3.4.17 \(\int \frac {1}{x^3 (a+b x^2)^2 (c+d x^2)^3} \, dx\) [317]

Optimal. Leaf size=215 \[ -\frac {1}{2 a^2 c^3 x^2}-\frac {b^4}{2 a^2 (b c-a d)^3 \left (a+b x^2\right )}-\frac {d^3}{4 c^2 (b c-a d)^2 \left (c+d x^2\right )^2}-\frac {d^3 (2 b c-a d)}{c^3 (b c-a d)^3 \left (c+d x^2\right )}-\frac {(2 b c+3 a d) \log (x)}{a^3 c^4}+\frac {b^4 (2 b c-5 a d) \log \left (a+b x^2\right )}{2 a^3 (b c-a d)^4}+\frac {d^3 \left (10 b^2 c^2-10 a b c d+3 a^2 d^2\right ) \log \left (c+d x^2\right )}{2 c^4 (b c-a d)^4} \]

[Out]

-1/2/a^2/c^3/x^2-1/2*b^4/a^2/(-a*d+b*c)^3/(b*x^2+a)-1/4*d^3/c^2/(-a*d+b*c)^2/(d*x^2+c)^2-d^3*(-a*d+2*b*c)/c^3/
(-a*d+b*c)^3/(d*x^2+c)-(3*a*d+2*b*c)*ln(x)/a^3/c^4+1/2*b^4*(-5*a*d+2*b*c)*ln(b*x^2+a)/a^3/(-a*d+b*c)^4+1/2*d^3
*(3*a^2*d^2-10*a*b*c*d+10*b^2*c^2)*ln(d*x^2+c)/c^4/(-a*d+b*c)^4

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Rubi [A]
time = 0.20, antiderivative size = 215, 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^4 (2 b c-5 a d) \log \left (a+b x^2\right )}{2 a^3 (b c-a d)^4}-\frac {\log (x) (3 a d+2 b c)}{a^3 c^4}-\frac {b^4}{2 a^2 \left (a+b x^2\right ) (b c-a d)^3}+\frac {d^3 \left (3 a^2 d^2-10 a b c d+10 b^2 c^2\right ) \log \left (c+d x^2\right )}{2 c^4 (b c-a d)^4}-\frac {1}{2 a^2 c^3 x^2}-\frac {d^3 (2 b c-a d)}{c^3 \left (c+d x^2\right ) (b c-a d)^3}-\frac {d^3}{4 c^2 \left (c+d x^2\right )^2 (b c-a d)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*1/(a^2*c^3*x^2) - b^4/(2*a^2*(b*c - a*d)^3*(a + b*x^2)) - d^3/(4*c^2*(b*c - a*d)^2*(c + d*x^2)^2) - (d^3*
(2*b*c - a*d))/(c^3*(b*c - a*d)^3*(c + d*x^2)) - ((2*b*c + 3*a*d)*Log[x])/(a^3*c^4) + (b^4*(2*b*c - 5*a*d)*Log
[a + b*x^2])/(2*a^3*(b*c - a*d)^4) + (d^3*(10*b^2*c^2 - 10*a*b*c*d + 3*a^2*d^2)*Log[c + d*x^2])/(2*c^4*(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^3 \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^2 (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^2}+\frac {-2 b c-3 a d}{a^3 c^4 x}-\frac {b^5}{a^2 (-b c+a d)^3 (a+b x)^2}-\frac {b^5 (-2 b c+5 a d)}{a^3 (-b c+a d)^4 (a+b x)}+\frac {d^4}{c^2 (b c-a d)^2 (c+d x)^3}+\frac {2 d^4 (2 b c-a d)}{c^3 (b c-a d)^3 (c+d x)^2}+\frac {d^4 \left (10 b^2 c^2-10 a b c d+3 a^2 d^2\right )}{c^4 (b c-a d)^4 (c+d x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {1}{2 a^2 c^3 x^2}-\frac {b^4}{2 a^2 (b c-a d)^3 \left (a+b x^2\right )}-\frac {d^3}{4 c^2 (b c-a d)^2 \left (c+d x^2\right )^2}-\frac {d^3 (2 b c-a d)}{c^3 (b c-a d)^3 \left (c+d x^2\right )}-\frac {(2 b c+3 a d) \log (x)}{a^3 c^4}+\frac {b^4 (2 b c-5 a d) \log \left (a+b x^2\right )}{2 a^3 (b c-a d)^4}+\frac {d^3 \left (10 b^2 c^2-10 a b c d+3 a^2 d^2\right ) \log \left (c+d x^2\right )}{2 c^4 (b c-a d)^4}\\ \end {align*}

