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

Optimal. Leaf size=271 \[ -\frac{5 a^2 d^2-4 a b c d+5 b^2 c^2}{6 a^2 c^2 x^3 (b c-a d)^2}+\frac{(a d+b c) \left (5 a^2 d^2-9 a b c d+5 b^2 c^2\right )}{2 a^3 c^3 x (b c-a d)^2}+\frac{b^{7/2} (5 b c-9 a d) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{7/2} (b c-a d)^3}+\frac{d^{7/2} (9 b c-5 a d) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{2 c^{7/2} (b c-a d)^3}+\frac{b}{2 a x^3 \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}+\frac{d (a d+b c)}{2 a c x^3 \left (c+d x^2\right ) (b c-a d)^2} \]

[Out]

-(5*b^2*c^2 - 4*a*b*c*d + 5*a^2*d^2)/(6*a^2*c^2*(b*c - a*d)^2*x^3) + ((b*c + a*d)*(5*b^2*c^2 - 9*a*b*c*d + 5*a
^2*d^2))/(2*a^3*c^3*(b*c - a*d)^2*x) + (d*(b*c + a*d))/(2*a*c*(b*c - a*d)^2*x^3*(c + d*x^2)) + b/(2*a*(b*c - a
*d)*x^3*(a + b*x^2)*(c + d*x^2)) + (b^(7/2)*(5*b*c - 9*a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(7/2)*(b*c - a*d
)^3) + (d^(7/2)*(9*b*c - 5*a*d)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(2*c^(7/2)*(b*c - a*d)^3)

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Rubi [A]  time = 0.448272, antiderivative size = 271, normalized size of antiderivative = 1., 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 = {472, 579, 583, 522, 205} \[ -\frac{5 a^2 d^2-4 a b c d+5 b^2 c^2}{6 a^2 c^2 x^3 (b c-a d)^2}+\frac{(a d+b c) \left (5 a^2 d^2-9 a b c d+5 b^2 c^2\right )}{2 a^3 c^3 x (b c-a d)^2}+\frac{b^{7/2} (5 b c-9 a d) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{7/2} (b c-a d)^3}+\frac{d^{7/2} (9 b c-5 a d) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{2 c^{7/2} (b c-a d)^3}+\frac{b}{2 a x^3 \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}+\frac{d (a d+b c)}{2 a c x^3 \left (c+d x^2\right ) (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(5*b^2*c^2 - 4*a*b*c*d + 5*a^2*d^2)/(6*a^2*c^2*(b*c - a*d)^2*x^3) + ((b*c + a*d)*(5*b^2*c^2 - 9*a*b*c*d + 5*a
^2*d^2))/(2*a^3*c^3*(b*c - a*d)^2*x) + (d*(b*c + a*d))/(2*a*c*(b*c - a*d)^2*x^3*(c + d*x^2)) + b/(2*a*(b*c - a
*d)*x^3*(a + b*x^2)*(c + d*x^2)) + (b^(7/2)*(5*b*c - 9*a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(7/2)*(b*c - a*d
)^3) + (d^(7/2)*(9*b*c - 5*a*d)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(2*c^(7/2)*(b*c - a*d)^3)

