3.318 \(\int \frac{1}{x^4 (a+b x^2)^2 (c+d x^2)^3} \, dx\)

Optimal. Leaf size=377 \[ \frac{d \left (-7 a^2 d^2+15 a b c d+4 b^2 c^2\right )}{8 a c^2 x^3 \left (c+d x^2\right ) (b c-a d)^3}-\frac{75 a^2 b c d^2-35 a^3 d^3-24 a b^2 c^2 d+20 b^3 c^3}{24 a^2 c^3 x^3 (b c-a d)^3}+\frac{-24 a^2 b^2 c^2 d^2+75 a^3 b c d^3-35 a^4 d^4-24 a b^3 c^3 d+20 b^4 c^4}{8 a^3 c^4 x (b c-a d)^3}+\frac{d^{7/2} \left (35 a^2 d^2-110 a b c d+99 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{8 c^{9/2} (b c-a d)^4}+\frac{b^{9/2} (5 b c-11 a d) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{7/2} (b c-a d)^4}+\frac{b}{2 a x^3 \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)}+\frac{d (a d+2 b c)}{4 a c x^3 \left (c+d x^2\right )^2 (b c-a d)^2} \]

[Out]

-(20*b^3*c^3 - 24*a*b^2*c^2*d + 75*a^2*b*c*d^2 - 35*a^3*d^3)/(24*a^2*c^3*(b*c - a*d)^3*x^3) + (20*b^4*c^4 - 24
*a*b^3*c^3*d - 24*a^2*b^2*c^2*d^2 + 75*a^3*b*c*d^3 - 35*a^4*d^4)/(8*a^3*c^4*(b*c - a*d)^3*x) + (d*(2*b*c + a*d
))/(4*a*c*(b*c - a*d)^2*x^3*(c + d*x^2)^2) + b/(2*a*(b*c - a*d)*x^3*(a + b*x^2)*(c + d*x^2)^2) + (d*(4*b^2*c^2
+ 15*a*b*c*d - 7*a^2*d^2))/(8*a*c^2*(b*c - a*d)^3*x^3*(c + d*x^2)) + (b^(9/2)*(5*b*c - 11*a*d)*ArcTan[(Sqrt[b
]*x)/Sqrt[a]])/(2*a^(7/2)*(b*c - a*d)^4) + (d^(7/2)*(99*b^2*c^2 - 110*a*b*c*d + 35*a^2*d^2)*ArcTan[(Sqrt[d]*x)
/Sqrt[c]])/(8*c^(9/2)*(b*c - a*d)^4)

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Rubi [A]  time = 0.68604, antiderivative size = 377, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {472, 579, 583, 522, 205} \[ \frac{d \left (-7 a^2 d^2+15 a b c d+4 b^2 c^2\right )}{8 a c^2 x^3 \left (c+d x^2\right ) (b c-a d)^3}-\frac{75 a^2 b c d^2-35 a^3 d^3-24 a b^2 c^2 d+20 b^3 c^3}{24 a^2 c^3 x^3 (b c-a d)^3}+\frac{-24 a^2 b^2 c^2 d^2+75 a^3 b c d^3-35 a^4 d^4-24 a b^3 c^3 d+20 b^4 c^4}{8 a^3 c^4 x (b c-a d)^3}+\frac{d^{7/2} \left (35 a^2 d^2-110 a b c d+99 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{8 c^{9/2} (b c-a d)^4}+\frac{b^{9/2} (5 b c-11 a d) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{7/2} (b c-a d)^4}+\frac{b}{2 a x^3 \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)}+\frac{d (a d+2 b c)}{4 a c x^3 \left (c+d x^2\right )^2 (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(20*b^3*c^3 - 24*a*b^2*c^2*d + 75*a^2*b*c*d^2 - 35*a^3*d^3)/(24*a^2*c^3*(b*c - a*d)^3*x^3) + (20*b^4*c^4 - 24
*a*b^3*c^3*d - 24*a^2*b^2*c^2*d^2 + 75*a^3*b*c*d^3 - 35*a^4*d^4)/(8*a^3*c^4*(b*c - a*d)^3*x) + (d*(2*b*c + a*d
))/(4*a*c*(b*c - a*d)^2*x^3*(c + d*x^2)^2) + b/(2*a*(b*c - a*d)*x^3*(a + b*x^2)*(c + d*x^2)^2) + (d*(4*b^2*c^2
+ 15*a*b*c*d - 7*a^2*d^2))/(8*a*c^2*(b*c - a*d)^3*x^3*(c + d*x^2)) + (b^(9/2)*(5*b*c - 11*a*d)*ArcTan[(Sqrt[b
]*x)/Sqrt[a]])/(2*a^(7/2)*(b*c - a*d)^4) + (d^(7/2)*(99*b^2*c^2 - 110*a*b*c*d + 35*a^2*d^2)*ArcTan[(Sqrt[d]*x)
/Sqrt[c]])/(8*c^(9/2)*(b*c - a*d)^4)

