\(\int \frac {x^3}{(a+b x^2)^2 (c+d x^2)^3} \, dx\) [311]

Optimal result

Integrand size = 22, antiderivative size = 142 \[ \int \frac {x^3}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx=\frac {a b}{2 (b c-a d)^3 \left (a+b x^2\right )}+\frac {c}{4 (b c-a d)^2 \left (c+d x^2\right )^2}+\frac {b c+a d}{2 (b c-a d)^3 \left (c+d x^2\right )}+\frac {b (b c+2 a d) \log \left (a+b x^2\right )}{2 (b c-a d)^4}-\frac {b (b c+2 a d) \log \left (c+d x^2\right )}{2 (b c-a d)^4} \]

[Out]

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

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 142, 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, 78} \[ \int \frac {x^3}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx=\frac {a b}{2 \left (a+b x^2\right ) (b c-a d)^3}+\frac {a d+b c}{2 \left (c+d x^2\right ) (b c-a d)^3}+\frac {c}{4 \left (c+d x^2\right )^2 (b c-a d)^2}+\frac {b (2 a d+b c) \log \left (a+b x^2\right )}{2 (b c-a d)^4}-\frac {b (2 a d+b c) \log \left (c+d x^2\right )}{2 (b c-a d)^4} \]

[In]

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

[Out]

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

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

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

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.85 \[ \int \frac {x^3}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx=\frac {\frac {2 a b (b c-a d)}{a+b x^2}+\frac {c (b c-a d)^2}{\left (c+d x^2\right )^2}+\frac {2 (b c-a d) (b c+a d)}{c+d x^2}+2 b (b c+2 a d) \log \left (a+b x^2\right )-2 b (b c+2 a d) \log \left (c+d x^2\right )}{4 (b c-a d)^4} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 2.78 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.15

