Optimal. Leaf size=55 \[ -\frac{(b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{3/2} b^{3/2}}-\frac{c^2}{a x}+\frac{d^2 x}{b} \]
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Rubi [A] time = 0.11726, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ -\frac{(b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{3/2} b^{3/2}}-\frac{c^2}{a x}+\frac{d^2 x}{b} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x^2)^2/(x^2*(a + b*x^2)),x]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ d^{2} \int \frac{1}{b}\, dx - \frac{c^{2}}{a x} - \frac{\left (a d - b c\right )^{2} \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{a^{\frac{3}{2}} b^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x**2+c)**2/x**2/(b*x**2+a),x)
[Out]
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Mathematica [A] time = 0.0937019, size = 55, normalized size = 1. \[ -\frac{(a d-b c)^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{3/2} b^{3/2}}-\frac{c^2}{a x}+\frac{d^2 x}{b} \]
Antiderivative was successfully verified.
[In] Integrate[(c + d*x^2)^2/(x^2*(a + b*x^2)),x]
[Out]
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Maple [A] time = 0.007, size = 85, normalized size = 1.6 \[{\frac{{d}^{2}x}{b}}-{\frac{a{d}^{2}}{b}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+2\,{\frac{cd}{\sqrt{ab}}\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ) }-{\frac{b{c}^{2}}{a}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{{c}^{2}}{ax}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x^2+c)^2/x^2/(b*x^2+a),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((d*x^2 + c)^2/((b*x^2 + a)*x^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.24246, size = 1, normalized size = 0.02 \[ \left [\frac{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x \log \left (-\frac{2 \, a b x -{\left (b x^{2} - a\right )} \sqrt{-a b}}{b x^{2} + a}\right ) + 2 \,{\left (a d^{2} x^{2} - b c^{2}\right )} \sqrt{-a b}}{2 \, \sqrt{-a b} a b x}, -\frac{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x \arctan \left (\frac{\sqrt{a b} x}{a}\right ) -{\left (a d^{2} x^{2} - b c^{2}\right )} \sqrt{a b}}{\sqrt{a b} a b x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)^2/((b*x^2 + a)*x^2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.76457, size = 165, normalized size = 3. \[ \frac{\sqrt{- \frac{1}{a^{3} b^{3}}} \left (a d - b c\right )^{2} \log{\left (- \frac{a^{2} b \sqrt{- \frac{1}{a^{3} b^{3}}} \left (a d - b c\right )^{2}}{a^{2} d^{2} - 2 a b c d + b^{2} c^{2}} + x \right )}}{2} - \frac{\sqrt{- \frac{1}{a^{3} b^{3}}} \left (a d - b c\right )^{2} \log{\left (\frac{a^{2} b \sqrt{- \frac{1}{a^{3} b^{3}}} \left (a d - b c\right )^{2}}{a^{2} d^{2} - 2 a b c d + b^{2} c^{2}} + x \right )}}{2} + \frac{d^{2} x}{b} - \frac{c^{2}}{a x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x**2+c)**2/x**2/(b*x**2+a),x)
[Out]
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GIAC/XCAS [A] time = 0.223369, size = 85, normalized size = 1.55 \[ \frac{d^{2} x}{b} - \frac{c^{2}}{a x} - \frac{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{\sqrt{a b} a b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)^2/((b*x^2 + a)*x^2),x, algorithm="giac")
[Out]