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

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

[Out]

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

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Rubi [A]  time = 0.530002, 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 \[ -\frac{b^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{3/2} (b c-a d)^2}+\frac{d^{3/2} (5 b c-3 a d) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{2 c^{5/2} (b c-a d)^2}-\frac{2 b c-3 a d}{2 a c^2 x (b c-a d)}-\frac{d}{2 c x \left (c+d x^2\right ) (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 102.365, size = 121, normalized size = 0.84 \[ \frac{d}{2 c x \left (c + d x^{2}\right ) \left (a d - b c\right )} - \frac{d^{\frac{3}{2}} \left (3 a d - 5 b c\right ) \operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{2 c^{\frac{5}{2}} \left (a d - b c\right )^{2}} - \frac{3 a d - 2 b c}{2 a c^{2} x \left (a d - b c\right )} - \frac{b^{\frac{5}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{a^{\frac{3}{2}} \left (a d - b c\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.480483, size = 123, normalized size = 0.85 \[ -\frac{b^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{3/2} (a d-b c)^2}+\frac{d^{3/2} (5 b c-3 a d) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{2 c^{5/2} (b c-a d)^2}+\frac{d^2 x}{2 c^2 \left (c+d x^2\right ) (b c-a d)}-\frac{1}{a c^2 x} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.02, size = 169, normalized size = 1.2 \[ -{\frac{1}{a{c}^{2}x}}-{\frac{{d}^{3}xa}{2\,{c}^{2} \left ( ad-bc \right ) ^{2} \left ( d{x}^{2}+c \right ) }}+{\frac{{d}^{2}xb}{2\,c \left ( ad-bc \right ) ^{2} \left ( d{x}^{2}+c \right ) }}-{\frac{3\,a{d}^{3}}{2\,{c}^{2} \left ( ad-bc \right ) ^{2}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}+{\frac{5\,b{d}^{2}}{2\,c \left ( ad-bc \right ) ^{2}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-{\frac{{b}^{3}}{a \left ( ad-bc \right ) ^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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)^2*x^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.614686, 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)*(d*x^2 + c)^2*x^2),x, algorithm="fricas")

[Out]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.283047, size = 221, normalized size = 1.53 \[ -\frac{b^{3} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{{\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )} \sqrt{a b}} + \frac{{\left (5 \, b c d^{2} - 3 \, a d^{3}\right )} \arctan \left (\frac{d x}{\sqrt{c d}}\right )}{2 \,{\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2}\right )} \sqrt{c d}} - \frac{2 \, b c d x^{2} - 3 \, a d^{2} x^{2} + 2 \, b c^{2} - 2 \, a c d}{2 \,{\left (a b c^{3} - a^{2} c^{2} d\right )}{\left (d x^{3} + c x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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