Optimal. Leaf size=74 \[ \frac{8 c \sqrt{b x^2+c x^4}}{3 b^3 x^2}-\frac{4 \sqrt{b x^2+c x^4}}{3 b^2 x^4}+\frac{1}{b x^2 \sqrt{b x^2+c x^4}} \]
[Out]
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Rubi [A] time = 0.215646, antiderivative size = 74, 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 \[ \frac{8 c \sqrt{b x^2+c x^4}}{3 b^3 x^2}-\frac{4 \sqrt{b x^2+c x^4}}{3 b^2 x^4}+\frac{1}{b x^2 \sqrt{b x^2+c x^4}} \]
Antiderivative was successfully verified.
[In] Int[1/(x*(b*x^2 + c*x^4)^(3/2)),x]
[Out]
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Rubi in Sympy [A] time = 21.7421, size = 68, normalized size = 0.92 \[ \frac{1}{b x^{2} \sqrt{b x^{2} + c x^{4}}} - \frac{4 \sqrt{b x^{2} + c x^{4}}}{3 b^{2} x^{4}} + \frac{8 c \sqrt{b x^{2} + c x^{4}}}{3 b^{3} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x/(c*x**4+b*x**2)**(3/2),x)
[Out]
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Mathematica [A] time = 0.0337585, size = 46, normalized size = 0.62 \[ \frac{-b^2+4 b c x^2+8 c^2 x^4}{3 b^3 x^2 \sqrt{x^2 \left (b+c x^2\right )}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x*(b*x^2 + c*x^4)^(3/2)),x]
[Out]
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Maple [A] time = 0.007, size = 45, normalized size = 0.6 \[ -{\frac{ \left ( c{x}^{2}+b \right ) \left ( -8\,{c}^{2}{x}^{4}-4\,bc{x}^{2}+{b}^{2} \right ) }{3\,{b}^{3}} \left ( c{x}^{4}+b{x}^{2} \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/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(1/((c*x^4 + b*x^2)^(3/2)*x),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.268223, size = 73, normalized size = 0.99 \[ \frac{{\left (8 \, c^{2} x^{4} + 4 \, b c x^{2} - b^{2}\right )} \sqrt{c x^{4} + b x^{2}}}{3 \,{\left (b^{3} c x^{6} + b^{4} x^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((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. \[ \int \frac{1}{x \left (x^{2} \left (b + c x^{2}\right )\right )^{\frac{3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/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. \[ \int \frac{1}{{\left (c x^{4} + b x^{2}\right )}^{\frac{3}{2}} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((c*x^4 + b*x^2)^(3/2)*x),x, algorithm="giac")
[Out]