Optimal. Leaf size=73 \[ b^{3/2} \left (-\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right )\right )+\frac{b \sqrt{b x^2+c x^4}}{x}+\frac{\left (b x^2+c x^4\right )^{3/2}}{3 x^3} \]
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Rubi [A] time = 0.166862, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ b^{3/2} \left (-\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right )\right )+\frac{b \sqrt{b x^2+c x^4}}{x}+\frac{\left (b x^2+c x^4\right )^{3/2}}{3 x^3} \]
Antiderivative was successfully verified.
[In] Int[(b*x^2 + c*x^4)^(3/2)/x^4,x]
[Out]
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Rubi in Sympy [A] time = 19.8416, size = 61, normalized size = 0.84 \[ - b^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{b x^{2} + c x^{4}}} \right )} + \frac{b \sqrt{b x^{2} + c x^{4}}}{x} + \frac{\left (b x^{2} + c x^{4}\right )^{\frac{3}{2}}}{3 x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**4+b*x**2)**(3/2)/x**4,x)
[Out]
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Mathematica [A] time = 0.108358, size = 89, normalized size = 1.22 \[ \frac{x \sqrt{b+c x^2} \left (-3 b^{3/2} \log \left (\sqrt{b} \sqrt{b+c x^2}+b\right )+3 b^{3/2} \log (x)+\sqrt{b+c x^2} \left (4 b+c x^2\right )\right )}{3 \sqrt{x^2 \left (b+c x^2\right )}} \]
Antiderivative was successfully verified.
[In] Integrate[(b*x^2 + c*x^4)^(3/2)/x^4,x]
[Out]
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Maple [A] time = 0.009, size = 78, normalized size = 1.1 \[ -{\frac{1}{3\,{x}^{3}} \left ( c{x}^{4}+b{x}^{2} \right ) ^{{\frac{3}{2}}} \left ( 3\,\ln \left ( 2\,{\frac{\sqrt{b}\sqrt{c{x}^{2}+b}+b}{x}} \right ){b}^{3/2}- \left ( c{x}^{2}+b \right ) ^{{\frac{3}{2}}}-3\,\sqrt{c{x}^{2}+b}b \right ) \left ( c{x}^{2}+b \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^4+b*x^2)^(3/2)/x^4,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((c*x^4 + b*x^2)^(3/2)/x^4,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.277891, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, b^{\frac{3}{2}} x \log \left (-\frac{c x^{3} + 2 \, b x - 2 \, \sqrt{c x^{4} + b x^{2}} \sqrt{b}}{x^{3}}\right ) + 2 \, \sqrt{c x^{4} + b x^{2}}{\left (c x^{2} + 4 \, b\right )}}{6 \, x}, -\frac{3 \, \sqrt{-b} b x \arctan \left (\frac{b x}{\sqrt{c x^{4} + b x^{2}} \sqrt{-b}}\right ) - \sqrt{c x^{4} + b x^{2}}{\left (c x^{2} + 4 \, b\right )}}{3 \, x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^4 + b*x^2)^(3/2)/x^4,x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (x^{2} \left (b + c x^{2}\right )\right )^{\frac{3}{2}}}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**4+b*x**2)**(3/2)/x**4,x)
[Out]
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GIAC/XCAS [A] time = 0.273278, size = 119, normalized size = 1.63 \[ \frac{1}{3} \,{\left (\frac{3 \, b^{2} \arctan \left (\frac{\sqrt{c x^{2} + b}}{\sqrt{-b}}\right )}{\sqrt{-b}} +{\left (c x^{2} + b\right )}^{\frac{3}{2}} + 3 \, \sqrt{c x^{2} + b} b\right )}{\rm sign}\left (x\right ) - \frac{{\left (3 \, b^{2} \arctan \left (\frac{\sqrt{b}}{\sqrt{-b}}\right ) + 4 \, \sqrt{-b} b^{\frac{3}{2}}\right )}{\rm sign}\left (x\right )}{3 \, \sqrt{-b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^4 + b*x^2)^(3/2)/x^4,x, algorithm="giac")
[Out]