3.303 \(\int \frac{x^2}{(a+b x^2)^2 (c+d x^2)^2} \, dx\)
Optimal. Leaf size=147 \[ -\frac{x}{2 \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}-\frac{d x}{\left (c+d x^2\right ) (b c-a d)^2}+\frac{\sqrt{b} (3 a d+b c) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 \sqrt{a} (b c-a d)^3}-\frac{\sqrt{d} (a d+3 b c) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{2 \sqrt{c} (b c-a d)^3} \]
[Out]
-((d*x)/((b*c - a*d)^2*(c + d*x^2))) - x/(2*(b*c - a*d)*(a + b*x^2)*(c + d*x^2)) + (Sqrt[b]*(b*c + 3*a*d)*ArcT
an[(Sqrt[b]*x)/Sqrt[a]])/(2*Sqrt[a]*(b*c - a*d)^3) - (Sqrt[d]*(3*b*c + a*d)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(2*Sq
rt[c]*(b*c - a*d)^3)
________________________________________________________________________________________
Rubi [A] time = 0.133562, antiderivative size = 147, 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 =
{471, 527, 522, 205} \[ -\frac{x}{2 \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}-\frac{d x}{\left (c+d x^2\right ) (b c-a d)^2}+\frac{\sqrt{b} (3 a d+b c) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 \sqrt{a} (b c-a d)^3}-\frac{\sqrt{d} (a d+3 b c) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{2 \sqrt{c} (b c-a d)^3} \]
Antiderivative was successfully verified.
[In]
Int[x^2/((a + b*x^2)^2*(c + d*x^2)^2),x]
[Out]
-((d*x)/((b*c - a*d)^2*(c + d*x^2))) - x/(2*(b*c - a*d)*(a + b*x^2)*(c + d*x^2)) + (Sqrt[b]*(b*c + 3*a*d)*ArcT
an[(Sqrt[b]*x)/Sqrt[a]])/(2*Sqrt[a]*(b*c - a*d)^3) - (Sqrt[d]*(3*b*c + a*d)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(2*Sq
rt[c]*(b*c - a*d)^3)
Rule 471
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(e^(n -
1)*(e*x)^(m - n + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(n*(b*c - a*d)*(p + 1)), x] - Dist[e^n/(n*(b*c -
a*d)*(p + 1)), Int[(e*x)^(m - n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(m - n + 1) + d*(m + n*(p + q + 1)
+ 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] && GeQ[n
, m - n + 1] && GtQ[m - n + 1, 0] && 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^2}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx &=-\frac{x}{2 (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )}+\frac{\int \frac{c-3 d x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx}{2 (b c-a d)}\\ &=-\frac{d x}{(b c-a d)^2 \left (c+d x^2\right )}-\frac{x}{2 (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )}+\frac{\int \frac{2 c (b c+a d)-4 b c d x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx}{4 c (b c-a d)^2}\\ &=-\frac{d x}{(b c-a d)^2 \left (c+d x^2\right )}-\frac{x}{2 (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )}-\frac{(d (3 b c+a d)) \int \frac{1}{c+d x^2} \, dx}{2 (b c-a d)^3}+\frac{(b (b c+3 a d)) \int \frac{1}{a+b x^2} \, dx}{2 (b c-a d)^3}\\ &=-\frac{d x}{(b c-a d)^2 \left (c+d x^2\right )}-\frac{x}{2 (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )}+\frac{\sqrt{b} (b c+3 a d) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 \sqrt{a} (b c-a d)^3}-\frac{\sqrt{d} (3 b c+a d) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{2 \sqrt{c} (b c-a d)^3}\\ \end{align*}
Mathematica [A] time = 0.178376, size = 137, normalized size = 0.93 \[ \frac{1}{2} \left (-\frac{b x}{\left (a+b x^2\right ) (b c-a d)^2}-\frac{d x}{\left (c+d x^2\right ) (b c-a d)^2}-\frac{\sqrt{b} (3 a d+b c) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} (a d-b c)^3}-\frac{\sqrt{d} (a d+3 b c) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{\sqrt{c} (b c-a d)^3}\right ) \]
Antiderivative was successfully verified.
