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

Optimal. Leaf size=167 \[ \frac{b^{3/2} (b c-5 a d) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{3/2} (b c-a d)^3}+\frac{d^{3/2} (5 b c-a d) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{2 c^{3/2} (b c-a d)^3}+\frac{b x}{2 a \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}+\frac{d x (a d+b c)}{2 a c \left (c+d x^2\right ) (b c-a d)^2} \]

[Out]

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

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Rubi [A]  time = 0.198204, antiderivative size = 167, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {414, 527, 522, 205} \[ \frac{b^{3/2} (b c-5 a d) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{3/2} (b c-a d)^3}+\frac{d^{3/2} (5 b c-a d) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{2 c^{3/2} (b c-a d)^3}+\frac{b x}{2 a \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}+\frac{d x (a d+b c)}{2 a c \left (c+d x^2\right ) (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 414

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Simp[(b*x*(a + b*x^n)^(p + 1)*(
c + d*x^n)^(q + 1))/(a*n*(p + 1)*(b*c - a*d)), x] + Dist[1/(a*n*(p + 1)*(b*c - a*d)), Int[(a + b*x^n)^(p + 1)*
(c + d*x^n)^q*Simp[b*c + n*(p + 1)*(b*c - a*d) + d*b*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d,
n, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &&  !( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomial
Q[a, b, c, d, 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{1}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx &=\frac{b x}{2 a (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )}-\frac{\int \frac{-b c+2 a d-3 b d x^2}{\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) x}{2 a c (b c-a d)^2 \left (c+d x^2\right )}+\frac{b x}{2 a (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )}-\frac{\int \frac{-2 \left (b^2 c^2-4 a b c d+a^2 d^2\right )-2 b d (b c+a d) x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx}{4 a c (b c-a d)^2}\\ &=\frac{d (b c+a d) x}{2 a c (b c-a d)^2 \left (c+d x^2\right )}+\frac{b x}{2 a (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )}+\frac{\left (b^2 (b c-5 a d)\right ) \int \frac{1}{a+b x^2} \, dx}{2 a (b c-a d)^3}+\frac{\left (d^2 (5 b c-a d)\right ) \int \frac{1}{c+d x^2} \, dx}{2 c (b c-a d)^3}\\ &=\frac{d (b c+a d) x}{2 a c (b c-a d)^2 \left (c+d x^2\right )}+\frac{b x}{2 a (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )}+\frac{b^{3/2} (b c-5 a d) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{3/2} (b c-a d)^3}+\frac{d^{3/2} (5 b c-a d) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{2 c^{3/2} (b c-a d)^3}\\ \end{align*}

