3.4.16 \(\int \frac {1}{x^2 (a+b x^2)^2 (c+d x^2)^3} \, dx\) [316]

Optimal. Leaf size=297 \[ -\frac {3 (2 b c-a d) \left (2 b^2 c^2-3 a b c d+5 a^2 d^2\right )}{8 a^2 c^3 (b c-a d)^3 x}+\frac {d (2 b c+a d)}{4 a c (b c-a d)^2 x \left (c+d x^2\right )^2}+\frac {b}{2 a (b c-a d) x \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac {d \left (4 b^2 c^2+13 a b c d-5 a^2 d^2\right )}{8 a c^2 (b c-a d)^3 x \left (c+d x^2\right )}-\frac {3 b^{7/2} (b c-3 a d) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{5/2} (b c-a d)^4}-\frac {3 d^{5/2} \left (21 b^2 c^2-18 a b c d+5 a^2 d^2\right ) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{8 c^{7/2} (b c-a d)^4} \]

[Out]

-3/8*(-a*d+2*b*c)*(5*a^2*d^2-3*a*b*c*d+2*b^2*c^2)/a^2/c^3/(-a*d+b*c)^3/x+1/4*d*(a*d+2*b*c)/a/c/(-a*d+b*c)^2/x/
(d*x^2+c)^2+1/2*b/a/(-a*d+b*c)/x/(b*x^2+a)/(d*x^2+c)^2+1/8*d*(-5*a^2*d^2+13*a*b*c*d+4*b^2*c^2)/a/c^2/(-a*d+b*c
)^3/x/(d*x^2+c)-3/2*b^(7/2)*(-3*a*d+b*c)*arctan(x*b^(1/2)/a^(1/2))/a^(5/2)/(-a*d+b*c)^4-3/8*d^(5/2)*(5*a^2*d^2
-18*a*b*c*d+21*b^2*c^2)*arctan(x*d^(1/2)/c^(1/2))/c^(7/2)/(-a*d+b*c)^4

________________________________________________________________________________________

Rubi [A]
time = 0.35, antiderivative size = 297, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {483, 593, 597, 536, 211} \begin {gather*} -\frac {3 b^{7/2} \text {ArcTan}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) (b c-3 a d)}{2 a^{5/2} (b c-a d)^4}-\frac {3 d^{5/2} \left (5 a^2 d^2-18 a b c d+21 b^2 c^2\right ) \text {ArcTan}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{8 c^{7/2} (b c-a d)^4}+\frac {d \left (-5 a^2 d^2+13 a b c d+4 b^2 c^2\right )}{8 a c^2 x \left (c+d x^2\right ) (b c-a d)^3}-\frac {3 (2 b c-a d) \left (5 a^2 d^2-3 a b c d+2 b^2 c^2\right )}{8 a^2 c^3 x (b c-a d)^3}+\frac {b}{2 a x \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)}+\frac {d (a d+2 b c)}{4 a c x \left (c+d x^2\right )^2 (b c-a d)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*(2*b*c - a*d)*(2*b^2*c^2 - 3*a*b*c*d + 5*a^2*d^2))/(8*a^2*c^3*(b*c - a*d)^3*x) + (d*(2*b*c + a*d))/(4*a*c*
(b*c - a*d)^2*x*(c + d*x^2)^2) + b/(2*a*(b*c - a*d)*x*(a + b*x^2)*(c + d*x^2)^2) + (d*(4*b^2*c^2 + 13*a*b*c*d
- 5*a^2*d^2))/(8*a*c^2*(b*c - a*d)^3*x*(c + d*x^2)) - (3*b^(7/2)*(b*c - 3*a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2
*a^(5/2)*(b*c - a*d)^4) - (3*d^(5/2)*(21*b^2*c^2 - 18*a*b*c*d + 5*a^2*d^2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(8*c^(
7/2)*(b*c - a*d)^4)

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 483

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-b)*(e*
x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*e*n*(b*c - a*d)*(p + 1))), x] + Dist[1/(a*n*(b*c - a*d)
*(p + 1)), Int[(e*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*b*(m + 1) + n*(b*c - a*d)*(p + 1) + d*b*(m + n
*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, m, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ
[p, -1] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]

Rule 536

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> Dist[(b*e - a*f
)/(b*c - a*d), Int[1/(a + b*x^n), x], x] - Dist[(d*e - c*f)/(b*c - a*d), Int[1/(c + d*x^n), x], x] /; FreeQ[{a
, b, c, d, e, f, n}, x]