[In]
Integrate[x^2/((a + b*x^2)^2*(c + d*x^2)^2),x]
[Out]
(-((b*x)/((b*c - a*d)^2*(a + b*x^2))) - (d*x)/((b*c - a*d)^2*(c + d*x^2)) - (Sqrt[b]*(b*c + 3*a*d)*ArcTan[(Sqr
t[b]*x)/Sqrt[a]])/(Sqrt[a]*(-(b*c) + a*d)^3) - (Sqrt[d]*(3*b*c + a*d)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(Sqrt[c]*(b
*c - a*d)^3))/2
________________________________________________________________________________________
Maple [A] time = 0.017, size = 222, normalized size = 1.5 \begin{align*} -{\frac{{d}^{2}xa}{2\, \left ( ad-bc \right ) ^{3} \left ( d{x}^{2}+c \right ) }}+{\frac{bdxc}{2\, \left ( ad-bc \right ) ^{3} \left ( d{x}^{2}+c \right ) }}+{\frac{a{d}^{2}}{2\, \left ( ad-bc \right ) ^{3}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}+{\frac{3\,bcd}{2\, \left ( ad-bc \right ) ^{3}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-{\frac{abdx}{2\, \left ( ad-bc \right ) ^{3} \left ( b{x}^{2}+a \right ) }}+{\frac{{b}^{2}cx}{2\, \left ( ad-bc \right ) ^{3} \left ( b{x}^{2}+a \right ) }}-{\frac{3\,abd}{2\, \left ( ad-bc \right ) ^{3}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{{b}^{2}c}{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^2/(b*x^2+a)^2/(d*x^2+c)^2,x)
[Out]
-1/2*d^2/(a*d-b*c)^3*x/(d*x^2+c)*a+1/2*d/(a*d-b*c)^3*x/(d*x^2+c)*b*c+1/2*d^2/(a*d-b*c)^3/(c*d)^(1/2)*arctan(x*
d/(c*d)^(1/2))*a+3/2*d/(a*d-b*c)^3/(c*d)^(1/2)*arctan(x*d/(c*d)^(1/2))*b*c-1/2*b/(a*d-b*c)^3*x/(b*x^2+a)*a*d+1
/2*b^2/(a*d-b*c)^3*x/(b*x^2+a)*c-3/2*b/(a*d-b*c)^3/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))*a*d-1/2*b^2/(a*d-b*c)^3
/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))*c
________________________________________________________________________________________
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^2/(b*x^2+a)^2/(d*x^2+c)^2,x, algorithm="maxima")
[Out]
Exception raised: ValueError
________________________________________________________________________________________
Fricas [B] time = 3.32104, size = 2805, normalized size = 19.08 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^2/(b*x^2+a)^2/(d*x^2+c)^2,x, algorithm="fricas")
[Out]
[-1/4*(4*(b^2*c*d - a*b*d^2)*x^3 + ((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(-b/a)*log((b*x^2 - 2*a*x*sqrt(-b/a) - a)/(b*x^2 + a)) + ((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(-d/c)*log((d*x^2 + 2*c*x*sqrt(-d/c) - c)/(d*x^2 +
c)) + 2*(b^2*c^2 - a^2*d^2)*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*(4*(b^2*c*d - a*b*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(d/c)*arctan(x*sqrt(d/c)) + ((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(-b/a)*log((b*x^2 - 2*a*x*sqrt(-b/a) - a)/(b*x^2 + a)) + 2*(b^2*c^2 - a^2*d^2)*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*(4*(b^2*c*d - a*b*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(b/a)*arctan(x
*sqrt(b/a)) + ((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(-