Mathematica [A]  time = 0.317483, size = 136, normalized size = 0.81 \[ \frac{1}{2} \left (\frac{x (b c-a d) \left (\frac{b^2}{a^2+a b x^2}+\frac{d^2}{c^2+c d x^2}\right )+\frac{d^{3/2} (5 b c-a d) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{c^{3/2}}}{(b c-a d)^3}+\frac{b^{3/2} (5 a d-b c) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{3/2} (a d-b c)^3}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0., size = 238, normalized size = 1.4 \begin{align*}{\frac{{d}^{3}xa}{2\, \left ( ad-bc \right ) ^{3}c \left ( d{x}^{2}+c \right ) }}-{\frac{b{d}^{2}x}{2\, \left ( ad-bc \right ) ^{3} \left ( d{x}^{2}+c \right ) }}+{\frac{{d}^{3}a}{2\, \left ( ad-bc \right ) ^{3}c}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-{\frac{5\,b{d}^{2}}{2\, \left ( ad-bc \right ) ^{3}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}+{\frac{{b}^{2}xd}{2\, \left ( ad-bc \right ) ^{3} \left ( b{x}^{2}+a \right ) }}-{\frac{{b}^{3}xc}{2\, \left ( ad-bc \right ) ^{3}a \left ( b{x}^{2}+a \right ) }}+{\frac{5\,{b}^{2}d}{2\, \left ( ad-bc \right ) ^{3}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{{b}^{3}c}{2\, \left ( ad-bc \right ) ^{3}a}\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/(b*x^2+a)^2/(d*x^2+c)^2,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 5.88019, size = 3294, normalized size = 19.72 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/4*(2*(b^3*c^2*d - a^2*b*d^3)*x^3 + (a*b^2*c^3 - 5*a^2*b*c^2*d + (b^3*c^2*d - 5*a*b^2*c*d^2)*x^4 + (b^3*c^3
- 4*a*b^2*c^2*d - 5*a^2*b*c*d^2)*x^2)*sqrt(-b/a)*log((b*x^2 + 2*a*x*sqrt(-b/a) - a)/(b*x^2 + a)) + (5*a^2*b*c^
2*d - a^3*c*d^2 + (5*a*b^2*c*d^2 - a^2*b*d^3)*x^4 + (5*a*b^2*c^2*d + 4*a^2*b*c*d^2 - a^3*d^3)*x^2)*sqrt(-d/c)*
log((d*x^2 + 2*c*x*sqrt(-d/c) - c)/(d*x^2 + c)) + 2*(b^3*c^3 - a*b^2*c^2*d + a^2*b*c*d^2 - a^3*d^3)*x)/(a^2*b^
3*c^5 - 3*a^3*b^2*c^4*d + 3*a^4*b*c^3*d^2 - a^5*c^2*d^3 + (a*b^4*c^4*d - 3*a^2*b^3*c^3*d^2 + 3*a^3*b^2*c^2*d^3
- a^4*b*c*d^4)*x^4 + (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), 1/4*(2*(b^3*c^2*d - a^
2*b*d^3)*x^3 + 2*(5*a^2*b*c^2*d - a^3*c*d^2 + (5*a*b^2*c*d^2 - a^2*b*d^3)*x^4 + (5*a*b^2*c^2*d + 4*a^2*b*c*d^2
- a^3*d^3)*x^2)*sqrt(d/c)*arctan(x*sqrt(d/c)) + (a*b^2*c^3 - 5*a^2*b*c^2*d + (b^3*c^2*d - 5*a*b^2*c*d^2)*x^4
+ (b^3*c^3 - 4*a*b^2*c^2*d - 5*a^2*b*c*d^2)*x^2)*sqrt(-b/a)*log((b*x^2 + 2*a*x*sqrt(-b/a) - a)/(b*x^2 + a)) +
