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

Optimal. Leaf size=81 \[ -\frac{b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{3/2} (b c-a d)}+\frac{d^{3/2} \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{c^{3/2} (b c-a d)}-\frac{1}{a c x} \]

[Out]

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

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Rubi [A]  time = 0.214359, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ -\frac{b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{3/2} (b c-a d)}+\frac{d^{3/2} \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{c^{3/2} (b c-a d)}-\frac{1}{a c x} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 42.5972, size = 66, normalized size = 0.81 \[ - \frac{d^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{c^{\frac{3}{2}} \left (a d - b c\right )} - \frac{1}{a c x} + \frac{b^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{a^{\frac{3}{2}} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-d**(3/2)*atan(sqrt(d)*x/sqrt(c))/(c**(3/2)*(a*d - b*c)) - 1/(a*c*x) + b**(3/2)*
atan(sqrt(b)*x/sqrt(a))/(a**(3/2)*(a*d - b*c))

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Mathematica [A]  time = 0.143093, size = 76, normalized size = 0.94 \[ \frac{-\frac{b^{3/2} x \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{3/2}}-\frac{b}{a}+\frac{d^{3/2} x \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{c^{3/2}}+\frac{d}{c}}{b c x-a d x} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.012, size = 76, normalized size = 0.9 \[ -{\frac{{d}^{2}}{c \left ( ad-bc \right ) }\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}+{\frac{{b}^{2}}{a \left ( ad-bc \right ) }\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{1}{acx}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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)*(d*x^2 + c)*x^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.257627, size = 1, normalized size = 0.01 \[ \left [-\frac{b c x \sqrt{-\frac{b}{a}} \log \left (\frac{b x^{2} + 2 \, a x \sqrt{-\frac{b}{a}} - a}{b x^{2} + a}\right ) + a d x \sqrt{-\frac{d}{c}} \log \left (\frac{d x^{2} - 2 \, c x \sqrt{-\frac{d}{c}} - c}{d x^{2} + c}\right ) + 2 \, b c - 2 \, a d}{2 \,{\left (a b c^{2} - a^{2} c d\right )} x}, \frac{2 \, a d x \sqrt{\frac{d}{c}} \arctan \left (\frac{d x}{c \sqrt{\frac{d}{c}}}\right ) - b c x \sqrt{-\frac{b}{a}} \log \left (\frac{b x^{2} + 2 \, a x \sqrt{-\frac{b}{a}} - a}{b x^{2} + a}\right ) - 2 \, b c + 2 \, a d}{2 \,{\left (a b c^{2} - a^{2} c d\right )} x}, -\frac{2 \, b c x \sqrt{\frac{b}{a}} \arctan \left (\frac{b x}{a \sqrt{\frac{b}{a}}}\right ) + a d x \sqrt{-\frac{d}{c}} \log \left (\frac{d x^{2} - 2 \, c x \sqrt{-\frac{d}{c}} - c}{d x^{2} + c}\right ) + 2 \, b c - 2 \, a d}{2 \,{\left (a b c^{2} - a^{2} c d\right )} x}, -\frac{b c x \sqrt{\frac{b}{a}} \arctan \left (\frac{b x}{a \sqrt{\frac{b}{a}}}\right ) - a d x \sqrt{\frac{d}{c}} \arctan \left (\frac{d x}{c \sqrt{\frac{d}{c}}}\right ) + b c - a d}{{\left (a b c^{2} - a^{2} c d\right )} x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/2*(b*c*x*sqrt(-b/a)*log((b*x^2 + 2*a*x*sqrt(-b/a) - a)/(b*x^2 + a)) + a*d*x*
sqrt(-d/c)*log((d*x^2 - 2*c*x*sqrt(-d/c) - c)/(d*x^2 + c)) + 2*b*c - 2*a*d)/((a*
b*c^2 - a^2*c*d)*x), 1/2*(2*a*d*x*sqrt(d/c)*arctan(d*x/(c*sqrt(d/c))) - b*c*x*sq
rt(-b/a)*log((b*x^2 + 2*a*x*sqrt(-b/a) - a)/(b*x^2 + a)) - 2*b*c + 2*a*d)/((a*b*
c^2 - a^2*c*d)*x), -1/2*(2*b*c*x*sqrt(b/a)*arctan(b*x/(a*sqrt(b/a))) + a*d*x*sqr
t(-d/c)*log((d*x^2 - 2*c*x*sqrt(-d/c) - c)/(d*x^2 + c)) + 2*b*c - 2*a*d)/((a*b*c
^2 - a^2*c*d)*x), -(b*c*x*sqrt(b/a)*arctan(b*x/(a*sqrt(b/a))) - a*d*x*sqrt(d/c)*
arctan(d*x/(c*sqrt(d/c))) + b*c - a*d)/((a*b*c^2 - a^2*c*d)*x)]

