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

Optimal. Leaf size=39 \[ \frac{(b c-a d) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} b^{3/2}}+\frac{d x}{b} \]

[Out]

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

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Rubi [A]  time = 0.0475927, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{(b c-a d) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} b^{3/2}}+\frac{d x}{b} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x^2)/(a + b*x^2),x]

[Out]

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

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Rubi in Sympy [A]  time = 8.52485, size = 34, normalized size = 0.87 \[ \frac{d x}{b} - \frac{\left (a d - b c\right ) \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{\sqrt{a} b^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0418032, size = 40, normalized size = 1.03 \[ \frac{d x}{b}-\frac{(a d-b c) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} b^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.004, size = 45, normalized size = 1.2 \[{\frac{dx}{b}}-{\frac{ad}{b}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{c\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x^2+c)/(b*x^2+a),x)

[Out]

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

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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, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.210324, size = 1, normalized size = 0.03 \[ \left [\frac{2 \, \sqrt{-a b} d x -{\left (b c - a d\right )} \log \left (-\frac{2 \, a b x -{\left (b x^{2} - a\right )} \sqrt{-a b}}{b x^{2} + a}\right )}{2 \, \sqrt{-a b} b}, \frac{\sqrt{a b} d x +{\left (b c - a d\right )} \arctan \left (\frac{\sqrt{a b} x}{a}\right )}{\sqrt{a b} b}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.60144, size = 82, normalized size = 2.1 \[ \frac{\sqrt{- \frac{1}{a b^{3}}} \left (a d - b c\right ) \log{\left (- a b \sqrt{- \frac{1}{a b^{3}}} + x \right )}}{2} - \frac{\sqrt{- \frac{1}{a b^{3}}} \left (a d - b c\right ) \log{\left (a b \sqrt{- \frac{1}{a b^{3}}} + x \right )}}{2} + \frac{d x}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.23033, size = 45, normalized size = 1.15 \[ \frac{d x}{b} + \frac{{\left (b c - a d\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{\sqrt{a b} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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