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

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

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Rubi [A]  time = 0.69, antiderivative size = 377, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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} \end {gather*}

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 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]

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 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 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]

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*}

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Mathematica [A]  time = 0.46, size = 230, normalized size = 0.61 \begin {gather*} \frac {1}{24} \left (\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 {12 b^5 x}{a^3 \left (a+b x^2\right ) (a d-b c)^3}+\frac {72 a d+48 b c}{a^3 c^4 x}+\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 {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 ) \end {gather*}

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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^4 \left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 47.78, size = 4225, normalized size = 11.21

result too large to display

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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giac [A]  time = 0.41, size = 367, normalized size = 0.97 \begin {gather*} \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 {gather*}

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)

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maple [A]  time = 0.03, size = 455, normalized size = 1.21 \begin {gather*} \frac {11 a^{2} d^{7} x^{3}}{8 \left (a d -b c \right )^{4} \left (d \,x^{2}+c \right )^{2} c^{4}}-\frac {15 a b \,d^{6} x^{3}}{4 \left (a d -b c \right )^{4} \left (d \,x^{2}+c \right )^{2} c^{3}}+\frac {19 b^{2} d^{5} x^{3}}{8 \left (a d -b c \right )^{4} \left (d \,x^{2}+c \right )^{2} c^{2}}+\frac {13 a^{2} d^{6} x}{8 \left (a d -b c \right )^{4} \left (d \,x^{2}+c \right )^{2} c^{3}}-\frac {17 a b \,d^{5} x}{4 \left (a d -b c \right )^{4} \left (d \,x^{2}+c \right )^{2} c^{2}}+\frac {21 b^{2} d^{4} x}{8 \left (a d -b c \right )^{4} \left (d \,x^{2}+c \right )^{2} c}+\frac {35 a^{2} d^{6} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{8 \left (a d -b c \right )^{4} \sqrt {c d}\, c^{4}}-\frac {55 a b \,d^{5} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{4 \left (a d -b c \right )^{4} \sqrt {c d}\, c^{3}}-\frac {b^{5} d x}{2 \left (a d -b c \right )^{4} \left (b \,x^{2}+a \right ) a^{2}}-\frac {11 b^{5} d \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \left (a d -b c \right )^{4} \sqrt {a b}\, a^{2}}+\frac {b^{6} c x}{2 \left (a d -b c \right )^{4} \left (b \,x^{2}+a \right ) a^{3}}+\frac {5 b^{6} c \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \left (a d -b c \right )^{4} \sqrt {a b}\, a^{3}}+\frac {99 b^{2} d^{4} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{8 \left (a d -b c \right )^{4} \sqrt {c d}\, c^{2}}+\frac {3 d}{a^{2} c^{4} x}+\frac {2 b}{a^{3} c^{3} x}-\frac {1}{3 a^{2} c^{3} x^{3}} \end {gather*}

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]

-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(1/(a*b)^(1/2)*b*x)*d+5/2*b^6/a^3/(a*d-b*c)^4/(a*b)^(1/2)*arctan(1/(a*b)^(1/2)*b*x)*c+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(1/(c*d)^(1/2)*d*x)*a^2-55/4*d^5/c^3/(a
*d-b*c)^4/(c*d)^(1/2)*arctan(1/(c*d)^(1/2)*d*x)*a*b+99/8*d^4/c^2/(a*d-b*c)^4/(c*d)^(1/2)*arctan(1/(c*d)^(1/2)*
d*x)*b^2-1/3/a^2/c^3/x^3+3/a^2/c^4/x*d+2/a^3/c^3/x*b