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Sympy [A]  time = 17.0518, size = 1093, normalized size = 13.49 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-b**3/a**3)*log(x + (-a**7*c**3*d**4*(-b**3/a**3)**(3/2)/(a*d - b*c)**3 +
2*a**6*b*c**4*d**3*(-b**3/a**3)**(3/2)/(a*d - b*c)**3 - 2*a**5*b**2*c**5*d**2*(-
b**3/a**3)**(3/2)/(a*d - b*c)**3 - a**5*d**5*sqrt(-b**3/a**3)/(a*d - b*c) + 2*a*
*4*b**3*c**6*d*(-b**3/a**3)**(3/2)/(a*d - b*c)**3 - a**3*b**4*c**7*(-b**3/a**3)*
*(3/2)/(a*d - b*c)**3 - b**5*c**5*sqrt(-b**3/a**3)/(a*d - b*c))/(a**2*b**2*d**4
+ a*b**3*c*d**3 + b**4*c**2*d**2))/(2*(a*d - b*c)) + sqrt(-b**3/a**3)*log(x + (a
**7*c**3*d**4*(-b**3/a**3)**(3/2)/(a*d - b*c)**3 - 2*a**6*b*c**4*d**3*(-b**3/a**
3)**(3/2)/(a*d - b*c)**3 + 2*a**5*b**2*c**5*d**2*(-b**3/a**3)**(3/2)/(a*d - b*c)
**3 + a**5*d**5*sqrt(-b**3/a**3)/(a*d - b*c) - 2*a**4*b**3*c**6*d*(-b**3/a**3)**
(3/2)/(a*d - b*c)**3 + a**3*b**4*c**7*(-b**3/a**3)**(3/2)/(a*d - b*c)**3 + b**5*
c**5*sqrt(-b**3/a**3)/(a*d - b*c))/(a**2*b**2*d**4 + a*b**3*c*d**3 + b**4*c**2*d
**2))/(2*(a*d - b*c)) - sqrt(-d**3/c**3)*log(x + (-a**7*c**3*d**4*(-d**3/c**3)**
(3/2)/(a*d - b*c)**3 + 2*a**6*b*c**4*d**3*(-d**3/c**3)**(3/2)/(a*d - b*c)**3 - 2
*a**5*b**2*c**5*d**2*(-d**3/c**3)**(3/2)/(a*d - b*c)**3 - a**5*d**5*sqrt(-d**3/c
**3)/(a*d - b*c) + 2*a**4*b**3*c**6*d*(-d**3/c**3)**(3/2)/(a*d - b*c)**3 - a**3*
b**4*c**7*(-d**3/c**3)**(3/2)/(a*d - b*c)**3 - b**5*c**5*sqrt(-d**3/c**3)/(a*d -
b*c))/(a**2*b**2*d**4 + a*b**3*c*d**3 + b**4*c**2*d**2))/(2*(a*d - b*c)) + sqrt
(-d**3/c**3)*log(x + (a**7*c**3*d**4*(-d**3/c**3)**(3/2)/(a*d - b*c)**3 - 2*a**6
*b*c**4*d**3*(-d**3/c**3)**(3/2)/(a*d - b*c)**3 + 2*a**5*b**2*c**5*d**2*(-d**3/c
**3)**(3/2)/(a*d - b*c)**3 + a**5*d**5*sqrt(-d**3/c**3)/(a*d - b*c) - 2*a**4*b**
3*c**6*d*(-d**3/c**3)**(3/2)/(a*d - b*c)**3 + a**3*b**4*c**7*(-d**3/c**3)**(3/2)
/(a*d - b*c)**3 + b**5*c**5*sqrt(-d**3/c**3)/(a*d - b*c))/(a**2*b**2*d**4 + a*b*
*3*c*d**3 + b**4*c**2*d**2))/(2*(a*d - b*c)) - 1/(a*c*x)