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Mathematica [A]
time = 0.22, size = 208, normalized size = 0.97 \begin {gather*} \frac {1}{4} \left (-\frac {2}{a^2 c^3 x^2}+\frac {2 b^4}{a^2 (-b c+a d)^3 \left (a+b x^2\right )}-\frac {d^3}{c^2 (b c-a d)^2 \left (c+d x^2\right )^2}+\frac {4 d^3 (-2 b c+a d)}{c^3 (b c-a d)^3 \left (c+d x^2\right )}-\frac {4 (2 b c+3 a d) \log (x)}{a^3 c^4}+\frac {2 b^4 (2 b c-5 a d) \log \left (a+b x^2\right )}{a^3 (b c-a d)^4}+\frac {2 d^3 \left (10 b^2 c^2-10 a b c d+3 a^2 d^2\right ) \log \left (c+d x^2\right )}{c^4 (b c-a d)^4}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2/(a^2*c^3*x^2) + (2*b^4)/(a^2*(-(b*c) + a*d)^3*(a + b*x^2)) - d^3/(c^2*(b*c - a*d)^2*(c + d*x^2)^2) + (4*d^
3*(-2*b*c + a*d))/(c^3*(b*c - a*d)^3*(c + d*x^2)) - (4*(2*b*c + 3*a*d)*Log[x])/(a^3*c^4) + (2*b^4*(2*b*c - 5*a
*d)*Log[a + b*x^2])/(a^3*(b*c - a*d)^4) + (2*d^3*(10*b^2*c^2 - 10*a*b*c*d + 3*a^2*d^2)*Log[c + d*x^2])/(c^4*(b
*c - a*d)^4))/4

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Maple [A]
time = 0.23, size = 223, normalized size = 1.04

method	result	size
default	\(-\frac {b^{5} \left (\frac {\left (5 a d -2 b c \right ) \ln \left (b \,x^{2}+a \right )}{b}-\frac {a \left (a d -b c \right )}{b \left (b \,x^{2}+a \right )}\right )}{2 a^{3} \left (a d -b c \right )^{4}}+\frac {d^{4} \left (-\frac {2 c \left (a^{2} d^{2}-3 a b c d +2 b^{2} c^{2}\right )}{d \left (d \,x^{2}+c \right )}+\frac {\left (3 a^{2} d^{2}-10 a b c d +10 b^{2} c^{2}\right ) \ln \left (d \,x^{2}+c \right )}{d}-\frac {c^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{2 d \left (d \,x^{2}+c \right )^{2}}\right )}{2 c^{4} \left (a d -b c \right )^{4}}-\frac {1}{2 a^{2} c^{3} x^{2}}+\frac {\left (-3 a d -2 b c \right ) \ln \left (x \right )}{a^{3} c^{4}}\)	\(223\)
norman	\(\frac {-\frac {1}{2 a c}+\frac {\left (6 a^{5} d^{5}-12 a^{4} b c \,d^{4}+4 a^{3} b^{2} c^{2} d^{3}+a \,b^{4} c^{4} d -2 b^{5} c^{5}\right ) x^{4}}{2 c^{3} a^{3} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {d \left (9 a^{5} d^{5}-7 a^{4} b c \,d^{4}-18 a^{3} b^{2} c^{2} d^{3}+8 a^{2} b^{3} c^{3} d^{2}+4 a \,b^{4} c^{4} d -8 b^{5} c^{5}\right ) x^{6}}{4 c^{4} a^{3} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {d^{2} b \left (9 a^{4} d^{4}-19 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}+2 a \,b^{3} c^{3} d -4 b^{4} c^{4}\right ) x^{8}}{4 