3.246 \(\int \frac{1}{(a+b x^2) (c+d x^2)^2} \, dx\)
Optimal. Leaf size=109 \[ \frac{b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} (b c-a d)^2}-\frac{\sqrt{d} (3 b c-a d) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{2 c^{3/2} (b c-a d)^2}-\frac{d x}{2 c \left (c+d x^2\right ) (b c-a d)} \]
[Out]
-(d*x)/(2*c*(b*c - a*d)*(c + d*x^2)) + (b^(3/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[a]*(b*c - a*d)^2) - (Sqrt[d
]*(3*b*c - a*d)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(2*c^(3/2)*(b*c - a*d)^2)
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Rubi [A] time = 0.0790209, antiderivative size = 109, 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} \[ \frac{b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} (b c-a d)^2}-\frac{\sqrt{d} (3 b c-a d) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{2 c^{3/2} (b c-a d)^2}-\frac{d x}{2 c \left (c+d x^2\right ) (b c-a d)} \]
Antiderivative was successfully verified.
[In]
Int[1/((a + b*x^2)*(c + d*x^2)^2),x]
[Out]
-(d*x)/(2*c*(b*c - a*d)*(c + d*x^2)) + (b^(3/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[a]*(b*c - a*d)^2) - (Sqrt[d
]*(3*b*c - a*d)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(2*c^(3/2)*(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 ) \left (c+d x^2\right )^2} \, dx &=-\frac{d x}{2 c (b c-a d) \left (c+d x^2\right )}+\frac{\int \frac{2 b c-a d-b d x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx}{2 c (b c-a d)}\\ &=-\frac{d x}{2 c (b c-a d) \left (c+d x^2\right )}+\frac{b^2 \int \frac{1}{a+b x^2} \, dx}{(b c-a d)^2}-\frac{(d (3 b c-a d)) \int \frac{1}{c+d x^2} \, dx}{2 c (b c-a d)^2}\\ &=-\frac{d x}{2 c (b c-a d) \left (c+d x^2\right )}+\frac{b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} (b c-a d)^2}-\frac{\sqrt{d} (3 b c-a d) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{2 c^{3/2} (b c-a d)^2}\\ \end{align*}
Mathematica [A] time = 0.163413, size = 95, normalized size = 0.87 \[ \frac{\frac{2 b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a}}+\frac{\sqrt{d} (a d-3 b c) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{c^{3/2}}+\frac{d x (a d-b c)}{c \left (c+d x^2\right )}}{2 (b c-a d)^2} \]
Antiderivative was successfully verified.
[In]
Integrate[1/((a + b*x^2)*(c + d*x^2)^2),x]
[Out]
((d*(-(b*c) + a*d)*x)/(c*(c + d*x^2)) + (2*b^(3/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/Sqrt[a] + (Sqrt[d]*(-3*b*c + a
*d)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/c^(3/2))/(2*(b*c - a*d)^2)
