Optimal. Leaf size=40 \[ \frac{b x}{d}-\frac{(b c-a d) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{\sqrt{c} d^{3/2}} \]
[Out]
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Rubi [A] time = 0.0524135, antiderivative size = 40, 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 \[ \frac{b x}{d}-\frac{(b c-a d) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{\sqrt{c} d^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x^2)/(c + d*x^2),x]
[Out]
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Rubi in Sympy [A] time = 8.76221, size = 34, normalized size = 0.85 \[ \frac{b x}{d} + \frac{\left (a d - b c\right ) \operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{\sqrt{c} d^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**2+a)/(d*x**2+c),x)
[Out]
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Mathematica [A] time = 0.038996, size = 40, normalized size = 1. \[ \frac{b x}{d}-\frac{(b c-a d) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{\sqrt{c} d^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x^2)/(c + d*x^2),x]
[Out]
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Maple [A] time = 0.008, size = 45, normalized size = 1.1 \[{\frac{bx}{d}}+{a\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-{\frac{bc}{d}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^2+a)/(d*x^2+c),x)
[Out]
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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)/(d*x^2 + c),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.206565, size = 1, normalized size = 0.02 \[ \left [\frac{2 \, \sqrt{-c d} b x -{\left (b c - a d\right )} \log \left (\frac{2 \, c d x +{\left (d x^{2} - c\right )} \sqrt{-c d}}{d x^{2} + c}\right )}{2 \, \sqrt{-c d} d}, \frac{\sqrt{c d} b x -{\left (b c - a d\right )} \arctan \left (\frac{\sqrt{c d} x}{c}\right )}{\sqrt{c d} d}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)/(d*x^2 + c),x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.56986, size = 82, normalized size = 2.05 \[ \frac{b x}{d} - \frac{\sqrt{- \frac{1}{c d^{3}}} \left (a d - b c\right ) \log{\left (- c d \sqrt{- \frac{1}{c d^{3}}} + x \right )}}{2} + \frac{\sqrt{- \frac{1}{c d^{3}}} \left (a d - b c\right ) \log{\left (c d \sqrt{- \frac{1}{c d^{3}}} + x \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**2+a)/(d*x**2+c),x)
[Out]
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GIAC/XCAS [A] time = 0.230823, size = 46, normalized size = 1.15 \[ \frac{b x}{d} - \frac{{\left (b c - a d\right )} \arctan \left (\frac{d x}{\sqrt{c d}}\right )}{\sqrt{c d} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)/(d*x^2 + c),x, algorithm="giac")
[Out]