Optimal. Leaf size=48 \[ -\frac{2 x^m \sqrt{a+b x} \left (-\frac{b x}{a}\right )^{-m} \, _2F_1\left (\frac{1}{2},1-m;\frac{3}{2};\frac{b x}{a}+1\right )}{a} \]
[Out]
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Rubi [A] time = 0.041306, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ -\frac{2 x^m \sqrt{a+b x} \left (-\frac{b x}{a}\right )^{-m} \, _2F_1\left (\frac{1}{2},1-m;\frac{3}{2};\frac{b x}{a}+1\right )}{a} \]
Antiderivative was successfully verified.
[In] Int[x^(-1 + m)/Sqrt[a + b*x],x]
[Out]
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Rubi in Sympy [A] time = 7.70562, size = 37, normalized size = 0.77 \[ - \frac{2 x^{m} \left (- \frac{b x}{a}\right )^{- m} \sqrt{a + b x}{{}_{2}F_{1}\left (\begin{matrix} - m + 1, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{1 + \frac{b x}{a}} \right )}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(-1+m)/(b*x+a)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0727344, size = 79, normalized size = 1.65 \[ \frac{x^m \sqrt{a+b x} \left (a (m+1) \, _2F_1\left (-\frac{1}{2},m;m+1;-\frac{b x}{a}\right )-b m x \, _2F_1\left (\frac{1}{2},m+1;m+2;-\frac{b x}{a}\right )\right )}{a^2 m (m+1) \sqrt{\frac{b x}{a}+1}} \]
Antiderivative was successfully verified.
[In] Integrate[x^(-1 + m)/Sqrt[a + b*x],x]
[Out]
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Maple [F] time = 0.024, size = 0, normalized size = 0. \[ \int{{x}^{-1+m}{\frac{1}{\sqrt{bx+a}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(-1+m)/(b*x+a)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{m - 1}}{\sqrt{b x + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(m - 1)/sqrt(b*x + a),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{m - 1}}{\sqrt{b x + a}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(m - 1)/sqrt(b*x + a),x, algorithm="fricas")
[Out]
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Sympy [A] time = 22.9459, size = 31, normalized size = 0.65 \[ \frac{x^{m} \Gamma \left (m\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, m \\ m + 1 \end{matrix}\middle |{\frac{b x e^{i \pi }}{a}} \right )}}{\sqrt{a} \Gamma \left (m + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**(-1+m)/(b*x+a)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{m - 1}}{\sqrt{b x + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(m - 1)/sqrt(b*x + a),x, algorithm="giac")
[Out]