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Maple [A] time = 0.009, size = 144, normalized size = 1.3 \begin{align*}{\frac{{d}^{2}xa}{2\, \left ( ad-bc \right ) ^{2}c \left ( d{x}^{2}+c \right ) }}-{\frac{bdx}{2\, \left ( ad-bc \right ) ^{2} \left ( d{x}^{2}+c \right ) }}+{\frac{a{d}^{2}}{2\, \left ( ad-bc \right ) ^{2}c}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-{\frac{3\,bd}{2\, \left ( ad-bc \right ) ^{2}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}+{\frac{{b}^{2}}{ \left ( ad-bc \right ) ^{2}}\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)/(d*x^2+c)^2,x)
[Out]
1/2*d^2/(a*d-b*c)^2/c*x/(d*x^2+c)*a-1/2*d/(a*d-b*c)^2*x/(d*x^2+c)*b+1/2*d^2/(a*d-b*c)^2/c/(c*d)^(1/2)*arctan(x
*d/(c*d)^(1/2))*a-3/2*d/(a*d-b*c)^2/(c*d)^(1/2)*arctan(x*d/(c*d)^(1/2))*b+b^2/(a*d-b*c)^2/(a*b)^(1/2)*arctan(b
*x/(a*b)^(1/2))
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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)/(d*x^2+c)^2,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 2.23271, size = 1451, normalized size = 13.31 \begin{align*} \left [\frac{2 \,{\left (b c d x^{2} + b c^{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 ) -{\left (3 \, b c^{2} - a c d +{\left (3 \, b c d - a d^{2}\right )} x^{2}\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 c d - a d^{2}\right )} x}{4 \,{\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2} +{\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3}\right )} x^{2}\right )}}, -\frac{{\left (3 \, b c^{2} - a c d +{\left (3 \, b c d - a d^{2}\right )} x^{2}\right )} \sqrt{\frac{d}{c}} \arctan \left (x \sqrt{\frac{d}{c}}\right ) -{\left (b c d x^{2} + b c^{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 ) +{\left (b c d - a d^{2}\right )} x}{2 \,{\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2} +{\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3}\right )} x^{2}\right )}}, \frac{4 \,{\left (b c d x^{2} + b c^{2}\right )} \sqrt{\frac{b}{a}} \arctan \left (x \sqrt{\frac{b}{a}}\right ) -{\left (3 \, b c^{2} - a c d +{\left (3 \, b c d - a d^{2}\right )} x^{2}\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 c d - a d^{2}\right )} x}{4 \,{\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2} +{\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3}\right )} x^{2}\right )}}, \frac{2 \,{\left (b c d x^{2} + b c^{2}\right )} \sqrt{\frac{b}{a}} \arctan \left (x \sqrt{\frac{b}{a}}\right ) -{\left (3 \, b c^{2} - a c d +{\left (3 \, b c d - a d^{2}\right )} x^{2}\right )} \sqrt{\frac{d}{c}} \arctan \left (x \sqrt{\frac{d}{c}}\right ) -{\left (b c d - a d^{2}\right )} x}{2 \,{\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2} +{\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3}\right )} x^{2}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(b*x^2+a)/(d*x^2+c)^2,x, algorithm="fricas")
[Out]