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 )^2} \, dx &=\frac{b}{2 a (b c-a d) x^3 \left (a+b x^2\right ) \left (c+d x^2\right )}-\frac{\int \frac{-5 b c+2 a d-7 b d x^2}{x^4 \left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx}{2 a (b c-a d)}\\ &=\frac{d (b c+a d)}{2 a c (b c-a d)^2 x^3 \left (c+d x^2\right )}+\frac{b}{2 a (b c-a d) x^3 \left (a+b x^2\right ) \left (c+d x^2\right )}-\frac{\int \frac{-2 \left (5 b^2 c^2-4 a b c d+5 a^2 d^2\right )-10 b d (b c+a d) x^2}{x^4 \left (a+b x^2\right ) \left (c+d x^2\right )} \, dx}{4 a c (b c-a d)^2}\\ &=-\frac{5 b^2 c^2-4 a b c d+5 a^2 d^2}{6 a^2 c^2 (b c-a d)^2 x^3}+\frac{d (b c+a d)}{2 a c (b c-a d)^2 x^3 \left (c+d x^2\right )}+\frac{b}{2 a (b c-a d) x^3 \left (a+b x^2\right ) \left (c+d x^2\right )}+\frac{\int \frac{-6 (b c+a d) \left (5 b^2 c^2-9 a b c d+5 a^2 d^2\right )-6 b d \left (5 b^2 c^2-4 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}{12 a^2 c^2 (b c-a d)^2}\\ &=-\frac{5 b^2 c^2-4 a b c d+5 a^2 d^2}{6 a^2 c^2 (b c-a d)^2 x^3}+\frac{(b c+a d) \left (5 b^2 c^2-9 a b c d+5 a^2 d^2\right )}{2 a^3 c^3 (b c-a d)^2 x}+\frac{d (b c+a d)}{2 a c (b c-a d)^2 x^3 \left (c+d x^2\right )}+\frac{b}{2 a (b c-a d) x^3 \left (a+b x^2\right ) \left (c+d x^2\right )}-\frac{\int \frac{-6 \left (5 b^4 c^4-4 a b^3 c^3 d-4 a^2 b^2 c^2 d^2-4 a^3 b c d^3+5 a^4 d^4\right )-6 b d (b c+a d) \left (5 b^2 c^2-9 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}{12 a^3 c^3 (b c-a d)^2}\\ &=-\frac{5 b^2 c^2-4 a b c d+5 a^2 d^2}{6 a^2 c^2 (b c-a d)^2 x^3}+\frac{(b c+a d) \left (5 b^2 c^2-9 a b c d+5 a^2 d^2\right )}{2 a^3 c^3 (b c-a d)^2 x}+\frac{d (b c+a d)}{2 a c (b c-a d)^2 x^3 \left (c+d x^2\right )}+\frac{b}{2 a (b c-a d) x^3 \left (a+b x^2\right ) \left (c+d x^2\right )}+\frac{\left (b^4 (5 b c-9 a d)\right ) \int \frac{1}{a+b x^2} \, dx}{2 a^3 (b c-a d)^3}+\frac{\left (d^4 (9 b c-5 a d)\right ) \int \frac{1}{c+d x^2} \, dx}{2 c^3 (b c-a d)^3}\\ &=-\frac{5 b^2 c^2-4 a b c d+5 a^2 d^2}{6 a^2 c^2 (b c-a d)^2 x^3}+\frac{(b c+a d) \left (5 b^2 c^2-9 a b c d+5 a^2 d^2\right )}{2 a^3 c^3 (b c-a d)^2 x}+\frac{d (b c+a d)}{2 a c (b c-a d)^2 x^3 \left (c+d x^2\right )}+\frac{b}{2 a (b c-a d) x^3 \left (a+b x^2\right ) \left (c+d x^2\right )}+\frac{b^{7/2} (5 b c-9 a d) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{7/2} (b c-a d)^3}+\frac{d^{7/2} (9 b c-5 a d) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{2 c^{7/2} (b c-a d)^3}\\ \end{align*}

Mathematica [A]  time = 0.371645, size = 178, normalized size = 0.66 \[ \frac{1}{6} \left (\frac{3 b^4 x}{a^3 \left (a+b x^2\right ) (b c-a d)^2}+\frac{3 b^{7/2} (9 a d-5 b c) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{7/2} (a d-b c)^3}+\frac{12 (a d+b c)}{a^3 c^3 x}-\frac{2}{a^2 c^2 x^3}+\frac{3 d^4 x}{c^3 \left (c+d x^2\right ) (b c-a d)^2}+\frac{3 d^{7/2} (9 b c-5 a d) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{c^{7/2} (b c-a d)^3}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2/(a^2*c^2*x^3) + (12*(b*c + a*d))/(a^3*c^3*x) + (3*b^4*x)/(a^3*(b*c - a*d)^2*(a + b*x^2)) + (3*d^4*x)/(c^3*
(b*c - a*d)^2*(c + d*x^2)) + (3*b^(7/2)*(-5*b*c + 9*a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(a^(7/2)*(-(b*c) + a*d)^
3) + (3*d^(7/2)*(9*b*c - 5*a*d)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(c^(7/2)*(b*c - a*d)^3))/6