c^{4} a^{3} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}}{x^{2} \left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )^{2}}-\frac {\left (3 a d +2 b c \right ) \ln \left (x \right )}{a^{3} c^{4}}-\frac {b^{4} \left (5 a d -2 b c \right ) \ln \left (b \,x^{2}+a \right )}{2 a^{3} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}+\frac {d^{3} \left (3 a^{2} d^{2}-10 a b c d +10 b^{2} c^{2}\right ) \ln \left (d \,x^{2}+c \right )}{2 c^{4} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}\)	\(536\)
risch	\(\frac {-\frac {b \,d^{2} \left (3 a^{3} d^{3}-7 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -2 b^{3} c^{3}\right ) x^{6}}{2 a^{2} c^{3} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}-\frac {d \left (6 a^{4} d^{4}-5 a^{3} b c \,d^{3}-15 a^{2} b^{2} c^{2} d^{2}+10 a \,b^{3} c^{3} d -8 b^{4} c^{4}\right ) x^{4}}{4 a^{2} c^{3} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}-\frac {\left (9 a^{4} d^{4}-19 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}+2 a \,b^{3} c^{3} d -4 b^{4} c^{4}\right ) x^{2}}{4 a^{2} c^{2} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}-\frac {1}{2 a c}}{x^{2} \left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )^{2}}-\frac {3 \ln \left (x \right ) d}{a^{2} c^{4}}-\frac {2 \ln \left (x \right ) b}{a^{3} c^{3}}-\frac {5 b^{4} \ln \left (b \,x^{2}+a \right ) d}{2 a^{2} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}+\frac {b^{5} \ln \left (b \,x^{2}+a \right ) c}{a^{3} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}+\frac {3 d^{5} \ln \left (-d \,x^{2}-c \right ) a^{2}}{2 c^{4} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}-\frac {5 d^{4} \ln \left (-d \,x^{2}-c \right ) a b}{c^{3} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}+\frac {5 d^{3} \ln \left (-d \,x^{2}-c \right ) b^{2}}{c^{2} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}\)	\(699\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*b^5/a^3/(a*d-b*c)^4*((5*a*d-2*b*c)/b*ln(b*x^2+a)-a*(a*d-b*c)/b/(b*x^2+a))+1/2*d^4/c^4/(a*d-b*c)^4*(-2*c*(
a^2*d^2-3*a*b*c*d+2*b^2*c^2)/d/(d*x^2+c)+(3*a^2*d^2-10*a*b*c*d+10*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)-1/2/a^2/c^3/x^2+(-3*a*d-2*b*c)/a^3/c^4*ln(x)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 651 vs. \(2 (205) = 410\).