[1/4*(2*(b*c*d*x^2 + b*c^2)*sqrt(-b/a)*log((b*x^2 + 2*a*x*sqrt(-b/a) - a)/(b*x^2 + a)) - (3*b*c^2 - a*c*d + (3
*b*c*d - a*d^2)*x^2)*sqrt(-d/c)*log((d*x^2 + 2*c*x*sqrt(-d/c) - c)/(d*x^2 + c)) - 2*(b*c*d - a*d^2)*x)/(b^2*c^
4 - 2*a*b*c^3*d + a^2*c^2*d^2 + (b^2*c^3*d - 2*a*b*c^2*d^2 + a^2*c*d^3)*x^2), -1/2*((3*b*c^2 - a*c*d + (3*b*c*
d - a*d^2)*x^2)*sqrt(d/c)*arctan(x*sqrt(d/c)) - (b*c*d*x^2 + b*c^2)*sqrt(-b/a)*log((b*x^2 + 2*a*x*sqrt(-b/a) -
a)/(b*x^2 + a)) + (b*c*d - a*d^2)*x)/(b^2*c^4 - 2*a*b*c^3*d + a^2*c^2*d^2 + (b^2*c^3*d - 2*a*b*c^2*d^2 + a^2*
c*d^3)*x^2), 1/4*(4*(b*c*d*x^2 + b*c^2)*sqrt(b/a)*arctan(x*sqrt(b/a)) - (3*b*c^2 - a*c*d + (3*b*c*d - a*d^2)*x
^2)*sqrt(-d/c)*log((d*x^2 + 2*c*x*sqrt(-d/c) - c)/(d*x^2 + c)) - 2*(b*c*d - a*d^2)*x)/(b^2*c^4 - 2*a*b*c^3*d +
a^2*c^2*d^2 + (b^2*c^3*d - 2*a*b*c^2*d^2 + a^2*c*d^3)*x^2), 1/2*(2*(b*c*d*x^2 + b*c^2)*sqrt(b/a)*arctan(x*sqr
t(b/a)) - (3*b*c^2 - a*c*d + (3*b*c*d - a*d^2)*x^2)*sqrt(d/c)*arctan(x*sqrt(d/c)) - (b*c*d - a*d^2)*x)/(b^2*c^
4 - 2*a*b*c^3*d + a^2*c^2*d^2 + (b^2*c^3*d - 2*a*b*c^2*d^2 + a^2*c*d^3)*x^2)]
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Sympy [B] time = 15.1212, size = 2033, normalized size = 18.65 \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)/(d*x**2+c)**2,x)
[Out]
d*x/(2*a*c**2*d - 2*b*c**3 + x**2*(2*a*c*d**2 - 2*b*c**2*d)) - sqrt(-b**3/a)*log(x + (-4*a**7*c**3*d**6*(-b**3
/a)**(3/2)/(a*d - b*c)**6 + 28*a**6*b*c**4*d**5*(-b**3/a)**(3/2)/(a*d - b*c)**6 - 64*a**5*b**2*c**5*d**4*(-b**
3/a)**(3/2)/(a*d - b*c)**6 - a**5*d**5*sqrt(-b**3/a)/(a*d - b*c)**2 + 56*a**4*b**3*c**6*d**3*(-b**3/a)**(3/2)/
(a*d - b*c)**6 + 9*a**4*b*c*d**4*sqrt(-b**3/a)/(a*d - b*c)**2 - 4*a**3*b**4*c**7*d**2*(-b**3/a)**(3/2)/(a*d -
b*c)**6 - 27*a**3*b**2*c**2*d**3*sqrt(-b**3/a)/(a*d - b*c)**2 - 20*a**2*b**5*c**8*d*(-b**3/a)**(3/2)/(a*d - b*
c)**6 + 27*a**2*b**3*c**3*d**2*sqrt(-b**3/a)/(a*d - b*c)**2 + 8*a*b**6*c**9*(-b**3/a)**(3/2)/(a*d - b*c)**6 +
8*b**5*c**5*sqrt(-b**3/a)/(a*d - b*c)**2)/(a**2*b**2*d**3 - 7*a*b**3*c*d**2 + 12*b**4*c**2*d))/(2*(a*d - b*c)*
*2) + sqrt(-b**3/a)*log(x + (4*a**7*c**3*d**6*(-b**3/a)**(3/2)/(a*d - b*c)**6 - 28*a**6*b*c**4*d**5*(-b**3/a)*
*(3/2)/(a*d - b*c)**6 + 64*a**5*b**2*c**5*d**4*(-b**3/a)**(3/2)/(a*d - b*c)**6 + a**5*d**5*sqrt(-b**3/a)/(a*d
- b*c)**2 - 56*a**4*b**3*c**6*d**3*(-b**3/a)**(3/2)/(a*d - b*c)**6 - 9*a**4*b*c*d**4*sqrt(-b**3/a)/(a*d - b*c)
**2 + 4*a**3*b**4*c**7*d**2*(-b**3/a)**(3/2)/(a*d - b*c)**6 + 27*a**3*b**2*c**2*d**3*sqrt(-b**3/a)/(a*d - b*c)
**2 + 20*a**2*b**5*c**8*d*(-b**3/a)**(3/2)/(a*d - b*c)**6 - 27*a**2*b**3*c**3*d**2*sqrt(-b**3/a)/(a*d - b*c)**
2 - 8*a*b**6*c**9*(-b**3/a)**(3/2)/(a*d - b*c)**6 - 8*b**5*c**5*sqrt(-b**3/a)/(a*d - b*c)**2)/(a**2*b**2*d**3