d/c)*log((d*x^2 + 2*c*x*sqrt(-d/c) - c)/(d*x^2 + c)) + 2*(b^2*c^2 - a^2*d^2)*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*(2*(b^2*c*d - a*b*d^2)*x^3 - ((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(b/a)*arctan(x*sqrt(b/a)) + ((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(d/c)*arctan(x*sqrt(d/c)) + (b^2*c^
2 - a^2*d^2)*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)]
________________________________________________________________________________________
Sympy [B] time = 34.427, size = 2399, normalized size = 16.32 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x**2/(b*x**2+a)**2/(d*x**2+c)**2,x)
[Out]
sqrt(-b/a)*(3*a*d + b*c)*log(x + (-a**9*c*d**8*(-b/a)**(3/2)*(3*a*d + b*c)**3/(a*d - b*c)**9 + 20*a**7*b**2*c*
*3*d**6*(-b/a)**(3/2)*(3*a*d + b*c)**3/(a*d - b*c)**9 - 64*a**6*b**3*c**4*d**5*(-b/a)**(3/2)*(3*a*d + b*c)**3/
(a*d - b*c)**9 + 90*a**5*b**4*c**5*d**4*(-b/a)**(3/2)*(3*a*d + b*c)**3/(a*d - b*c)**9 - a**5*d**5*sqrt(-b/a)*(
3*a*d + b*c)/(a*d - b*c)**3 - 64*a**4*b**5*c**6*d**3*(-b/a)**(3/2)*(3*a*d + b*c)**3/(a*d - b*c)**9 - 9*a**4*b*
c*d**4*sqrt(-b/a)*(3*a*d + b*c)/(a*d - b*c)**3 + 20*a**3*b**6*c**7*d**2*(-b/a)**(3/2)*(3*a*d + b*c)**3/(a*d -
b*c)**9 - 54*a**3*b**2*c**2*d**3*sqrt(-b/a)*(3*a*d + b*c)/(a*d - b*c)**3 - 54*a**2*b**3*c**3*d**2*sqrt(-b/a)*(
3*a*d + b*c)/(a*d - b*c)**3 - a*b**8*c**9*(-b/a)**(3/2)*(3*a*d + b*c)**3/(a*d - b*c)**9 - 9*a*b**4*c**4*d*sqrt
(-b/a)*(3*a*d + b*c)/(a*d - b*c)**3 - b**5*c**5*sqrt(-b/a)*(3*a*d + b*c)/(a*d - b*c)**3)/(3*a**2*b*d**3 + 10*a
*b**2*c*d**2 + 3*b**3*c**2*d))/(4*(a*d - b*c)**3) - sqrt(-b/a)*(3*a*d + b*c)*log(x + (a**9*c*d**8*(-b/a)**(3/2
)*(3*a*d + b*c)**3/(a*d - b*c)**9 - 20*a**7*b**2*c**3*d**6*(-b/a)**(3/2)*(3*a*d + b*c)**3/(a*d - b*c)**9 + 64*
a**6*b**3*c**4*d**5*(-b/a)**(3/2)*(3*a*d + b*c)**3/(a*d - b*c)**9 - 90*a**5*b**4*c**5*d**4*(-b/a)**(3/2)*(3*a*
d + b*c)**3/(a*d - b*c)**9 + a**5*d**5*sqrt(-b/a)*(3*a*d + b*c)/(a*d - b*c)**3 + 64*a**4*b**5*c**6*d**3*(-b/a)
**(3/2)*(3*a*d + b*c)**3/(a*d - b*c)**9 + 9*a**4*b*c*d**4*sqrt(-b/a)*(3*a*d + b*c)/(a*d - b*c)**3 - 20*a**3*b*
*6*c**7*d**2*(-b/a)**(3/2)*(3*a*d + b*c)**3/(a*d - b*c)**9 + 54*a**3*b**2*c**2*d**3*sqrt(-b/a)*(3*a*d + b*c)/(
a*d - b*c)**3 + 54*a**2*b**3*c**3*d**2*sqrt(-b/a)*(3*a*d + b*c)/(a*d - b*c)**3 + a*b**8*c**9*(-b/a)**(3/2)*(3*
a*d + b*c)**3/(a*d - b*c)**9 + 9*a*b**4*c**4*d*sqrt(-b/a)*(3*a*d + b*c)/(a*d - b*c)**3 + b**5*c**5*sqrt(-b/a)*
(3*a*d + b*c)/(a*d - b*c)**3)/(3*a**2*b*d**3 + 10*a*b**2*c*d**2 + 3*b**3*c**2*d))/(4*(a*d - b*c)**3) + sqrt(-d
/c)*(a*d + 3*b*c)*log(x + (-a**9*c*d**8*(-d/c)**(3/2)*(a*d + 3*b*c)**3/(a*d - b*c)**9 + 20*a**7*b**2*c**3*d**6
*(-d/c)**(3/2)*(a*d + 3*b*c)**3/(a*d - b*c)**9 - 64*a**6*b**3*c**4*d**5*(-d/c)**(3/2)*(a*d + 3*b*c)**3/(a*d -
b*c)**9 + 90*a**5*b**4*c**5*d**4*(-d/c)**(3/2)*(a*d + 3*b*c)**3/(a*d - b*c)**9 - a**5*d**5*sqrt(-d/c)*(a*d + 3