Rule 472

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 579

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 583

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]

Rule 522

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 205

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

Rubi steps

\begin{align*} \int \frac{1}{x^4 \left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx &=\frac{b}{2 a (b c-a d) x^3 \left (a+b x^2\right ) \left (c+d x^2\right )^2}-\frac{\int \frac{-5 b c+2 a d-9 b d x^2}{x^4 \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^3 \left (c+d x^2\right )^2}+\frac{b}{2 a (b c-a d) x^3 \left (a+b x^2\right ) \left (c+d x^2\right )^2}-\frac{\int \frac{-2 \left (10 b^2 c^2-8 a b c d+7 a^2 d^2\right )-14 b d (2 b c+a d) x^2}{x^4 \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^3 \left (c+d x^2\right )^2}+\frac{b}{2 a (b c-a d) x^3 \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac{d \left (4 b^2 c^2+15 a b c d-7 a^2 d^2\right )}{8 a c^2 (b c-a d)^3 x^3 \left (c+d x^2\right )}-\frac{\int \frac{-2 \left (20 b^3 c^3-24 a b^2 c^2 d+75 a^2 b c d^2-35 a^3 d^3\right )-10 b d \left (4 b^2 c^2+15 a b c d-7 a^2 d^2\right ) x^2}{x^4 \left (a+b x^2\right ) \left (c+d x^2\right )} \, dx}{16 a c^2 (b c-a d)^3}\\ &=-\frac{20 b^3 c^3-24 a b^2 c^2 d+75 a^2 b c d^2-35 a^3 d^3}{24 a^2 c^3 (b c-a d)^3 x^3}+\frac{d (2 b c+a d)}{4 a c (b c-a d)^2 x^3 \left (c+d x^2\right )^2}+\frac{b}{2 a (b c-a d) x^3 \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac{d \left (4 b^2 c^2+15 a b c d-7 a^2 d^2\right )}{8 a c^2 (b c-a d)^3 x^3 \left (c+d x^2\right )}+\frac{\int \frac{-6 \left (20 b^4 c^4-24 a b^3 c^3 d-24 a^2 b^2 c^2 d^2+75 a^3 b c d^3-35 a^4 d^4\right )-6 b d \left (20 b^3 c^3-24 a b^2 c^2 d+75 a^2 b c d^2-35 a^3 d^3\right ) x^2}{x^2 \left (a+b x^2\right ) \left (c+d x^2\right )} \, dx}{48 a^2 c^3 (b c-a d)^3}\\ &=-\frac{20 b^3 c^3-24 a b^2 c^2 d+75 a^2 b c d^2-35 a^3 d^3}{24 a^2 c^3 (b c-a d)^3 x^3}+\frac{20 b^4 c^4-24 a b^3 c^3 d-24 a^2 b^2 c^2 d^2+75 a^3 b c d^3-35 a^4 d^4}{8 a^3 c^4 (b c-a d)^3 x}+\frac{d (2 b c+a d)}{4 a c (b c-a d)^2 x^3 \left (c+d x^2\right )^2}+\frac{b}{2 a (b c-a d) x^3 \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac{d \left (4 b^2 c^2+15 a b c d-7 a^2 d^2\right )}{8 a c^2 (b c-a d)^3 x^3 \left (c+d x^2\right )}-\frac{\int \frac{-6 \left (20 b^5 c^5-24 a b^4 c^4 d-24 a^2 b^3 c^3 d^2-24 a^3 b^2 c^2 d^3+75 a^4 b c d^4-35 a^5 d^5\right )-6 b d \left (20 b^4 c^4-24 a b^3 c^3 d-24 a^2 b^2 c^2 d^2+75 a^3 b c d^3-35 a^4 d^4\right ) x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx}{48 a^3 c^4 (b c-a d)^3}\\ &=-\frac{20 b^3 c^3-24 a b^2 c^2 d+75 a^2 b c d^2-35 a^3 d^3}{24 a^2 c^3 (b c-a d)^3 x^3}+\frac{20 b^4 c^4-24 a b^3 c^3 d-24 a^2 b^2 c^2 d^2+75 a^3 b c d^3-35 a^4 d^4}{8 a^3 c^4 (b c-a d)^3 x}+\frac{d (2 b c+a d)}{4 a c (b c-a d)^2 x^3 \left (c+d x^2\right )^2}+\frac{b}{2 a (b c-a d) x^3 \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac{d \left (4 b^2 c^2+15 a b c d-7 a^2 d^2\right )}{8 a c^2 (b c-a d)^3 x^3 \left (c+d x^2\right )}+\frac{\left (b^5 (5 b c-11 a d)\right ) \int \frac{1}{a+b x^2} \, dx}{2 a^3 (b c-a d)^4}+\frac{\left (d^4 \left (99 b^2 c^2-110 a b c d+35 a^2 d^2\right )\right ) \int \frac{1}{c+d x^2} \, dx}{8 c^4 (b c-a d)^4}\\ &=-\frac{20 b^3 c^3-24 a b^2 c^2 d+75 a^2 b c d^2-35 a^3 d^3}{24 a^2 c^3 (b c-a d)^3 x^3}+\frac{20 b^4 c^4-24 a b^3 c^3 d-24 a^2 b^2 c^2 d^2+75 a^3 b c d^3-35 a^4 d^4}{8 a^3 c^4 (b c-a d)^3 x}+\frac{d (2 b c+a d)}{4 a c (b c-a d)^2 x^3 \left (c+d x^2\right )^2}+\frac{b}{2 a (b c-a d) x^3 \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac{d \left (4 b^2 c^2+15 a b c d-7 a^2 d^2\right )}{8 a c^2 (b c-a d)^3 x^3 \left (c+d x^2\right )}+\frac{b^{9/2} (5 b c-11 a d) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{7/2} (b c-a d)^4}+\frac{d^{7/2} \left (99 b^2 c^2-110 a b c d+35 a^2 d^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{8 c^{9/2} (b c-a d)^4}\\ \end{align*}