time = 0.33, size = 651, normalized size = 3.03 \begin {gather*} \frac {{\left (2 \, b^{5} c - 5 \, a b^{4} d\right )} \log \left (b x^{2} + a\right )}{2 \, {\left (a^{3} b^{4} c^{4} - 4 \, a^{4} b^{3} c^{3} d + 6 \, a^{5} b^{2} c^{2} d^{2} - 4 \, a^{6} b c d^{3} + a^{7} d^{4}\right )}} + \frac {{\left (10 \, b^{2} c^{2} d^{3} - 10 \, a b c d^{4} + 3 \, a^{2} d^{5}\right )} \log \left (d x^{2} + c\right )}{2 \, {\left (b^{4} c^{8} - 4 \, a b^{3} c^{7} d + 6 \, a^{2} b^{2} c^{6} d^{2} - 4 \, a^{3} b c^{5} d^{3} + a^{4} c^{4} d^{4}\right )}} - \frac {2 \, a b^{3} c^{5} - 6 \, a^{2} b^{2} c^{4} d + 6 \, a^{3} b c^{3} d^{2} - 2 \, a^{4} c^{2} d^{3} + 2 \, {\left (2 \, b^{4} c^{3} d^{2} - 3 \, a b^{3} c^{2} d^{3} + 7 \, a^{2} b^{2} c d^{4} - 3 \, a^{3} b d^{5}\right )} x^{6} + {\left (8 \, b^{4} c^{4} d - 10 \, a b^{3} c^{3} d^{2} + 15 \, a^{2} b^{2} c^{2} d^{3} + 5 \, a^{3} b c d^{4} - 6 \, a^{4} d^{5}\right )} x^{4} + {\left (4 \, b^{4} c^{5} - 2 \, a b^{3} c^{4} d - 6 \, a^{2} b^{2} c^{3} d^{2} + 19 \, a^{3} b c^{2} d^{3} - 9 \, a^{4} c d^{4}\right )} x^{2}}{4 \, {\left ({\left (a^{2} b^{4} c^{6} d^{2} - 3 \, a^{3} b^{3} c^{5} d^{3} + 3 \, a^{4} b^{2} c^{4} d^{4} - a^{5} b c^{3} d^{5}\right )} x^{8} + {\left (2 \, a^{2} b^{4} c^{7} d - 5 \, a^{3} b^{3} c^{6} d^{2} + 3 \, a^{4} b^{2} c^{5} d^{3} + a^{5} b c^{4} d^{4} - a^{6} c^{3} d^{5}\right )} x^{6} + {\left (a^{2} b^{4} c^{8} - a^{3} b^{3} c^{7} d - 3 \, a^{4} b^{2} c^{6} d^{2} + 5 \, a^{5} b c^{5} d^{3} - 2 \, a^{6} c^{4} d^{4}\right )} x^{4} + {\left (a^{3} b^{3} c^{8} - 3 \, a^{4} b^{2} c^{7} d + 3 \, a^{5} b c^{6} d^{2} - a^{6} c^{5} d^{3}\right )} x^{2}\right )}} - \frac {{\left (2 \, b c + 3 \, a d\right )} \log \left (x^{2}\right )}{2 \, a^{3} c^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2*b^5*c - 5*a*b^4*d)*log(b*x^2 + a)/(a^3*b^4*c^4 - 4*a^4*b^3*c^3*d + 6*a^5*b^2*c^2*d^2 - 4*a^6*b*c*d^3 +
a^7*d^4) + 1/2*(10*b^2*c^2*d^3 - 10*a*b*c*d^4 + 3*a^2*d^5)*log(d*x^2 + c)/(b^4*c^8 - 4*a*b^3*c^7*d + 6*a^2*b^2
*c^6*d^2 - 4*a^3*b*c^5*d^3 + a^4*c^4*d^4) - 1/4*(2*a*b^3*c^5 - 6*a^2*b^2*c^4*d + 6*a^3*b*c^3*d^2 - 2*a^4*c^2*d
^3 + 2*(2*b^4*c^3*d^2 - 3*a*b^3*c^2*d^3 + 7*a^2*b^2*c*d^4 - 3*a^3*b*d^5)*x^6 + (8*b^4*c^4*d - 10*a*b^3*c^3*d^2
+ 15*a^2*b^2*c^2*d^3 + 5*a^3*b*c*d^4 - 6*a^4*d^5)*x^4 + (4*b^4*c^5 - 2*a*b^3*c^4*d - 6*a^2*b^2*c^3*d^2 + 19*a
^3*b*c^2*d^3 - 9*a^4*c*d^4)*x^2)/((a^2*b^4*c^6*d^2 - 3*a^3*b^3*c^5*d^3 + 3*a^4*b^2*c^4*d^4 - a^5*b*c^3*d^5)*x^
8 + (2*a^2*b^4*c^7*d - 5*a^3*b^3*c^6*d^2 + 3*a^4*b^2*c^5*d^3 + a^5*b*c^4*d^4 - a^6*c^3*d^5)*x^6 + (a^2*b^4*c^8
- a^3*b^3*c^7*d - 3*a^4*b^2*c^6*d^2 + 5*a^5*b*c^5*d^3 - 2*a^6*c^4*d^4)*x^4 + (a^3*b^3*c^8 - 3*a^4*b^2*c^7*d +
3*a^5*b*c^6*d^2 - a^6*c^5*d^3)*x^2) - 1/2*(2*b*c + 3*a*d)*log(x^2)/(a^3*c^4)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1227 vs. \(2 (205) = 410\).