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Maple [A]  time = 0.02, size = 285, normalized size = 1.1 \begin{align*}{\frac{{d}^{5}xa}{2\,{c}^{3} \left ( ad-bc \right ) ^{3} \left ( d{x}^{2}+c \right ) }}-{\frac{{d}^{4}xb}{2\,{c}^{2} \left ( ad-bc \right ) ^{3} \left ( d{x}^{2}+c \right ) }}+{\frac{5\,{d}^{5}a}{2\,{c}^{3} \left ( ad-bc \right ) ^{3}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-{\frac{9\,{d}^{4}b}{2\,{c}^{2} \left ( ad-bc \right ) ^{3}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-{\frac{1}{3\,{a}^{2}{c}^{2}{x}^{3}}}+2\,{\frac{d}{{a}^{2}{c}^{3}x}}+2\,{\frac{b}{{a}^{3}{c}^{2}x}}+{\frac{{b}^{4}xd}{2\,{a}^{2} \left ( ad-bc \right ) ^{3} \left ( b{x}^{2}+a \right ) }}-{\frac{{b}^{5}xc}{2\,{a}^{3} \left ( ad-bc \right ) ^{3} \left ( b{x}^{2}+a \right ) }}+{\frac{9\,{b}^{4}d}{2\,{a}^{2} \left ( ad-bc \right ) ^{3}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{5\,{b}^{5}c}{2\,{a}^{3} \left ( ad-bc \right ) ^{3}}\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)^2,x)

[Out]

1/2*d^5/c^3/(a*d-b*c)^3*x/(d*x^2+c)*a-1/2*d^4/c^2/(a*d-b*c)^3*x/(d*x^2+c)*b+5/2*d^5/c^3/(a*d-b*c)^3/(c*d)^(1/2
)*arctan(x*d/(c*d)^(1/2))*a-9/2*d^4/c^2/(a*d-b*c)^3/(c*d)^(1/2)*arctan(x*d/(c*d)^(1/2))*b-1/3/a^2/c^2/x^3+2/a^
2/c^3/x*d+2/a^3/c^2/x*b+1/2*b^4/a^2/(a*d-b*c)^3*x/(b*x^2+a)*d-1/2*b^5/a^3/(a*d-b*c)^3*x/(b*x^2+a)*c+9/2*b^4/a^
2/(a*d-b*c)^3/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))*d-5/2*b^5/a^3/(a*d-b*c)^3/(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)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 27.7042, size = 4880, normalized size = 18.01 \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)^2,x, algorithm="fricas")

[Out]

