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

Optimal. Leaf size=162 \[ \frac{x (a d+b c)}{2 b \left (c+d x^2\right ) (b c-a d)^2}+\frac{a x}{2 b \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}-\frac{\sqrt{a} (a d+3 b c) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 \sqrt{b} (b c-a d)^3}+\frac{\sqrt{c} (3 a d+b c) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{2 \sqrt{d} (b c-a d)^3} \]

[Out]

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

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Rubi [A]  time = 0.163833, antiderivative size = 162, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {470, 527, 522, 205} \[ \frac{x (a d+b c)}{2 b \left (c+d x^2\right ) (b c-a d)^2}+\frac{a x}{2 b \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}-\frac{\sqrt{a} (a d+3 b c) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 \sqrt{b} (b c-a d)^3}+\frac{\sqrt{c} (3 a d+b c) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{2 \sqrt{d} (b c-a d)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 470

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

Rule 527

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> -Simp[
((b*e - a*f)*x*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*n*(b*c - a*d)*(p + 1)), x] + Dist[1/(a*n*(b*c - a*d
)*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*f)
*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, q}, x] && LtQ[p, -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{x^4}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx &=\frac{a x}{2 b (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )}-\frac{\int \frac{a c+(-2 b c-a d) x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx}{2 b (b c-a d)}\\ &=\frac{(b c+a d) x}{2 b (b c-a d)^2 \left (c+d x^2\right )}+\frac{a x}{2 b (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )}-\frac{\int \frac{4 a b c^2-2 b c (b c+a d) x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx}{4 b c (b c-a d)^2}\\ &=\frac{(b c+a d) x}{2 b (b c-a d)^2 \left (c+d x^2\right )}+\frac{a x}{2 b (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )}-\frac{(a (3 b c+a d)) \int \frac{1}{a+b x^2} \, dx}{2 (b c-a d)^3}+\frac{(c (b c+3 a d)) \int \frac{1}{c+d x^2} \, dx}{2 (b c-a d)^3}\\ &=\frac{(b c+a d) x}{2 b (b c-a d)^2 \left (c+d x^2\right )}+\frac{a x}{2 b (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )}-\frac{\sqrt{a} (3 b c+a d) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 \sqrt{b} (b c-a d)^3}+\frac{\sqrt{c} (b c+3 a d) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{2 \sqrt{d} (b c-a d)^3}\\ \end{align*}

Mathematica [A]  time = 0.189935, size = 133, normalized size = 0.82 \[ \frac{1}{2} \left (\frac{a x}{\left (a+b x^2\right ) (b c-a d)^2}+\frac{c x}{\left (c+d x^2\right ) (b c-a d)^2}+\frac{\sqrt{a} (a d+3 b c) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{b} (a d-b c)^3}+\frac{\sqrt{c} (3 a d+b c) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{\sqrt{d} (b c-a d)^3}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.013, size = 222, normalized size = 1.4 \begin{align*}{\frac{cxad}{2\, \left ( ad-bc \right ) ^{3} \left ( d{x}^{2}+c \right ) }}-{\frac{{c}^{2}xb}{2\, \left ( ad-bc \right ) ^{3} \left ( d{x}^{2}+c \right ) }}-{\frac{3\,acd}{2\, \left ( ad-bc \right ) ^{3}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-{\frac{b{c}^{2}}{2\, \left ( ad-bc \right ) ^{3}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}+{\frac{{a}^{2}dx}{2\, \left ( ad-bc \right ) ^{3} \left ( b{x}^{2}+a \right ) }}-{\frac{abcx}{2\, \left ( ad-bc \right ) ^{3} \left ( b{x}^{2}+a \right ) }}+{\frac{{a}^{2}d}{2\, \left ( ad-bc \right ) ^{3}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{3\,abc}{2\, \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(x^4/(b*x^2+a)^2/(d*x^2+c)^2,x)

[Out]