time = 25.42, size = 1227, normalized size = 5.71 \begin {gather*} -\frac {2 \, a^{2} b^{4} c^{7} - 8 \, a^{3} b^{3} c^{6} d + 12 \, a^{4} b^{2} c^{5} d^{2} - 8 \, a^{5} b c^{4} d^{3} + 2 \, a^{6} c^{3} d^{4} + 2 \, {\left (2 \, a b^{5} c^{5} d^{2} - 5 \, a^{2} b^{4} c^{4} d^{3} + 10 \, a^{3} b^{3} c^{3} d^{4} - 10 \, a^{4} b^{2} c^{2} d^{5} + 3 \, a^{5} b c d^{6}\right )} x^{6} + {\left (8 \, a b^{5} c^{6} d - 18 \, a^{2} b^{4} c^{5} d^{2} + 25 \, a^{3} b^{3} c^{4} d^{3} - 10 \, a^{4} b^{2} c^{3} d^{4} - 11 \, a^{5} b c^{2} d^{5} + 6 \, a^{6} c d^{6}\right )} x^{4} + {\left (4 \, a b^{5} c^{7} - 6 \, a^{2} b^{4} c^{6} d - 4 \, a^{3} b^{3} c^{5} d^{2} + 25 \, a^{4} b^{2} c^{4} d^{3} - 28 \, a^{5} b c^{3} d^{4} + 9 \, a^{6} c^{2} d^{5}\right )} x^{2} - 2 \, {\left ({\left (2 \, b^{6} c^{5} d^{2} - 5 \, a b^{5} c^{4} d^{3}\right )} x^{8} + {\left (4 \, b^{6} c^{6} d - 8 \, a b^{5} c^{5} d^{2} - 5 \, a^{2} b^{4} c^{4} d^{3}\right )} x^{6} + {\left (2 \, b^{6} c^{7} - a b^{5} c^{6} d - 10 \, a^{2} b^{4} c^{5} d^{2}\right )} x^{4} + {\left (2 \, a b^{5} c^{7} - 5 \, a^{2} b^{4} c^{6} d\right )} x^{2}\right )} \log \left (b x^{2} + a\right ) - 2 \, {\left ({\left (10 \, a^{3} b^{3} c^{2} d^{5} - 10 \, a^{4} b^{2} c d^{6} + 3 \, a^{5} b d^{7}\right )} x^{8} + {\left (20 \, a^{3} b^{3} c^{3} d^{4} - 10 \, a^{4} b^{2} c^{2} d^{5} - 4 \, a^{5} b c d^{6} + 3 \, a^{6} d^{7}\right )} x^{6} + {\left (10 \, a^{3} b^{3} c^{4} d^{3} + 10 \, a^{4} b^{2} c^{3} d^{4} - 17 \, a^{5} b c^{2} d^{5} + 6 \, a^{6} c d^{6}\right )} x^{4} + {\left (10 \, a^{4} b^{2} c^{4} d^{3} - 10 \, a^{5} b c^{3} d^{4} + 3 \, a^{6} c^{2} d^{5}\right )} x^{2}\right )} \log \left (d x^{2} + c\right ) + 4 \, {\left ({\left (2 \, b^{6} c^{5} d^{2} - 5 \, a b^{5} c^{4} d^{3} + 10 \, a^{3} b^{3} c^{2} d^{5} - 10 \, a^{4} b^{2} c d^{6} + 3 \, a^{5} b d^{7}\right )} x^{8} + {\left (4 \, b^{6} c^{6} d - 8 \, a b^{5} c^{5} d^{2} - 5 \, a^{2} b^{4} c^{4} d^{3} + 20 \, a^{3} b^{3} c^{3} d^{4} - 10 \, a^{4} b^{2} c^{2} d^{5} - 4 \, a^{5} b c d^{6} + 3 \, a^{6} d^{7}\right )} x^{6} + {\left (2 \, b^{6} c^{7} - a b^{5} c^{6} d - 10 \, a^{2} b^{4} c^{5} d^{2} + 10 \, a^{3} b^{3} c^{4} d^{3} + 10 \, a^{4} b^{2} c^{3} d^{4} - 17 \, a^{5} b c^{2} d^{5} + 6 \, a^{6} c d^{6}\right )} x^{4} + {\left (2 \, a b^{5} c^{7} - 5 \, a^{2} b^{4} c^{6} d + 10 \, a^{4} b^{2} c^{4} d^{3} - 10 \, a^{5} b c^{3} d^{4} + 3 \, a^{6} c^{2} d^{5}\right )} x^{2}\right )} \log \left (x\right )}{4 \, {\left ({\left (a^{3} b^{5} c^{8} d^{2} - 4 \, a^{4} b^{4} c^{7} d^{3} + 6 \, a^{5} b^{3} c^{6} d^{4} - 4 \, a^{6} b^{2} c^{5} d^{5} + a^{7} b