3.293 \(\int \frac{1}{(a+b x^2)^2 (c+d x^2)} \, dx\)
Optimal. Leaf size=108 \[ \frac{\sqrt{b} (b c-3 a d) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{3/2} (b c-a d)^2}+\frac{d^{3/2} \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{\sqrt{c} (b c-a d)^2}+\frac{b x}{2 a \left (a+b x^2\right ) (b c-a d)} \]
[Out]
(b*x)/(2*a*(b*c - a*d)*(a + b*x^2)) + (Sqrt[b]*(b*c - 3*a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(3/2)*(b*c - a*
d)^2) + (d^(3/2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(Sqrt[c]*(b*c - a*d)^2)
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Rubi [A] time = 0.0804604, antiderivative size = 108, normalized size of antiderivative = 1.,
number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used =
{414, 522, 205} \[ \frac{\sqrt{b} (b c-3 a d) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{3/2} (b c-a d)^2}+\frac{d^{3/2} \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{\sqrt{c} (b c-a d)^2}+\frac{b x}{2 a \left (a+b x^2\right ) (b c-a d)} \]
Antiderivative was successfully verified.
[In]
Int[1/((a + b*x^2)^2*(c + d*x^2)),x]
[Out]
(b*x)/(2*a*(b*c - a*d)*(a + b*x^2)) + (Sqrt[b]*(b*c - 3*a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(3/2)*(b*c - a*
d)^2) + (d^(3/2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(Sqrt[c]*(b*c - a*d)^2)
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 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 )} \, dx &=\frac{b x}{2 a (b c-a d) \left (a+b x^2\right )}-\frac{\int \frac{-b c+2 a d-b d x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx}{2 a (b c-a d)}\\ &=\frac{b x}{2 a (b c-a d) \left (a+b x^2\right )}+\frac{d^2 \int \frac{1}{c+d x^2} \, dx}{(b c-a d)^2}+\frac{(b (b c-3 a d)) \int \frac{1}{a+b x^2} \, dx}{2 a (b c-a d)^2}\\ &=\frac{b x}{2 a (b c-a d) \left (a+b x^2\right )}+\frac{\sqrt{b} (b c-3 a d) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{3/2} (b c-a d)^2}+\frac{d^{3/2} \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{\sqrt{c} (b c-a d)^2}\\ \end{align*}
Mathematica [A] time = 0.14189, size = 109, normalized size = 1.01 \[ -\frac{\sqrt{b} (3 a d-b c) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{3/2} (a d-b c)^2}+\frac{d^{3/2} \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{\sqrt{c} (b c-a d)^2}-\frac{b x}{2 a \left (a+b x^2\right ) (a d-b c)} \]
Antiderivative was successfully verified.
[In]
Integrate[1/((a + b*x^2)^2*(c + d*x^2)),x]
[Out]
-(b*x)/(2*a*(-(b*c) + a*d)*(a + b*x^2)) - (Sqrt[b]*(-(b*c) + 3*a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(3/2)*(-
(b*c) + a*d)^2) + (d^(3/2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(Sqrt[c]*(b*c - a*d)^2)