Mathematica [A]  time = 0.450843, size = 230, normalized size = 0.61 \[ \frac{1}{24} \left (\frac{3 d^{7/2} \left (35 a^2 d^2-110 a b c d+99 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{c^{9/2} (b c-a d)^4}-\frac{12 b^5 x}{a^3 \left (a+b x^2\right ) (a d-b c)^3}+\frac{12 b^{9/2} (5 b c-11 a d) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{7/2} (b c-a d)^4}+\frac{72 a d+48 b c}{a^3 c^4 x}-\frac{8}{a^2 c^3 x^3}+\frac{3 d^4 x (19 b c-11 a d)}{c^4 \left (c+d x^2\right ) (b c-a d)^3}+\frac{6 d^4 x}{c^3 \left (c+d x^2\right )^2 (b c-a d)^2}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-8/(a^2*c^3*x^3) + (48*b*c + 72*a*d)/(a^3*c^4*x) - (12*b^5*x)/(a^3*(-(b*c) + a*d)^3*(a + b*x^2)) + (6*d^4*x)/
(c^3*(b*c - a*d)^2*(c + d*x^2)^2) + (3*d^4*(19*b*c - 11*a*d)*x)/(c^4*(b*c - a*d)^3*(c + d*x^2)) + (12*b^(9/2)*
(5*b*c - 11*a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(a^(7/2)*(b*c - a*d)^4) + (3*d^(7/2)*(99*b^2*c^2 - 110*a*b*c*d +
35*a^2*d^2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(c^(9/2)*(b*c - a*d)^4))/24