2*(b^3*c^3 - a*b^2*c^2*d + a^2*b*c*d^2 - a^3*d^3)*x)/(a^2*b^3*c^5 - 3*a^3*b^2*c^4*d + 3*a^4*b*c^3*d^2 - a^5*c^
2*d^3 + (a*b^4*c^4*d - 3*a^2*b^3*c^3*d^2 + 3*a^3*b^2*c^2*d^3 - a^4*b*c*d^4)*x^4 + (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), 1/4*(2*(b^3*c^2*d - a^2*b*d^3)*x^3 + 2*(a*b^2*c^3 - 5*a^2*b*c^2*d + (b^3
*c^2*d - 5*a*b^2*c*d^2)*x^4 + (b^3*c^3 - 4*a*b^2*c^2*d - 5*a^2*b*c*d^2)*x^2)*sqrt(b/a)*arctan(x*sqrt(b/a)) + (
5*a^2*b*c^2*d - a^3*c*d^2 + (5*a*b^2*c*d^2 - a^2*b*d^3)*x^4 + (5*a*b^2*c^2*d + 4*a^2*b*c*d^2 - a^3*d^3)*x^2)*s
qrt(-d/c)*log((d*x^2 + 2*c*x*sqrt(-d/c) - c)/(d*x^2 + c)) + 2*(b^3*c^3 - a*b^2*c^2*d + a^2*b*c*d^2 - a^3*d^3)*
x)/(a^2*b^3*c^5 - 3*a^3*b^2*c^4*d + 3*a^4*b*c^3*d^2 - a^5*c^2*d^3 + (a*b^4*c^4*d - 3*a^2*b^3*c^3*d^2 + 3*a^3*b
^2*c^2*d^3 - a^4*b*c*d^4)*x^4 + (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), 1/2*((b^3*c^
2*d - a^2*b*d^3)*x^3 + (a*b^2*c^3 - 5*a^2*b*c^2*d + (b^3*c^2*d - 5*a*b^2*c*d^2)*x^4 + (b^3*c^3 - 4*a*b^2*c^2*d
- 5*a^2*b*c*d^2)*x^2)*sqrt(b/a)*arctan(x*sqrt(b/a)) + (5*a^2*b*c^2*d - a^3*c*d^2 + (5*a*b^2*c*d^2 - a^2*b*d^3
)*x^4 + (5*a*b^2*c^2*d + 4*a^2*b*c*d^2 - a^3*d^3)*x^2)*sqrt(d/c)*arctan(x*sqrt(d/c)) + (b^3*c^3 - a*b^2*c^2*d
+ a^2*b*c*d^2 - a^3*d^3)*x)/(a^2*b^3*c^5 - 3*a^3*b^2*c^4*d + 3*a^4*b*c^3*d^2 - a^5*c^2*d^3 + (a*b^4*c^4*d - 3*
a^2*b^3*c^3*d^2 + 3*a^3*b^2*c^2*d^3 - a^4*b*c*d^4)*x^4 + (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)]

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Sympy [B]  time = 143.969, size = 3662, normalized size = 21.93 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(-b**3/a**3)*(5*a*d - b*c)*log(x + (-a**12*c**3*d**9*(-b**3/a**3)**(3/2)*(5*a*d - b*c)**3/(a*d - b*c)**9
+ 11*a**11*b*c**4*d**8*(-b**3/a**3)**(3/2)*(5*a*d - b*c)**3/(a*d - b*c)**9 - 40*a**10*b**2*c**5*d**7*(-b**3/a*
*3)**(3/2)*(5*a*d - b*c)**3/(a*d - b*c)**9 + 64*a**9*b**3*c**6*d**6*(-b**3/a**3)**(3/2)*(5*a*d - b*c)**3/(a*d
- b*c)**9 - 34*a**8*b**4*c**7*d**5*(-b**3/a**3)**(3/2)*(5*a*d - b*c)**3/(a*d - b*c)**9 - a**8*d**8*sqrt(-b**3/
a**3)*(5*a*d - b*c)/(a*d - b*c)**3 - 34*a**7*b**5*c**8*d**4*(-b**3/a**3)**(3/2)*(5*a*d - b*c)**3/(a*d - b*c)**
9 + 15*a**7*b*c*d**7*sqrt(-b**3/a**3)*(5*a*d - b*c)/(a*d - b*c)**3 + 64*a**6*b**6*c**9*d**3*(-b**3/a**3)**(3/2
)*(5*a*d - b*c)**3/(a*d - b*c)**9 - 75*a**6*b**2*c**2*d**6*sqrt(-b**3/a**3)*(5*a*d - b*c)/(a*d - b*c)**3 - 40*
a**5*b**7*c**10*d**2*(-b**3/a**3)**(3/2)*(5*a*d - b*c)**3/(a*d - b*c)**9 + 125*a**5*b**3*c**3*d**5*sqrt(-b**3/
