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

Optimal. Leaf size=43 \[ -\frac{(b c-a d) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{3/2} \sqrt{b}}-\frac{c}{a x} \]

[Out]

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

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Rubi [A]  time = 0.0643476, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ -\frac{(b c-a d) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{3/2} \sqrt{b}}-\frac{c}{a x} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 9.92165, size = 34, normalized size = 0.79 \[ - \frac{c}{a x} + \frac{\left (a d - b c\right ) \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{a^{\frac{3}{2}} \sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0424333, size = 42, normalized size = 0.98 \[ \frac{(a d-b c) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{3/2} \sqrt{b}}-\frac{c}{a x} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.005, size = 48, normalized size = 1.1 \[{d\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{bc}{a}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{c}{ax}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x^2+c)/x^2/(b*x^2+a),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.246497, size = 1, normalized size = 0.02 \[ \left [-\frac{{\left (b c - a d\right )} x \log \left (\frac{2 \, a b x +{\left (b x^{2} - a\right )} \sqrt{-a b}}{b x^{2} + a}\right ) + 2 \, \sqrt{-a b} c}{2 \, \sqrt{-a b} a x}, -\frac{{\left (b c - a d\right )} x \arctan \left (\frac{\sqrt{a b} x}{a}\right ) + \sqrt{a b} c}{\sqrt{a b} a x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.82574, size = 82, normalized size = 1.91 \[ - \frac{\sqrt{- \frac{1}{a^{3} b}} \left (a d - b c\right ) \log{\left (- a^{2} \sqrt{- \frac{1}{a^{3} b}} + x \right )}}{2} + \frac{\sqrt{- \frac{1}{a^{3} b}} \left (a d - b c\right ) \log{\left (a^{2} \sqrt{- \frac{1}{a^{3} b}} + x \right )}}{2} - \frac{c}{a x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.230791, size = 50, normalized size = 1.16 \[ -\frac{{\left (b c - a d\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{\sqrt{a b} a} - \frac{c}{a x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(b*c - a*d)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a) - c/(a*x)