Optimal. Leaf size=51 \[ -\frac{2 b^2 x^m \sqrt{a+b x} \left (-\frac{b x}{a}\right )^{-m} \, _2F_1\left (\frac{1}{2},3-m;\frac{3}{2};\frac{b x}{a}+1\right )}{a^3} \]
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Rubi [A] time = 0.0458692, antiderivative size = 51, 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 \[ -\frac{2 b^2 x^m \sqrt{a+b x} \left (-\frac{b x}{a}\right )^{-m} \, _2F_1\left (\frac{1}{2},3-m;\frac{3}{2};\frac{b x}{a}+1\right )}{a^3} \]
Antiderivative was successfully verified.
[In] Int[x^(-3 + m)/Sqrt[a + b*x],x]
[Out]
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Rubi in Sympy [A] time = 8.37523, size = 42, normalized size = 0.82 \[ - \frac{2 b^{2} x^{m} \left (- \frac{b x}{a}\right )^{- m} \sqrt{a + b x}{{}_{2}F_{1}\left (\begin{matrix} - m + 3, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{1 + \frac{b x}{a}} \right )}}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(-3+m)/(b*x+a)**(1/2),x)
[Out]
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Mathematica [B] time = 0.205362, size = 156, normalized size = 3.06 \[ \frac{x^{m-2} \sqrt{\frac{b x}{a}+1} \left (a^3 m \left (m^2-1\right ) \, _2F_1\left (-\frac{1}{2},m-2;m-1;-\frac{b x}{a}\right )-b (m-2) x \left (a^2 m (m+1) \, _2F_1\left (-\frac{1}{2},m-1;m;-\frac{b x}{a}\right )+b (m-1) x \left (b m x \, _2F_1\left (\frac{1}{2},m+1;m+2;-\frac{b x}{a}\right )-a (m+1) \, _2F_1\left (-\frac{1}{2},m;m+1;-\frac{b x}{a}\right )\right )\right )\right )}{a^3 (m-2) (m-1) m (m+1) \sqrt{a+b x}} \]
Antiderivative was successfully verified.
[In] Integrate[x^(-3 + m)/Sqrt[a + b*x],x]
[Out]
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Maple [F] time = 0.03, size = 0, normalized size = 0. \[ \int{{x}^{-3+m}{\frac{1}{\sqrt{bx+a}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(-3+m)/(b*x+a)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{m - 3}}{\sqrt{b x + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(m - 3)/sqrt(b*x + a),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{m - 3}}{\sqrt{b x + a}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(m - 3)/sqrt(b*x + a),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(x**(-3+m)/(b*x+a)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{m - 3}}{\sqrt{b x + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(m - 3)/sqrt(b*x + a),x, algorithm="giac")
[Out]