Rule 593

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_)*((e_) + (f_.)*(x_)^(n_)), x
_Symbol] :> Simp[(-(b*e - a*f))*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*g*n*(b*c - a*d)*(p +
1))), x] + Dist[1/(a*n*(b*c - a*d)*(p + 1)), Int[(g*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f)
*(m + 1) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*f)*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c,
d, e, f, g, m, q}, x] && IGtQ[n, 0] && LtQ[p, -1]

Rule 597

Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*c*g*(m + 1))), x] + Dist[1/(a*c*
g^n*(m + 1)), Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + n + 1) - e
*n*(b*c*p + a*d*q) - b*e*d*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] &&
IGtQ[n, 0] && LtQ[m, -1]

Rubi steps

\begin {align*} \int \frac {1}{x^2 \left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx &=\frac {b}{2 a (b c-a d) x \left (a+b x^2\right ) \left (c+d x^2\right )^2}-\frac {\int \frac {-3 b c+2 a d-7 b d x^2}{x^2 \left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx}{2 a (b c-a d)}\\ &=\frac {d (2 b c+a d)}{4 a c (b c-a d)^2 x \left (c+d x^2\right )^2}+\frac {b}{2 a (b c-a d) x \left (a+b x^2\right ) \left (c+d x^2\right )^2}-\frac {\int \frac {-2 \left (6 b^2 c^2-8 a b c d+5 a^2 d^2\right )-10 b d (2 b c+a d) x^2}{x^2 \left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx}{8 a c (b c-a d)^2}\\ &=\frac {d (2 b c+a d)}{4 a c (b c-a d)^2 x \left (c+d x^2\right )^2}+\frac {b}{2 a (b c-a d) x \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac {d \left (4 b^2 c^2+13 a b c d-5 a^2 d^2\right )}{8 a c^2 (b c-a d)^3 x \left (c+d x^2\right )}-\frac {\int \frac {-6 (2 b c-a d) \left (2 b^2 c^2-3 a b c d+5 a^2 d^2\right )-6 b d \left (4 b^2 c^2+13 a b c d-5 a^2 d^2\right ) x^2}{x^2 \left (a+b x^2\right ) \left (c+d x^2\right )} \, dx}{16 a c^2 (b c-a d)^3}\\ &=-\frac {3 (2 b c-a d) \left (2 b^2 c^2-3 a b c d+5 a^2 d^2\right )}{8 a^2 c^3 (b c-a d)^3 x}+\frac {d (2 b c+a d)}{4 a c (b c-a d)^2 x \left (c+d x^2\right )^2}+\frac {b}{2 a (b c-a d) x \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac {d \left (4 b^2 c^2+13 a b c d-5 a^2 d^2\right )}{8 a c^2 (b c-a d)^3 x \left (c+d x^2\right )}+\frac {\int \frac {-6 \left (4 b^4 c^4-8 a b^3 c^3 d-8 a^2 b^2 c^2 d^2+13 a^3 b c d^3-5 a^4 d^4\right )-6 b d (2 b c-a d) \left (2 b^2 c^2-3 a b c d+5 a^2 d^2\right ) x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx}{16 a^2 c^3 (b c-a d)^3}\\ &=-\frac {3 (2 b c-a d) \left (2 b^2 c^2-3 a b c d+5 a^2 d^2\right )}{8 a^2 c^3 (b c-a d)^3 x}+\frac {d (2 b c+a d)}{4 a c (b c-a d)^2 x \left (c+d x^2\right )^2}+\frac {b}{2 a (b c-a d) x \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac {d \left (4 b^2 c^2+13 a b c d-5 a^2 d^2\right )}{8 a c^2 (b c-a d)^3 x \left (c+d x^2\right )}-\frac {\left (3 b^4 (b c-3 a d)\right ) \int \frac {1}{a+b x^2} \, dx}{2 a^2 (b c-a d)^4}-\frac {\left (3 d^3 \left (21 b^2 c^2-18 a b c d+5 a^2 d^2\right )\right ) \int \frac {1}{c+d x^2} \, dx}{8 c^3 (b c-a d)^4}\\ &=-\frac {3 (2 b c-a d) \left (2 b^2 c^2-3 a b c d+5 a^2 d^2\right )}{8 a^2 c^3 (b c-a d)^3 x}+\frac {d (2 b c+a d)}{4 a c (b c-a d)^2 x \left (c+d x^2\right )^2}+\frac {b}{2 a (b c-a d) x \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac {d \left (4 b^2 c^2+13 a b c d-5 a^2 d^2\right )}{8 a c^2 (b c-a d)^3 x \left (c+d x^2\right )}-\frac {3 b^{7/2} (b c-3 a d) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{5/2} (b c-a d)^4}-\frac {3 d^{5/2} \left (21 b^2 c^2-18 a b c d+5 a^2 d^2\right ) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{8 c^{7/2} (b c-a d)^4}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.32, size = 210, normalized size = 0.71 \begin {gather*} \frac {1}{8} \left (-\frac {8}{a^2 c^3 x}+\frac {4 b^4 x}{a^2 (-b c+a d)^3 \left (a+b x^2\right )}-\frac {2 d^3 x}{c^2 (b c-a d)^2 \left (c+d x^2\right )^2}+\frac {d^3 (-15 b c+7 a d) x}{c^3 (b c-a d)^3 \left (c+d x^2\right )}+\frac {12 b^{7/2} (-b c+3 a d) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{5/2} (b c-a d)^4}-\frac {3 d^{5/2} \left (21 b^2 c^2-18 a b c d+5 a^2 d^2\right ) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{c^{7/2} (b c-a d)^4}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-8/(a^2*c^3*x) + (4*b^4*x)/(a^2*(-(b*c) + a*d)^3*(a + b*x^2)) - (2*d^3*x)/(c^2*(b*c - a*d)^2*(c + d*x^2)^2) +
(d^3*(-15*b*c + 7*a*d)*x)/(c^3*(b*c - a*d)^3*(c + d*x^2)) + (12*b^(7/2)*(-(b*c) + 3*a*d)*ArcTan[(Sqrt[b]*x)/S
qrt[a]])/(a^(5/2)*(b*c - a*d)^4) - (3*d^(5/2)*(21*b^2*c^2 - 18*a*b*c*d + 5*a^2*d^2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]
])/(c^(7/2)*(b*c - a*d)^4))/8

