Optimal. Leaf size=49 \[ -\frac{2 a x^m \sqrt{a+b x} \left (-\frac{b x}{a}\right )^{-m} \, _2F_1\left (\frac{1}{2},-m-1;\frac{3}{2};\frac{b x}{a}+1\right )}{b^2} \]
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Rubi [A] time = 0.0421082, 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 \[ -\frac{2 a x^m \sqrt{a+b x} \left (-\frac{b x}{a}\right )^{-m} \, _2F_1\left (\frac{1}{2},-m-1;\frac{3}{2};\frac{b x}{a}+1\right )}{b^2} \]
Antiderivative was successfully verified.
[In] Int[x^(1 + m)/Sqrt[a + b*x],x]
[Out]
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Rubi in Sympy [A] time = 7.29981, size = 42, normalized size = 0.86 \[ - \frac{2 a x^{m} \left (- \frac{b x}{a}\right )^{- m} \sqrt{a + b x}{{}_{2}F_{1}\left (\begin{matrix} - m - 1, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{1 + \frac{b x}{a}} \right )}}{b^{2}} \]
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 = 0.0538295, size = 72, normalized size = 1.47 \[ \frac{x^{m+1} \sqrt{a+b x} \left (\, _2F_1\left (-\frac{1}{2},m+1;m+2;-\frac{b x}{a}\right )-\, _2F_1\left (\frac{1}{2},m+1;m+2;-\frac{b x}{a}\right )\right )}{b (m+1) \sqrt{\frac{b x}{a}+1}} \]
Antiderivative was successfully verified.
[In] Integrate[x^(1 + m)/Sqrt[a + b*x],x]
[Out]
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Maple [F] time = 0.036, 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 = 10.1239, size = 37, normalized size = 0.76 \[ \frac{x^{2} x^{m} \Gamma \left (m + 2\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, m + 2 \\ m + 3 \end{matrix}\middle |{\frac{b x e^{i \pi }}{a}} \right )}}{\sqrt{a} \Gamma \left (m + 3\right )} \]
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]