a**3)*(5*a*d - b*c)/(a*d - b*c)**3 + 11*a**4*b**8*c**11*d*(-b**3/a**3)**(3/2)*(5*a*d - b*c)**3/(a*d - b*c)**9
- a**3*b**9*c**12*(-b**3/a**3)**(3/2)*(5*a*d - b*c)**3/(a*d - b*c)**9 + 125*a**3*b**5*c**5*d**3*sqrt(-b**3/a**
3)*(5*a*d - b*c)/(a*d - b*c)**3 - 75*a**2*b**6*c**6*d**2*sqrt(-b**3/a**3)*(5*a*d - b*c)/(a*d - b*c)**3 + 15*a*
b**7*c**7*d*sqrt(-b**3/a**3)*(5*a*d - b*c)/(a*d - b*c)**3 - b**8*c**8*sqrt(-b**3/a**3)*(5*a*d - b*c)/(a*d - b*
c)**3)/(5*a**4*b**2*d**6 - 61*a**3*b**3*c*d**5 + 192*a**2*b**4*c**2*d**4 - 61*a*b**5*c**3*d**3 + 5*b**6*c**4*d
**2))/(4*(a*d - b*c)**3) + sqrt(-b**3/a**3)*(5*a*d - b*c)*log(x + (a**12*c**3*d**9*(-b**3/a**3)**(3/2)*(5*a*d
- b*c)**3/(a*d - b*c)**9 - 11*a**11*b*c**4*d**8*(-b**3/a**3)**(3/2)*(5*a*d - b*c)**3/(a*d - b*c)**9 + 40*a**10
*b**2*c**5*d**7*(-b**3/a**3)**(3/2)*(5*a*d - b*c)**3/(a*d - b*c)**9 - 64*a**9*b**3*c**6*d**6*(-b**3/a**3)**(3/
2)*(5*a*d - b*c)**3/(a*d - b*c)**9 + 34*a**8*b**4*c**7*d**5*(-b**3/a**3)**(3/2)*(5*a*d - b*c)**3/(a*d - b*c)**
9 + a**8*d**8*sqrt(-b**3/a**3)*(5*a*d - b*c)/(a*d - b*c)**3 + 34*a**7*b**5*c**8*d**4*(-b**3/a**3)**(3/2)*(5*a*
d - b*c)**3/(a*d - b*c)**9 - 15*a**7*b*c*d**7*sqrt(-b**3/a**3)*(5*a*d - b*c)/(a*d - b*c)**3 - 64*a**6*b**6*c**
9*d**3*(-b**3/a**3)**(3/2)*(5*a*d - b*c)**3/(a*d - b*c)**9 + 75*a**6*b**2*c**2*d**6*sqrt(-b**3/a**3)*(5*a*d -
b*c)/(a*d - b*c)**3 + 40*a**5*b**7*c**10*d**2*(-b**3/a**3)**(3/2)*(5*a*d - b*c)**3/(a*d - b*c)**9 - 125*a**5*b
**3*c**3*d**5*sqrt(-b**3/a**3)*(5*a*d - b*c)/(a*d - b*c)**3 - 11*a**4*b**8*c**11*d*(-b**3/a**3)**(3/2)*(5*a*d
- b*c)**3/(a*d - b*c)**9 + a**3*b**9*c**12*(-b**3/a**3)**(3/2)*(5*a*d - b*c)**3/(a*d - b*c)**9 - 125*a**3*b**5
*c**5*d**3*sqrt(-b**3/a**3)*(5*a*d - b*c)/(a*d - b*c)**3 + 75*a**2*b**6*c**6*d**2*sqrt(-b**3/a**3)*(5*a*d - b*
c)/(a*d - b*c)**3 - 15*a*b**7*c**7*d*sqrt(-b**3/a**3)*(5*a*d - b*c)/(a*d - b*c)**3 + b**8*c**8*sqrt(-b**3/a**3
)*(5*a*d - b*c)/(a*d - b*c)**3)/(5*a**4*b**2*d**6 - 61*a**3*b**3*c*d**5 + 192*a**2*b**4*c**2*d**4 - 61*a*b**5*
c**3*d**3 + 5*b**6*c**4*d**2))/(4*(a*d - b*c)**3) - sqrt(-d**3/c**3)*(a*d - 5*b*c)*log(x + (-a**12*c**3*d**9*(
-d**3/c**3)**(3/2)*(a*d - 5*b*c)**3/(a*d - b*c)**9 + 11*a**11*b*c**4*d**8*(-d**3/c**3)**(3/2)*(a*d - 5*b*c)**3
/(a*d - b*c)**9 - 40*a**10*b**2*c**5*d**7*(-d**3/c**3)**(3/2)*(a*d - 5*b*c)**3/(a*d - b*c)**9 + 64*a**9*b**3*c
**6*d**6*(-d**3/c**3)**(3/2)*(a*d - 5*b*c)**3/(a*d - b*c)**9 - 34*a**8*b**4*c**7*d**5*(-d**3/c**3)**(3/2)*(a*d
- 5*b*c)**3/(a*d - b*c)**9 - a**8*d**8*sqrt(-d**3/c**3)*(a*d - 5*b*c)/(a*d - b*c)**3 - 34*a**7*b**5*c**8*d**4
