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

Optimal. Leaf size=106 \[ -\frac{x \left (3 a^2 d^2-2 a b c d+b^2 c^2\right )}{2 c^2 d \left (c+d x^2\right )}-\frac{a^2}{c x \left (c+d x^2\right )}+\frac{(b c-a d) (3 a d+b c) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{2 c^{5/2} d^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.190138, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ -\frac{x \left (3 a^2 d^2-2 a b c d+b^2 c^2\right )}{2 c^2 d \left (c+d x^2\right )}-\frac{a^2}{c x \left (c+d x^2\right )}+\frac{(b c-a d) (3 a d+b c) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{2 c^{5/2} d^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 24.3111, size = 90, normalized size = 0.85 \[ - \frac{a^{2}}{c x \left (c + d x^{2}\right )} - \frac{x \left (a d \left (3 a d - 2 b c\right ) + b^{2} c^{2}\right )}{2 c^{2} d \left (c + d x^{2}\right )} - \frac{\left (a d - b c\right ) \left (3 a d + b c\right ) \operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{2 c^{\frac{5}{2}} d^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0984207, size = 91, normalized size = 0.86 \[ \frac{\left (-3 a^2 d^2+2 a b c d+b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{2 c^{5/2} d^{3/2}}-\frac{a^2}{c^2 x}-\frac{x (b c-a d)^2}{2 c^2 d \left (c+d x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.016, size = 131, normalized size = 1.2 \[ -{\frac{{a}^{2}}{{c}^{2}x}}-{\frac{x{a}^{2}d}{2\,{c}^{2} \left ( d{x}^{2}+c \right ) }}+{\frac{abx}{c \left ( d{x}^{2}+c \right ) }}-{\frac{x{b}^{2}}{2\,d \left ( d{x}^{2}+c \right ) }}-{\frac{3\,{a}^{2}d}{2\,{c}^{2}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}+{\frac{ab}{c}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}+{\frac{{b}^{2}}{2\,d}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.240258, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left ({\left (b^{2} c^{2} d + 2 \, a b c d^{2} - 3 \, a^{2} d^{3}\right )} x^{3} +{\left (b^{2} c^{3} + 2 \, a b c^{2} d - 3 \, a^{2} c d^{2}\right )} x\right )} \log \left (-\frac{2 \, c d x -{\left (d x^{2} - c\right )} \sqrt{-c d}}{d x^{2} + c}\right ) + 2 \,{\left (2 \, a^{2} c d +{\left (b^{2} c^{2} - 2 \, a b c d + 3 \, a^{2} d^{2}\right )} x^{2}\right )} \sqrt{-c d}}{4 \,{\left (c^{2} d^{2} x^{3} + c^{3} d x\right )} \sqrt{-c d}}, \frac{{\left ({\left (b^{2} c^{2} d + 2 \, a b c d^{2} - 3 \, a^{2} d^{3}\right )} x^{3} +{\left (b^{2} c^{3} + 2 \, a b c^{2} d - 3 \, a^{2} c d^{2}\right )} x\right )} \arctan \left (\frac{\sqrt{c d} x}{c}\right ) -{\left (2 \, a^{2} c d +{\left (b^{2} c^{2} - 2 \, a b c d + 3 \, a^{2} d^{2}\right )} x^{2}\right )} \sqrt{c d}}{2 \,{\left (c^{2} d^{2} x^{3} + c^{3} d x\right )} \sqrt{c d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 3.97024, size = 238, normalized size = 2.25 \[ \frac{\sqrt{- \frac{1}{c^{5} d^{3}}} \left (a d - b c\right ) \left (3 a d + b c\right ) \log{\left (- \frac{c^{3} d \sqrt{- \frac{1}{c^{5} d^{3}}} \left (a d - b c\right ) \left (3 a d + b c\right )}{3 a^{2} d^{2} - 2 a b c d - b^{2} c^{2}} + x \right )}}{4} - \frac{\sqrt{- \frac{1}{c^{5} d^{3}}} \left (a d - b c\right ) \left (3 a d + b c\right ) \log{\left (\frac{c^{3} d \sqrt{- \frac{1}{c^{5} d^{3}}} \left (a d - b c\right ) \left (3 a d + b c\right )}{3 a^{2} d^{2} - 2 a b c d - b^{2} c^{2}} + x \right )}}{4} - \frac{2 a^{2} c d + x^{2} \left (3 a^{2} d^{2} - 2 a b c d + b^{2} c^{2}\right )}{2 c^{3} d x + 2 c^{2} d^{2} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.224013, size = 138, normalized size = 1.3 \[ \frac{{\left (b^{2} c^{2} + 2 \, a b c d - 3 \, a^{2} d^{2}\right )} \arctan \left (\frac{d x}{\sqrt{c d}}\right )}{2 \, \sqrt{c d} c^{2} d} - \frac{b^{2} c^{2} x^{2} - 2 \, a b c d x^{2} + 3 \, a^{2} d^{2} x^{2} + 2 \, a^{2} c d}{2 \,{\left (d x^{3} + c x\right )} c^{2} d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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