Optimal. Leaf size=66 \[ -\frac{\left (b+2 c x^2\right ) (3 b B-4 A c)}{3 b^3 \sqrt{b x^2+c x^4}}-\frac{A}{3 b x^2 \sqrt{b x^2+c x^4}} \]
[Out]
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Rubi [A] time = 0.310948, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ -\frac{\left (b+2 c x^2\right ) (3 b B-4 A c)}{3 b^3 \sqrt{b x^2+c x^4}}-\frac{A}{3 b x^2 \sqrt{b x^2+c x^4}} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x^2)/(x*(b*x^2 + c*x^4)^(3/2)),x]
[Out]
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Rubi in Sympy [A] time = 19.0681, size = 60, normalized size = 0.91 \[ - \frac{A}{3 b x^{2} \sqrt{b x^{2} + c x^{4}}} + \frac{\left (2 b + 4 c x^{2}\right ) \left (4 A c - 3 B b\right )}{6 b^{3} \sqrt{b x^{2} + c x^{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x**2+A)/x/(c*x**4+b*x**2)**(3/2),x)
[Out]
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Mathematica [A] time = 0.0817957, size = 64, normalized size = 0.97 \[ \frac{A \left (-b^2+4 b c x^2+8 c^2 x^4\right )-3 b B x^2 \left (b+2 c x^2\right )}{3 b^3 x^2 \sqrt{x^2 \left (b+c x^2\right )}} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x^2)/(x*(b*x^2 + c*x^4)^(3/2)),x]
[Out]
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Maple [A] time = 0.009, size = 66, normalized size = 1. \[ -{\frac{ \left ( c{x}^{2}+b \right ) \left ( -8\,A{c}^{2}{x}^{4}+6\,B{x}^{4}bc-4\,Abc{x}^{2}+3\,B{b}^{2}{x}^{2}+{b}^{2}A \right ) }{3\,{b}^{3}} \left ( c{x}^{4}+b{x}^{2} \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x^2+A)/x/(c*x^4+b*x^2)^(3/2),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*x^2 + A)/((c*x^4 + b*x^2)^(3/2)*x),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.234245, size = 97, normalized size = 1.47 \[ -\frac{{\left (2 \,{\left (3 \, B b c - 4 \, A c^{2}\right )} x^{4} + A b^{2} +{\left (3 \, B b^{2} - 4 \, A b c\right )} x^{2}\right )} \sqrt{c x^{4} + b x^{2}}}{3 \,{\left (b^{3} c x^{6} + b^{4} x^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)/((c*x^4 + b*x^2)^(3/2)*x),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{A + B x^{2}}{x \left (x^{2} \left (b + c x^{2}\right )\right )^{\frac{3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x**2+A)/x/(c*x**4+b*x**2)**(3/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{B x^{2} + A}{{\left (c x^{4} + b x^{2}\right )}^{\frac{3}{2}} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)/((c*x^4 + b*x^2)^(3/2)*x),x, algorithm="giac")
[Out]