________________________________________________________________________________________

Maple [A]
time = 0.23, size = 202, normalized size = 0.68

method	result	size
default	\(\frac {b^{4} \left (\frac {\left (\frac {a d}{2}-\frac {b c}{2}\right ) x}{b \,x^{2}+a}+\frac {3 \left (3 a d -b c \right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{a^{2} \left (a d -b c \right )^{4}}-\frac {d^{3} \left (\frac {\left (\frac {7}{8} a^{2} d^{3}-\frac {11}{4} a b c \,d^{2}+\frac {15}{8} b^{2} c^{2} d \right ) x^{3}+\frac {c \left (9 a^{2} d^{2}-26 a b c d +17 b^{2} c^{2}\right ) x}{8}}{\left (d \,x^{2}+c \right )^{2}}+\frac {3 \left (5 a^{2} d^{2}-18 a b c d +21 b^{2} c^{2}\right ) \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{8 \sqrt {c d}}\right )}{c^{3} \left (a d -b c \right )^{4}}-\frac {1}{a^{2} c^{3} x}\)	\(202\)
risch	\(\text {Expression too large to display}\)	\(2501\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b^4/a^2/(a*d-b*c)^4*((1/2*a*d-1/2*b*c)*x/(b*x^2+a)+3/2*(3*a*d-b*c)/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2)))-d^3/c^
3/(a*d-b*c)^4*(((7/8*a^2*d^3-11/4*a*b*c*d^2+15/8*b^2*c^2*d)*x^3+1/8*c*(9*a^2*d^2-26*a*b*c*d+17*b^2*c^2)*x)/(d*
x^2+c)^2+3/8*(5*a^2*d^2-18*a*b*c*d+21*b^2*c^2)/(c*d)^(1/2)*arctan(d*x/(c*d)^(1/2)))-1/a^2/c^3/x