- 7*a*b**3*c*d**2 + 12*b**4*c**2*d))/(2*(a*d - b*c)**2) - sqrt(-d/c**3)*(a*d - 3*b*c)*log(x + (-a**7*c**3*d**6
*(-d/c**3)**(3/2)*(a*d - 3*b*c)**3/(2*(a*d - b*c)**6) + 7*a**6*b*c**4*d**5*(-d/c**3)**(3/2)*(a*d - 3*b*c)**3/(
2*(a*d - b*c)**6) - 8*a**5*b**2*c**5*d**4*(-d/c**3)**(3/2)*(a*d - 3*b*c)**3/(a*d - b*c)**6 - a**5*d**5*sqrt(-d
/c**3)*(a*d - 3*b*c)/(2*(a*d - b*c)**2) + 7*a**4*b**3*c**6*d**3*(-d/c**3)**(3/2)*(a*d - 3*b*c)**3/(a*d - b*c)*
*6 + 9*a**4*b*c*d**4*sqrt(-d/c**3)*(a*d - 3*b*c)/(2*(a*d - b*c)**2) - a**3*b**4*c**7*d**2*(-d/c**3)**(3/2)*(a*
d - 3*b*c)**3/(2*(a*d - b*c)**6) - 27*a**3*b**2*c**2*d**3*sqrt(-d/c**3)*(a*d - 3*b*c)/(2*(a*d - b*c)**2) - 5*a
**2*b**5*c**8*d*(-d/c**3)**(3/2)*(a*d - 3*b*c)**3/(2*(a*d - b*c)**6) + 27*a**2*b**3*c**3*d**2*sqrt(-d/c**3)*(a
*d - 3*b*c)/(2*(a*d - b*c)**2) + a*b**6*c**9*(-d/c**3)**(3/2)*(a*d - 3*b*c)**3/(a*d - b*c)**6 + 4*b**5*c**5*sq
rt(-d/c**3)*(a*d - 3*b*c)/(a*d - b*c)**2)/(a**2*b**2*d**3 - 7*a*b**3*c*d**2 + 12*b**4*c**2*d))/(4*(a*d - b*c)*
*2) + sqrt(-d/c**3)*(a*d - 3*b*c)*log(x + (a**7*c**3*d**6*(-d/c**3)**(3/2)*(a*d - 3*b*c)**3/(2*(a*d - b*c)**6)
- 7*a**6*b*c**4*d**5*(-d/c**3)**(3/2)*(a*d - 3*b*c)**3/(2*(a*d - b*c)**6) + 8*a**5*b**2*c**5*d**4*(-d/c**3)**
(3/2)*(a*d - 3*b*c)**3/(a*d - b*c)**6 + a**5*d**5*sqrt(-d/c**3)*(a*d - 3*b*c)/(2*(a*d - b*c)**2) - 7*a**4*b**3
*c**6*d**3*(-d/c**3)**(3/2)*(a*d - 3*b*c)**3/(a*d - b*c)**6 - 9*a**4*b*c*d**4*sqrt(-d/c**3)*(a*d - 3*b*c)/(2*(
a*d - b*c)**2) + a**3*b**4*c**7*d**2*(-d/c**3)**(3/2)*(a*d - 3*b*c)**3/(2*(a*d - b*c)**6) + 27*a**3*b**2*c**2*
d**3*sqrt(-d/c**3)*(a*d - 3*b*c)/(2*(a*d - b*c)**2) + 5*a**2*b**5*c**8*d*(-d/c**3)**(3/2)*(a*d - 3*b*c)**3/(2*
(a*d - b*c)**6) - 27*a**2*b**3*c**3*d**2*sqrt(-d/c**3)*(a*d - 3*b*c)/(2*(a*d - b*c)**2) - a*b**6*c**9*(-d/c**3
)**(3/2)*(a*d - 3*b*c)**3/(a*d - b*c)**6 - 4*b**5*c**5*sqrt(-d/c**3)*(a*d - 3*b*c)/(a*d - b*c)**2)/(a**2*b**2*
d**3 - 7*a*b**3*c*d**2 + 12*b**4*c**2*d))/(4*(a*d - b*c)**2)
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Giac [A] time = 1.1655, size = 165, normalized size = 1.51 \begin{align*} \frac{b^{2} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt{a b}} - \frac{{\left (3 \, b c d - a d^{2}\right )} \arctan \left (\frac{d x}{\sqrt{c d}}\right )}{2 \,{\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} \sqrt{c d}} - \frac{d x}{2 \,{\left (b c^{2} - a c d\right )}{\left (d x^{2} + c\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(b*x^2+a)/(d*x^2+c)^2,x, algorithm="giac")
[Out]
b^2*arctan(b*x/sqrt(a*b))/((b^2*c^2 - 2*a*b*c*d + a^2*d^2)*sqrt(a*b)) - 1/2*(3*b*c*d - a*d^2)*arctan(d*x/sqrt(
c*d))/((b^2*c^3 - 2*a*b*c^2*d + a^2*c*d^2)*sqrt(c*d)) - 1/2*d*x/((b*c^2 - a*c*d)*(d*x^2 + c))