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

Optimal. Leaf size=55 \[ -\frac{a^2}{c x}-\frac{(b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{c^{3/2} d^{3/2}}+\frac{b^2 x}{d} \]

[Out]

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

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Rubi [A]  time = 0.115327, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ -\frac{a^2}{c x}-\frac{(b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{c^{3/2} d^{3/2}}+\frac{b^2 x}{d} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{a^{2}}{c x} + \frac{\int b^{2}\, dx}{d} - \frac{\left (a d - b c\right )^{2} \operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{c^{\frac{3}{2}} d^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0739474, size = 55, normalized size = 1. \[ -\frac{a^2}{c x}-\frac{(b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{c^{3/2} d^{3/2}}+\frac{b^2 x}{d} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.007, size = 85, normalized size = 1.6 \[{\frac{{b}^{2}x}{d}}-{\frac{{a}^{2}d}{c}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}+2\,{\frac{ab}{\sqrt{cd}}\arctan \left ({\frac{dx}{\sqrt{cd}}} \right ) }-{\frac{{b}^{2}c}{d}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-{\frac{{a}^{2}}{cx}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b^2*x/d-1/c*d/(c*d)^(1/2)*arctan(x*d/(c*d)^(1/2))*a^2+2/(c*d)^(1/2)*arctan(x*d/(
c*d)^(1/2))*a*b-c/d/(c*d)^(1/2)*arctan(x*d/(c*d)^(1/2))*b^2-a^2/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((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 = 0.231899, size = 1, normalized size = 0.02 \[ \left [\frac{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x \log \left (-\frac{2 \, c d x -{\left (d x^{2} - c\right )} \sqrt{-c d}}{d x^{2} + c}\right ) + 2 \,{\left (b^{2} c x^{2} - a^{2} d\right )} \sqrt{-c d}}{2 \, \sqrt{-c d} c d x}, -\frac{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x \arctan \left (\frac{\sqrt{c d} x}{c}\right ) -{\left (b^{2} c x^{2} - a^{2} d\right )} \sqrt{c d}}{\sqrt{c d} c d x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 2.88532, size = 165, normalized size = 3. \[ - \frac{a^{2}}{c x} + \frac{b^{2} x}{d} + \frac{\sqrt{- \frac{1}{c^{3} d^{3}}} \left (a d - b c\right )^{2} \log{\left (- \frac{c^{2} d \sqrt{- \frac{1}{c^{3} d^{3}}} \left (a d - b c\right )^{2}}{a^{2} d^{2} - 2 a b c d + b^{2} c^{2}} + x \right )}}{2} - \frac{\sqrt{- \frac{1}{c^{3} d^{3}}} \left (a d - b c\right )^{2} \log{\left (\frac{c^{2} d \sqrt{- \frac{1}{c^{3} d^{3}}} \left (a d - b c\right )^{2}}{a^{2} d^{2} - 2 a b c d + b^{2} c^{2}} + x \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.234251, size = 85, normalized size = 1.55 \[ \frac{b^{2} x}{d} - \frac{a^{2}}{c x} - \frac{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \arctan \left (\frac{d x}{\sqrt{c d}}\right )}{\sqrt{c d} c d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b^2*x/d - a^2/(c*x) - (b^2*c^2 - 2*a*b*c*d + a^2*d^2)*arctan(d*x/sqrt(c*d))/(sqr
t(c*d)*c*d)