Optimal. Leaf size=54 \[ -\frac{2 \sqrt{x} \tanh ^{-1}\left (\frac{\sqrt{a}}{\sqrt{x} \sqrt{\frac{a}{x}+b x^n}}\right )}{\sqrt{a} c (n+1) \sqrt{c x}} \]
[Out]
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Rubi [A] time = 0.186412, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13 \[ -\frac{2 \sqrt{x} \tanh ^{-1}\left (\frac{\sqrt{a}}{\sqrt{x} \sqrt{\frac{a}{x}+b x^n}}\right )}{\sqrt{a} c (n+1) \sqrt{c x}} \]
Antiderivative was successfully verified.
[In] Int[1/((c*x)^(3/2)*Sqrt[a/x + b*x^n]),x]
[Out]
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Rubi in Sympy [A] time = 15.734, size = 49, normalized size = 0.91 \[ - \frac{2 \sqrt{c x} \operatorname{atanh}{\left (\frac{\sqrt{a}}{\sqrt{x} \sqrt{\frac{a}{x} + b x^{n}}} \right )}}{\sqrt{a} c^{2} \sqrt{x} \left (n + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(c*x)**(3/2)/(a/x+b*x**n)**(1/2),x)
[Out]
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Mathematica [A] time = 0.149559, size = 83, normalized size = 1.54 \[ \frac{2 x \sqrt{a+b x^{n+1}} \left (\log \left (x^{\frac{n+1}{2}}\right )-\log \left (\sqrt{a} \sqrt{a+b x^{n+1}}+a\right )\right )}{\sqrt{a} (n+1) (c x)^{3/2} \sqrt{\frac{a}{x}+b x^n}} \]
Antiderivative was successfully verified.
[In] Integrate[1/((c*x)^(3/2)*Sqrt[a/x + b*x^n]),x]
[Out]
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Maple [F] time = 0.06, size = 0, normalized size = 0. \[ \int{1 \left ( cx \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{{\frac{a}{x}}+b{x}^{n}}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(c*x)^(3/2)/(a/x+b*x^n)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{b x^{n} + \frac{a}{x}} \left (c x\right )^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(b*x^n + a/x)*(c*x)^(3/2)),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(1/(sqrt(b*x^n + a/x)*(c*x)^(3/2)),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(c*x)**(3/2)/(a/x+b*x**n)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{b x^{n} + \frac{a}{x}} \left (c x\right )^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(b*x^n + a/x)*(c*x)^(3/2)),x, algorithm="giac")
[Out]