Optimal. Leaf size=49 \[ \frac{2 b x^m \sqrt{a+b x} \left (-\frac{b x}{a}\right )^{-m} \, _2F_1\left (\frac{1}{2},2-m;\frac{3}{2};\frac{b x}{a}+1\right )}{a^2} \]
[Out]
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Rubi [A] time = 0.0446194, antiderivative size = 49, 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 \[ \frac{2 b x^m \sqrt{a+b x} \left (-\frac{b x}{a}\right )^{-m} \, _2F_1\left (\frac{1}{2},2-m;\frac{3}{2};\frac{b x}{a}+1\right )}{a^2} \]
Antiderivative was successfully verified.
[In] Int[x^(-2 + m)/Sqrt[a + b*x],x]
[Out]
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Rubi in Sympy [A] time = 7.60031, size = 39, normalized size = 0.8 \[ \frac{2 b x^{m} \left (- \frac{b x}{a}\right )^{- m} \sqrt{a + b x}{{}_{2}F_{1}\left (\begin{matrix} - m + 2, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{1 + \frac{b x}{a}} \right )}}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(-2+m)/(b*x+a)**(1/2),x)
[Out]
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Mathematica [B] time = 0.13131, size = 114, normalized size = 2.33 \[ \frac{x^{m-1} \sqrt{a+b x} \left (a^2 m (m+1) \, _2F_1\left (-\frac{1}{2},m-1;m;-\frac{b x}{a}\right )-b (m-1) x \left (a (m+1) \, _2F_1\left (-\frac{1}{2},m;m+1;-\frac{b x}{a}\right )-b m x \, _2F_1\left (\frac{1}{2},m+1;m+2;-\frac{b x}{a}\right )\right )\right )}{a^3 m \left (m^2-1\right ) \sqrt{\frac{b x}{a}+1}} \]
Antiderivative was successfully verified.
[In] Integrate[x^(-2 + m)/Sqrt[a + b*x],x]
[Out]
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Maple [F] time = 0.03, size = 0, normalized size = 0. \[ \int{{x}^{-2+m}{\frac{1}{\sqrt{bx+a}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(-2+m)/(b*x+a)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{m - 2}}{\sqrt{b x + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(m - 2)/sqrt(b*x + a),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{m - 2}}{\sqrt{b x + a}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(m - 2)/sqrt(b*x + a),x, algorithm="fricas")
[Out]
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Sympy [A] time = 148.243, size = 32, normalized size = 0.65 \[ \frac{x^{m} \Gamma \left (m - 1\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, m - 1 \\ m \end{matrix}\middle |{\frac{b x e^{i \pi }}{a}} \right )}}{\sqrt{a} x \Gamma \left (m\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**(-2+m)/(b*x+a)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{m - 2}}{\sqrt{b x + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(m - 2)/sqrt(b*x + a),x, algorithm="giac")
[Out]