1/2*c/(a*d-b*c)^3*x/(d*x^2+c)*a*d-1/2*c^2/(a*d-b*c)^3*x/(d*x^2+c)*b-3/2*c/(a*d-b*c)^3/(c*d)^(1/2)*arctan(x*d/(
c*d)^(1/2))*a*d-1/2*c^2/(a*d-b*c)^3/(c*d)^(1/2)*arctan(x*d/(c*d)^(1/2))*b+1/2*a^2/(a*d-b*c)^3*x/(b*x^2+a)*d-1/
2*a/(a*d-b*c)^3*x/(b*x^2+a)*b*c+1/2*a^2/(a*d-b*c)^3/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))*d+3/2*a/(a*d-b*c)^3/(a
*b)^(1/2)*arctan(b*x/(a*b)^(1/2))*b*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(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 = 2.63473, size = 2822, normalized size = 17.42 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/4*(2*(b^2*c^2 - a^2*d^2)*x^3 - ((3*b^2*c*d + a*b*d^2)*x^4 + 3*a*b*c^2 + a^2*c*d + (3*b^2*c^2 + 4*a*b*c*d +
a^2*d^2)*x^2)*sqrt(-a/b)*log((b*x^2 + 2*b*x*sqrt(-a/b) - a)/(b*x^2 + a)) - ((b^2*c*d + 3*a*b*d^2)*x^4 + a*b*c^
2 + 3*a^2*c*d + (b^2*c^2 + 4*a*b*c*d + 3*a^2*d^2)*x^2)*sqrt(-c/d)*log((d*x^2 - 2*d*x*sqrt(-c/d) - c)/(d*x^2 +
c)) + 4*(a*b*c^2 - a^2*c*d)*x)/(a*b^3*c^4 - 3*a^2*b^2*c^3*d + 3*a^3*b*c^2*d^2 - a^4*c*d^3 + (b^4*c^3*d - 3*a*b
^3*c^2*d^2 + 3*a^2*b^2*c*d^3 - a^3*b*d^4)*x^4 + (b^4*c^4 - 2*a*b^3*c^3*d + 2*a^3*b*c*d^3 - a^4*d^4)*x^2), 1/4*
(2*(b^2*c^2 - a^2*d^2)*x^3 - 2*((3*b^2*c*d + a*b*d^2)*x^4 + 3*a*b*c^2 + a^2*c*d + (3*b^2*c^2 + 4*a*b*c*d + a^2
*d^2)*x^2)*sqrt(a/b)*arctan(b*x*sqrt(a/b)/a) - ((b^2*c*d + 3*a*b*d^2)*x^4 + a*b*c^2 + 3*a^2*c*d + (b^2*c^2 + 4
*a*b*c*d + 3*a^2*d^2)*x^2)*sqrt(-c/d)*log((d*x^2 - 2*d*x*sqrt(-c/d) - c)/(d*x^2 + c)) + 4*(a*b*c^2 - a^2*c*d)*
x)/(a*b^3*c^4 - 3*a^2*b^2*c^3*d + 3*a^3*b*c^2*d^2 - a^4*c*d^3 + (b^4*c^3*d - 3*a*b^3*c^2*d^2 + 3*a^2*b^2*c*d^3
- a^3*b*d^4)*x^4 + (b^4*c^4 - 2*a*b^3*c^3*d + 2*a^3*b*c*d^3 - a^4*d^4)*x^2), 1/4*(2*(b^2*c^2 - a^2*d^2)*x^3 +
2*((b^2*c*d + 3*a*b*d^2)*x^4 + a*b*c^2 + 3*a^2*c*d + (b^2*c^2 + 4*a*b*c*d + 3*a^2*d^2)*x^2)*sqrt(c/d)*arctan(
d*x*sqrt(c/d)/c) - ((3*b^2*c*d + a*b*d^2)*x^4 + 3*a*b*c^2 + a^2*c*d + (3*b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^2)*s
qrt(-a/b)*log((b*x^2 + 2*b*x*sqrt(-a/b) - a)/(b*x^2 + a)) + 4*(a*b*c^2 - a^2*c*d)*x)/(a*b^3*c^4 - 3*a^2*b^2*c^
3*d + 3*a^3*b*c^2*d^2 - a^4*c*d^3 + (b^4*c^3*d - 3*a*b^3*c^2*d^2 + 3*a^2*b^2*c*d^3 - a^3*b*d^4)*x^4 + (b^4*c^4
- 2*a*b^3*c^3*d + 2*a^3*b*c*d^3 - a^4*d^4)*x^2), 1/2*((b^2*c^2 - a^2*d^2)*x^3 - ((3*b^2*c*d + a*b*d^2)*x^4 +
3*a*b*c^2 + a^2*c*d + (3*b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^2)*sqrt(a/b)*arctan(b*x*sqrt(a/b)/a) + ((b^2*c*d + 3
*a*b*d^2)*x^4 + a*b*c^2 + 3*a^2*c*d + (b^2*c^2 + 4*a*b*c*d + 3*a^2*d^2)*x^2)*sqrt(c/d)*arctan(d*x*sqrt(c/d)/c)
+ 2*(a*b*c^2 - a^2*c*d)*x)/(a*b^3*c^4 - 3*a^2*b^2*c^3*d + 3*a^3*b*c^2*d^2 - a^4*c*d^3 + (b^4*c^3*d - 3*a*b^3*
c^2*d^2 + 3*a^2*b^2*c*d^3 - a^3*b*d^4)*x^4 + (b^4*c^4 - 2*a*b^3*c^3*d + 2*a^3*b*c*d^3 - a^4*d^4)*x^2)]