________________________________________________________________________________________

Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 639 vs. \(2 (269) = 538\).
time = 0.54, size = 639, normalized size = 2.15 \begin {gather*} -\frac {3 \, {\left (b^{5} c - 3 \, a b^{4} d\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\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 )} \sqrt {a b}} - \frac {3 \, {\left (21 \, b^{2} c^{2} d^{3} - 18 \, a b c d^{4} + 5 \, a^{2} d^{5}\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{8 \, {\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 )} \sqrt {c d}} - \frac {8 \, a b^{3} c^{5} - 24 \, a^{2} b^{2} c^{4} d + 24 \, a^{3} b c^{3} d^{2} - 8 \, a^{4} c^{2} d^{3} + 3 \, {\left (4 \, b^{4} c^{3} d^{2} - 8 \, a b^{3} c^{2} d^{3} + 13 \, a^{2} b^{2} c d^{4} - 5 \, a^{3} b d^{5}\right )} x^{6} + {\left (24 \, b^{4} c^{4} d - 40 \, a b^{3} c^{3} d^{2} + 41 \, a^{2} b^{2} c^{2} d^{3} + 14 \, a^{3} b c d^{4} - 15 \, a^{4} d^{5}\right )} x^{4} + {\left (12 \, b^{4} c^{5} - 8 \, a b^{3} c^{4} d - 24 \, a^{2} b^{2} c^{3} d^{2} + 57 \, a^{3} b c^{2} d^{3} - 25 \, a^{4} c d^{4}\right )} x^{2}}{8 \, {\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^{7} + {\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^{5} + {\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^{3} + {\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\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/2*(b^5*c - 3*a*b^4*d)*arctan(b*x/sqrt(a*b))/((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)*sqrt(a*b)) - 3/8*(21*b^2*c^2*d^3 - 18*a*b*c*d^4 + 5*a^2*d^5)*arctan(d*x/sqrt(c*d))/((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)*sqrt(c*d)) - 1/8*(8*a*b^3*c^5 - 24*a^2*b^2
*c^4*d + 24*a^3*b*c^3*d^2 - 8*a^4*c^2*d^3 + 3*(4*b^4*c^3*d^2 - 8*a*b^3*c^2*d^3 + 13*a^2*b^2*c*d^4 - 5*a^3*b*d^
5)*x^6 + (24*b^4*c^4*d - 40*a*b^3*c^3*d^2 + 41*a^2*b^2*c^2*d^3 + 14*a^3*b*c*d^4 - 15*a^4*d^5)*x^4 + (12*b^4*c^
5 - 8*a*b^3*c^4*d - 24*a^2*b^2*c^3*d^2 + 57*a^3*b*c^2*d^3 - 25*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^7 + (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^5 + (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^3 + (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)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 914 vs. \(2 (269) = 538\).
time = 7.41, size = 3753, normalized size = 12.64 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/16*(16*a*b^4*c^6 - 64*a^2*b^3*c^5*d + 96*a^3*b^2*c^4*d^2 - 64*a^4*b*c^3*d^3 + 16*a^5*c^2*d^4 + 6*(4*b^5*c^
4*d^2 - 12*a*b^4*c^3*d^3 + 21*a^2*b^3*c^2*d^4 - 18*a^3*b^2*c*d^5 + 5*a^4*b*d^6)*x^6 + 2*(24*b^5*c^5*d - 64*a*b
^4*c^4*d^2 + 81*a^2*b^3*c^3*d^3 - 27*a^3*b^2*c^2*d^4 - 29*a^4*b*c*d^5 + 15*a^5*d^6)*x^4 + 2*(12*b^5*c^6 - 20*a
*b^4*c^5*d - 16*a^2*b^3*c^4*d^2 + 81*a^3*b^2*c^3*d^3 - 82*a^4*b*c^2*d^4 + 25*a^5*c*d^5)*x^2 + 12*((b^5*c^4*d^2
- 3*a*b^4*c^3*d^3)*x^7 + (2*b^5*c^5*d - 5*a*b^4*c^4*d^2 - 3*a^2*b^3*c^3*d^3)*x^5 + (b^5*c^6 - a*b^4*c^5*d - 6
*a^2*b^3*c^4*d^2)*x^3 + (a*b^4*c^6 - 3*a^2*b^3*c^5*d)*x)*sqrt(-b/a)*log((b*x^2 + 2*a*x*sqrt(-b/a) - a)/(b*x^2
+ a)) - 3*((21*a^2*b^3*c^2*d^4 - 18*a^3*b^2*c*d^5 + 5*a^4*b*d^6)*x^7 + (42*a^2*b^3*c^3*d^3 - 15*a^3*b^2*c^2*d^
4 - 8*a^4*b*c*d^5 + 5*a^5*d^6)*x^5 + (21*a^2*b^3*c^4*d^2 + 24*a^3*b^2*c^3*d^3 - 31*a^4*b*c^2*d^4 + 10*a^5*c*d^
5)*x^3 + (21*a^3*b^2*c^4*d^2 - 18*a^4*b*c^3*d^3 + 5*a^5*c^2*d^4)*x)*sqrt(-d/c)*log((d*x^2 - 2*c*x*sqrt(-d/c) -