[-1/12*(4*a^2*b^3*c^5 - 12*a^3*b^2*c^4*d + 12*a^4*b*c^3*d^2 - 4*a^5*c^2*d^3 - 6*(5*b^5*c^4*d - 9*a*b^4*c^3*d^2
+ 9*a^3*b^2*c*d^4 - 5*a^4*b*d^5)*x^6 - 2*(15*b^5*c^5 - 17*a*b^4*c^4*d - 18*a^2*b^3*c^3*d^2 + 18*a^3*b^2*c^2*d
^3 + 17*a^4*b*c*d^4 - 15*a^5*d^5)*x^4 - 20*(a*b^4*c^5 - 2*a^2*b^3*c^4*d + 2*a^4*b*c^2*d^3 - a^5*c*d^4)*x^2 - 3
*((5*b^5*c^4*d - 9*a*b^4*c^3*d^2)*x^7 + (5*b^5*c^5 - 4*a*b^4*c^4*d - 9*a^2*b^3*c^3*d^2)*x^5 + (5*a*b^4*c^5 - 9
*a^2*b^3*c^4*d)*x^3)*sqrt(-b/a)*log((b*x^2 + 2*a*x*sqrt(-b/a) - a)/(b*x^2 + a)) - 3*((9*a^3*b^2*c*d^4 - 5*a^4*
b*d^5)*x^7 + (9*a^3*b^2*c^2*d^3 + 4*a^4*b*c*d^4 - 5*a^5*d^5)*x^5 + (9*a^4*b*c^2*d^3 - 5*a^5*c*d^4)*x^3)*sqrt(-
d/c)*log((d*x^2 + 2*c*x*sqrt(-d/c) - c)/(d*x^2 + c)))/((a^3*b^4*c^6*d - 3*a^4*b^3*c^5*d^2 + 3*a^5*b^2*c^4*d^3
- a^6*b*c^3*d^4)*x^7 + (a^3*b^4*c^7 - 2*a^4*b^3*c^6*d + 2*a^6*b*c^4*d^3 - a^7*c^3*d^4)*x^5 + (a^4*b^3*c^7 - 3*
a^5*b^2*c^6*d + 3*a^6*b*c^5*d^2 - a^7*c^4*d^3)*x^3), -1/12*(4*a^2*b^3*c^5 - 12*a^3*b^2*c^4*d + 12*a^4*b*c^3*d^
2 - 4*a^5*c^2*d^3 - 6*(5*b^5*c^4*d - 9*a*b^4*c^3*d^2 + 9*a^3*b^2*c*d^4 - 5*a^4*b*d^5)*x^6 - 2*(15*b^5*c^5 - 17
*a*b^4*c^4*d - 18*a^2*b^3*c^3*d^2 + 18*a^3*b^2*c^2*d^3 + 17*a^4*b*c*d^4 - 15*a^5*d^5)*x^4 - 20*(a*b^4*c^5 - 2*
a^2*b^3*c^4*d + 2*a^4*b*c^2*d^3 - a^5*c*d^4)*x^2 - 6*((9*a^3*b^2*c*d^4 - 5*a^4*b*d^5)*x^7 + (9*a^3*b^2*c^2*d^3
+ 4*a^4*b*c*d^4 - 5*a^5*d^5)*x^5 + (9*a^4*b*c^2*d^3 - 5*a^5*c*d^4)*x^3)*sqrt(d/c)*arctan(x*sqrt(d/c)) - 3*((5
*b^5*c^4*d - 9*a*b^4*c^3*d^2)*x^7 + (5*b^5*c^5 - 4*a*b^4*c^4*d - 9*a^2*b^3*c^3*d^2)*x^5 + (5*a*b^4*c^5 - 9*a^2
*b^3*c^4*d)*x^3)*sqrt(-b/a)*log((b*x^2 + 2*a*x*sqrt(-b/a) - a)/(b*x^2 + a)))/((a^3*b^4*c^6*d - 3*a^4*b^3*c^5*d
^2 + 3*a^5*b^2*c^4*d^3 - a^6*b*c^3*d^4)*x^7 + (a^3*b^4*c^7 - 2*a^4*b^3*c^6*d + 2*a^6*b*c^4*d^3 - a^7*c^3*d^4)*
x^5 + (a^4*b^3*c^7 - 3*a^5*b^2*c^6*d + 3*a^6*b*c^5*d^2 - a^7*c^4*d^3)*x^3), -1/12*(4*a^2*b^3*c^5 - 12*a^3*b^2*