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Sympy [B]  time = 28.3417, size = 2378, normalized size = 14.68 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(-a/b)*(a*d + 3*b*c)*log(x + (-4*a**7*b*d**8*(-a/b)**(3/2)*(a*d + 3*b*c)**3/(a*d - b*c)**9 + 20*a**6*b**2*
c*d**7*(-a/b)**(3/2)*(a*d + 3*b*c)**3/(a*d - b*c)**9 - 36*a**5*b**3*c**2*d**6*(-a/b)**(3/2)*(a*d + 3*b*c)**3/(
a*d - b*c)**9 + 20*a**4*b**4*c**3*d**5*(-a/b)**(3/2)*(a*d + 3*b*c)**3/(a*d - b*c)**9 - a**4*d**4*sqrt(-a/b)*(a
*d + 3*b*c)/(a*d - b*c)**3 + 20*a**3*b**5*c**4*d**4*(-a/b)**(3/2)*(a*d + 3*b*c)**3/(a*d - b*c)**9 - 36*a**3*b*
c*d**3*sqrt(-a/b)*(a*d + 3*b*c)/(a*d - b*c)**3 - 36*a**2*b**6*c**5*d**3*(-a/b)**(3/2)*(a*d + 3*b*c)**3/(a*d -
b*c)**9 - 54*a**2*b**2*c**2*d**2*sqrt(-a/b)*(a*d + 3*b*c)/(a*d - b*c)**3 + 20*a*b**7*c**6*d**2*(-a/b)**(3/2)*(
a*d + 3*b*c)**3/(a*d - b*c)**9 - 36*a*b**3*c**3*d*sqrt(-a/b)*(a*d + 3*b*c)/(a*d - b*c)**3 - 4*b**8*c**7*d*(-a/
b)**(3/2)*(a*d + 3*b*c)**3/(a*d - b*c)**9 - b**4*c**4*sqrt(-a/b)*(a*d + 3*b*c)/(a*d - b*c)**3)/(3*a**2*d**2 +
10*a*b*c*d + 3*b**2*c**2))/(4*(a*d - b*c)**3) - sqrt(-a/b)*(a*d + 3*b*c)*log(x + (4*a**7*b*d**8*(-a/b)**(3/2)*
(a*d + 3*b*c)**3/(a*d - b*c)**9 - 20*a**6*b**2*c*d**7*(-a/b)**(3/2)*(a*d + 3*b*c)**3/(a*d - b*c)**9 + 36*a**5*
b**3*c**2*d**6*(-a/b)**(3/2)*(a*d + 3*b*c)**3/(a*d - b*c)**9 - 20*a**4*b**4*c**3*d**5*(-a/b)**(3/2)*(a*d + 3*b
*c)**3/(a*d - b*c)**9 + a**4*d**4*sqrt(-a/b)*(a*d + 3*b*c)/(a*d - b*c)**3 - 20*a**3*b**5*c**4*d**4*(-a/b)**(3/
2)*(a*d + 3*b*c)**3/(a*d - b*c)**9 + 36*a**3*b*c*d**3*sqrt(-a/b)*(a*d + 3*b*c)/(a*d - b*c)**3 + 36*a**2*b**6*c
**5*d**3*(-a/b)**(3/2)*(a*d + 3*b*c)**3/(a*d - b*c)**9 + 54*a**2*b**2*c**2*d**2*sqrt(-a/b)*(a*d + 3*b*c)/(a*d
- b*c)**3 - 20*a*b**7*c**6*d**2*(-a/b)**(3/2)*(a*d + 3*b*c)**3/(a*d - b*c)**9 + 36*a*b**3*c**3*d*sqrt(-a/b)*(a
*d + 3*b*c)/(a*d - b*c)**3 + 4*b**8*c**7*d*(-a/b)**(3/2)*(a*d + 3*b*c)**3/(a*d - b*c)**9 + b**4*c**4*sqrt(-a/b
)*(a*d + 3*b*c)/(a*d - b*c)**3)/(3*a**2*d**2 + 10*a*b*c*d + 3*b**2*c**2))/(4*(a*d - b*c)**3) + sqrt(-c/d)*(3*a