c^{4} d^{6}\right )} x^{8} + {\left (2 \, a^{3} b^{5} c^{9} d - 7 \, a^{4} b^{4} c^{8} d^{2} + 8 \, a^{5} b^{3} c^{7} d^{3} - 2 \, a^{6} b^{2} c^{6} d^{4} - 2 \, a^{7} b c^{5} d^{5} + a^{8} c^{4} d^{6}\right )} x^{6} + {\left (a^{3} b^{5} c^{10} - 2 \, a^{4} b^{4} c^{9} d - 2 \, a^{5} b^{3} c^{8} d^{2} + 8 \, a^{6} b^{2} c^{7} d^{3} - 7 \, a^{7} b c^{6} d^{4} + 2 \, a^{8} c^{5} d^{5}\right )} x^{4} + {\left (a^{4} b^{4} c^{10} - 4 \, a^{5} b^{3} c^{9} d + 6 \, a^{6} b^{2} c^{8} d^{2} - 4 \, a^{7} b c^{7} d^{3} + a^{8} c^{6} d^{4}\right )} x^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(2*a^2*b^4*c^7 - 8*a^3*b^3*c^6*d + 12*a^4*b^2*c^5*d^2 - 8*a^5*b*c^4*d^3 + 2*a^6*c^3*d^4 + 2*(2*a*b^5*c^5*
d^2 - 5*a^2*b^4*c^4*d^3 + 10*a^3*b^3*c^3*d^4 - 10*a^4*b^2*c^2*d^5 + 3*a^5*b*c*d^6)*x^6 + (8*a*b^5*c^6*d - 18*a
^2*b^4*c^5*d^2 + 25*a^3*b^3*c^4*d^3 - 10*a^4*b^2*c^3*d^4 - 11*a^5*b*c^2*d^5 + 6*a^6*c*d^6)*x^4 + (4*a*b^5*c^7
- 6*a^2*b^4*c^6*d - 4*a^3*b^3*c^5*d^2 + 25*a^4*b^2*c^4*d^3 - 28*a^5*b*c^3*d^4 + 9*a^6*c^2*d^5)*x^2 - 2*((2*b^6
*c^5*d^2 - 5*a*b^5*c^4*d^3)*x^8 + (4*b^6*c^6*d - 8*a*b^5*c^5*d^2 - 5*a^2*b^4*c^4*d^3)*x^6 + (2*b^6*c^7 - a*b^5
*c^6*d - 10*a^2*b^4*c^5*d^2)*x^4 + (2*a*b^5*c^7 - 5*a^2*b^4*c^6*d)*x^2)*log(b*x^2 + a) - 2*((10*a^3*b^3*c^2*d^
5 - 10*a^4*b^2*c*d^6 + 3*a^5*b*d^7)*x^8 + (20*a^3*b^3*c^3*d^4 - 10*a^4*b^2*c^2*d^5 - 4*a^5*b*c*d^6 + 3*a^6*d^7
)*x^6 + (10*a^3*b^3*c^4*d^3 + 10*a^4*b^2*c^3*d^4 - 17*a^5*b*c^2*d^5 + 6*a^6*c*d^6)*x^4 + (10*a^4*b^2*c^4*d^3 -
10*a^5*b*c^3*d^4 + 3*a^6*c^2*d^5)*x^2)*log(d*x^2 + c) + 4*((2*b^6*c^5*d^2 - 5*a*b^5*c^4*d^3 + 10*a^3*b^3*c^2*
d^5 - 10*a^4*b^2*c*d^6 + 3*a^5*b*d^7)*x^8 + (4*b^6*c^6*d - 8*a*b^5*c^5*d^2 - 5*a^2*b^4*c^4*d^3 + 20*a^3*b^3*c^
3*d^4 - 10*a^4*b^2*c^2*d^5 - 4*a^5*b*c*d^6 + 3*a^6*d^7)*x^6 + (2*b^6*c^7 - a*b^5*c^6*d - 10*a^2*b^4*c^5*d^2 +
10*a^3*b^3*c^4*d^3 + 10*a^4*b^2*c^3*d^4 - 17*a^5*b*c^2*d^5 + 6*a^6*c*d^6)*x^4 + (2*a*b^5*c^7 - 5*a^2*b^4*c^6*d
+ 10*a^4*b^2*c^4*d^3 - 10*a^5*b*c^3*d^4 + 3*a^6*c^2*d^5)*x^2)*log(x))/((a^3*b^5*c^8*d^2 - 4*a^4*b^4*c^7*d^3 +
6*a^5*b^3*c^6*d^4 - 4*a^6*b^2*c^5*d^5 + a^7*b*c^4*d^6)*x^8 + (2*a^3*b^5*c^9*d - 7*a^4*b^4*c^8*d^2 + 8*a^5*b^3
*c^7*d^3 - 2*a^6*b^2*c^6*d^4 - 2*a^7*b*c^5*d^5 + a^8*c^4*d^6)*x^6 + (a^3*b^5*c^10 - 2*a^4*b^4*c^9*d - 2*a^5*b^
3*c^8*d^2 + 8*a^6*b^2*c^7*d^3 - 7*a^7*b*c^6*d^4 + 2*a^8*c^5*d^5)*x^4 + (a^4*b^4*c^10 - 4*a^5*b^3*c^9*d + 6*a^6
*b^2*c^8*d^2 - 4*a^7*b*c^7*d^3 + a^8*c^6*d^4)*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**3/(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. 638 vs. \(2 (205) = 410\).
