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

Optimal. Leaf size=71 \[ -\frac{(3 b c-a d) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{5/2} \sqrt{b}}-\frac{x (b c-a d)}{2 a^2 \left (a+b x^2\right )}-\frac{c}{a^2 x} \]

[Out]

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

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Rubi [A]  time = 0.145078, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ -\frac{(3 b c-a d) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{5/2} \sqrt{b}}-\frac{x (b c-a d)}{2 a^2 \left (a+b x^2\right )}-\frac{c}{a^2 x} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 20.0822, size = 60, normalized size = 0.85 \[ - \frac{c}{a^{2} x} + \frac{x \left (a d - b c\right )}{2 a^{2} \left (a + b x^{2}\right )} + \frac{\left (a d - 3 b c\right ) \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2 a^{\frac{5}{2}} \sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0553049, size = 70, normalized size = 0.99 \[ \frac{(a d-3 b c) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{5/2} \sqrt{b}}+\frac{x (a d-b c)}{2 a^2 \left (a+b x^2\right )}-\frac{c}{a^2 x} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.014, size = 85, normalized size = 1.2 \[ -{\frac{c}{{a}^{2}x}}+{\frac{dx}{2\,a \left ( b{x}^{2}+a \right ) }}-{\frac{bcx}{2\,{a}^{2} \left ( b{x}^{2}+a \right ) }}+{\frac{d}{2\,a}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{3\,bc}{2\,{a}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.241225, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left ({\left (3 \, b^{2} c - a b d\right )} x^{3} +{\left (3 \, a b c - a^{2} d\right )} x\right )} \log \left (\frac{2 \, a b x +{\left (b x^{2} - a\right )} \sqrt{-a b}}{b x^{2} + a}\right ) + 2 \,{\left ({\left (3 \, b c - a d\right )} x^{2} + 2 \, a c\right )} \sqrt{-a b}}{4 \,{\left (a^{2} b x^{3} + a^{3} x\right )} \sqrt{-a b}}, -\frac{{\left ({\left (3 \, b^{2} c - a b d\right )} x^{3} +{\left (3 \, a b c - a^{2} d\right )} x\right )} \arctan \left (\frac{\sqrt{a b} x}{a}\right ) +{\left ({\left (3 \, b c - a d\right )} x^{2} + 2 \, a c\right )} \sqrt{a b}}{2 \,{\left (a^{2} b x^{3} + a^{3} x\right )} \sqrt{a b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 2.49772, size = 114, normalized size = 1.61 \[ - \frac{\sqrt{- \frac{1}{a^{5} b}} \left (a d - 3 b c\right ) \log{\left (- a^{3} \sqrt{- \frac{1}{a^{5} b}} + x \right )}}{4} + \frac{\sqrt{- \frac{1}{a^{5} b}} \left (a d - 3 b c\right ) \log{\left (a^{3} \sqrt{- \frac{1}{a^{5} b}} + x \right )}}{4} + \frac{- 2 a c + x^{2} \left (a d - 3 b c\right )}{2 a^{3} x + 2 a^{2} b x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.254733, size = 86, normalized size = 1.21 \[ -\frac{{\left (3 \, b c - a d\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} a^{2}} - \frac{3 \, b c x^{2} - a d x^{2} + 2 \, a c}{2 \,{\left (b x^{3} + a x\right )} a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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