*d + b*c)*log(x + (-4*a**7*b*d**8*(-c/d)**(3/2)*(3*a*d + b*c)**3/(a*d - b*c)**9 + 20*a**6*b**2*c*d**7*(-c/d)**
(3/2)*(3*a*d + b*c)**3/(a*d - b*c)**9 - 36*a**5*b**3*c**2*d**6*(-c/d)**(3/2)*(3*a*d + b*c)**3/(a*d - b*c)**9 +
20*a**4*b**4*c**3*d**5*(-c/d)**(3/2)*(3*a*d + b*c)**3/(a*d - b*c)**9 - a**4*d**4*sqrt(-c/d)*(3*a*d + b*c)/(a*
d - b*c)**3 + 20*a**3*b**5*c**4*d**4*(-c/d)**(3/2)*(3*a*d + b*c)**3/(a*d - b*c)**9 - 36*a**3*b*c*d**3*sqrt(-c/
d)*(3*a*d + b*c)/(a*d - b*c)**3 - 36*a**2*b**6*c**5*d**3*(-c/d)**(3/2)*(3*a*d + b*c)**3/(a*d - b*c)**9 - 54*a*
*2*b**2*c**2*d**2*sqrt(-c/d)*(3*a*d + b*c)/(a*d - b*c)**3 + 20*a*b**7*c**6*d**2*(-c/d)**(3/2)*(3*a*d + b*c)**3
/(a*d - b*c)**9 - 36*a*b**3*c**3*d*sqrt(-c/d)*(3*a*d + b*c)/(a*d - b*c)**3 - 4*b**8*c**7*d*(-c/d)**(3/2)*(3*a*
d + b*c)**3/(a*d - b*c)**9 - b**4*c**4*sqrt(-c/d)*(3*a*d + b*c)/(a*d - b*c)**3)/(3*a**2*d**2 + 10*a*b*c*d + 3*
b**2*c**2))/(4*(a*d - b*c)**3) - sqrt(-c/d)*(3*a*d + b*c)*log(x + (4*a**7*b*d**8*(-c/d)**(3/2)*(3*a*d + b*c)**
3/(a*d - b*c)**9 - 20*a**6*b**2*c*d**7*(-c/d)**(3/2)*(3*a*d + b*c)**3/(a*d - b*c)**9 + 36*a**5*b**3*c**2*d**6*
(-c/d)**(3/2)*(3*a*d + b*c)**3/(a*d - b*c)**9 - 20*a**4*b**4*c**3*d**5*(-c/d)**(3/2)*(3*a*d + b*c)**3/(a*d - b
*c)**9 + a**4*d**4*sqrt(-c/d)*(3*a*d + b*c)/(a*d - b*c)**3 - 20*a**3*b**5*c**4*d**4*(-c/d)**(3/2)*(3*a*d + b*c
)**3/(a*d - b*c)**9 + 36*a**3*b*c*d**3*sqrt(-c/d)*(3*a*d + b*c)/(a*d - b*c)**3 + 36*a**2*b**6*c**5*d**3*(-c/d)
**(3/2)*(3*a*d + b*c)**3/(a*d - b*c)**9 + 54*a**2*b**2*c**2*d**2*sqrt(-c/d)*(3*a*d + b*c)/(a*d - b*c)**3 - 20*
a*b**7*c**6*d**2*(-c/d)**(3/2)*(3*a*d + b*c)**3/(a*d - b*c)**9 + 36*a*b**3*c**3*d*sqrt(-c/d)*(3*a*d + b*c)/(a*
d - b*c)**3 + 4*b**8*c**7*d*(-c/d)**(3/2)*(3*a*d + b*c)**3/(a*d - b*c)**9 + b**4*c**4*sqrt(-c/d)*(3*a*d + b*c)
/(a*d - b*c)**3)/(3*a**2*d**2 + 10*a*b*c*d + 3*b**2*c**2))/(4*(a*d - b*c)**3) + (2*a*c*x + x**3*(a*d + b*c))/(
2*a**3*c*d**2 - 4*a**2*b*c**2*d + 2*a*b**2*c**3 + x**4*(2*a**2*b*d**3 - 4*a*b**2*c*d**2 + 2*b**3*c**2*d) + x**
2*(2*a**3*d**3 - 2*a**2*b*c*d**2 - 2*a*b**2*c**2*d + 2*b**3*c**3))