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Maple [A]  time = 0.025, size = 455, normalized size = 1.2 \begin{align*}{\frac{11\,{d}^{7}{x}^{3}{a}^{2}}{8\,{c}^{4} \left ( ad-bc \right ) ^{4} \left ( d{x}^{2}+c \right ) ^{2}}}-{\frac{15\,{d}^{6}{x}^{3}ab}{4\,{c}^{3} \left ( ad-bc \right ) ^{4} \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{19\,{d}^{5}{x}^{3}{b}^{2}}{8\,{c}^{2} \left ( ad-bc \right ) ^{4} \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{13\,{d}^{6}{a}^{2}x}{8\,{c}^{3} \left ( ad-bc \right ) ^{4} \left ( d{x}^{2}+c \right ) ^{2}}}-{\frac{17\,{d}^{5}abx}{4\,{c}^{2} \left ( ad-bc \right ) ^{4} \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{21\,{b}^{2}{d}^{4}x}{8\,c \left ( ad-bc \right ) ^{4} \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{35\,{d}^{6}{a}^{2}}{8\,{c}^{4} \left ( ad-bc \right ) ^{4}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-{\frac{55\,{d}^{5}ab}{4\,{c}^{3} \left ( ad-bc \right ) ^{4}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}+{\frac{99\,{b}^{2}{d}^{4}}{8\,{c}^{2} \left ( ad-bc \right ) ^{4}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-{\frac{1}{3\,{a}^{2}{c}^{3}{x}^{3}}}+3\,{\frac{d}{{a}^{2}{c}^{4}x}}+2\,{\frac{b}{{a}^{3}{c}^{3}x}}-{\frac{{b}^{5}xd}{2\,{a}^{2} \left ( ad-bc \right ) ^{4} \left ( b{x}^{2}+a \right ) }}+{\frac{{b}^{6}xc}{2\,{a}^{3} \left ( ad-bc \right ) ^{4} \left ( b{x}^{2}+a \right ) }}-{\frac{11\,{b}^{5}d}{2\,{a}^{2} \left ( ad-bc \right ) ^{4}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{5\,{b}^{6}c}{2\,{a}^{3} \left ( ad-bc \right ) ^{4}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^4/(b*x^2+a)^2/(d*x^2+c)^3,x)

[Out]

11/8*d^7/c^4/(a*d-b*c)^4/(d*x^2+c)^2*x^3*a^2-15/4*d^6/c^3/(a*d-b*c)^4/(d*x^2+c)^2*x^3*a*b+19/8*d^5/c^2/(a*d-b*
c)^4/(d*x^2+c)^2*x^3*b^2+13/8*d^6/c^3/(a*d-b*c)^4/(d*x^2+c)^2*a^2*x-17/4*d^5/c^2/(a*d-b*c)^4/(d*x^2+c)^2*a*b*x
+21/8*d^4/c/(a*d-b*c)^4/(d*x^2+c)^2*b^2*x+35/8*d^6/c^4/(a*d-b*c)^4/(c*d)^(1/2)*arctan(x*d/(c*d)^(1/2))*a^2-55/
4*d^5/c^3/(a*d-b*c)^4/(c*d)^(1/2)*arctan(x*d/(c*d)^(1/2))*a*b+99/8*d^4/c^2/(a*d-b*c)^4/(c*d)^(1/2)*arctan(x*d/
(c*d)^(1/2))*b^2-1/3/a^2/c^3/x^3+3/a^2/c^4/x*d+2/a^3/c^3/x*b-1/2*b^5/a^2/(a*d-b*c)^4*x/(b*x^2+a)*d+1/2*b^6/a^3
/(a*d-b*c)^4*x/(b*x^2+a)*c-11/2*b^5/a^2/(a*d-b*c)^4/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))*d+5/2*b^6/a^3/(a*d-b*c
)^4/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))*c

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(b*x^2+a)^2/(d*x^2+c)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 66.8721, size = 8631, normalized size = 22.89 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(b*x^2+a)^2/(d*x^2+c)^3,x, algorithm="fricas")

[Out]