c)/(d*x^2 + c)))/((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^7 + (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^5 + (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^3 + (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)*x), -1/8*
(8*a*b^4*c^6 - 32*a^2*b^3*c^5*d + 48*a^3*b^2*c^4*d^2 - 32*a^4*b*c^3*d^3 + 8*a^5*c^2*d^4 + 3*(4*b^5*c^4*d^2 - 1
2*a*b^4*c^3*d^3 + 21*a^2*b^3*c^2*d^4 - 18*a^3*b^2*c*d^5 + 5*a^4*b*d^6)*x^6 + (24*b^5*c^5*d - 64*a*b^4*c^4*d^2
+ 81*a^2*b^3*c^3*d^3 - 27*a^3*b^2*c^2*d^4 - 29*a^4*b*c*d^5 + 15*a^5*d^6)*x^4 + (12*b^5*c^6 - 20*a*b^4*c^5*d -
16*a^2*b^3*c^4*d^2 + 81*a^3*b^2*c^3*d^3 - 82*a^4*b*c^2*d^4 + 25*a^5*c*d^5)*x^2 + 3*((21*a^2*b^3*c^2*d^4 - 18*a
^3*b^2*c*d^5 + 5*a^4*b*d^6)*x^7 + (42*a^2*b^3*c^3*d^3 - 15*a^3*b^2*c^2*d^4 - 8*a^4*b*c*d^5 + 5*a^5*d^6)*x^5 +
(21*a^2*b^3*c^4*d^2 + 24*a^3*b^2*c^3*d^3 - 31*a^4*b*c^2*d^4 + 10*a^5*c*d^5)*x^3 + (21*a^3*b^2*c^4*d^2 - 18*a^4
*b*c^3*d^3 + 5*a^5*c^2*d^4)*x)*sqrt(d/c)*arctan(x*sqrt(d/c)) + 6*((b^5*c^4*d^2 - 3*a*b^4*c^3*d^3)*x^7 + (2*b^5
*c^5*d - 5*a*b^4*c^4*d^2 - 3*a^2*b^3*c^3*d^3)*x^5 + (b^5*c^6 - a*b^4*c^5*d - 6*a^2*b^3*c^4*d^2)*x^3 + (a*b^4*c
^6 - 3*a^2*b^3*c^5*d)*x)*sqrt(-b/a)*log((b*x^2 + 2*a*x*sqrt(-b/a) - a)/(b*x^2 + a)))/((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^7 + (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^5 + (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^3 + (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)*x), -1/16*(16*a*b^4*c^6 - 64*a^2*b^3*c^5*d + 96*a^3*
b^2*c^4*d^2 - 64*a^4*b*c^3*d^3 + 16*a^5*c^2*d^4 + 6*(4*b^5*c^4*d^2 - 12*a*b^4*c^3*d^3 + 21*a^2*b^3*c^2*d^4 - 1
8*a^3*b^2*c*d^5 + 5*a^4*b*d^6)*x^6 + 2*(24*b^5*c^5*d - 64*a*b^4*c^4*d^2 + 81*a^2*b^3*c^3*d^3 - 27*a^3*b^2*c^2*
d^4 - 29*a^4*b*c*d^5 + 15*a^5*d^6)*x^4 + 2*(12*b^5*c^6 - 20*a*b^4*c^5*d - 16*a^2*b^3*c^4*d^2 + 81*a^3*b^2*c^3*
d^3 - 82*a^4*b*c^2*d^4 + 25*a^5*c*d^5)*x^2 + 24*((b^5*c^4*d^2 - 3*a*b^4*c^3*d^3)*x^7 + (2*b^5*c^5*d - 5*a*b^4*
c^4*d^2 - 3*a^2*b^3*c^3*d^3)*x^5 + (b^5*c^6 - a*b^4*c^5*d - 6*a^2*b^3*c^4*d^2)*x^3 + (a*b^4*c^6 - 3*a^2*b^3*c^
5*d)*x)*sqrt(b/a)*arctan(x*sqrt(b/a)) - 3*((21*a^2*b^3*c^2*d^4 - 18*a^3*b^2*c*d^5 + 5*a^4*b*d^6)*x^7 + (42*a^2
*b^3*c^3*d^3 - 15*a^3*b^2*c^2*d^4 - 8*a^4*b*c*d^5 + 5*a^5*d^6)*x^5 + (21*a^2*b^3*c^4*d^2 + 24*a^3*b^2*c^3*d^3
- 31*a^4*b*c^2*d^4 + 10*a^5*c*d^5)*x^3 + (21*a^3*b^2*c^4*d^2 - 18*a^4*b*c^3*d^3 + 5*a^5*c^2*d^4)*x)*sqrt(-d/c)
*log((d*x^2 - 2*c*x*sqrt(-d/c) - c)/(d*x^2 + c)))/((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^7 + (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^5 + (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^3 + (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)*x), -1/8*(8*a*b^4*c^6 - 32*a^2*b^3*c^5*d + 48*a^3*b^2*c^4*d^2 - 32*a^4*b*c^3*d^3 + 8*a^
5*c^2*d^4 + 3*(4*b^5*c^4*d^2 - 12*a*b^4*c^3*d^3 + 21*a^2*b^3*c^2*d^4 - 18*a^3*b^2*c*d^5 + 5*a^4*b*d^6)*x^6 + (
24*b^5*c^5*d - 64*a*b^4*c^4*d^2 + 81*a^2*b^3*c^3*d^3 - 27*a^3*b^2*c^2*d^4 - 29*a^4*b*c*d^5 + 15*a^5*d^6)*x^4 +
(12*b^5*c^6 - 20*a*b^4*c^5*d - 16*a^2*b^3*c^4*d^2 + 81*a^3*b^2*c^3*d^3 - 82*a^4*b*c^2*d^4 + 25*a^5*c*d^5)*x^2
+ 12*((b^5*c^4*d^2 - 3*a*b^4*c^3*d^3)*x^7 + (2*b^5*c^5*d - 5*a*b^4*c^4*d^2 - 3*a^2*b^3*c^3*d^3)*x^5 + (b^5*c^
6 - a*b^4*c^5*d - 6*a^2*b^3*c^4*d^2)*x^3 + (a*b^4*c^6 - 3*a^2*b^3*c^5*d)*x)*sqrt(b/a)*arctan(x*sqrt(b/a)) + 3*
((21*a^2*b^3*c^2*d^4 - 18*a^3*b^2*c*d^5 + 5*a^4*b*d^6)*x^7 + (42*a^2*b^3*c^3*d^3 - 15*a^3*b^2*c^2*d^4 - 8*a^4*
b*c*d^5 + 5*a^5*d^6)*x^5 + (21*a^2*b^3*c^4*d^2 ...

