Optimal. Leaf size=98 \[ -\frac{2 a^{3/2} c^2 \tanh ^{-1}\left (\frac{\sqrt{a}}{x \sqrt{\frac{a}{x^2}+b x^n}}\right )}{n+2}+\frac{2 a c^2 x \sqrt{\frac{a}{x^2}+b x^n}}{n+2}+\frac{2 c^2 x^3 \left (\frac{a}{x^2}+b x^n\right )^{3/2}}{3 (n+2)} \]
[Out]
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Rubi [A] time = 0.268372, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ -\frac{2 a^{3/2} c^2 \tanh ^{-1}\left (\frac{\sqrt{a}}{x \sqrt{\frac{a}{x^2}+b x^n}}\right )}{n+2}+\frac{2 a c^2 x \sqrt{\frac{a}{x^2}+b x^n}}{n+2}+\frac{2 c^2 x^3 \left (\frac{a}{x^2}+b x^n\right )^{3/2}}{3 (n+2)} \]
Antiderivative was successfully verified.
[In] Int[c^2*x^2*(a/x^2 + b*x^n)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 21.755, size = 87, normalized size = 0.89 \[ - \frac{2 a^{\frac{3}{2}} c^{2} \operatorname{atanh}{\left (\frac{\sqrt{a}}{x \sqrt{\frac{a}{x^{2}} + b x^{n}}} \right )}}{n + 2} + \frac{2 a c^{2} x \sqrt{\frac{a}{x^{2}} + b x^{n}}}{n + 2} + \frac{2 c^{2} x^{3} \left (\frac{a}{x^{2}} + b x^{n}\right )^{\frac{3}{2}}}{3 \left (n + 2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(c**2*x**2*(a/x**2+b*x**n)**(3/2),x)
[Out]
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Mathematica [A] time = 0.225331, size = 113, normalized size = 1.15 \[ \frac{2 c^2 x \sqrt{\frac{a}{x^2}+b x^n} \left (-3 a^{3/2} \log \left (\sqrt{a} \sqrt{a+b x^{n+2}}+a\right )+3 a^{3/2} \log \left (x^{\frac{n+2}{2}}\right )+\sqrt{a+b x^{n+2}} \left (4 a+b x^{n+2}\right )\right )}{3 (n+2) \sqrt{a+b x^{n+2}}} \]
Antiderivative was successfully verified.
[In] Integrate[c^2*x^2*(a/x^2 + b*x^n)^(3/2),x]
[Out]
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Maple [F] time = 0.052, size = 0, normalized size = 0. \[ \int{c}^{2}{x}^{2} \left ({\frac{a}{{x}^{2}}}+b{x}^{n} \right ) ^{{\frac{3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(c^2*x^2*(a/x^2+b*x^n)^(3/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ c^{2} \int{\left (b x^{n} + \frac{a}{x^{2}}\right )}^{\frac{3}{2}} x^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^n + a/x^2)^(3/2)*c^2*x^2,x, algorithm="maxima")
[Out]
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Fricas [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^n + a/x^2)^(3/2)*c^2*x^2,x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ c^{2} \left (\int a \sqrt{\frac{a}{x^{2}} + b x^{n}}\, dx + \int b x^{2} x^{n} \sqrt{\frac{a}{x^{2}} + b x^{n}}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(c**2*x**2*(a/x**2+b*x**n)**(3/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{n} + \frac{a}{x^{2}}\right )}^{\frac{3}{2}} c^{2} x^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^n + a/x^2)^(3/2)*c^2*x^2,x, algorithm="giac")
[Out]