[-1/48*(16*a^2*b^4*c^7 - 64*a^3*b^3*c^6*d + 96*a^4*b^2*c^5*d^2 - 64*a^5*b*c^4*d^3 + 16*a^6*c^3*d^4 - 6*(20*b^6
*c^5*d^2 - 44*a*b^5*c^4*d^3 + 99*a^3*b^3*c^2*d^5 - 110*a^4*b^2*c*d^6 + 35*a^5*b*d^7)*x^8 - 2*(120*b^6*c^6*d -
224*a*b^5*c^5*d^2 - 88*a^2*b^4*c^4*d^3 + 495*a^3*b^3*c^3*d^4 - 253*a^4*b^2*c^2*d^5 - 155*a^5*b*c*d^6 + 105*a^6
*d^7)*x^6 - 2*(60*b^6*c^7 - 52*a*b^5*c^6*d - 184*a^2*b^4*c^5*d^2 + 176*a^3*b^3*c^4*d^3 + 319*a^4*b^2*c^3*d^4 -
494*a^5*b*c^2*d^5 + 175*a^6*c*d^6)*x^4 - 16*(5*a*b^5*c^7 - 13*a^2*b^4*c^6*d + 2*a^3*b^3*c^5*d^2 + 22*a^4*b^2*
c^4*d^3 - 23*a^5*b*c^3*d^4 + 7*a^6*c^2*d^5)*x^2 + 12*((5*b^6*c^5*d^2 - 11*a*b^5*c^4*d^3)*x^9 + (10*b^6*c^6*d -
17*a*b^5*c^5*d^2 - 11*a^2*b^4*c^4*d^3)*x^7 + (5*b^6*c^7 - a*b^5*c^6*d - 22*a^2*b^4*c^5*d^2)*x^5 + (5*a*b^5*c^
7 - 11*a^2*b^4*c^6*d)*x^3)*sqrt(-b/a)*log((b*x^2 - 2*a*x*sqrt(-b/a) - a)/(b*x^2 + a)) - 3*((99*a^3*b^3*c^2*d^5
- 110*a^4*b^2*c*d^6 + 35*a^5*b*d^7)*x^9 + (198*a^3*b^3*c^3*d^4 - 121*a^4*b^2*c^2*d^5 - 40*a^5*b*c*d^6 + 35*a^
6*d^7)*x^7 + (99*a^3*b^3*c^4*d^3 + 88*a^4*b^2*c^3*d^4 - 185*a^5*b*c^2*d^5 + 70*a^6*c*d^6)*x^5 + (99*a^4*b^2*c^
4*d^3 - 110*a^5*b*c^3*d^4 + 35*a^6*c^2*d^5)*x^3)*sqrt(-d/c)*log((d*x^2 + 2*c*x*sqrt(-d/c) - c)/(d*x^2 + c)))/(
(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^9 + (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^7 + (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^5 + (
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^3), -1/24*(8*a^2*b^4*c^7
- 32*a^3*b^3*c^6*d + 48*a^4*b^2*c^5*d^2 - 32*a^5*b*c^4*d^3 + 8*a^6*c^3*d^4 - 3*(20*b^6*c^5*d^2 - 44*a*b^5*c^4
*d^3 + 99*a^3*b^3*c^2*d^5 - 110*a^4*b^2*c*d^6 + 35*a^5*b*d^7)*x^8 - (120*b^6*c^6*d - 224*a*b^5*c^5*d^2 - 88*a^
2*b^4*c^4*d^3 + 495*a^3*b^3*c^3*d^4 - 253*a^4*b^2*c^2*d^5 - 155*a^5*b*c*d^6 + 105*a^6*d^7)*x^6 - (60*b^6*c^7 -
52*a*b^5*c^6*d - 184*a^2*b^4*c^5*d^2 + 176*a^3*b^3*c^4*d^3 + 319*a^4*b^2*c^3*d^4 - 494*a^5*b*c^2*d^5 + 175*a^
6*c*d^6)*x^4 - 8*(5*a*b^5*c^7 - 13*a^2*b^4*c^6*d + 2*a^3*b^3*c^5*d^2 + 22*a^4*b^2*c^4*d^3 - 23*a^5*b*c^3*d^4 +
7*a^6*c^2*d^5)*x^2 - 3*((99*a^3*b^3*c^2*d^5 - 110*a^4*b^2*c*d^6 + 35*a^5*b*d^7)*x^9 + (198*a^3*b^3*c^3*d^4 -
121*a^4*b^2*c^2*d^5 - 40*a^5*b*c*d^6 + 35*a^6*d^7)*x^7 + (99*a^3*b^3*c^4*d^3 + 88*a^4*b^2*c^3*d^4 - 185*a^5*b*
c^2*d^5 + 70*a^6*c*d^6)*x^5 + (99*a^4*b^2*c^4*d^3 - 110*a^5*b*c^3*d^4 + 35*a^6*c^2*d^5)*x^3)*sqrt(d/c)*arctan(
x*sqrt(d/c)) + 6*((5*b^6*c^5*d^2 - 11*a*b^5*c^4*d^3)*x^9 + (10*b^6*c^6*d - 17*a*b^5*c^5*d^2 - 11*a^2*b^4*c^4*d
^3)*x^7 + (5*b^6*c^7 - a*b^5*c^6*d - 22*a^2*b^4*c^5*d^2)*x^5 + (5*a*b^5*c^7 - 11*a^2*b^4*c^6*d)*x^3)*sqrt(-b/a
)*log((b*x^2 - 2*a*x*sqrt(-b/a) - a)/(b*x^2 + a)))/((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^9 + (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^7 + (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^5 + (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^3), -1/48*(16*a^2*b^4*c^7 - 64*a^3*b^3*c^6*d + 96*a^4*b^2*c^5*d^2 - 64*a^5*b*c^4*d