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GIAC/XCAS [A]  time = 0.299003, size = 520, normalized size = 6.42 \[ \frac{{\left (\sqrt{c d} a b^{2} c^{2}{\left | d \right |} + \sqrt{c d} a^{2} b c d{\left | d \right |} - \sqrt{c d} b{\left | a b c^{2} - a^{2} c d \right |}{\left | d \right |}\right )} \arctan \left (\frac{2 \, \sqrt{\frac{1}{2}} x}{\sqrt{\frac{a b c^{2} + a^{2} c d + \sqrt{-4 \, a^{3} b c^{3} d +{\left (a b c^{2} + a^{2} c d\right )}^{2}}}{a b c d}}}\right )}{a b c^{2} d{\left | a b c^{2} - a^{2} c d \right |} + a^{2} c d^{2}{\left | a b c^{2} - a^{2} c d \right |} +{\left (a b c^{2} - a^{2} c d\right )}^{2} d} - \frac{{\left (\sqrt{a b} a b c^{2} d{\left | b \right |} + \sqrt{a b} a^{2} c d^{2}{\left | b \right |} + \sqrt{a b} d{\left | a b c^{2} - a^{2} c d \right |}{\left | b \right |}\right )} \arctan \left (\frac{2 \, \sqrt{\frac{1}{2}} x}{\sqrt{\frac{a b c^{2} + a^{2} c d - \sqrt{-4 \, a^{3} b c^{3} d +{\left (a b c^{2} + a^{2} c d\right )}^{2}}}{a b c d}}}\right )}{a b^{2} c^{2}{\left | a b c^{2} - a^{2} c d \right |} + a^{2} b c d{\left | a b c^{2} - a^{2} c d \right |} -{\left (a b c^{2} - a^{2} c d\right )}^{2} b} - \frac{1}{a c x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(sqrt(c*d)*a*b^2*c^2*abs(d) + sqrt(c*d)*a^2*b*c*d*abs(d) - sqrt(c*d)*b*abs(a*b*c
^2 - a^2*c*d)*abs(d))*arctan(2*sqrt(1/2)*x/sqrt((a*b*c^2 + a^2*c*d + sqrt(-4*a^3
*b*c^3*d + (a*b*c^2 + a^2*c*d)^2))/(a*b*c*d)))/(a*b*c^2*d*abs(a*b*c^2 - a^2*c*d)
+ a^2*c*d^2*abs(a*b*c^2 - a^2*c*d) + (a*b*c^2 - a^2*c*d)^2*d) - (sqrt(a*b)*a*b*
c^2*d*abs(b) + sqrt(a*b)*a^2*c*d^2*abs(b) + sqrt(a*b)*d*abs(a*b*c^2 - a^2*c*d)*a
bs(b))*arctan(2*sqrt(1/2)*x/sqrt((a*b*c^2 + a^2*c*d - sqrt(-4*a^3*b*c^3*d + (a*b
*c^2 + a^2*c*d)^2))/(a*b*c*d)))/(a*b^2*c^2*abs(a*b*c^2 - a^2*c*d) + a^2*b*c*d*ab
s(a*b*c^2 - a^2*c*d) - (a*b*c^2 - a^2*c*d)^2*b) - 1/(a*c*x)