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

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]])/(Sqr
t[c]*(b*c - a*d)^2)

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Rubi [A]  time = 0.195759, 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 \[ \frac{\sqrt{b} (b c-3 a d) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{3/2} (b c-a d)^2}+\frac{d^{3/2} \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{\sqrt{c} (b c-a d)^2}+\frac{b x}{2 a \left (a+b x^2\right ) (b c-a d)} \]

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]])/(Sqr
t[c]*(b*c - a*d)^2)

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Rubi in Sympy [A]  time = 41.6443, size = 94, normalized size = 0.87 \[ \frac{d^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{\sqrt{c} \left (a d - b c\right )^{2}} - \frac{b x}{2 a \left (a + b x^{2}\right ) \left (a d - b c\right )} - \frac{\sqrt{b} \left (3 a d - b c\right ) \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2 a^{\frac{3}{2}} \left (a d - b c\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x**2+a)**2/(d*x**2+c),x)

[Out]

d**(3/2)*atan(sqrt(d)*x/sqrt(c))/(sqrt(c)*(a*d - b*c)**2) - b*x/(2*a*(a + b*x**2
)*(a*d - b*c)) - sqrt(b)*(3*a*d - b*c)*atan(sqrt(b)*x/sqrt(a))/(2*a**(3/2)*(a*d
- b*c)**2)

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Mathematica [A]  time = 0.235244, 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., size = 144, normalized size = 1.3 \[{\frac{{d}^{2}}{ \left ( ad-bc \right ) ^{2}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-{\frac{bxd}{2\, \left ( ad-bc \right ) ^{2} \left ( b{x}^{2}+a \right ) }}+{\frac{x{b}^{2}c}{2\, \left ( ad-bc \right ) ^{2}a \left ( b{x}^{2}+a \right ) }}-{\frac{3\,bd}{2\, \left ( ad-bc \right ) ^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{{b}^{2}c}{2\, \left ( ad-bc \right ) ^{2}a}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

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*x/a/(b*x^2+a)*c-3/2*b/(a*d-b*c)^2/(a*b)^(1/2)*arctan(x*b
/(a*b)^(1/2))*d+1/2*b^2/(a*d-b*c)^2/a/(a*b)^(1/2)*arctan(x*b/(a*b)^(1/2))*c

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

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 = 0.346089, size = 1, normalized size = 0.01 \[ \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 (\frac{d x}{c \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 (\frac{b x}{a \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 (\frac{b x}{a \sqrt{\frac{b}{a}}}\right ) + 2 \,{\left (a b d x^{2} + a^{2} d\right )} \sqrt{\frac{d}{c}} \arctan \left (\frac{d x}{c \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 ] \]

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*s
qrt(-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(d*x/(c*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(b*x
/(a*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)*sqrt(b/a)*arctan(b*x/(a*sqrt(b/a))) + 2*(a*b*d*x^2 + a^2*d)*sqrt(d/c)*
arctan(d*x/(c*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 [A]  time = 47.2932, size = 2033, normalized size = 18.82 \[ \text{result too large to display} \]

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*s
qrt(-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*sq
rt(-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/XCAS [A]  time = 0.235098, size = 163, normalized size = 1.51 \[ \frac{d^{2} \arctan \left (\frac{d x}{\sqrt{c d}}\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt{c d}} + \frac{{\left (b^{2} c - 3 \, a b d\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \,{\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )} \sqrt{a b}} + \frac{b x}{2 \,{\left (a b c - a^{2} d\right )}{\left (b x^{2} + a\right )}} \]

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))