Optimal. Leaf size=46 \[ \frac{2 x^m \sqrt{a+b x} \left (-\frac{b x}{a}\right )^{-m} \, _2F_1\left (\frac{1}{2},-m;\frac{3}{2};\frac{b x}{a}+1\right )}{b} \]
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Rubi [A] time = 0.0396741, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{2 x^m \sqrt{a+b x} \left (-\frac{b x}{a}\right )^{-m} \, _2F_1\left (\frac{1}{2},-m;\frac{3}{2};\frac{b x}{a}+1\right )}{b} \]
Antiderivative was successfully verified.
[In] Int[x^m/Sqrt[a + b*x],x]
[Out]
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Rubi in Sympy [A] time = 6.01463, size = 36, normalized size = 0.78 \[ \frac{2 x^{m} \left (- \frac{b x}{a}\right )^{- m} \sqrt{a + b x}{{}_{2}F_{1}\left (\begin{matrix} - m, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{1 + \frac{b x}{a}} \right )}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**m/(b*x+a)**(1/2),x)
[Out]
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Mathematica [A] time = 0.00828212, size = 50, normalized size = 1.09 \[ \frac{x^{m+1} \sqrt{\frac{a+b x}{a}} \, _2F_1\left (\frac{1}{2},m+1;m+2;-\frac{b x}{a}\right )}{(m+1) \sqrt{a+b x}} \]
Antiderivative was successfully verified.
[In] Integrate[x^m/Sqrt[a + b*x],x]
[Out]
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Maple [F] time = 0., size = 0, normalized size = 0. \[ \int{{x}^{m}{\frac{1}{\sqrt{bx+a}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^m/(b*x+a)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{m}}{\sqrt{b x + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^m/sqrt(b*x + a),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{m}}{\sqrt{b x + a}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^m/sqrt(b*x + a),x, algorithm="fricas")
[Out]
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Sympy [A] time = 4.18694, size = 36, normalized size = 0.78 \[ \frac{x x^{m} \Gamma \left (m + 1\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{b x e^{i \pi }}{a}} \right )}}{\sqrt{a} \Gamma \left (m + 2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**m/(b*x+a)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{m}}{\sqrt{b x + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^m/sqrt(b*x + a),x, algorithm="giac")
[Out]