method	result	size
default	\(\frac {b^{2} \left (\frac {\left (2 a d +b c \right ) \ln \left (b \,x^{2}+a \right )}{b}-\frac {\left (a d -b c \right ) a}{b \left (b \,x^{2}+a \right )}\right )}{2 \left (a d -b c \right )^{4}}+\frac {d \left (\frac {c \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{2 d \left (d \,x^{2}+c \right )^{2}}-\frac {b \left (2 a d +b c \right ) \ln \left (d \,x^{2}+c \right )}{d}-\frac {a^{2} d^{2}-b^{2} c^{2}}{d \left (d \,x^{2}+c \right )}\right )}{2 \left (a d -b c \right )^{4}}\)	\(163\)
norman	\(\frac {\frac {\left (-2 a \,b^{2} d^{3}-b^{3} c \,d^{2}\right ) x^{4}}{2 d b \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {a c \left (-a b \,d^{3}-5 b^{2} c \,d^{2}\right )}{4 d^{2} b \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {\left (2 a d +b c \right ) \left (-a b \,d^{3}-3 b^{2} c \,d^{2}\right ) x^{2}}{4 d^{2} b \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}}{\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )^{2}}+\frac {b \left (2 a d +b c \right ) \ln \left (b \,x^{2}+a \right )}{2 a^{4} d^{4}-8 a^{3} b c \,d^{3}+12 a^{2} b^{2} c^{2} d^{2}-8 a \,b^{3} c^{3} d +2 b^{4} c^{4}}-\frac {b \left (2 a d +b c \right ) \ln \left (d \,x^{2}+c \right )}{2 \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}\)	\(369\)
risch	\(\frac {-\frac {b d \left (2 a d +b c \right ) x^{4}}{2 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}-\frac {\left (a d +3 b c \right ) \left (2 a d +b c \right ) x^{2}}{4 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}-\frac {a c \left (a d +5 b c \right )}{4 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}}{\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )^{2}}-\frac {b \ln \left (-d \,x^{2}-c \right ) a d}{a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}}-\frac {b^{2} \ln \left (-d \,x^{2}-c \right ) c}{2 \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}+\frac {b \ln \left (b \,x^{2}+a \right ) a d}{a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}}+\frac {b^{2} \ln \left (b \,x^{2}+a \right ) c}{2 a^{4} d^{4}-8 a^{3} b c \,d^{3}+12 a^{2} b^{2} c^{2} d^{2}-8 a \,b^{3} c^{3} d +2 b^{4} c^{4}}\)	\(447\)
parallelrisch	\(\frac {-a^{3} b c \,d^{4}-4 a^{2} b^{2} c^{2} d^{3}+5 a \,b^{3} c^{3} d^{2}+10 \ln \left (b \,x^{2}+a \right ) x^{4} a \,b^{3} c \,d^{4}-10 \ln \left (d \,x^{2}+c \right ) x^{4} a \,b^{3} c \,d^{4}+8 \ln \left (b \,x^{2}+a \right ) x^{2} a^{2} b^{2} c \,d^{4}+8 \ln \left (b \,x^{2}+a \right ) x^{2} a \,b^{3} c^{2} d^{3}-8 \ln \left (d \,x^{2}+c \right ) x^{2} a^{2} b^{2} c \,d^{4}-8 \ln \left (d \,x^{2}+c \right ) x^{2} a \,b^{3} c^{2} d^{3}-4 x^{4} a^{2} b^{2} d^{5}+2 x^{4} b^{4} c^{2} d^{3}-2 x^{2} a^{3} b \,d^{5}+3 x^{2} b^{4} c^{3} d^{2}+4 \ln \left (b \,x^{2}+a \right ) a^{2} b^{2} c^{2} d^{3}+2 \ln \left (b \,x^{2}+a \right ) a \,b^{3} c^{3} d^{2}-4 \ln \left (d \,x^{2}+c \right ) a^{2} b^{2} c^{2} d^{3}-2 \ln \left (d \,x^{2}+c \right ) a \,b^{3} c^{3} d^{2}+2 x^{4} a \,b^{3} c \,d^{4}-5 x^{2} a^{2} b^{2} c \,d^{4}+4 x^{2} a \,b^{3} c^{2} d^{3}+4 \ln \left (b \,x^{2}+a \right ) x^{6} a \,b^{3} d^{5}+2 \ln \left (b \,x^{2}+a \right ) x^{6} b^{4} c \,d^{4}-4 \ln \left (d \,x^{2}+c \right ) x^{6} a \,b^{3} d^{5}-2 \ln \left (d \,x^{2}+c \right ) x^{6} b^{4} c \,d^{4}+4 \ln \left (b \,x^{2}+a \right ) x^{4} a^{2} b^{2} d^{5}+4 \ln \left (b \,x^{2}+a \right ) x^{4} b^{4} c^{2} d^{3}-4 \ln \left (d \,x^{2}+c \right ) x^{4} a^{2} b^{2} d^{5}-4 \ln \left (d \,x^{2}+c \right ) x^{4} b^{4} c^{2} d^{3}+2 \ln \left (b \,x^{2}+a \right ) x^{2} b^{4} c^{3} d^{2}-2 \ln \left (d \,x^{2}+c \right ) x^{2} b^{4} c^{3} d^{2}}{4 \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) \left (d \,x^{2}+c \right )^{2} \left (b \,x^{2}+a \right ) b \,d^{2}}\)	\(642\)

[In]

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

[Out]

1/2*b^2/(a*d-b*c)^4*((2*a*d+b*c)/b*ln(b*x^2+a)-(a*d-b*c)*a/b/(b*x^2+a))+1/2*d/(a*d-b*c)^4*(1/2*c*(a^2*d^2-2*a*
b*c*d+b^2*c^2)/d/(d*x^2+c)^2-b*(2*a*d+b*c)/d*ln(d*x^2+c)-(a^2*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. 598 vs. \(2 (132) = 264\).