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Maple [A] time = 0.002, size = 144, normalized size = 1.3 \begin{align*}{\frac{{d}^{2}}{ \left ( ad-bc \right ) ^{2}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-{\frac{bdx}{2\, \left ( ad-bc \right ) ^{2} \left ( b{x}^{2}+a \right ) }}+{\frac{{b}^{2}cx}{2\, \left ( ad-bc \right ) ^{2}a \left ( b{x}^{2}+a \right ) }}-{\frac{3\,bd}{2\, \left ( ad-bc \right ) ^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{{b}^{2}c}{2\, \left ( ad-bc \right ) ^{2}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),x)
[Out]
d^2/(a*d-b*c)^2/(c*d)^(1/2)*arctan(x*d/(c*d)^(1/2))-1/2*b/(a*d-b*c)^2*x/(b*x^2+a)*d+1/2*b^2/(a*d-b*c)^2/a*x/(b
*x^2+a)*c-3/2*b/(a*d-b*c)^2/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))*d+1/2*b^2/(a*d-b*c)^2/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),x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 2.22051, size = 1451, normalized size = 13.44 \begin{align*} \left [-\frac{{\left (a b c - 3 \, a^{2} d +{\left (b^{2} c - 3 \, a b d\right )} x^{2}\right )} \sqrt{-\frac{b}{a}} \log \left (\frac{b x^{2} - 2 \, a x \sqrt{-\frac{b}{a}} - a}{b x^{2} + a}\right ) - 2 \,{\left (a b d x^{2} + a^{2} d\right )} \sqrt{-\frac{d}{c}} \log \left (\frac{d x^{2} + 2 \, c x \sqrt{-\frac{d}{c}} - c}{d x^{2} + c}\right ) - 2 \,{\left (b^{2} c - a b d\right )} x}{4 \,{\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2} +{\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x^{2}\right )}}, \frac{4 \,{\left (a b d x^{2} + a^{2} d\right )} \sqrt{\frac{d}{c}} \arctan \left (x \sqrt{\frac{d}{c}}\right ) -{\left (a b c - 3 \, a^{2} d +{\left (b^{2} c - 3 \, a b d\right )} x^{2}\right )} \sqrt{-\frac{b}{a}} \log \left (\frac{b x^{2} - 2 \, a x \sqrt{-\frac{b}{a}} - a}{b x^{2} + a}\right ) + 2 \,{\left (b^{2} c - a b d\right )} x}{4 \,{\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2} +{\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x^{2}\right )}}, \frac{{\left (a b c - 3 \, a^{2} d +{\left (b^{2} c - 3 \, a b d\right )} x^{2}\right )} \sqrt{\frac{b}{a}} \arctan \left (x \sqrt{\frac{b}{a}}\right ) +{\left (a b d x^{2} + a^{2} d\right )} \sqrt{-\frac{d}{c}} \log \left (\frac{d x^{2} + 2 \, c x \sqrt{-\frac{d}{c}} - c}{d x^{2} + c}\right ) +{\left (b^{2} c - a b d\right )} x}{2 \,{\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2} +{\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x^{2}\right )}}, \frac{{\left (a b c - 3 \, a^{2} d +{\left (b^{2} c - 3 \, a b d\right )} x^{2}\right )} \sqrt{\frac{b}{a}} \arctan \left (x \sqrt{\frac{b}{a}}\right ) + 2 \,{\left (a b d x^{2} + a^{2} d\right )} \sqrt{\frac{d}{c}} \arctan \left (x \sqrt{\frac{d}{c}}\right ) +{\left (b^{2} c - a b d\right )} x}{2 \,{\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2} +{\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x^{2}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(b*x^2+a)^2/(d*x^2+c),x, algorithm="fricas")