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

Giac [A]
time = 0.88, size = 430, normalized size = 1.45 \begin {gather*} -\frac {3 \, {\left (b^{5} c - 3 \, a b^{4} d\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\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 )} \sqrt {a b}} - \frac {3 \, {\left (21 \, b^{2} c^{2} d^{3} - 18 \, a b c d^{4} + 5 \, a^{2} d^{5}\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{8 \, {\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 )} \sqrt {c d}} - \frac {3 \, b^{4} c^{3} x^{2} - 6 \, a b^{3} c^{2} d x^{2} + 6 \, a^{2} b^{2} c d^{2} x^{2} - 2 \, a^{3} b d^{3} x^{2} + 2 \, a b^{3} c^{3} - 6 \, a^{2} b^{2} c^{2} d + 6 \, a^{3} b c d^{2} - 2 \, a^{4} d^{3}}{2 \, {\left (a^{2} b^{3} c^{6} - 3 \, a^{3} b^{2} c^{5} d + 3 \, a^{4} b c^{4} d^{2} - a^{5} c^{3} d^{3}\right )} {\left (b x^{3} + a x\right )}} - \frac {15 \, b c d^{4} x^{3} - 7 \, a d^{5} x^{3} + 17 \, b c^{2} d^{3} x - 9 \, a c d^{4} x}{8 \, {\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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/2*(b^5*c - 3*a*b^4*d)*arctan(b*x/sqrt(a*b))/((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)*sqrt(a*b)) - 3/8*(21*b^2*c^2*d^3 - 18*a*b*c*d^4 + 5*a^2*d^5)*arctan(d*x/sqrt(c*d))/((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)*sqrt(c*d)) - 1/2*(3*b^4*c^3*x^2 - 6*a*b^3*
c^2*d*x^2 + 6*a^2*b^2*c*d^2*x^2 - 2*a^3*b*d^3*x^2 + 2*a*b^3*c^3 - 6*a^2*b^2*c^2*d + 6*a^3*b*c*d^2 - 2*a^4*d^3)
/((a^2*b^3*c^6 - 3*a^3*b^2*c^5*d + 3*a^4*b*c^4*d^2 - a^5*c^3*d^3)*(b*x^3 + a*x)) - 1/8*(15*b*c*d^4*x^3 - 7*a*d
^5*x^3 + 17*b*c^2*d^3*x - 9*a*c*d^4*x)/((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)