Time = 0.26 (sec) , antiderivative size = 598, normalized size of antiderivative = 4.21 \[ \int \frac {x^3}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx=\frac {5 \, a b^{2} c^{3} - 4 \, a^{2} b c^{2} d - a^{3} c d^{2} + 2 \, {\left (b^{3} c^{2} d + a b^{2} c d^{2} - 2 \, a^{2} b d^{3}\right )} x^{4} + {\left (3 \, b^{3} c^{3} + 4 \, a b^{2} c^{2} d - 5 \, a^{2} b c d^{2} - 2 \, a^{3} d^{3}\right )} x^{2} + 2 \, {\left ({\left (b^{3} c d^{2} + 2 \, a b^{2} d^{3}\right )} x^{6} + a b^{2} c^{3} + 2 \, a^{2} b c^{2} d + {\left (2 \, b^{3} c^{2} d + 5 \, a b^{2} c d^{2} + 2 \, a^{2} b d^{3}\right )} x^{4} + {\left (b^{3} c^{3} + 4 \, a b^{2} c^{2} d + 4 \, a^{2} b c d^{2}\right )} x^{2}\right )} \log \left (b x^{2} + a\right ) - 2 \, {\left ({\left (b^{3} c d^{2} + 2 \, a b^{2} d^{3}\right )} x^{6} + a b^{2} c^{3} + 2 \, a^{2} b c^{2} d + {\left (2 \, b^{3} c^{2} d + 5 \, a b^{2} c d^{2} + 2 \, a^{2} b d^{3}\right )} x^{4} + {\left (b^{3} c^{3} + 4 \, a b^{2} c^{2} d + 4 \, a^{2} b c d^{2}\right )} x^{2}\right )} \log \left (d x^{2} + c\right )}{4 \, {\left (a b^{4} c^{6} - 4 \, a^{2} b^{3} c^{5} d + 6 \, a^{3} b^{2} c^{4} d^{2} - 4 \, a^{4} b c^{3} d^{3} + a^{5} c^{2} d^{4} + {\left (b^{5} c^{4} d^{2} - 4 \, a b^{4} c^{3} d^{3} + 6 \, a^{2} b^{3} c^{2} d^{4} - 4 \, a^{3} b^{2} c d^{5} + a^{4} b d^{6}\right )} x^{6} + {\left (2 \, b^{5} c^{5} d - 7 \, a b^{4} c^{4} d^{2} + 8 \, a^{2} b^{3} c^{3} d^{3} - 2 \, a^{3} b^{2} c^{2} d^{4} - 2 \, a^{4} b c d^{5} + a^{5} d^{6}\right )} x^{4} + {\left (b^{5} c^{6} - 2 \, a b^{4} c^{5} d - 2 \, a^{2} b^{3} c^{4} d^{2} + 8 \, a^{3} b^{2} c^{3} d^{3} - 7 \, a^{4} b c^{2} d^{4} + 2 \, a^{5} c d^{5}\right )} x^{2}\right )}} \]

[In]

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

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 784 vs. \(2 (122) = 244\).

Time = 29.00 (sec) , antiderivative size = 784, normalized size of antiderivative = 5.52 \[ \int \frac {x^3}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx=- \frac {b \left (2 a d + b c\right ) \log {\left (x^{2} + \frac {- \frac {a^{5} b d^{5} \cdot \left (2 a d + b c\right )}{\left (a d - b c\right )^{4}} + \frac {5 a^{4} b^{2} c d^{4} \cdot \left (2 a d + b c\right )}{\left (a d - b c\right )^{4}} - \frac {10 a^{3} b^{3} c^{2} d^{3} \cdot \left (2 a d + b c\right )}{\left (a d - b c\right )^{4}} + \frac {10 a^{2} b^{4} c^{3} d^{2} \cdot \left (2 a d + b c\right )}{\left (a d - b c\right )^{4}} + 2 a^{2} b d^{2} - \frac {5 a b^{5} c^{4} d \left (2 a d + b c\right )}{\left (a d - b c\right )^{4}} + 3 a b^{2} c d + \frac {b^{6} c^{5} \cdot \left (2 a d + b c\right )}{\left (a d - b c\right )^{4}} + b^{3} c^{2}}{4 a b^{2} d^{2} + 2 b^{3} c d} \right )}}{2 \left (a d - b c\right )^{4}} + \frac {b \left (2 a d + b c\right ) \log {\left (x^{2} + \frac {\frac {a^{5} b d^{5} \cdot \left (2 a d + b c\right )}{\left (a d - b c\right )^{4}} - \frac {5 a^{4} b^{2} c d^{4} \cdot \left (2 a d + b c\right )}{\left (a d - b c\right )^{4}} + \frac {10 a^{3} b^{3} c^{2} d^{3} \cdot \left (2 a d + b c\right )}{\left (a d - b c\right )^{4}} - \frac {10 a^{2} b^{4} c^{3} d^{2} \cdot \left (2 a d + b c\right )}{\left (a d - b c\right )^{4}} + 2 a^{2} b d^{2} + \frac {5 a b^{5} c^{4} d \left (2 a d + b c\right )}{\left (a d - b c\right )^{4}} + 3 a b^{2} c d - \frac {b^{6} c^{5} \cdot \left (2 a d + b c\right )}{\left (a d - b c\right )^{4}} + b^{3} c^{2}}{4 a b^{2} d^{2} + 2 b^{3} c d} \right )}}{2 \left (a d - b c\right )^{4}} + \frac {- a^{2} c d - 5 a b c^{2} + x^{4} \left (- 4 a b d^{2} - 2 b^{2} c d\right ) + x^{2} \left (- 2 a^{2} d^{2} - 7 a b c d - 3 b^{2} c^{2}\right )}{4 a^{4} c^{2} d^{3} - 12 a^{3} b c^{3} d^{2} + 12 a^{2} b^{2} c^{4} d - 4 a b^{3} c^{5} + x^{6} \cdot \left (4 a^{3} b d^{5} - 12 a^{2} b^{2} c d^{4} + 12 a b^{3} c^{2} d^{3} - 4 b^{4} c^{3} d^{2}\right ) + x^{4} \cdot \left (4 a^{4} d^{5} - 4 a^{3} b c d^{4} - 12 a^{2} b^{2} c^{2} d^{3} + 20 a b^{3} c^{3} d^{2} - 8 b^{4} c^{4} d\right ) + x^{2} \cdot \left (8 a^{4} c d^{4} - 20 a^{3} b c^{2} d^{3} + 12 a^{2} b^{2} c^{3} d^{2} + 4 a b^{3} c^{4} d - 4 b^{4} c^{5}\right )} \]

