*c^2*d^4*x^2 + 8*a^2*c*d^5*x^2 +
25*b^2*c^4*d^2 - 22*a*b*c^3*d^3 + 6*a^2*c^2*d^4)/((b^4*c^7 - 4*a*b^3*c^6*d + 6*a^2*b^2*c^5*d^2 - 4*a^3*b*c^4*d
^3 + a^4*c^3*d^4)*(d*x^2 + c)^2) + 1/2*log(x^2)/(a^2*c^3)

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Mupad [B]
time = 1.11, size = 472, normalized size = 2.46 \begin {gather*} \frac {\ln \left (x\right )}{a^2\,c^3}-\frac {\ln \left (b\,x^2+a\right )\,\left (b^4\,c-4\,a\,b^3\,d\right )}{2\,a^6\,d^4-8\,a^5\,b\,c\,d^3+12\,a^4\,b^2\,c^2\,d^2-8\,a^3\,b^3\,c^3\,d+2\,a^2\,b^4\,c^4}-\frac {\ln \left (d\,x^2+c\right )\,\left (a^2\,d^4-4\,a\,b\,c\,d^3+6\,b^2\,c^2\,d^2\right )}{2\,a^4\,c^3\,d^4-8\,a^3\,b\,c^4\,d^3+12\,a^2\,b^2\,c^5\,d^2-8\,a\,b^3\,c^6\,d+2\,b^4\,c^7}-\frac {\frac {-3\,a^3\,d^3+7\,a^2\,b\,c\,d^2+2\,b^3\,c^3}{4\,a\,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 )}+\frac {x^2\,\left (-2\,a^3\,d^4+3\,a^2\,b\,c\,d^3+7\,a\,b^2\,c^2\,d^2+4\,b^3\,c^3\,d\right )}{4\,a\,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 {b\,d^2\,x^4\,\left (-a^2\,d^2+3\,a\,b\,c\,d+b^2\,c^2\right )}{2\,a\,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 )}}{a\,c^2+x^2\,\left (b\,c^2+2\,a\,d\,c\right )+x^4\,\left (a\,d^2+2\,b\,c\,d\right )+b\,d^2\,x^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x)/(a^2*c^3) - (log(a + b*x^2)*(b^4*c - 4*a*b^3*d))/(2*a^6*d^4 + 2*a^2*b^4*c^4 - 8*a^3*b^3*c^3*d + 12*a^4*
b^2*c^2*d^2 - 8*a^5*b*c*d^3) - (log(c + d*x^2)*(a^2*d^4 + 6*b^2*c^2*d^2 - 4*a*b*c*d^3))/(2*b^4*c^7 + 2*a^4*c^3
*d^4 - 8*a^3*b*c^4*d^3 + 12*a^2*b^2*c^5*d^2 - 8*a*b^3*c^6*d) - ((2*b^3*c^3 - 3*a^3*d^3 + 7*a^2*b*c*d^2)/(4*a*c
*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)) + (x^2*(4*b^3*c^3*d - 2*a^3*d^4 + 7*a*b^2*c^2*d^2 + 3*a^
2*b*c*d^3))/(4*a*c^2*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)) + (b*d^2*x^4*(b^2*c^2 - a^2*d^2 + 3*
a*b*c*d))/(2*a*c^2*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)))/(a*c^2 + x^2*(b*c^2 + 2*a*c*d) + x^4*
(a*d^2 + 2*b*c*d) + b*d^2*x^6)

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