________________________________________________________________________________________

Mupad [B]
time = 1.55, size = 2500, normalized size = 8.42 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(atan((a^9*d^5*x*(-c^7*d^5)^(3/2)*25i + b^9*c^16*d*x*(-c^7*d^5)^(1/2)*16i - a^6*b^3*c^3*d^2*x*(-c^7*d^5)^(3/2)
*756i + a^7*b^2*c^2*d^3*x*(-c^7*d^5)^(3/2)*534i + a^2*b^7*c^14*d^3*x*(-c^7*d^5)^(1/2)*144i - a^8*b*c*d^4*x*(-c
^7*d^5)^(3/2)*180i + a^5*b^4*c^4*d*x*(-c^7*d^5)^(3/2)*441i - a*b^8*c^15*d^2*x*(-c^7*d^5)^(1/2)*96i)/(25*a^9*c^
11*d^12 - 16*b^9*c^20*d^3 + 96*a*b^8*c^19*d^4 - 180*a^8*b*c^12*d^11 - 144*a^2*b^7*c^18*d^5 + 441*a^5*b^4*c^15*
d^8 - 756*a^6*b^3*c^14*d^9 + 534*a^7*b^2*c^13*d^10))*(-c^7*d^5)^(1/2)*(5*a^2*d^2 + 21*b^2*c^2 - 18*a*b*c*d)*3i
)/(8*(b^4*c^11 + a^4*c^7*d^4 - 4*a^3*b*c^8*d^3 + 6*a^2*b^2*c^9*d^2 - 4*a*b^3*c^10*d)) - (atan((((x*(147456*a^6
*b^20*c^26*d^3 - 2211840*a^7*b^19*c^25*d^4 + 14598144*a^8*b^18*c^24*d^5 - 56180736*a^9*b^17*c^23*d^6 + 1447372
80*a^10*b^16*c^22*d^7 - 285078528*a^11*b^15*c^21*d^8 + 505018368*a^12*b^14*c^20*d^9 - 885012480*a^13*b^13*c^19
*d^10 + 1434332160*a^14*b^12*c^18*d^11 - 1921047552*a^15*b^11*c^17*d^12 + 1999835136*a^16*b^10*c^16*d^13 - 158
1355008*a^17*b^9*c^15*d^14 + 938843136*a^18*b^8*c^14*d^15 - 412314624*a^19*b^7*c^13*d^16 + 130332672*a^20*b^6*
c^12*d^17 - 28145664*a^21*b^5*c^11*d^18 + 3732480*a^22*b^4*c^10*d^19 - 230400*a^23*b^3*c^9*d^20) + (3*(3*a*d -
b*c)*(-a^5*b^7)^(1/2)*(3145728*a^9*b^19*c^29*d^3 - 196608*a^8*b^20*c^30*d^2 - 23003136*a^10*b^18*c^28*d^4 + 1
01203968*a^11*b^17*c^27*d^5 - 294961152*a^12*b^16*c^26*d^6 + 582500352*a^13*b^15*c^25*d^7 - 729071616*a^14*b^1
4*c^24*d^8 + 339296256*a^15*b^13*c^23*d^9 + 766132224*a^16*b^12*c^22*d^10 - 2185936896*a^17*b^11*c^21*d^11 + 3
127787520*a^18*b^10*c^20*d^12 - 3084337152*a^19*b^9*c^19*d^13 + 2249834496*a^20*b^8*c^18*d^14 - 1236221952*a^2
1*b^7*c^17*d^15 + 508674048*a^22*b^6*c^16*d^16 - 152715264*a^23*b^5*c^15*d^17 + 31703040*a^24*b^4*c^14*d^18 -
4079616*a^25*b^3*c^13*d^19 + 245760*a^26*b^2*c^12*d^20 + (3*x*(3*a*d - b*c)*(-a^5*b^7)^(1/2)*(262144*a^10*b^20
*c^33*d^2 - 4194304*a^11*b^19*c^32*d^3 + 31195136*a^12*b^18*c^31*d^4 - 142606336*a^13*b^17*c^30*d^5 + 44564480
0*a^14*b^16*c^29*d^6 - 998244352*a^15*b^15*c^28*d^7 + 1622147072*a^16*b^14*c^27*d^8 - 1853882368*a^17*b^13*c^2
6*d^9 + 1274544128*a^18*b^12*c^25*d^10 - 1274544128*a^20*b^10*c^23*d^12 + 1853882368*a^21*b^9*c^22*d^13 - 1622
147072*a^22*b^8*c^21*d^14 + 998244352*a^23*b^7*c^20*d^15 - 445644800*a^24*b^6*c^19*d^16 + 142606336*a^25*b^5*c
^18*d^17 - 31195136*a^26*b^4*c^17*d^18 + 4194304*a^27*b^3*c^16*d^19 - 262144*a^28*b^2*c^15*d^20))/(4*(a^9*d^4