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Giac [B]  time = 1.46013, size = 1670, normalized size = 10.31 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(sqrt(c*d)*b^4*c^4*abs(d) + 4*sqrt(c*d)*a*b^3*c^3*d*abs(d) - 10*sqrt(c*d)*a^2*b^2*c^2*d^2*abs(d) + 4*sqrt(
c*d)*a^3*b*c*d^3*abs(d) + sqrt(c*d)*a^4*d^4*abs(d) + sqrt(c*d)*b*c*abs(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2
- a^3*d^3)*abs(d) + sqrt(c*d)*a*d*abs(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*abs(d))*arctan(2*sqr
t(1/2)*x/sqrt((b^3*c^3 - a*b^2*c^2*d - a^2*b*c*d^2 + a^3*d^3 + sqrt((b^3*c^3 - a*b^2*c^2*d - a^2*b*c*d^2 + a^3
*d^3)^2 - 4*(a*b^2*c^3 - 2*a^2*b*c^2*d + a^3*c*d^2)*(b^3*c^2*d - 2*a*b^2*c*d^2 + a^2*b*d^3)))/(b^3*c^2*d - 2*a
*b^2*c*d^2 + a^2*b*d^3)))/(b^3*c^3*d^2*abs(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3) - a*b^2*c^2*d^3*
abs(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3) - a^2*b*c*d^4*abs(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d
^2 - a^3*d^3) + a^3*d^5*abs(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3) + (b^3*c^3 - 3*a*b^2*c^2*d + 3*
a^2*b*c*d^2 - a^3*d^3)^2*d^2) - 1/2*(sqrt(a*b)*b^4*c^4*abs(b) + 4*sqrt(a*b)*a*b^3*c^3*d*abs(b) - 10*sqrt(a*b)*
a^2*b^2*c^2*d^2*abs(b) + 4*sqrt(a*b)*a^3*b*c*d^3*abs(b) + sqrt(a*b)*a^4*d^4*abs(b) - sqrt(a*b)*b*c*abs(b^3*c^3
- 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*abs(b) - sqrt(a*b)*a*d*abs(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2
- a^3*d^3)*abs(b))*arctan(2*sqrt(1/2)*x/sqrt((b^3*c^3 - a*b^2*c^2*d - a^2*b*c*d^2 + a^3*d^3 - sqrt((b^3*c^3 -
a*b^2*c^2*d - a^2*b*c*d^2 + a^3*d^3)^2 - 4*(a*b^2*c^3 - 2*a^2*b*c^2*d + a^3*c*d^2)*(b^3*c^2*d - 2*a*b^2*c*d^2
+ a^2*b*d^3)))/(b^3*c^2*d - 2*a*b^2*c*d^2 + a^2*b*d^3)))/(b^5*c^3*abs(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2
- a^3*d^3) - a*b^4*c^2*d*abs(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3) - a^2*b^3*c*d^2*abs(b^3*c^3 -
3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3) + a^3*b^2*d^3*abs(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)
- (b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)^2*b^2) + 1/2*(b*c*x^3 + a*d*x^3 + 2*a*c*x)/((b*d*x^4 +
b*c*x^2 + a*d*x^2 + a*c)*(b^2*c^2 - 2*a*b*c*d + a^2*d^2))