*(-d**3/c**3)**(3/2)*(a*d - 5*b*c)**3/(a*d - b*c)**9 + 15*a**7*b*c*d**7*sqrt(-d**3/c**3)*(a*d - 5*b*c)/(a*d -
b*c)**3 + 64*a**6*b**6*c**9*d**3*(-d**3/c**3)**(3/2)*(a*d - 5*b*c)**3/(a*d - b*c)**9 - 75*a**6*b**2*c**2*d**6*
sqrt(-d**3/c**3)*(a*d - 5*b*c)/(a*d - b*c)**3 - 40*a**5*b**7*c**10*d**2*(-d**3/c**3)**(3/2)*(a*d - 5*b*c)**3/(
a*d - b*c)**9 + 125*a**5*b**3*c**3*d**5*sqrt(-d**3/c**3)*(a*d - 5*b*c)/(a*d - b*c)**3 + 11*a**4*b**8*c**11*d*(
-d**3/c**3)**(3/2)*(a*d - 5*b*c)**3/(a*d - b*c)**9 - a**3*b**9*c**12*(-d**3/c**3)**(3/2)*(a*d - 5*b*c)**3/(a*d
- b*c)**9 + 125*a**3*b**5*c**5*d**3*sqrt(-d**3/c**3)*(a*d - 5*b*c)/(a*d - b*c)**3 - 75*a**2*b**6*c**6*d**2*sq
rt(-d**3/c**3)*(a*d - 5*b*c)/(a*d - b*c)**3 + 15*a*b**7*c**7*d*sqrt(-d**3/c**3)*(a*d - 5*b*c)/(a*d - b*c)**3 -
b**8*c**8*sqrt(-d**3/c**3)*(a*d - 5*b*c)/(a*d - b*c)**3)/(5*a**4*b**2*d**6 - 61*a**3*b**3*c*d**5 + 192*a**2*b
**4*c**2*d**4 - 61*a*b**5*c**3*d**3 + 5*b**6*c**4*d**2))/(4*(a*d - b*c)**3) + sqrt(-d**3/c**3)*(a*d - 5*b*c)*l
og(x + (a**12*c**3*d**9*(-d**3/c**3)**(3/2)*(a*d - 5*b*c)**3/(a*d - b*c)**9 - 11*a**11*b*c**4*d**8*(-d**3/c**3
)**(3/2)*(a*d - 5*b*c)**3/(a*d - b*c)**9 + 40*a**10*b**2*c**5*d**7*(-d**3/c**3)**(3/2)*(a*d - 5*b*c)**3/(a*d -
b*c)**9 - 64*a**9*b**3*c**6*d**6*(-d**3/c**3)**(3/2)*(a*d - 5*b*c)**3/(a*d - b*c)**9 + 34*a**8*b**4*c**7*d**5
*(-d**3/c**3)**(3/2)*(a*d - 5*b*c)**3/(a*d - b*c)**9 + a**8*d**8*sqrt(-d**3/c**3)*(a*d - 5*b*c)/(a*d - b*c)**3
+ 34*a**7*b**5*c**8*d**4*(-d**3/c**3)**(3/2)*(a*d - 5*b*c)**3/(a*d - b*c)**9 - 15*a**7*b*c*d**7*sqrt(-d**3/c*
*3)*(a*d - 5*b*c)/(a*d - b*c)**3 - 64*a**6*b**6*c**9*d**3*(-d**3/c**3)**(3/2)*(a*d - 5*b*c)**3/(a*d - b*c)**9
+ 75*a**6*b**2*c**2*d**6*sqrt(-d**3/c**3)*(a*d - 5*b*c)/(a*d - b*c)**3 + 40*a**5*b**7*c**10*d**2*(-d**3/c**3)*
*(3/2)*(a*d - 5*b*c)**3/(a*d - b*c)**9 - 125*a**5*b**3*c**3*d**5*sqrt(-d**3/c**3)*(a*d - 5*b*c)/(a*d - b*c)**3
- 11*a**4*b**8*c**11*d*(-d**3/c**3)**(3/2)*(a*d - 5*b*c)**3/(a*d - b*c)**9 + a**3*b**9*c**12*(-d**3/c**3)**(3
/2)*(a*d - 5*b*c)**3/(a*d - b*c)**9 - 125*a**3*b**5*c**5*d**3*sqrt(-d**3/c**3)*(a*d - 5*b*c)/(a*d - b*c)**3 +
75*a**2*b**6*c**6*d**2*sqrt(-d**3/c**3)*(a*d - 5*b*c)/(a*d - b*c)**3 - 15*a*b**7*c**7*d*sqrt(-d**3/c**3)*(a*d
- 5*b*c)/(a*d - b*c)**3 + b**8*c**8*sqrt(-d**3/c**3)*(a*d - 5*b*c)/(a*d - b*c)**3)/(5*a**4*b**2*d**6 - 61*a**3
*b**3*c*d**5 + 192*a**2*b**4*c**2*d**4 - 61*a*b**5*c**3*d**3 + 5*b**6*c**4*d**2))/(4*(a*d - b*c)**3) + (x**3*(