+ a^5*b^4*c^4 - 4*a^6*b^3*c^3*d + 6*a^7*b^2*c^2*d^2 - 4*a^8*b*c*d^3))))/(4*(a^9*d^4 + a^5*b^4*c^4 - 4*a^6*b^3*
c^3*d + 6*a^7*b^2*c^2*d^2 - 4*a^8*b*c*d^3)))*(3*a*d - b*c)*(-a^5*b^7)^(1/2)*3i)/(4*(a^9*d^4 + a^5*b^4*c^4 - 4*
a^6*b^3*c^3*d + 6*a^7*b^2*c^2*d^2 - 4*a^8*b*c*d^3)) + ((x*(147456*a^6*b^20*c^26*d^3 - 2211840*a^7*b^19*c^25*d^
4 + 14598144*a^8*b^18*c^24*d^5 - 56180736*a^9*b^17*c^23*d^6 + 144737280*a^10*b^16*c^22*d^7 - 285078528*a^11*b^
15*c^21*d^8 + 505018368*a^12*b^14*c^20*d^9 - 885012480*a^13*b^13*c^19*d^10 + 1434332160*a^14*b^12*c^18*d^11 -
1921047552*a^15*b^11*c^17*d^12 + 1999835136*a^16*b^10*c^16*d^13 - 1581355008*a^17*b^9*c^15*d^14 + 938843136*a^
18*b^8*c^14*d^15 - 412314624*a^19*b^7*c^13*d^16 + 130332672*a^20*b^6*c^12*d^17 - 28145664*a^21*b^5*c^11*d^18 +
3732480*a^22*b^4*c^10*d^19 - 230400*a^23*b^3*c^9*d^20) + (3*(3*a*d - b*c)*(-a^5*b^7)^(1/2)*(196608*a^8*b^20*c
^30*d^2 - 3145728*a^9*b^19*c^29*d^3 + 23003136*a^10*b^18*c^28*d^4 - 101203968*a^11*b^17*c^27*d^5 + 294961152*a
^12*b^16*c^26*d^6 - 582500352*a^13*b^15*c^25*d^7 + 729071616*a^14*b^14*c^24*d^8 - 339296256*a^15*b^13*c^23*d^9
- 766132224*a^16*b^12*c^22*d^10 + 2185936896*a^17*b^11*c^21*d^11 - 3127787520*a^18*b^10*c^20*d^12 + 308433715
2*a^19*b^9*c^19*d^13 - 2249834496*a^20*b^8*c^18*d^14 + 1236221952*a^21*b^7*c^17*d^15 - 508674048*a^22*b^6*c^16
*d^16 + 152715264*a^23*b^5*c^15*d^17 - 31703040*a^24*b^4*c^14*d^18 + 4079616*a^25*b^3*c^13*d^19 - 245760*a^26*
b^2*c^12*d^20 + (3*x*(3*a*d - b*c)*(-a^5*b^7)^(1/2)*(262144*a^10*b^20*c^33*d^2 - 4194304*a^11*b^19*c^32*d^3 +
31195136*a^12*b^18*c^31*d^4 - 142606336*a^13*b^17*c^30*d^5 + 445644800*a^14*b^16*c^29*d^6 - 998244352*a^15*b^1
5*c^28*d^7 + 1622147072*a^16*b^14*c^27*d^8 - 1853882368*a^17*b^13*c^26*d^9 + 1274544128*a^18*b^12*c^25*d^10 -
1274544128*a^20*b^10*c^23*d^12 + 1853882368*a^21*b^9*c^22*d^13 - 1622147072*a^22*b^8*c^21*d^14 + 998244352*a^2
3*b^7*c^20*d^15 - 445644800*a^24*b^6*c^19*d^16 + 142606336*a^25*b^5*c^18*d^17 - 31195136*a^26*b^4*c^17*d^18 +
4194304*a^27*b^3*c^16*d^19 - 262144*a^28*b^2*c^15*d^20))/(4*(a^9*d^4 + a^5*b^4*c^4 - 4*a^6*b^3*c^3*d + 6*a^7*b
^2*c^2*d^2 - 4*a^8*b*c*d^3))))/(4*(a^9*d^4 + a^5*b^4*c^4 - 4*a^6*b^3*c^3*d + 6*a^7*b^2*c^2*d^2 - 4*a^8*b*c*d^3
)))*(3*a*d - b*c)*(-a^5*b^7)^(1/2)*3i)/(4*(a^9*d^4 + a^5*b^4*c^4 - 4*a^6*b^3*c^3*d + 6*a^7*b^2*c^2*d^2 - 4*a^8
*b*c*d^3)))/(1161216*a^6*b^18*c^21*d^5 - 13768704*a^7*b^17*c^20*d^6 + 74221056*a^8*b^16*c^19*d^7 - 244574208*a
^9*b^15*c^18*d^8 + 551397888*a^10*b^14*c^17*d^9 - 893251584*a^11*b^13*c^16*d^10 + 1058724864*a^12*b^12*c^15*d^
11 - 918245376*a^13*b^11*c^14*d^12 + 575106048*...

________________________________________________________________________________________