[In]

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

[Out]

-b*(2*a*d + b*c)*log(x**2 + (-a**5*b*d**5*(2*a*d + b*c)/(a*d - b*c)**4 + 5*a**4*b**2*c*d**4*(2*a*d + b*c)/(a*d
- b*c)**4 - 10*a**3*b**3*c**2*d**3*(2*a*d + b*c)/(a*d - b*c)**4 + 10*a**2*b**4*c**3*d**2*(2*a*d + b*c)/(a*d -
b*c)**4 + 2*a**2*b*d**2 - 5*a*b**5*c**4*d*(2*a*d + b*c)/(a*d - b*c)**4 + 3*a*b**2*c*d + b**6*c**5*(2*a*d + b*
c)/(a*d - b*c)**4 + b**3*c**2)/(4*a*b**2*d**2 + 2*b**3*c*d))/(2*(a*d - b*c)**4) + b*(2*a*d + b*c)*log(x**2 + (
a**5*b*d**5*(2*a*d + b*c)/(a*d - b*c)**4 - 5*a**4*b**2*c*d**4*(2*a*d + b*c)/(a*d - b*c)**4 + 10*a**3*b**3*c**2
*d**3*(2*a*d + b*c)/(a*d - b*c)**4 - 10*a**2*b**4*c**3*d**2*(2*a*d + b*c)/(a*d - b*c)**4 + 2*a**2*b*d**2 + 5*a
*b**5*c**4*d*(2*a*d + b*c)/(a*d - b*c)**4 + 3*a*b**2*c*d - b**6*c**5*(2*a*d + b*c)/(a*d - b*c)**4 + b**3*c**2)
/(4*a*b**2*d**2 + 2*b**3*c*d))/(2*(a*d - b*c)**4) + (-a**2*c*d - 5*a*b*c**2 + x**4*(-4*a*b*d**2 - 2*b**2*c*d)
+ x**2*(-2*a**2*d**2 - 7*a*b*c*d - 3*b**2*c**2))/(4*a**4*c**2*d**3 - 12*a**3*b*c**3*d**2 + 12*a**2*b**2*c**4*d
- 4*a*b**3*c**5 + x**6*(4*a**3*b*d**5 - 12*a**2*b**2*c*d**4 + 12*a*b**3*c**2*d**3 - 4*b**4*c**3*d**2) + x**4*
(4*a**4*d**5 - 4*a**3*b*c*d**4 - 12*a**2*b**2*c**2*d**3 + 20*a*b**3*c**3*d**2 - 8*b**4*c**4*d) + x**2*(8*a**4*
c*d**4 - 20*a**3*b*c**2*d**3 + 12*a**2*b**2*c**3*d**2 + 4*a*b**3*c**4*d - 4*b**4*c**5))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 415 vs. \(2 (132) = 264\).