a*b*d**2 + b**2*c*d) + x*(a**2*d**2 + b**2*c**2))/(2*a**4*c**2*d**2 - 4*a**3*b*c**3*d + 2*a**2*b**2*c**4 + x**
4*(2*a**3*b*c*d**3 - 4*a**2*b**2*c**2*d**2 + 2*a*b**3*c**3*d) + x**2*(2*a**4*c*d**3 - 2*a**3*b*c**2*d**2 - 2*a
**2*b**2*c**3*d + 2*a*b**3*c**4))

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Giac [B]  time = 1.47755, size = 1928, normalized size = 11.54 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(sqrt(c*d)*a*b^5*c^5*abs(d) - 12*sqrt(c*d)*a^2*b^4*c^4*d*abs(d) + 22*sqrt(c*d)*a^3*b^3*c^3*d^2*abs(d) - 1
2*sqrt(c*d)*a^4*b^2*c^2*d^3*abs(d) + sqrt(c*d)*a^5*b*c*d^4*abs(d) - sqrt(c*d)*b^2*c*abs(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)*abs(d) - sqrt(c*d)*a*b*d*abs(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)*abs(d))*arctan(2*sqrt(1/2)*x/sqrt((a*b^3*c^4 - a^2*b^2*c^3*d - a^3*b*c^2*d^2 + a^4*c*d^3 + sqrt
((a*b^3*c^4 - a^2*b^2*c^3*d - a^3*b*c^2*d^2 + a^4*c*d^3)^2 - 4*(a^2*b^2*c^4 - 2*a^3*b*c^3*d + a^4*c^2*d^2)*(a*
b^3*c^3*d - 2*a^2*b^2*c^2*d^2 + a^3*b*c*d^3)))/(a*b^3*c^3*d - 2*a^2*b^2*c^2*d^2 + a^3*b*c*d^3)))/(a*b^3*c^4*d*
abs(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) - a^2*b^2*c^3*d^2*abs(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) - a^3*b*c^2*d^3*abs(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) + a^4*c*d^4*abs(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) + (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)^2*d) + 1/2*(sqrt(a*b)*a*b^4*c^5*d*abs(b) - 12*sqrt(a*b)*a^2*b^3*c^4*d^2*abs(b) +
22*sqrt(a*b)*a^3*b^2*c^3*d^3*abs(b) - 12*sqrt(a*b)*a^4*b*c^2*d^4*abs(b) + sqrt(a*b)*a^5*c*d^5*abs(b) + sqrt(a
*b)*b*c*d*abs(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)*abs(b) + sqrt(a*b)*a*d^2*abs(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)*abs(b))*arctan(2*sqrt(1/2)*x/sqrt((a*b^3*c^4 - a^2*b^2*c^3*
d - a^3*b*c^2*d^2 + a^4*c*d^3 - sqrt((a*b^3*c^4 - a^2*b^2*c^3*d - a^3*b*c^2*d^2 + a^4*c*d^3)^2 - 4*(a^2*b^2*c^
4 - 2*a^3*b*c^3*d + a^4*c^2*d^2)*(a*b^3*c^3*d - 2*a^2*b^2*c^2*d^2 + a^3*b*c*d^3)))/(a*b^3*c^3*d - 2*a^2*b^2*c^
2*d^2 + a^3*b*c*d^3)))/(a*b^4*c^4*abs(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) - a^2*b^3*c^3
*d*abs(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) - a^3*b^2*c^2*d^2*abs(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) + a^4*b*c*d^3*abs(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) - (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)^2*b) + 1/2*(b^2*c*d*x^3 + a*b*d^2*x^3 + b^2*
c^2*x + a^2*d^2*x)/((a*b^2*c^3 - 2*a^2*b*c^2*d + a^3*c*d^2)*(b*d*x^4 + b*c*x^2 + a*d*x^2 + a*c))