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

Optimal. Leaf size=144 \[ -\frac{b^{3/2} (3 b c-5 a d) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{5/2} (b c-a d)^2}-\frac{3 b c-2 a d}{2 a^2 c x (b c-a d)}-\frac{d^{5/2} \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{c^{3/2} (b c-a d)^2}+\frac{b}{2 a x \left (a+b x^2\right ) (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 0.516638, antiderivative size = 144, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{b^{3/2} (3 b c-5 a d) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{5/2} (b c-a d)^2}-\frac{3 b c-2 a d}{2 a^2 c x (b c-a d)}-\frac{d^{5/2} \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{c^{3/2} (b c-a d)^2}+\frac{b}{2 a x \left (a+b x^2\right ) (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 94.9558, size = 121, normalized size = 0.84 \[ - \frac{d^{\frac{5}{2}} \operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{c^{\frac{3}{2}} \left (a d - b c\right )^{2}} - \frac{b}{2 a x \left (a + b x^{2}\right ) \left (a d - b c\right )} - \frac{2 a d - 3 b c}{2 a^{2} c x \left (a d - b c\right )} + \frac{b^{\frac{3}{2}} \left (5 a d - 3 b c\right ) \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2 a^{\frac{5}{2}} \left (a d - b c\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-d**(5/2)*atan(sqrt(d)*x/sqrt(c))/(c**(3/2)*(a*d - b*c)**2) - b/(2*a*x*(a + b*x*
*2)*(a*d - b*c)) - (2*a*d - 3*b*c)/(2*a**2*c*x*(a*d - b*c)) + b**(3/2)*(5*a*d -
3*b*c)*atan(sqrt(b)*x/sqrt(a))/(2*a**(5/2)*(a*d - b*c)**2)

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Mathematica [A]  time = 0.321635, size = 123, normalized size = 0.85 \[ \frac{b^{3/2} (5 a d-3 b c) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{5/2} (a d-b c)^2}+\frac{b^2 x}{2 a^2 \left (a+b x^2\right ) (a d-b c)}-\frac{1}{a^2 c x}-\frac{d^{5/2} \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{c^{3/2} (b c-a d)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.019, size = 169, normalized size = 1.2 \[ -{\frac{1}{{a}^{2}cx}}-{\frac{{d}^{3}}{c \left ( ad-bc \right ) ^{2}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}+{\frac{{b}^{2}xd}{2\,a \left ( ad-bc \right ) ^{2} \left ( b{x}^{2}+a \right ) }}-{\frac{{b}^{3}xc}{2\,{a}^{2} \left ( ad-bc \right ) ^{2} \left ( b{x}^{2}+a \right ) }}+{\frac{5\,{b}^{2}d}{2\,a \left ( ad-bc \right ) ^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{3\,{b}^{3}c}{2\,{a}^{2} \left ( ad-bc \right ) ^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.620206, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/4*(4*a*b^2*c^2 - 8*a^2*b*c*d + 4*a^3*d^2 + 2*(3*b^3*c^2 - 5*a*b^2*c*d + 2*a^
2*b*d^2)*x^2 + ((3*b^3*c^2 - 5*a*b^2*c*d)*x^3 + (3*a*b^2*c^2 - 5*a^2*b*c*d)*x)*s
qrt(-b/a)*log((b*x^2 + 2*a*x*sqrt(-b/a) - a)/(b*x^2 + a)) - 2*(a^2*b*d^2*x^3 + a
^3*d^2*x)*sqrt(-d/c)*log((d*x^2 - 2*c*x*sqrt(-d/c) - c)/(d*x^2 + c)))/((a^2*b^3*
c^3 - 2*a^3*b^2*c^2*d + a^4*b*c*d^2)*x^3 + (a^3*b^2*c^3 - 2*a^4*b*c^2*d + a^5*c*
d^2)*x), -1/4*(4*a*b^2*c^2 - 8*a^2*b*c*d + 4*a^3*d^2 + 2*(3*b^3*c^2 - 5*a*b^2*c*
d + 2*a^2*b*d^2)*x^2 + 4*(a^2*b*d^2*x^3 + a^3*d^2*x)*sqrt(d/c)*arctan(d*x/(c*sqr
t(d/c))) + ((3*b^3*c^2 - 5*a*b^2*c*d)*x^3 + (3*a*b^2*c^2 - 5*a^2*b*c*d)*x)*sqrt(
-b/a)*log((b*x^2 + 2*a*x*sqrt(-b/a) - a)/(b*x^2 + a)))/((a^2*b^3*c^3 - 2*a^3*b^2
*c^2*d + a^4*b*c*d^2)*x^3 + (a^3*b^2*c^3 - 2*a^4*b*c^2*d + a^5*c*d^2)*x), -1/2*(
2*a*b^2*c^2 - 4*a^2*b*c*d + 2*a^3*d^2 + (3*b^3*c^2 - 5*a*b^2*c*d + 2*a^2*b*d^2)*
x^2 + ((3*b^3*c^2 - 5*a*b^2*c*d)*x^3 + (3*a*b^2*c^2 - 5*a^2*b*c*d)*x)*sqrt(b/a)*
arctan(b*x/(a*sqrt(b/a))) - (a^2*b*d^2*x^3 + a^3*d^2*x)*sqrt(-d/c)*log((d*x^2 -
2*c*x*sqrt(-d/c) - c)/(d*x^2 + c)))/((a^2*b^3*c^3 - 2*a^3*b^2*c^2*d + a^4*b*c*d^
2)*x^3 + (a^3*b^2*c^3 - 2*a^4*b*c^2*d + a^5*c*d^2)*x), -1/2*(2*a*b^2*c^2 - 4*a^2
*b*c*d + 2*a^3*d^2 + (3*b^3*c^2 - 5*a*b^2*c*d + 2*a^2*b*d^2)*x^2 + ((3*b^3*c^2 -
5*a*b^2*c*d)*x^3 + (3*a*b^2*c^2 - 5*a^2*b*c*d)*x)*sqrt(b/a)*arctan(b*x/(a*sqrt(
b/a))) + 2*(a^2*b*d^2*x^3 + a^3*d^2*x)*sqrt(d/c)*arctan(d*x/(c*sqrt(d/c))))/((a^
2*b^3*c^3 - 2*a^3*b^2*c^2*d + a^4*b*c*d^2)*x^3 + (a^3*b^2*c^3 - 2*a^4*b*c^2*d +
a^5*c*d^2)*x)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.234968, size = 221, normalized size = 1.53 \[ -\frac{d^{3} \arctan \left (\frac{d x}{\sqrt{c d}}\right )}{{\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} \sqrt{c d}} - \frac{{\left (3 \, b^{3} c - 5 \, a b^{2} d\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \,{\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2}\right )} \sqrt{a b}} - \frac{3 \, b^{2} c x^{2} - 2 \, a b d x^{2} + 2 \, a b c - 2 \, a^{2} d}{2 \,{\left (a^{2} b c^{2} - a^{3} c d\right )}{\left (b x^{3} + a x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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