c^4*d + 12*a^4*b*c^3*d^2 - 4*a^5*c^2*d^3 - 6*(5*b^5*c^4*d - 9*a*b^4*c^3*d^2 + 9*a^3*b^2*c*d^4 - 5*a^4*b*d^5)*x
^6 - 2*(15*b^5*c^5 - 17*a*b^4*c^4*d - 18*a^2*b^3*c^3*d^2 + 18*a^3*b^2*c^2*d^3 + 17*a^4*b*c*d^4 - 15*a^5*d^5)*x
^4 - 20*(a*b^4*c^5 - 2*a^2*b^3*c^4*d + 2*a^4*b*c^2*d^3 - a^5*c*d^4)*x^2 - 6*((5*b^5*c^4*d - 9*a*b^4*c^3*d^2)*x
^7 + (5*b^5*c^5 - 4*a*b^4*c^4*d - 9*a^2*b^3*c^3*d^2)*x^5 + (5*a*b^4*c^5 - 9*a^2*b^3*c^4*d)*x^3)*sqrt(b/a)*arct
an(x*sqrt(b/a)) - 3*((9*a^3*b^2*c*d^4 - 5*a^4*b*d^5)*x^7 + (9*a^3*b^2*c^2*d^3 + 4*a^4*b*c*d^4 - 5*a^5*d^5)*x^5
+ (9*a^4*b*c^2*d^3 - 5*a^5*c*d^4)*x^3)*sqrt(-d/c)*log((d*x^2 + 2*c*x*sqrt(-d/c) - c)/(d*x^2 + c)))/((a^3*b^4*
c^6*d - 3*a^4*b^3*c^5*d^2 + 3*a^5*b^2*c^4*d^3 - a^6*b*c^3*d^4)*x^7 + (a^3*b^4*c^7 - 2*a^4*b^3*c^6*d + 2*a^6*b*
c^4*d^3 - a^7*c^3*d^4)*x^5 + (a^4*b^3*c^7 - 3*a^5*b^2*c^6*d + 3*a^6*b*c^5*d^2 - a^7*c^4*d^3)*x^3), -1/6*(2*a^2
*b^3*c^5 - 6*a^3*b^2*c^4*d + 6*a^4*b*c^3*d^2 - 2*a^5*c^2*d^3 - 3*(5*b^5*c^4*d - 9*a*b^4*c^3*d^2 + 9*a^3*b^2*c*
d^4 - 5*a^4*b*d^5)*x^6 - (15*b^5*c^5 - 17*a*b^4*c^4*d - 18*a^2*b^3*c^3*d^2 + 18*a^3*b^2*c^2*d^3 + 17*a^4*b*c*d
^4 - 15*a^5*d^5)*x^4 - 10*(a*b^4*c^5 - 2*a^2*b^3*c^4*d + 2*a^4*b*c^2*d^3 - a^5*c*d^4)*x^2 - 3*((5*b^5*c^4*d -
9*a*b^4*c^3*d^2)*x^7 + (5*b^5*c^5 - 4*a*b^4*c^4*d - 9*a^2*b^3*c^3*d^2)*x^5 + (5*a*b^4*c^5 - 9*a^2*b^3*c^4*d)*x
^3)*sqrt(b/a)*arctan(x*sqrt(b/a)) - 3*((9*a^3*b^2*c*d^4 - 5*a^4*b*d^5)*x^7 + (9*a^3*b^2*c^2*d^3 + 4*a^4*b*c*d^
4 - 5*a^5*d^5)*x^5 + (9*a^4*b*c^2*d^3 - 5*a^5*c*d^4)*x^3)*sqrt(d/c)*arctan(x*sqrt(d/c)))/((a^3*b^4*c^6*d - 3*a
^4*b^3*c^5*d^2 + 3*a^5*b^2*c^4*d^3 - a^6*b*c^3*d^4)*x^7 + (a^3*b^4*c^7 - 2*a^4*b^3*c^6*d + 2*a^6*b*c^4*d^3 - a
^7*c^3*d^4)*x^5 + (a^4*b^3*c^7 - 3*a^5*b^2*c^6*d + 3*a^6*b*c^5*d^2 - a^7*c^4*d^3)*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)**2,x)