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maxima [B]  time = 2.78, size = 738, normalized size = 1.96 \begin {gather*} \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 {8 \, a^{2} b^{3} c^{6} - 24 \, a^{3} b^{2} c^{5} d + 24 \, a^{4} b c^{4} d^{2} - 8 \, a^{5} c^{3} d^{3} - 3 \, {\left (20 \, b^{5} c^{4} d^{2} - 24 \, a b^{4} c^{3} d^{3} - 24 \, a^{2} b^{3} c^{2} d^{4} + 75 \, a^{3} b^{2} c d^{5} - 35 \, a^{4} b d^{6}\right )} x^{8} - {\left (120 \, b^{5} c^{5} d - 104 \, a b^{4} c^{4} d^{2} - 192 \, a^{2} b^{3} c^{3} d^{3} + 303 \, a^{3} b^{2} c^{2} d^{4} + 50 \, a^{4} b c d^{5} - 105 \, a^{5} d^{6}\right )} x^{6} - {\left (60 \, b^{5} c^{6} + 8 \, a b^{4} c^{5} d - 176 \, a^{2} b^{3} c^{4} d^{2} + 319 \, a^{4} b c^{2} d^{4} - 175 \, a^{5} c d^{5}\right )} x^{4} - 8 \, {\left (5 \, a b^{4} c^{6} - 8 \, a^{2} b^{3} c^{5} d - 6 \, a^{3} b^{2} c^{4} d^{2} + 16 \, a^{4} b c^{3} d^{3} - 7 \, a^{5} c^{2} d^{4}\right )} x^{2}}{24 \, {\left ({\left (a^{3} b^{4} c^{7} d^{2} - 3 \, a^{4} b^{3} c^{6} d^{3} + 3 \, a^{5} b^{2} c^{5} d^{4} - a^{6} b c^{4} d^{5}\right )} x^{9} + {\left (2 \, a^{3} b^{4} c^{8} d - 5 \, a^{4} b^{3} c^{7} d^{2} + 3 \, a^{5} b^{2} c^{6} d^{3} + a^{6} b c^{5} d^{4} - a^{7} c^{4} d^{5}\right )} x^{7} + {\left (a^{3} b^{4} c^{9} - a^{4} b^{3} c^{8} d - 3 \, a^{5} b^{2} c^{7} d^{2} + 5 \, a^{6} b c^{6} d^{3} - 2 \, a^{7} c^{5} d^{4}\right )} x^{5} + {\left (a^{4} b^{3} c^{9} - 3 \, a^{5} b^{2} c^{8} d + 3 \, a^{6} b c^{7} d^{2} - a^{7} c^{6} d^{3}\right )} x^{3}\right )}} \end {gather*}

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]

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)) - 1/24*(8*a^2*b^3*c^6 - 24*
a^3*b^2*c^5*d + 24*a^4*b*c^4*d^2 - 8*a^5*c^3*d^3 - 3*(20*b^5*c^4*d^2 - 24*a*b^4*c^3*d^3 - 24*a^2*b^3*c^2*d^4 +
75*a^3*b^2*c*d^5 - 35*a^4*b*d^6)*x^8 - (120*b^5*c^5*d - 104*a*b^4*c^4*d^2 - 192*a^2*b^3*c^3*d^3 + 303*a^3*b^2
*c^2*d^4 + 50*a^4*b*c*d^5 - 105*a^5*d^6)*x^6 - (60*b^5*c^6 + 8*a*b^4*c^5*d - 176*a^2*b^3*c^4*d^2 + 319*a^4*b*c
^2*d^4 - 175*a^5*c*d^5)*x^4 - 8*(5*a*b^4*c^6 - 8*a^2*b^3*c^5*d - 6*a^3*b^2*c^4*d^2 + 16*a^4*b*c^3*d^3 - 7*a^5*
c^2*d^4)*x^2)/((a^3*b^4*c^7*d^2 - 3*a^4*b^3*c^6*d^3 + 3*a^5*b^2*c^5*d^4 - a^6*b*c^4*d^5)*x^9 + (2*a^3*b^4*c^8*
d - 5*a^4*b^3*c^7*d^2 + 3*a^5*b^2*c^6*d^3 + a^6*b*c^5*d^4 - a^7*c^4*d^5)*x^7 + (a^3*b^4*c^9 - a^4*b^3*c^8*d -
3*a^5*b^2*c^7*d^2 + 5*a^6*b*c^6*d^3 - 2*a^7*c^5*d^4)*x^5 + (a^4*b^3*c^9 - 3*a^5*b^2*c^8*d + 3*a^6*b*c^7*d^2 -
a^7*c^6*d^3)*x^3)