Time = 0.22 (sec) , antiderivative size = 415, normalized size of antiderivative = 2.92 \[ \int \frac {x^3}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx=\frac {{\left (b^{2} c + 2 \, a b d\right )} \log \left (b x^{2} + a\right )}{2 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )}} - \frac {{\left (b^{2} c + 2 \, a b d\right )} \log \left (d x^{2} + c\right )}{2 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )}} + \frac {2 \, {\left (b^{2} c d + 2 \, a b d^{2}\right )} x^{4} + 5 \, a b c^{2} + a^{2} c d + {\left (3 \, b^{2} c^{2} + 7 \, a b c d + 2 \, a^{2} d^{2}\right )} x^{2}}{4 \, {\left (a b^{3} c^{5} - 3 \, a^{2} b^{2} c^{4} d + 3 \, a^{3} b c^{3} d^{2} - a^{4} c^{2} d^{3} + {\left (b^{4} c^{3} d^{2} - 3 \, a b^{3} c^{2} d^{3} + 3 \, a^{2} b^{2} c d^{4} - a^{3} b d^{5}\right )} x^{6} + {\left (2 \, b^{4} c^{4} d - 5 \, a b^{3} c^{3} d^{2} + 3 \, a^{2} b^{2} c^{2} d^{3} + a^{3} b c d^{4} - a^{4} d^{5}\right )} x^{4} + {\left (b^{4} c^{5} - a b^{3} c^{4} d - 3 \, a^{2} b^{2} c^{3} d^{2} + 5 \, a^{3} b c^{2} d^{3} - 2 \, a^{4} c d^{4}\right )} x^{2}\right )}} \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 267 vs. \(2 (132) = 264\).

Time = 0.28 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.88 \[ \int \frac {x^3}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx=\frac {\frac {2 \, a b^{5}}{{\left (b^{6} c^{3} - 3 \, a b^{5} c^{2} d + 3 \, a^{2} b^{4} c d^{2} - a^{3} b^{3} d^{3}\right )} {\left (b x^{2} + a\right )}} - \frac {2 \, {\left (b^{4} c + 2 \, a b^{3} d\right )} \log \left ({\left | \frac {b c}{b x^{2} + a} - \frac {a d}{b x^{2} + a} + d \right |}\right )}{b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}} - \frac {3 \, b^{3} c d^{2} + 2 \, a b^{2} d^{3} + \frac {2 \, {\left (2 \, b^{5} c^{2} d - a b^{4} c d^{2} - a^{2} b^{3} d^{3}\right )}}{{\left (b x^{2} + a\right )} b}}{{\left (b c - a d\right )}^{4} {\left (\frac {b c}{b x^{2} + a} - \frac {a d}{b x^{2} + a} + d\right )}^{2}}}{4 \, b} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 5.31 (sec) , antiderivative size = 926, normalized size of antiderivative = 6.52 \[ \int \frac {x^3}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx=\frac {5\,a\,b^2\,c^3-a^3\,c\,d^2-2\,a^3\,d^3\,x^2+3\,b^3\,c^3\,x^2-4\,a^2\,b\,d^3\,x^4+2\,b^3\,c^2\,d\,x^4+a\,b^2\,c^3\,\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}-4\,a^2\,b\,c^2\,d+b^3\,c^3\,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^2\,b\,d^3\,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 )\,8{}\mathrm {i}+a\,b^2\,d^3\,x^6\,\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 )\,8{}\mathrm {i}+b^3\,c^2\,d\,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 )\,8{}\mathrm {i}+b^3\,c\,d^2\,x^6\,\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}+4\,a\,b^2\,c^2\,d\,x^2-5\,a^2\,b\,c\,d^2\,x^2+2\,a\,b^2\,c\,d^2\,x^4+a^2\,b\,c^2\,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 )\,8{}\mathrm {i}+a\,b^2\,c^2\,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 )\,16{}\mathrm {i}+a^2\,b\,c\,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 )\,16{}\mathrm {i}+a\,b^2\,c\,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 )\,20{}\mathrm {i}}{4\,a^5\,c^2\,d^4+8\,a^5\,c\,d^5\,x^2+4\,a^5\,d^6\,x^4-16\,a^4\,b\,c^3\,d^3-28\,a^4\,b\,c^2\,d^4\,x^2-8\,a^4\,b\,c\,d^5\,x^4+4\,a^4\,b\,d^6\,x^6+24\,a^3\,b^2\,c^4\,d^2+32\,a^3\,b^2\,c^3\,d^3\,x^2-8\,a^3\,b^2\,c^2\,d^4\,x^4-16\,a^3\,b^2\,c\,d^5\,x^6-16\,a^2\,b^3\,c^5\,d-8\,a^2\,b^3\,c^4\,d^2\,x^2+32\,a^2\,b^3\,c^3\,d^3\,x^4+24\,a^2\,b^3\,c^2\,d^4\,x^6+4\,a\,b^4\,c^6-8\,a\,b^4\,c^5\,d\,x^2-28\,a\,b^4\,c^4\,d^2\,x^4-16\,a\,b^4\,c^3\,d^3\,x^6+4\,b^5\,c^6\,x^2+8\,b^5\,c^5\,d\,x^4+4\,b^5\,c^4\,d^2\,x^6} \]