time = 0.79, size = 638, normalized size = 2.97 \begin {gather*} \frac {{\left (2 \, b^{6} c - 5 \, a b^{5} d\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, {\left (a^{3} b^{5} c^{4} - 4 \, a^{4} b^{4} c^{3} d + 6 \, a^{5} b^{3} c^{2} d^{2} - 4 \, a^{6} b^{2} c d^{3} + a^{7} b d^{4}\right )}} + \frac {{\left (10 \, b^{2} c^{2} d^{4} - 10 \, a b c d^{5} + 3 \, a^{2} d^{6}\right )} \log \left ({\left | d x^{2} + c \right |}\right )}{2 \, {\left (b^{4} c^{8} d - 4 \, a b^{3} c^{7} d^{2} + 6 \, a^{2} b^{2} c^{6} d^{3} - 4 \, a^{3} b c^{5} d^{4} + a^{4} c^{4} d^{5}\right )}} + \frac {10 \, a^{2} b^{3} c^{2} d^{3} x^{4} - 10 \, a^{3} b^{2} c d^{4} x^{4} + 3 \, a^{4} b d^{5} x^{4} - 4 \, b^{5} c^{5} x^{2} + 10 \, a b^{4} c^{4} d x^{2} - 12 \, a^{2} b^{3} c^{3} d^{2} x^{2} + 18 \, a^{3} b^{2} c^{2} d^{3} x^{2} - 12 \, a^{4} b c d^{4} x^{2} + 3 \, a^{5} d^{5} x^{2} - 2 \, a b^{4} c^{5} + 8 \, a^{2} b^{3} c^{4} d - 12 \, a^{3} b^{2} c^{3} d^{2} + 8 \, a^{4} b c^{2} d^{3} - 2 \, a^{5} c d^{4}}{4 \, {\left (a^{2} b^{4} c^{8} - 4 \, a^{3} b^{3} c^{7} d + 6 \, a^{4} b^{2} c^{6} d^{2} - 4 \, a^{5} b c^{5} d^{3} + a^{6} c^{4} d^{4}\right )} {\left (b x^{4} + a x^{2}\right )}} - \frac {30 \, b^{2} c^{2} d^{5} x^{4} - 30 \, a b c d^{6} x^{4} + 9 \, a^{2} d^{7} x^{4} + 68 \, b^{2} c^{3} d^{4} x^{2} - 72 \, a b c^{2} d^{5} x^{2} + 22 \, a^{2} c d^{6} x^{2} + 39 \, b^{2} c^{4} d^{3} - 44 \, a b c^{3} d^{4} + 14 \, a^{2} c^{2} d^{5}}{4 \, {\left (b^{4} c^{8} - 4 \, a b^{3} c^{7} d + 6 \, a^{2} b^{2} c^{6} d^{2} - 4 \, a^{3} b c^{5} d^{3} + a^{4} c^{4} d^{4}\right )} {\left (d x^{2} + c\right )}^{2}} - \frac {{\left (2 \, b c + 3 \, a d\right )} \log \left (x^{2}\right )}{2 \, a^{3} c^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2*b^6*c - 5*a*b^5*d)*log(abs(b*x^2 + a))/(a^3*b^5*c^4 - 4*a^4*b^4*c^3*d + 6*a^5*b^3*c^2*d^2 - 4*a^6*b^2*c
*d^3 + a^7*b*d^4) + 1/2*(10*b^2*c^2*d^4 - 10*a*b*c*d^5 + 3*a^2*d^6)*log(abs(d*x^2 + c))/(b^4*c^8*d - 4*a*b^3*c
^7*d^2 + 6*a^2*b^2*c^6*d^3 - 4*a^3*b*c^5*d^4 + a^4*c^4*d^5) + 1/4*(10*a^2*b^3*c^2*d^3*x^4 - 10*a^3*b^2*c*d^4*x
^4 + 3*a^4*b*d^5*x^4 - 4*b^5*c^5*x^2 + 10*a*b^4*c^4*d*x^2 - 12*a^2*b^3*c^3*d^2*x^2 + 18*a^3*b^2*c^2*d^3*x^2 -
12*a^4*b*c*d^4*x^2 + 3*a^5*d^5*x^2 - 2*a*b^4*c^5 + 8*a^2*b^3*c^4*d - 12*a^3*b^2*c^3*d^2 + 8*a^4*b*c^2*d^3 - 2*
a^5*c*d^4)/((a^2*b^4*c^8 - 4*a^3*b^3*c^7*d + 6*a^4*b^2*c^6*d^2 - 4*a^5*b*c^5*d^3 + a^6*c^4*d^4)*(b*x^4 + a*x^2
