Optimal. Leaf size=70 \[ \frac{\sqrt{c} \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{\sqrt{d} (b c-a d)}-\frac{\sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{b} (b c-a d)} \]
[Out]
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Rubi [A] time = 0.0979148, antiderivative size = 70, 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 \[ \frac{\sqrt{c} \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{\sqrt{d} (b c-a d)}-\frac{\sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{b} (b c-a d)} \]
Antiderivative was successfully verified.
[In] Int[x^2/((a + b*x^2)*(c + d*x^2)),x]
[Out]
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Rubi in Sympy [A] time = 19.0047, size = 60, normalized size = 0.86 \[ \frac{\sqrt{a} \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{\sqrt{b} \left (a d - b c\right )} - \frac{\sqrt{c} \operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{\sqrt{d} \left (a d - b c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2/(b*x**2+a)/(d*x**2+c),x)
[Out]
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Mathematica [A] time = 0.0688101, size = 61, normalized size = 0.87 \[ \frac{\frac{\sqrt{c} \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{\sqrt{d}}-\frac{\sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{b}}}{b c-a d} \]
Antiderivative was successfully verified.
[In] Integrate[x^2/((a + b*x^2)*(c + d*x^2)),x]
[Out]
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Maple [A] time = 0.008, size = 55, normalized size = 0.8 \[ -{\frac{c}{ad-bc}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}+{\frac{a}{ad-bc}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2/(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(x^2/((b*x^2 + a)*(d*x^2 + c)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.256699, size = 1, normalized size = 0.01 \[ \left [-\frac{\sqrt{-\frac{a}{b}} \log \left (\frac{b x^{2} + 2 \, b x \sqrt{-\frac{a}{b}} - a}{b x^{2} + a}\right ) + \sqrt{-\frac{c}{d}} \log \left (\frac{d x^{2} - 2 \, d x \sqrt{-\frac{c}{d}} - c}{d x^{2} + c}\right )}{2 \,{\left (b c - a d\right )}}, -\frac{2 \, \sqrt{\frac{a}{b}} \arctan \left (\frac{x}{\sqrt{\frac{a}{b}}}\right ) + \sqrt{-\frac{c}{d}} \log \left (\frac{d x^{2} - 2 \, d x \sqrt{-\frac{c}{d}} - c}{d x^{2} + c}\right )}{2 \,{\left (b c - a d\right )}}, \frac{2 \, \sqrt{\frac{c}{d}} \arctan \left (\frac{x}{\sqrt{\frac{c}{d}}}\right ) - \sqrt{-\frac{a}{b}} \log \left (\frac{b x^{2} + 2 \, b x \sqrt{-\frac{a}{b}} - a}{b x^{2} + a}\right )}{2 \,{\left (b c - a d\right )}}, -\frac{\sqrt{\frac{a}{b}} \arctan \left (\frac{x}{\sqrt{\frac{a}{b}}}\right ) - \sqrt{\frac{c}{d}} \arctan \left (\frac{x}{\sqrt{\frac{c}{d}}}\right )}{b c - a d}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/((b*x^2 + a)*(d*x^2 + c)),x, algorithm="fricas")
[Out]
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Sympy [A] time = 7.46214, size = 570, normalized size = 8.14 \[ \frac{\sqrt{- \frac{a}{b}} \log{\left (- \frac{2 a^{2} b d^{3} \left (- \frac{a}{b}\right )^{\frac{3}{2}}}{\left (a d - b c\right )^{3}} + \frac{4 a b^{2} c d^{2} \left (- \frac{a}{b}\right )^{\frac{3}{2}}}{\left (a d - b c\right )^{3}} - \frac{a d \sqrt{- \frac{a}{b}}}{a d - b c} - \frac{2 b^{3} c^{2} d \left (- \frac{a}{b}\right )^{\frac{3}{2}}}{\left (a d - b c\right )^{3}} - \frac{b c \sqrt{- \frac{a}{b}}}{a d - b c} + x \right )}}{2 \left (a d - b c\right )} - \frac{\sqrt{- \frac{a}{b}} \log{\left (\frac{2 a^{2} b d^{3} \left (- \frac{a}{b}\right )^{\frac{3}{2}}}{\left (a d - b c\right )^{3}} - \frac{4 a b^{2} c d^{2} \left (- \frac{a}{b}\right )^{\frac{3}{2}}}{\left (a d - b c\right )^{3}} + \frac{a d \sqrt{- \frac{a}{b}}}{a d - b c} + \frac{2 b^{3} c^{2} d \left (- \frac{a}{b}\right )^{\frac{3}{2}}}{\left (a d - b c\right )^{3}} + \frac{b c \sqrt{- \frac{a}{b}}}{a d - b c} + x \right )}}{2 \left (a d - b c\right )} + \frac{\sqrt{- \frac{c}{d}} \log{\left (- \frac{2 a^{2} b d^{3} \left (- \frac{c}{d}\right )^{\frac{3}{2}}}{\left (a d - b c\right )^{3}} + \frac{4 a b^{2} c d^{2} \left (- \frac{c}{d}\right )^{\frac{3}{2}}}{\left (a d - b c\right )^{3}} - \frac{a d \sqrt{- \frac{c}{d}}}{a d - b c} - \frac{2 b^{3} c^{2} d \left (- \frac{c}{d}\right )^{\frac{3}{2}}}{\left (a d - b c\right )^{3}} - \frac{b c \sqrt{- \frac{c}{d}}}{a d - b c} + x \right )}}{2 \left (a d - b c\right )} - \frac{\sqrt{- \frac{c}{d}} \log{\left (\frac{2 a^{2} b d^{3} \left (- \frac{c}{d}\right )^{\frac{3}{2}}}{\left (a d - b c\right )^{3}} - \frac{4 a b^{2} c d^{2} \left (- \frac{c}{d}\right )^{\frac{3}{2}}}{\left (a d - b c\right )^{3}} + \frac{a d \sqrt{- \frac{c}{d}}}{a d - b c} + \frac{2 b^{3} c^{2} d \left (- \frac{c}{d}\right )^{\frac{3}{2}}}{\left (a d - b c\right )^{3}} + \frac{b c \sqrt{- \frac{c}{d}}}{a d - b c} + x \right )}}{2 \left (a d - b c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**2/(b*x**2+a)/(d*x**2+c),x)
[Out]
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GIAC/XCAS [A] time = 0.259343, size = 176, normalized size = 2.51 \[ \frac{\sqrt{c d}{\left | d \right |} \arctan \left (\frac{2 \, \sqrt{\frac{1}{2}} x}{\sqrt{\frac{b c + a d + \sqrt{-4 \, a b c d +{\left (b c + a d\right )}^{2}}}{b d}}}\right )}{d^{2}{\left | b c - a d \right |}} - \frac{\sqrt{a b}{\left | b \right |} \arctan \left (\frac{2 \, \sqrt{\frac{1}{2}} x}{\sqrt{\frac{b c + a d - \sqrt{-4 \, a b c d +{\left (b c + a d\right )}^{2}}}{b d}}}\right )}{b^{2}{\left | b c - a d \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/((b*x^2 + a)*(d*x^2 + c)),x, algorithm="giac")
[Out]