[Out]
[-1/4*((a*b*c - 3*a^2*d + (b^2*c - 3*a*b*d)*x^2)*sqrt(-b/a)*log((b*x^2 - 2*a*x*sqrt(-b/a) - a)/(b*x^2 + a)) -
2*(a*b*d*x^2 + a^2*d)*sqrt(-d/c)*log((d*x^2 + 2*c*x*sqrt(-d/c) - c)/(d*x^2 + c)) - 2*(b^2*c - a*b*d)*x)/(a^2*b
^2*c^2 - 2*a^3*b*c*d + a^4*d^2 + (a*b^3*c^2 - 2*a^2*b^2*c*d + a^3*b*d^2)*x^2), 1/4*(4*(a*b*d*x^2 + a^2*d)*sqrt
(d/c)*arctan(x*sqrt(d/c)) - (a*b*c - 3*a^2*d + (b^2*c - 3*a*b*d)*x^2)*sqrt(-b/a)*log((b*x^2 - 2*a*x*sqrt(-b/a)
- a)/(b*x^2 + a)) + 2*(b^2*c - a*b*d)*x)/(a^2*b^2*c^2 - 2*a^3*b*c*d + a^4*d^2 + (a*b^3*c^2 - 2*a^2*b^2*c*d +
a^3*b*d^2)*x^2), 1/2*((a*b*c - 3*a^2*d + (b^2*c - 3*a*b*d)*x^2)*sqrt(b/a)*arctan(x*sqrt(b/a)) + (a*b*d*x^2 + a
^2*d)*sqrt(-d/c)*log((d*x^2 + 2*c*x*sqrt(-d/c) - c)/(d*x^2 + c)) + (b^2*c - a*b*d)*x)/(a^2*b^2*c^2 - 2*a^3*b*c
*d + a^4*d^2 + (a*b^3*c^2 - 2*a^2*b^2*c*d + a^3*b*d^2)*x^2), 1/2*((a*b*c - 3*a^2*d + (b^2*c - 3*a*b*d)*x^2)*sq
rt(b/a)*arctan(x*sqrt(b/a)) + 2*(a*b*d*x^2 + a^2*d)*sqrt(d/c)*arctan(x*sqrt(d/c)) + (b^2*c - a*b*d)*x)/(a^2*b^
2*c^2 - 2*a^3*b*c*d + a^4*d^2 + (a*b^3*c^2 - 2*a^2*b^2*c*d + a^3*b*d^2)*x^2)]
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Sympy [B] time = 15.6387, size = 2033, normalized size = 18.82 \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),x)
[Out]
-b*x/(2*a**3*d - 2*a**2*b*c + x**2*(2*a**2*b*d - 2*a*b**2*c)) + sqrt(-b/a**3)*(3*a*d - b*c)*log(x + (-a**9*c*d
**6*(-b/a**3)**(3/2)*(3*a*d - b*c)**3/(a*d - b*c)**6 + 5*a**8*b*c**2*d**5*(-b/a**3)**(3/2)*(3*a*d - b*c)**3/(2
*(a*d - b*c)**6) + a**7*b**2*c**3*d**4*(-b/a**3)**(3/2)*(3*a*d - b*c)**3/(2*(a*d - b*c)**6) - 7*a**6*b**3*c**4
*d**3*(-b/a**3)**(3/2)*(3*a*d - b*c)**3/(a*d - b*c)**6 + 8*a**5*b**4*c**5*d**2*(-b/a**3)**(3/2)*(3*a*d - b*c)*
*3/(a*d - b*c)**6 - 4*a**5*d**5*sqrt(-b/a**3)*(3*a*d - b*c)/(a*d - b*c)**2 - 7*a**4*b**5*c**6*d*(-b/a**3)**(3/
2)*(3*a*d - b*c)**3/(2*(a*d - b*c)**6) + a**3*b**6*c**7*(-b/a**3)**(3/2)*(3*a*d - b*c)**3/(2*(a*d - b*c)**6) -
27*a**3*b**2*c**2*d**3*sqrt(-b/a**3)*(3*a*d - b*c)/(2*(a*d - b*c)**2) + 27*a**2*b**3*c**3*d**2*sqrt(-b/a**3)*
(3*a*d - b*c)/(2*(a*d - b*c)**2) - 9*a*b**4*c**4*d*sqrt(-b/a**3)*(3*a*d - b*c)/(2*(a*d - b*c)**2) + b**5*c**5*
sqrt(-b/a**3)*(3*a*d - b*c)/(2*(a*d - b*c)**2))/(12*a**2*b*d**4 - 7*a*b**2*c*d**3 + b**3*c**2*d**2))/(4*(a*d -
b*c)**2) - sqrt(-b/a**3)*(3*a*d - b*c)*log(x + (a**9*c*d**6*(-b/a**3)**(3/2)*(3*a*d - b*c)**3/(a*d - b*c)**6
- 5*a**8*b*c**2*d**5*(-b/a**3)**(3/2)*(3*a*d - b*c)**3/(2*(a*d - b*c)**6) - a**7*b**2*c**3*d**4*(-b/a**3)**(3/
2)*(3*a*d - b*c)**3/(2*(a*d - b*c)**6) + 7*a**6*b**3*c**4*d**3*(-b/a**3)**(3/2)*(3*a*d - b*c)**3/(a*d - b*c)**
6 - 8*a**5*b**4*c**5*d**2*(-b/a**3)**(3/2)*(3*a*d - b*c)**3/(a*d - b*c)**6 + 4*a**5*d**5*sqrt(-b/a**3)*(3*a*d