)) - 1/4*(30*b^2*c^2*d^5*x^4 - 30*a*b*c*d^6*x^4 + 9*a^2*d^7*x^4 + 68*b^2*c^3*d^4*x^2 - 72*a*b*c^2*d^5*x^2 + 22
*a^2*c*d^6*x^2 + 39*b^2*c^4*d^3 - 44*a*b*c^3*d^4 + 14*a^2*c^2*d^5)/((b^4*c^8 - 4*a*b^3*c^7*d + 6*a^2*b^2*c^6*d
^2 - 4*a^3*b*c^5*d^3 + a^4*c^4*d^4)*(d*x^2 + c)^2) - 1/2*(2*b*c + 3*a*d)*log(x^2)/(a^3*c^4)

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Mupad [B]
time = 1.32, size = 549, normalized size = 2.55 \begin {gather*} \frac {\ln \left (b\,x^2+a\right )\,\left (2\,b^5\,c-5\,a\,b^4\,d\right )}{2\,a^7\,d^4-8\,a^6\,b\,c\,d^3+12\,a^5\,b^2\,c^2\,d^2-8\,a^4\,b^3\,c^3\,d+2\,a^3\,b^4\,c^4}-\frac {\frac {1}{2\,a\,c}-\frac {x^4\,\left (-6\,a^4\,d^5+5\,a^3\,b\,c\,d^4+15\,a^2\,b^2\,c^2\,d^3-10\,a\,b^3\,c^3\,d^2+8\,b^4\,c^4\,d\right )}{4\,a^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 {x^2\,\left (9\,a^4\,d^4-19\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2+2\,a\,b^3\,c^3\,d-4\,b^4\,c^4\right )}{4\,a^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 {b\,d^2\,x^6\,\left (3\,a^3\,d^3-7\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-2\,b^3\,c^3\right )}{2\,a^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 )}}{x^4\,\left (b\,c^2+2\,a\,d\,c\right )+x^6\,\left (a\,d^2+2\,b\,c\,d\right )+a\,c^2\,x^2+b\,d^2\,x^8}+\frac {\ln \left (d\,x^2+c\right )\,\left (3\,a^2\,d^5-10\,a\,b\,c\,d^4+10\,b^2\,c^2\,d^3\right )}{2\,a^4\,c^4\,d^4-8\,a^3\,b\,c^5\,d^3+12\,a^2\,b^2\,c^6\,d^2-8\,a\,b^3\,c^7\,d+2\,b^4\,c^8}-\frac {\ln \left (x\right )\,\left (3\,a\,d+2\,b\,c\right )}{a^3\,c^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(log(a + b*x^2)*(2*b^5*c - 5*a*b^4*d))/(2*a^7*d^4 + 2*a^3*b^4*c^4 - 8*a^4*b^3*c^3*d + 12*a^5*b^2*c^2*d^2 - 8*a
^6*b*c*d^3) - (1/(2*a*c) - (x^4*(8*b^4*c^4*d - 6*a^4*d^5 - 10*a*b^3*c^3*d^2 + 15*a^2*b^2*c^2*d^3 + 5*a^3*b*c*d
^4))/(4*a^2*c^3*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)) + (x^2*(9*a^4*d^4 - 4*b^4*c^4 + 6*a^2*b^2
*c^2*d^2 + 2*a*b^3*c^3*d - 19*a^3*b*c*d^3))/(4*a^2*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^6*(3*a^3*d^3 - 2*b^3*c^3 + 3*a*b^2*c^2*d - 7*a^2*b*c*d^2))/(2*a^2*c^3*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^
2*d - 3*a^2*b*c*d^2)))/(x^4*(b*c^2 + 2*a*c*d) + x^6*(a*d^2 + 2*b*c*d) + a*c^2*x^2 + b*d^2*x^8) + (log(c + d*x^
2)*(3*a^2*d^5 + 10*b^2*c^2*d^3 - 10*a*b*c*d^4))/(2*b^4*c^8 + 2*a^4*c^4*d^4 - 8*a^3*b*c^5*d^3 + 12*a^2*b^2*c^6*
d^2 - 8*a*b^3*c^7*d) - (log(x)*(3*a*d + 2*b*c))/(a^3*c^4)

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