Optimal. Leaf size=45 \[ -\frac{c (c x)^{m-1} \, _2F_1\left (1,\frac{m-1}{2};\frac{m+1}{2};-\frac{c x^2}{b}\right )}{b (1-m)} \]
[Out]
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Rubi [A] time = 0.0633662, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ -\frac{c (c x)^{m-1} \, _2F_1\left (1,\frac{m-1}{2};\frac{m+1}{2};-\frac{c x^2}{b}\right )}{b (1-m)} \]
Antiderivative was successfully verified.
[In] Int[(c*x)^m/(b*x^2 + c*x^4),x]
[Out]
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Rubi in Sympy [A] time = 8.41014, size = 34, normalized size = 0.76 \[ - \frac{c \left (c x\right )^{m - 1}{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{m}{2} - \frac{1}{2} \\ \frac{m}{2} + \frac{1}{2} \end{matrix}\middle |{- \frac{c x^{2}}{b}} \right )}}{b \left (- m + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x)**m/(c*x**4+b*x**2),x)
[Out]
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Mathematica [A] time = 0.0712378, size = 56, normalized size = 1.24 \[ \frac{(c x)^m \left (\frac{b}{m-1}-\frac{c x^2 \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{c x^2}{b}\right )}{m+1}\right )}{b^2 x} \]
Antiderivative was successfully verified.
[In] Integrate[(c*x)^m/(b*x^2 + c*x^4),x]
[Out]
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Maple [F] time = 0.058, size = 0, normalized size = 0. \[ \int{\frac{ \left ( cx \right ) ^{m}}{c{x}^{4}+b{x}^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x)^m/(c*x^4+b*x^2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (c x\right )^{m}}{c x^{4} + b x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x)^m/(c*x^4 + b*x^2),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\left (c x\right )^{m}}{c x^{4} + b x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x)^m/(c*x^4 + b*x^2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (c x\right )^{m}}{x^{2} \left (b + c x^{2}\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x)**m/(c*x**4+b*x**2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (c x\right )^{m}}{c x^{4} + b x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x)^m/(c*x^4 + b*x^2),x, algorithm="giac")
[Out]