Optimal. Leaf size=36 \[ -\frac{\sqrt{b} c \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{3/2}}-\frac{c}{a x} \]
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Rubi [A] time = 0.0412215, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13 \[ -\frac{\sqrt{b} c \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{3/2}}-\frac{c}{a x} \]
Antiderivative was successfully verified.
[In] Int[(a*c + b*c*x^2)/(x^2*(a + b*x^2)^2),x]
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Rubi in Sympy [A] time = 9.88082, size = 31, normalized size = 0.86 \[ - \frac{c}{a x} - \frac{\sqrt{b} c \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{a^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*c*x**2+a*c)/x**2/(b*x**2+a)**2,x)
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Mathematica [A] time = 0.0219419, size = 36, normalized size = 1. \[ c \left (-\frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{3/2}}-\frac{1}{a x}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(a*c + b*c*x^2)/(x^2*(a + b*x^2)^2),x]
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Maple [A] time = 0.006, size = 32, normalized size = 0.9 \[ -{\frac{bc}{a}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{c}{ax}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*c*x^2+a*c)/x^2/(b*x^2+a)^2,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*c*x^2 + a*c)/((b*x^2 + a)^2*x^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.248306, size = 1, normalized size = 0.03 \[ \left [\frac{c x \sqrt{-\frac{b}{a}} \log \left (\frac{b x^{2} - 2 \, a x \sqrt{-\frac{b}{a}} - a}{b x^{2} + a}\right ) - 2 \, c}{2 \, a x}, -\frac{c x \sqrt{\frac{b}{a}} \arctan \left (\frac{b x}{a \sqrt{\frac{b}{a}}}\right ) + c}{a x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*c*x^2 + a*c)/((b*x^2 + a)^2*x^2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.32918, size = 66, normalized size = 1.83 \[ c \left (\frac{\sqrt{- \frac{b}{a^{3}}} \log{\left (- \frac{a^{2} \sqrt{- \frac{b}{a^{3}}}}{b} + x \right )}}{2} - \frac{\sqrt{- \frac{b}{a^{3}}} \log{\left (\frac{a^{2} \sqrt{- \frac{b}{a^{3}}}}{b} + x \right )}}{2} - \frac{1}{a x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*c*x**2+a*c)/x**2/(b*x**2+a)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.221115, size = 42, normalized size = 1.17 \[ -\frac{b c \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{\sqrt{a b} a} - \frac{c}{a x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*c*x^2 + a*c)/((b*x^2 + a)^2*x^2),x, algorithm="giac")
[Out]