*b*c)/(a*d - b*c)**3 - 64*a**4*b**5*c**6*d**3*(-d/c)**(3/2)*(a*d + 3*b*c)**3/(a*d - b*c)**9 - 9*a**4*b*c*d**4*
sqrt(-d/c)*(a*d + 3*b*c)/(a*d - b*c)**3 + 20*a**3*b**6*c**7*d**2*(-d/c)**(3/2)*(a*d + 3*b*c)**3/(a*d - b*c)**9
- 54*a**3*b**2*c**2*d**3*sqrt(-d/c)*(a*d + 3*b*c)/(a*d - b*c)**3 - 54*a**2*b**3*c**3*d**2*sqrt(-d/c)*(a*d + 3
*b*c)/(a*d - b*c)**3 - a*b**8*c**9*(-d/c)**(3/2)*(a*d + 3*b*c)**3/(a*d - b*c)**9 - 9*a*b**4*c**4*d*sqrt(-d/c)*
(a*d + 3*b*c)/(a*d - b*c)**3 - b**5*c**5*sqrt(-d/c)*(a*d + 3*b*c)/(a*d - b*c)**3)/(3*a**2*b*d**3 + 10*a*b**2*c
*d**2 + 3*b**3*c**2*d))/(4*(a*d - b*c)**3) - sqrt(-d/c)*(a*d + 3*b*c)*log(x + (a**9*c*d**8*(-d/c)**(3/2)*(a*d
+ 3*b*c)**3/(a*d - b*c)**9 - 20*a**7*b**2*c**3*d**6*(-d/c)**(3/2)*(a*d + 3*b*c)**3/(a*d - b*c)**9 + 64*a**6*b*
*3*c**4*d**5*(-d/c)**(3/2)*(a*d + 3*b*c)**3/(a*d - b*c)**9 - 90*a**5*b**4*c**5*d**4*(-d/c)**(3/2)*(a*d + 3*b*c
)**3/(a*d - b*c)**9 + a**5*d**5*sqrt(-d/c)*(a*d + 3*b*c)/(a*d - b*c)**3 + 64*a**4*b**5*c**6*d**3*(-d/c)**(3/2)
*(a*d + 3*b*c)**3/(a*d - b*c)**9 + 9*a**4*b*c*d**4*sqrt(-d/c)*(a*d + 3*b*c)/(a*d - b*c)**3 - 20*a**3*b**6*c**7
*d**2*(-d/c)**(3/2)*(a*d + 3*b*c)**3/(a*d - b*c)**9 + 54*a**3*b**2*c**2*d**3*sqrt(-d/c)*(a*d + 3*b*c)/(a*d - b
*c)**3 + 54*a**2*b**3*c**3*d**2*sqrt(-d/c)*(a*d + 3*b*c)/(a*d - b*c)**3 + a*b**8*c**9*(-d/c)**(3/2)*(a*d + 3*b
*c)**3/(a*d - b*c)**9 + 9*a*b**4*c**4*d*sqrt(-d/c)*(a*d + 3*b*c)/(a*d - b*c)**3 + b**5*c**5*sqrt(-d/c)*(a*d +
3*b*c)/(a*d - b*c)**3)/(3*a**2*b*d**3 + 10*a*b**2*c*d**2 + 3*b**3*c**2*d))/(4*(a*d - b*c)**3) - (2*b*d*x**3 +
x*(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))
________________________________________________________________________________________
Giac [B] time = 1.40361, size = 1481, normalized size = 10.07 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^2/(b*x^2+a)^2/(d*x^2+c)^2,x, algorithm="giac")
[Out]
-(2*sqrt(c*d)*b^4*c^3*abs(d) - 2*sqrt(c*d)*a*b^3*c^2*d*abs(d) - 2*sqrt(c*d)*a^2*b^2*c*d^2*abs(d) + 2*sqrt(c*d)
*a^3*b*d^3*abs(d) + sqrt(c*d)*b*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*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^3*c^3*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*b^2*c^2*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^2*b*c*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^3*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) + (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*sqrt(a*b)*b^3*c^3*d*abs(b) - 2*sqrt(a*b)*a*b^2*c^2*d^2*abs(b) - 2*sqrt(a*b)*a^2*b*
c*d^3*abs(b) + 2*sqrt(a*b)*a^3*d^4*abs(b) - sqrt(a*b)*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^4*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^3*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^2*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*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) - 1/2*(2*b*d*x^3 + b*c*x + a*d*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))