- b*c)/(a*d - b*c)**2 + 7*a**4*b**5*c**6*d*(-b/a**3)**(3/2)*(3*a*d - b*c)**3/(2*(a*d - b*c)**6) - a**3*b**6*c*
*7*(-b/a**3)**(3/2)*(3*a*d - b*c)**3/(2*(a*d - b*c)**6) + 27*a**3*b**2*c**2*d**3*sqrt(-b/a**3)*(3*a*d - b*c)/(
2*(a*d - b*c)**2) - 27*a**2*b**3*c**3*d**2*sqrt(-b/a**3)*(3*a*d - b*c)/(2*(a*d - b*c)**2) + 9*a*b**4*c**4*d*sq
rt(-b/a**3)*(3*a*d - b*c)/(2*(a*d - b*c)**2) - b**5*c**5*sqrt(-b/a**3)*(3*a*d - b*c)/(2*(a*d - b*c)**2))/(12*a
**2*b*d**4 - 7*a*b**2*c*d**3 + b**3*c**2*d**2))/(4*(a*d - b*c)**2) + sqrt(-d**3/c)*log(x + (-8*a**9*c*d**6*(-d
**3/c)**(3/2)/(a*d - b*c)**6 + 20*a**8*b*c**2*d**5*(-d**3/c)**(3/2)/(a*d - b*c)**6 + 4*a**7*b**2*c**3*d**4*(-d
**3/c)**(3/2)/(a*d - b*c)**6 - 56*a**6*b**3*c**4*d**3*(-d**3/c)**(3/2)/(a*d - b*c)**6 + 64*a**5*b**4*c**5*d**2
*(-d**3/c)**(3/2)/(a*d - b*c)**6 - 8*a**5*d**5*sqrt(-d**3/c)/(a*d - b*c)**2 - 28*a**4*b**5*c**6*d*(-d**3/c)**(
3/2)/(a*d - b*c)**6 + 4*a**3*b**6*c**7*(-d**3/c)**(3/2)/(a*d - b*c)**6 - 27*a**3*b**2*c**2*d**3*sqrt(-d**3/c)/
(a*d - b*c)**2 + 27*a**2*b**3*c**3*d**2*sqrt(-d**3/c)/(a*d - b*c)**2 - 9*a*b**4*c**4*d*sqrt(-d**3/c)/(a*d - b*
c)**2 + b**5*c**5*sqrt(-d**3/c)/(a*d - b*c)**2)/(12*a**2*b*d**4 - 7*a*b**2*c*d**3 + b**3*c**2*d**2))/(2*(a*d -
b*c)**2) - sqrt(-d**3/c)*log(x + (8*a**9*c*d**6*(-d**3/c)**(3/2)/(a*d - b*c)**6 - 20*a**8*b*c**2*d**5*(-d**3/
c)**(3/2)/(a*d - b*c)**6 - 4*a**7*b**2*c**3*d**4*(-d**3/c)**(3/2)/(a*d - b*c)**6 + 56*a**6*b**3*c**4*d**3*(-d*
*3/c)**(3/2)/(a*d - b*c)**6 - 64*a**5*b**4*c**5*d**2*(-d**3/c)**(3/2)/(a*d - b*c)**6 + 8*a**5*d**5*sqrt(-d**3/
c)/(a*d - b*c)**2 + 28*a**4*b**5*c**6*d*(-d**3/c)**(3/2)/(a*d - b*c)**6 - 4*a**3*b**6*c**7*(-d**3/c)**(3/2)/(a
*d - b*c)**6 + 27*a**3*b**2*c**2*d**3*sqrt(-d**3/c)/(a*d - b*c)**2 - 27*a**2*b**3*c**3*d**2*sqrt(-d**3/c)/(a*d
- b*c)**2 + 9*a*b**4*c**4*d*sqrt(-d**3/c)/(a*d - b*c)**2 - b**5*c**5*sqrt(-d**3/c)/(a*d - b*c)**2)/(12*a**2*b
*d**4 - 7*a*b**2*c*d**3 + b**3*c**2*d**2))/(2*(a*d - b*c)**2)
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Giac [A] time = 1.14282, size = 163, normalized size = 1.51 \begin{align*} \frac{d^{2} \arctan \left (\frac{d x}{\sqrt{c d}}\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt{c d}} + \frac{{\left (b^{2} c - 3 \, a b d\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \,{\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )} \sqrt{a b}} + \frac{b x}{2 \,{\left (a b c - a^{2} d\right )}{\left (b x^{2} + a\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(b*x^2+a)^2/(d*x^2+c),x, algorithm="giac")
[Out]
d^2*arctan(d*x/sqrt(c*d))/((b^2*c^2 - 2*a*b*c*d + a^2*d^2)*sqrt(c*d)) + 1/2*(b^2*c - 3*a*b*d)*arctan(b*x/sqrt(
a*b))/((a*b^2*c^2 - 2*a^2*b*c*d + a^3*d^2)*sqrt(a*b)) + 1/2*b*x/((a*b*c - a^2*d)*(b*x^2 + a))