^3 + 16*a^6*c^3*d^4 - 6*(20*b^6*c^5*d^2 - 44*a*b^5*c^4*d^3 + 99*a^3*b^3*c^2*d^5 - 110*a^4*b^2*c*d^6 + 35*a^5*b
*d^7)*x^8 - 2*(120*b^6*c^6*d - 224*a*b^5*c^5*d^2 - 88*a^2*b^4*c^4*d^3 + 495*a^3*b^3*c^3*d^4 - 253*a^4*b^2*c^2*
d^5 - 155*a^5*b*c*d^6 + 105*a^6*d^7)*x^6 - 2*(60*b^6*c^7 - 52*a*b^5*c^6*d - 184*a^2*b^4*c^5*d^2 + 176*a^3*b^3*
c^4*d^3 + 319*a^4*b^2*c^3*d^4 - 494*a^5*b*c^2*d^5 + 175*a^6*c*d^6)*x^4 - 16*(5*a*b^5*c^7 - 13*a^2*b^4*c^6*d +
2*a^3*b^3*c^5*d^2 + 22*a^4*b^2*c^4*d^3 - 23*a^5*b*c^3*d^4 + 7*a^6*c^2*d^5)*x^2 - 24*((5*b^6*c^5*d^2 - 11*a*b^5
*c^4*d^3)*x^9 + (10*b^6*c^6*d - 17*a*b^5*c^5*d^2 - 11*a^2*b^4*c^4*d^3)*x^7 + (5*b^6*c^7 - a*b^5*c^6*d - 22*a^2
*b^4*c^5*d^2)*x^5 + (5*a*b^5*c^7 - 11*a^2*b^4*c^6*d)*x^3)*sqrt(b/a)*arctan(x*sqrt(b/a)) - 3*((99*a^3*b^3*c^2*d
^5 - 110*a^4*b^2*c*d^6 + 35*a^5*b*d^7)*x^9 + (198*a^3*b^3*c^3*d^4 - 121*a^4*b^2*c^2*d^5 - 40*a^5*b*c*d^6 + 35*
a^6*d^7)*x^7 + (99*a^3*b^3*c^4*d^3 + 88*a^4*b^2*c^3*d^4 - 185*a^5*b*c^2*d^5 + 70*a^6*c*d^6)*x^5 + (99*a^4*b^2*
c^4*d^3 - 110*a^5*b*c^3*d^4 + 35*a^6*c^2*d^5)*x^3)*sqrt(-d/c)*log((d*x^2 + 2*c*x*sqrt(-d/c) - c)/(d*x^2 + c)))
/((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^9 + (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^7 + (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^5 +
(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^3), -1/24*(8*a^2*b^4*c
^7 - 32*a^3*b^3*c^6*d + 48*a^4*b^2*c^5*d^2 - 32*a^5*b*c^4*d^3 + 8*a^6*c^3*d^4 - 3*(20*b^6*c^5*d^2 - 44*a*b^5*c
^4*d^3 + 99*a^3*b^3*c^2*d^5 - 110*a^4*b^2*c*d^6 + 35*a^5*b*d^7)*x^8 - (120*b^6*c^6*d - 224*a*b^5*c^5*d^2 - 88*
a^2*b^4*c^4*d^3 + 495*a^3*b^3*c^3*d^4 - 253*a^4*b^2*c^2*d^5 - 155*a^5*b*c*d^6 + 105*a^6*d^7)*x^6 - (60*b^6*c^7
- 52*a*b^5*c^6*d - 184*a^2*b^4*c^5*d^2 + 176*a^3*b^3*c^4*d^3 + 319*a^4*b^2*c^3*d^4 - 494*a^5*b*c^2*d^5 + 175*
a^6*c*d^6)*x^4 - 8*(5*a*b^5*c^7 - 13*a^2*b^4*c^6*d + 2*a^3*b^3*c^5*d^2 + 22*a^4*b^2*c^4*d^3 - 23*a^5*b*c^3*d^4
+ 7*a^6*c^2*d^5)*x^2 - 12*((5*b^6*c^5*d^2 - 11*a*b^5*c^4*d^3)*x^9 + (10*b^6*c^6*d - 17*a*b^5*c^5*d^2 - 11*a^2
*b^4*c^4*d^3)*x^7 + (5*b^6*c^7 - a*b^5*c^6*d - 22*a^2*b^4*c^5*d^2)*x^5 + (5*a*b^5*c^7 - 11*a^2*b^4*c^6*d)*x^3)
*sqrt(b/a)*arctan(x*sqrt(b/a)) - 3*((99*a^3*b^3*c^2*d^5 - 110*a^4*b^2*c*d^6 + 35*a^5*b*d^7)*x^9 + (198*a^3*b^3
*c^3*d^4 - 121*a^4*b^2*c^2*d^5 - 40*a^5*b*c*d^6 + 35*a^6*d^7)*x^7 + (99*a^3*b^3*c^4*d^3 + 88*a^4*b^2*c^3*d^4 -
185*a^5*b*c^2*d^5 + 70*a^6*c*d^6)*x^5 + (99*a^4*b^2*c^4*d^3 - 110*a^5*b*c^3*d^4 + 35*a^6*c^2*d^5)*x^3)*sqrt(d
/c)*arctan(x*sqrt(d/c)))/((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^9 + (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^7 + (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^5 + (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^3)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**4/(b*x**2+a)**2/(d*x**2+c)**3,x)