[Out]

Timed out

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Giac [B]  time = 1.64454, size = 2608, normalized size = 9.62 \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)^2,x, algorithm="giac")

[Out]

-1/2*(5*sqrt(c*d)*a^3*b^7*c^9*abs(d) - 19*sqrt(c*d)*a^4*b^6*c^8*d*abs(d) + 23*sqrt(c*d)*a^5*b^5*c^7*d^2*abs(d)
- 18*sqrt(c*d)*a^6*b^4*c^6*d^3*abs(d) + 23*sqrt(c*d)*a^7*b^3*c^5*d^4*abs(d) - 19*sqrt(c*d)*a^8*b^2*c^4*d^5*ab
s(d) + 5*sqrt(c*d)*a^9*b*c^3*d^6*abs(d) - 5*sqrt(c*d)*b^4*c^3*abs(a^3*b^3*c^6 - 3*a^4*b^2*c^5*d + 3*a^5*b*c^4*
d^2 - a^6*c^3*d^3)*abs(d) + 4*sqrt(c*d)*a*b^3*c^2*d*abs(a^3*b^3*c^6 - 3*a^4*b^2*c^5*d + 3*a^5*b*c^4*d^2 - a^6*
c^3*d^3)*abs(d) + 4*sqrt(c*d)*a^2*b^2*c*d^2*abs(a^3*b^3*c^6 - 3*a^4*b^2*c^5*d + 3*a^5*b*c^4*d^2 - a^6*c^3*d^3)
*abs(d) - 5*sqrt(c*d)*a^3*b*d^3*abs(a^3*b^3*c^6 - 3*a^4*b^2*c^5*d + 3*a^5*b*c^4*d^2 - a^6*c^3*d^3)*abs(d))*arc
tan(2*sqrt(1/2)*x/sqrt((a^3*b^3*c^6 - a^4*b^2*c^5*d - a^5*b*c^4*d^2 + a^6*c^3*d^3 + sqrt((a^3*b^3*c^6 - a^4*b^
2*c^5*d - a^5*b*c^4*d^2 + a^6*c^3*d^3)^2 - 4*(a^4*b^2*c^6 - 2*a^5*b*c^5*d + a^6*c^4*d^2)*(a^3*b^3*c^5*d - 2*a^
4*b^2*c^4*d^2 + a^5*b*c^3*d^3)))/(a^3*b^3*c^5*d - 2*a^4*b^2*c^4*d^2 + a^5*b*c^3*d^3)))/(a^3*b^3*c^6*d*abs(a^3*
b^3*c^6 - 3*a^4*b^2*c^5*d + 3*a^5*b*c^4*d^2 - a^6*c^3*d^3) - a^4*b^2*c^5*d^2*abs(a^3*b^3*c^6 - 3*a^4*b^2*c^5*d
+ 3*a^5*b*c^4*d^2 - a^6*c^3*d^3) - a^5*b*c^4*d^3*abs(a^3*b^3*c^6 - 3*a^4*b^2*c^5*d + 3*a^5*b*c^4*d^2 - a^6*c^
3*d^3) + a^6*c^3*d^4*abs(a^3*b^3*c^6 - 3*a^4*b^2*c^5*d + 3*a^5*b*c^4*d^2 - a^6*c^3*d^3) + (a^3*b^3*c^6 - 3*a^4
*b^2*c^5*d + 3*a^5*b*c^4*d^2 - a^6*c^3*d^3)^2*d) + 1/2*(5*sqrt(a*b)*a^3*b^6*c^9*d*abs(b) - 19*sqrt(a*b)*a^4*b^
5*c^8*d^2*abs(b) + 23*sqrt(a*b)*a^5*b^4*c^7*d^3*abs(b) - 18*sqrt(a*b)*a^6*b^3*c^6*d^4*abs(b) + 23*sqrt(a*b)*a^
7*b^2*c^5*d^5*abs(b) - 19*sqrt(a*b)*a^8*b*c^4*d^6*abs(b) + 5*sqrt(a*b)*a^9*c^3*d^7*abs(b) + 5*sqrt(a*b)*b^3*c^
3*d*abs(a^3*b^3*c^6 - 3*a^4*b^2*c^5*d + 3*a^5*b*c^4*d^2 - a^6*c^3*d^3)*abs(b) - 4*sqrt(a*b)*a*b^2*c^2*d^2*abs(
a^3*b^3*c^6 - 3*a^4*b^2*c^5*d + 3*a^5*b*c^4*d^2 - a^6*c^3*d^3)*abs(b) - 4*sqrt(a*b)*a^2*b*c*d^3*abs(a^3*b^3*c^
6 - 3*a^4*b^2*c^5*d + 3*a^5*b*c^4*d^2 - a^6*c^3*d^3)*abs(b) + 5*sqrt(a*b)*a^3*d^4*abs(a^3*b^3*c^6 - 3*a^4*b^2*
c^5*d + 3*a^5*b*c^4*d^2 - a^6*c^3*d^3)*abs(b))*arctan(2*sqrt(1/2)*x/sqrt((a^3*b^3*c^6 - a^4*b^2*c^5*d - a^5*b*
c^4*d^2 + a^6*c^3*d^3 - sqrt((a^3*b^3*c^6 - a^4*b^2*c^5*d - a^5*b*c^4*d^2 + a^6*c^3*d^3)^2 - 4*(a^4*b^2*c^6 -
2*a^5*b*c^5*d + a^6*c^4*d^2)*(a^3*b^3*c^5*d - 2*a^4*b^2*c^4*d^2 + a^5*b*c^3*d^3)))/(a^3*b^3*c^5*d - 2*a^4*b^2*
c^4*d^2 + a^5*b*c^3*d^3)))/(a^3*b^4*c^6*abs(a^3*b^3*c^6 - 3*a^4*b^2*c^5*d + 3*a^5*b*c^4*d^2 - a^6*c^3*d^3) - a
^4*b^3*c^5*d*abs(a^3*b^3*c^6 - 3*a^4*b^2*c^5*d + 3*a^5*b*c^4*d^2 - a^6*c^3*d^3) - a^5*b^2*c^4*d^2*abs(a^3*b^3*
c^6 - 3*a^4*b^2*c^5*d + 3*a^5*b*c^4*d^2 - a^6*c^3*d^3) + a^6*b*c^3*d^3*abs(a^3*b^3*c^6 - 3*a^4*b^2*c^5*d + 3*a
^5*b*c^4*d^2 - a^6*c^3*d^3) - (a^3*b^3*c^6 - 3*a^4*b^2*c^5*d + 3*a^5*b*c^4*d^2 - a^6*c^3*d^3)^2*b) + 1/2*(b^4*
c^3*d*x^3 + a^3*b*d^4*x^3 + b^4*c^4*x + a^4*d^4*x)/((a^3*b^2*c^5 - 2*a^4*b*c^4*d + a^5*c^3*d^2)*(b*d*x^4 + b*c
*x^2 + a*d*x^2 + a*c)) + 1/3*(6*b*c*x^2 + 6*a*d*x^2 - a*c)/(a^3*c^3*x^3)