[In]

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

[Out]

(5*a*b^2*c^3 - a^3*c*d^2 - 2*a^3*d^3*x^2 + 3*b^3*c^3*x^2 - 4*a^2*b*d^3*x^4 + 2*b^3*c^2*d*x^4 + a*b^2*c^3*atan(
(a*d*x^2*1i - b*c*x^2*1i)/(2*a*c + a*d*x^2 + b*c*x^2))*4i - 4*a^2*b*c^2*d + b^3*c^3*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^2*b*d^3*x^4*atan((a*d*x^2*1i - b*c*x^2*1i)/(2*a*c + a*d*x^2 + b*c
*x^2))*8i + a*b^2*d^3*x^6*atan((a*d*x^2*1i - b*c*x^2*1i)/(2*a*c + a*d*x^2 + b*c*x^2))*8i + b^3*c^2*d*x^4*atan(
(a*d*x^2*1i - b*c*x^2*1i)/(2*a*c + a*d*x^2 + b*c*x^2))*8i + b^3*c*d^2*x^6*atan((a*d*x^2*1i - b*c*x^2*1i)/(2*a*
c + a*d*x^2 + b*c*x^2))*4i + 4*a*b^2*c^2*d*x^2 - 5*a^2*b*c*d^2*x^2 + 2*a*b^2*c*d^2*x^4 + a^2*b*c^2*d*atan((a*d
*x^2*1i - b*c*x^2*1i)/(2*a*c + a*d*x^2 + b*c*x^2))*8i + a*b^2*c^2*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))*16i + a^2*b*c*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))*16i +
a*b^2*c*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))*20i)/(4*a*b^4*c^6 + 4*a^5*c^2*d^4
+ 4*b^5*c^6*x^2 + 4*a^5*d^6*x^4 - 16*a^2*b^3*c^5*d - 16*a^4*b*c^3*d^3 + 4*a^4*b*d^6*x^6 + 8*a^5*c*d^5*x^2 + 8*
b^5*c^5*d*x^4 + 24*a^3*b^2*c^4*d^2 + 4*b^5*c^4*d^2*x^6 - 8*a^2*b^3*c^4*d^2*x^2 + 32*a^3*b^2*c^3*d^3*x^2 + 32*a
^2*b^3*c^3*d^3*x^4 - 8*a^3*b^2*c^2*d^4*x^4 + 24*a^2*b^3*c^2*d^4*x^6 - 8*a*b^4*c^5*d*x^2 - 8*a^4*b*c*d^5*x^4 -
28*a^4*b*c^2*d^4*x^2 - 28*a*b^4*c^4*d^2*x^4 - 16*a*b^4*c^3*d^3*x^6 - 16*a^3*b^2*c*d^5*x^6)