Optimal. Leaf size=234 \[ -\frac{\left (\sqrt{a} B+A \sqrt{c}\right ) \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 a^{3/4} c^{3/4}}+\frac{\left (\sqrt{a} B+A \sqrt{c}\right ) \tan ^{-1}\left (\frac{2 \sqrt [4]{c} x}{\sqrt [4]{a}}+\sqrt{3}\right )}{2 a^{3/4} c^{3/4}}-\frac{\left (A-\frac{\sqrt{a} B}{\sqrt{c}}\right ) \log \left (-\sqrt{3} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{3} a^{3/4} \sqrt [4]{c}}+\frac{\left (A-\frac{\sqrt{a} B}{\sqrt{c}}\right ) \log \left (\sqrt{3} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{3} a^{3/4} \sqrt [4]{c}} \]
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Rubi [A] time = 0.39161, antiderivative size = 234, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.156 \[ -\frac{\left (\sqrt{a} B+A \sqrt{c}\right ) \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 a^{3/4} c^{3/4}}+\frac{\left (\sqrt{a} B+A \sqrt{c}\right ) \tan ^{-1}\left (\frac{2 \sqrt [4]{c} x}{\sqrt [4]{a}}+\sqrt{3}\right )}{2 a^{3/4} c^{3/4}}-\frac{\left (A-\frac{\sqrt{a} B}{\sqrt{c}}\right ) \log \left (-\sqrt{3} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{3} a^{3/4} \sqrt [4]{c}}+\frac{\left (A-\frac{\sqrt{a} B}{\sqrt{c}}\right ) \log \left (\sqrt{3} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{3} a^{3/4} \sqrt [4]{c}} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x^2)/(a - Sqrt[a]*Sqrt[c]*x^2 + c*x^4),x]
[Out]
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Rubi in Sympy [A] time = 88.3607, size = 240, normalized size = 1.03 \[ - \frac{\sqrt{3} \left (A \sqrt{c} - B \sqrt{a}\right ) \log{\left (- \frac{\sqrt{3} \sqrt [4]{a} x}{\sqrt [4]{c}} + \frac{\sqrt{a}}{\sqrt{c}} + x^{2} \right )}}{12 a^{\frac{3}{4}} c^{\frac{3}{4}}} + \frac{\sqrt{3} \left (A \sqrt{c} - B \sqrt{a}\right ) \log{\left (\frac{\sqrt{3} \sqrt [4]{a} x}{\sqrt [4]{c}} + \frac{\sqrt{a}}{\sqrt{c}} + x^{2} \right )}}{12 a^{\frac{3}{4}} c^{\frac{3}{4}}} - \frac{\left (A \sqrt{c} + B \sqrt{a}\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\sqrt [4]{a} - \frac{2 \sqrt{3} \sqrt [4]{c} x}{3}\right )}{\sqrt [4]{a}} \right )}}{2 a^{\frac{3}{4}} c^{\frac{3}{4}}} + \frac{\left (A \sqrt{c} + B \sqrt{a}\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\sqrt [4]{a} + \frac{2 \sqrt{3} \sqrt [4]{c} x}{3}\right )}{\sqrt [4]{a}} \right )}}{2 a^{\frac{3}{4}} c^{\frac{3}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x**2+A)/(a+c*x**4-x**2*a**(1/2)*c**(1/2)),x)
[Out]