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mupad [B]  time = 2.29, size = 1161, normalized size = 3.08 \begin {gather*} \frac {\frac {x^2\,\left (7\,a\,d+5\,b\,c\right )}{3\,a^2\,c^2}-\frac {1}{3\,a\,c}+\frac {x^8\,\left (35\,a^4\,b\,d^6-75\,a^3\,b^2\,c\,d^5+24\,a^2\,b^3\,c^2\,d^4+24\,a\,b^4\,c^3\,d^3-20\,b^5\,c^4\,d^2\right )}{8\,a^3\,c^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 {x^4\,\left (-175\,a^5\,d^5+319\,a^4\,b\,c\,d^4-176\,a^2\,b^3\,c^3\,d^2+8\,a\,b^4\,c^4\,d+60\,b^5\,c^5\right )}{24\,a^3\,c^3\,\left (a\,d-b\,c\right )\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {d\,x^6\,\left (105\,a^5\,d^5-50\,a^4\,b\,c\,d^4-303\,a^3\,b^2\,c^2\,d^3+192\,a^2\,b^3\,c^3\,d^2+104\,a\,b^4\,c^4\,d-120\,b^5\,c^5\right )}{24\,a^3\,c^4\,\left (a\,d-b\,c\right )\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}}{x^5\,\left (b\,c^2+2\,a\,d\,c\right )+x^7\,\left (a\,d^2+2\,b\,c\,d\right )+a\,c^2\,x^3+b\,d^2\,x^9}+\frac {\mathrm {atan}\left (\frac {b^3\,c^{11}\,x\,{\left (-a^7\,b^9\right )}^{3/2}\,400{}\mathrm {i}+a^{18}\,b\,d^{11}\,x\,\sqrt {-a^7\,b^9}\,1225{}\mathrm {i}+a^{14}\,b^5\,c^4\,d^7\,x\,\sqrt {-a^7\,b^9}\,9801{}\mathrm {i}-a^{15}\,b^4\,c^3\,d^8\,x\,\sqrt {-a^7\,b^9}\,21780{}\mathrm {i}+a^{16}\,b^3\,c^2\,d^9\,x\,\sqrt {-a^7\,b^9}\,19030{}\mathrm {i}-a\,b^2\,c^{10}\,d\,x\,{\left (-a^7\,b^9\right )}^{3/2}\,1760{}\mathrm {i}+a^2\,b\,c^9\,d^2\,x\,{\left (-a^7\,b^9\right )}^{3/2}\,1936{}\mathrm {i}-a^{17}\,b^2\,c\,d^{10}\,x\,\sqrt {-a^7\,b^9}\,7700{}\mathrm {i}}{-1225\,a^{22}\,b^5\,d^{11}+7700\,a^{21}\,b^6\,c\,d^{10}-19030\,a^{20}\,b^7\,c^2\,d^9+21780\,a^{19}\,b^8\,c^3\,d^8-9801\,a^{18}\,b^9\,c^4\,d^7+1936\,a^{13}\,b^{14}\,c^9\,d^2-1760\,a^{12}\,b^{15}\,c^{10}\,d+400\,a^{11}\,b^{16}\,c^{11}}\right )\,\left (11\,a\,d-5\,b\,c\right )\,\sqrt {-a^7\,b^9}\,1{}\mathrm {i}}{2\,\left (a^{11}\,d^4-4\,a^{10}\,b\,c\,d^3+6\,a^9\,b^2\,c^2\,d^2-4\,a^8\,b^3\,c^3\,d+a^7\,b^4\,c^4\right )}-\frac {\mathrm {atan}\left (\frac {a^{11}\,d^5\,x\,{\left (-c^9\,d^7\right )}^{3/2}\,1225{}\mathrm {i}+b^{11}\,c^{20}\,d\,x\,\sqrt {-c^9\,d^7}\,400{}\mathrm {i}-a^8\,b^3\,c^3\,d^2\,x\,{\left (-c^9\,d^7\right )}^{3/2}\,21780{}\mathrm {i}+a^9\,b^2\,c^2\,d^3\,x\,{\left (-c^9\,d^7\right )}^{3/2}\,19030{}\mathrm {i}+a^2\,b^9\,c^{18}\,d^3\,x\,\sqrt {-c^9\,d^7}\,1936{}\mathrm {i}-a^{10}\,b\,c\,d^4\,x\,{\left (-c^9\,d^7\right )}^{3/2}\,7700{}\mathrm {i}+a^7\,b^4\,c^4\,d\,x\,{\left (-c^9\,d^7\right )}^{3/2}\,9801{}\mathrm {i}-a\,b^{10}\,c^{19}\,d^2\,x\,\sqrt {-c^9\,d^7}\,1760{}\mathrm {i}}{1225\,a^{11}\,c^{14}\,d^{15}-7700\,a^{10}\,b\,c^{15}\,d^{14}+19030\,a^9\,b^2\,c^{16}\,d^{13}-21780\,a^8\,b^3\,c^{17}\,d^{12}+9801\,a^7\,b^4\,c^{18}\,d^{11}-1936\,a^2\,b^9\,c^{23}\,d^6+1760\,a\,b^{10}\,c^{24}\,d^5-400\,b^{11}\,c^{25}\,d^4}\right )\,\sqrt {-c^9\,d^7}\,\left (35\,a^2\,d^2-110\,a\,b\,c\,d+99\,b^2\,c^2\right )\,1{}\mathrm {i}}{8\,\left (a^4\,c^9\,d^4-4\,a^3\,b\,c^{10}\,d^3+6\,a^2\,b^2\,c^{11}\,d^2-4\,a\,b^3\,c^{12}\,d+b^4\,c^{13}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((x^2*(7*a*d + 5*b*c))/(3*a^2*c^2) - 1/(3*a*c) + (x^8*(35*a^4*b*d^6 - 20*b^5*c^4*d^2 + 24*a*b^4*c^3*d^3 - 75*a
^3*b^2*c*d^5 + 24*a^2*b^3*c^2*d^4))/(8*a^3*c^4*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)) - (x^4*(60
*b^5*c^5 - 175*a^5*d^5 - 176*a^2*b^3*c^3*d^2 + 8*a*b^4*c^4*d + 319*a^4*b*c*d^4))/(24*a^3*c^3*(a*d - b*c)*(a^2*
d^2 + b^2*c^2 - 2*a*b*c*d)) + (d*x^6*(105*a^5*d^5 - 120*b^5*c^5 + 192*a^2*b^3*c^3*d^2 - 303*a^3*b^2*c^2*d^3 +
104*a*b^4*c^4*d - 50*a^4*b*c*d^4))/(24*a^3*c^4*(a*d - b*c)*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)))/(x^5*(b*c^2 + 2*a
*c*d) + x^7*(a*d^2 + 2*b*c*d) + a*c^2*x^3 + b*d^2*x^9) + (atan((b^3*c^11*x*(-a^7*b^9)^(3/2)*400i + a^18*b*d^11
*x*(-a^7*b^9)^(1/2)*1225i + a^14*b^5*c^4*d^7*x*(-a^7*b^9)^(1/2)*9801i - a^15*b^4*c^3*d^8*x*(-a^7*b^9)^(1/2)*21
780i + a^16*b^3*c^2*d^9*x*(-a^7*b^9)^(1/2)*19030i - a*b^2*c^10*d*x*(-a^7*b^9)^(3/2)*1760i + a^2*b*c^9*d^2*x*(-
a^7*b^9)^(3/2)*1936i - a^17*b^2*c*d^10*x*(-a^7*b^9)^(1/2)*7700i)/(400*a^11*b^16*c^11 - 1225*a^22*b^5*d^11 - 17
60*a^12*b^15*c^10*d + 7700*a^21*b^6*c*d^10 + 1936*a^13*b^14*c^9*d^2 - 9801*a^18*b^9*c^4*d^7 + 21780*a^19*b^8*c
^3*d^8 - 19030*a^20*b^7*c^2*d^9))*(11*a*d - 5*b*c)*(-a^7*b^9)^(1/2)*1i)/(2*(a^11*d^4 + a^7*b^4*c^4 - 4*a^8*b^3
*c^3*d + 6*a^9*b^2*c^2*d^2 - 4*a^10*b*c*d^3)) - (atan((a^11*d^5*x*(-c^9*d^7)^(3/2)*1225i + b^11*c^20*d*x*(-c^9
*d^7)^(1/2)*400i - a^8*b^3*c^3*d^2*x*(-c^9*d^7)^(3/2)*21780i + a^9*b^2*c^2*d^3*x*(-c^9*d^7)^(3/2)*19030i + a^2
*b^9*c^18*d^3*x*(-c^9*d^7)^(1/2)*1936i - a^10*b*c*d^4*x*(-c^9*d^7)^(3/2)*7700i + a^7*b^4*c^4*d*x*(-c^9*d^7)^(3
/2)*9801i - a*b^10*c^19*d^2*x*(-c^9*d^7)^(1/2)*1760i)/(1225*a^11*c^14*d^15 - 400*b^11*c^25*d^4 + 1760*a*b^10*c
^24*d^5 - 7700*a^10*b*c^15*d^14 - 1936*a^2*b^9*c^23*d^6 + 9801*a^7*b^4*c^18*d^11 - 21780*a^8*b^3*c^17*d^12 + 1
9030*a^9*b^2*c^16*d^13))*(-c^9*d^7)^(1/2)*(35*a^2*d^2 + 99*b^2*c^2 - 110*a*b*c*d)*1i)/(8*(b^4*c^13 + a^4*c^9*d
^4 - 4*a^3*b*c^10*d^3 + 6*a^2*b^2*c^11*d^2 - 4*a*b^3*c^12*d))

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sympy [F(-1)]  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**4/(b*x**2+a)**2/(d*x**2+c)**3,x)

[Out]

Timed out

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