[Out]

Timed out

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Giac [A]  time = 1.17197, size = 495, normalized size = 1.31 \begin{align*} \frac{b^{5} x}{2 \,{\left (a^{3} b^{3} c^{3} - 3 \, a^{4} b^{2} c^{2} d + 3 \, a^{5} b c d^{2} - a^{6} d^{3}\right )}{\left (b x^{2} + a\right )}} + \frac{{\left (5 \, b^{6} c - 11 \, a b^{5} d\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\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 )} \sqrt{a b}} + \frac{{\left (99 \, b^{2} c^{2} d^{4} - 110 \, a b c d^{5} + 35 \, a^{2} d^{6}\right )} \arctan \left (\frac{d x}{\sqrt{c d}}\right )}{8 \,{\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 )} \sqrt{c d}} + \frac{19 \, b c d^{5} x^{3} - 11 \, a d^{6} x^{3} + 21 \, b c^{2} d^{4} x - 13 \, a c d^{5} x}{8 \,{\left (b^{3} c^{7} - 3 \, a b^{2} c^{6} d + 3 \, a^{2} b c^{5} d^{2} - a^{3} c^{4} d^{3}\right )}{\left (d x^{2} + c\right )}^{2}} + \frac{6 \, b c x^{2} + 9 \, a d x^{2} - a c}{3 \, a^{3} c^{4} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(b*x^2+a)^2/(d*x^2+c)^3,x, algorithm="giac")

[Out]

1/2*b^5*x/((a^3*b^3*c^3 - 3*a^4*b^2*c^2*d + 3*a^5*b*c*d^2 - a^6*d^3)*(b*x^2 + a)) + 1/2*(5*b^6*c - 11*a*b^5*d)
*arctan(b*x/sqrt(a*b))/((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)*sqrt(a*b
)) + 1/8*(99*b^2*c^2*d^4 - 110*a*b*c*d^5 + 35*a^2*d^6)*arctan(d*x/sqrt(c*d))/((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)*sqrt(c*d)) + 1/8*(19*b*c*d^5*x^3 - 11*a*d^6*x^3 + 21*b*c^2*d^4*x
- 13*a*c*d^5*x)/((b^3*c^7 - 3*a*b^2*c^6*d + 3*a^2*b*c^5*d^2 - a^3*c^4*d^3)*(d*x^2 + c)^2) + 1/3*(6*b*c*x^2 +
9*a*d*x^2 - a*c)/(a^3*c^4*x^3)