Optimal. Leaf size=93 \[ \frac{2 x \sqrt{\frac{a x^{j-n}}{b}+1} \, _2F_1\left (\frac{1}{2},\frac{2-n}{2 (j-n)};\frac{1-\frac{n}{2}}{j-n}+1;-\frac{a x^{j-n}}{b}\right )}{(2-n) \sqrt{a x^j+b x^n}} \]
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Rubi [A] time = 0.12887, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{2 x \sqrt{\frac{a x^{j-n}}{b}+1} \, _2F_1\left (\frac{1}{2},\frac{2-n}{2 (j-n)};\frac{1-\frac{n}{2}}{j-n}+1;-\frac{a x^{j-n}}{b}\right )}{(2-n) \sqrt{a x^j+b x^n}} \]
Antiderivative was successfully verified.
[In] Int[1/Sqrt[a*x^j + b*x^n],x]
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Rubi in Sympy [A] time = 13.1735, size = 78, normalized size = 0.84 \[ \frac{2 x^{- \frac{n}{2}} x^{- \frac{n}{2} + 1} \sqrt{a x^{j} + b x^{n}}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, - \frac{n - 2}{2 \left (j - n\right )} \\ \frac{j - \frac{3 n}{2} + 1}{j - n} \end{matrix}\middle |{- \frac{a x^{j - n}}{b}} \right )}}{b \left (- n + 2\right ) \sqrt{\frac{a x^{j - n}}{b} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(a*x**j+b*x**n)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0750133, size = 88, normalized size = 0.95 \[ -\frac{2 x \sqrt{\frac{a x^{j-n}}{b}+1} \, _2F_1\left (\frac{1}{2},\frac{n-2}{2 (n-j)};\frac{n-2}{2 (n-j)}+1;-\frac{a x^{j-n}}{b}\right )}{(n-2) \sqrt{a x^j+b x^n}} \]
Antiderivative was successfully verified.
[In] Integrate[1/Sqrt[a*x^j + b*x^n],x]
[Out]
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Maple [F] time = 0.068, size = 0, normalized size = 0. \[ \int{\frac{1}{\sqrt{a{x}^{j}+b{x}^{n}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(a*x^j+b*x^n)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{a x^{j} + b x^{n}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/sqrt(a*x^j + b*x^n),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(a*x^j + b*x^n),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{a x^{j} + b x^{n}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(a*x**j+b*x**n)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{a x^{j} + b x^{n}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/sqrt(a*x^j + b*x^n),x, algorithm="giac")
[Out]