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Mathematica [C] time = 0.309243, size = 163, normalized size = 0.7 \[ \frac{\sqrt [4]{-1} \left (\frac{\left (\left (\sqrt{3}-i\right ) \sqrt{a} B-2 i A \sqrt{c}\right ) \tan ^{-1}\left (\frac{(1+i) \sqrt [4]{c} x}{\sqrt{\sqrt{3}-i} \sqrt [4]{a}}\right )}{\sqrt{\sqrt{3}-i}}-\frac{\left (\left (\sqrt{3}+i\right ) \sqrt{a} B+2 i A \sqrt{c}\right ) \tanh ^{-1}\left (\frac{(1+i) \sqrt [4]{c} x}{\sqrt{\sqrt{3}+i} \sqrt [4]{a}}\right )}{\sqrt{\sqrt{3}+i}}\right )}{\sqrt{6} a^{3/4} c^{3/4}} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x^2)/(a - Sqrt[a]*Sqrt[c]*x^2 + c*x^4),x]
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Maple [A] time = 0.07, size = 320, normalized size = 1.4 \[ -{\frac{A\sqrt{3}}{12}\ln \left ( \sqrt [4]{a}\sqrt [4]{c}x\sqrt{3}-{x}^{2}\sqrt{c}-\sqrt{a} \right ){\frac{1}{\sqrt [4]{c}}}{a}^{-{\frac{3}{4}}}}+{\frac{B\sqrt{3}}{12}\ln \left ( \sqrt [4]{a}\sqrt [4]{c}x\sqrt{3}-{x}^{2}\sqrt{c}-\sqrt{a} \right ){c}^{-{\frac{3}{4}}}{\frac{1}{\sqrt [4]{a}}}}-{\frac{A}{2}\arctan \left ({1 \left ( \sqrt{3}\sqrt [4]{c}\sqrt [4]{a}-2\,x\sqrt{c} \right ){\frac{1}{\sqrt{\sqrt{a}\sqrt{c}}}}} \right ){\frac{1}{\sqrt{a}}}{\frac{1}{\sqrt{\sqrt{a}\sqrt{c}}}}}-{\frac{B}{2}\arctan \left ({1 \left ( \sqrt{3}\sqrt [4]{c}\sqrt [4]{a}-2\,x\sqrt{c} \right ){\frac{1}{\sqrt{\sqrt{a}\sqrt{c}}}}} \right ){\frac{1}{\sqrt{c}}}{\frac{1}{\sqrt{\sqrt{a}\sqrt{c}}}}}+{\frac{A\sqrt{3}}{12}\ln \left ( \sqrt [4]{a}\sqrt [4]{c}x\sqrt{3}+\sqrt{a}+{x}^{2}\sqrt{c} \right ){\frac{1}{\sqrt [4]{c}}}{a}^{-{\frac{3}{4}}}}-{\frac{B\sqrt{3}}{12}\ln \left ( \sqrt [4]{a}\sqrt [4]{c}x\sqrt{3}+\sqrt{a}+{x}^{2}\sqrt{c} \right ){c}^{-{\frac{3}{4}}}{\frac{1}{\sqrt [4]{a}}}}+{\frac{A}{2}\arctan \left ({1 \left ( 2\,x\sqrt{c}+\sqrt{3}\sqrt [4]{c}\sqrt [4]{a} \right ){\frac{1}{\sqrt{\sqrt{a}\sqrt{c}}}}} \right ){\frac{1}{\sqrt{a}}}{\frac{1}{\sqrt{\sqrt{a}\sqrt{c}}}}}+{\frac{B}{2}\arctan \left ({1 \left ( 2\,x\sqrt{c}+\sqrt{3}\sqrt [4]{c}\sqrt [4]{a} \right ){\frac{1}{\sqrt{\sqrt{a}\sqrt{c}}}}} \right ){\frac{1}{\sqrt{c}}}{\frac{1}{\sqrt{\sqrt{a}\sqrt{c}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x^2+A)/(a+c*x^4-x^2*a^(1/2)*c^(1/2)),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{B x^{2} + A}{c x^{4} - \sqrt{a} \sqrt{c} x^{2} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)/(c*x^4 - sqrt(a)*sqrt(c)*x^2 + a),x, algorithm="maxima")
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)/(c*x^4 - sqrt(a)*sqrt(c)*x^2 + a),x, algorithm="fricas")
[Out]
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: PolynomialError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x**2+A)/(a+c*x**4-x**2*a**(1/2)*c**(1/2)),x)
[Out]
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GIAC/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)/(c*x^4 - sqrt(a)*sqrt(c)*x^2 + a),x, algorithm="giac")
[Out]