Optimal. Leaf size=13 \[ \frac{x^m}{\sqrt{a+b x}} \]
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Rubi [C] time = 0.111998, antiderivative size = 92, normalized size of antiderivative = 7.08, number of steps used = 5, number of rules used = 2, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059 \[ \frac{x^m \left (-\frac{b x}{a}\right )^{-m} \, _2F_1\left (-\frac{1}{2},-m;\frac{1}{2};\frac{b x}{a}+1\right )}{\sqrt{a+b x}}-\frac{2 m x^m \sqrt{a+b x} \left (-\frac{b x}{a}\right )^{-m} \, _2F_1\left (\frac{1}{2},1-m;\frac{3}{2};\frac{b x}{a}+1\right )}{a} \]
Antiderivative was successfully verified.
[In] Int[-(b*x^m)/(2*(a + b*x)^(3/2)) + (m*x^(-1 + m))/Sqrt[a + b*x],x]
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Rubi in Sympy [A] time = 14.4416, size = 73, normalized size = 5.62 \[ \frac{x^{m} \left (- \frac{b x}{a}\right )^{- m}{{}_{2}F_{1}\left (\begin{matrix} - m, - \frac{1}{2} \\ \frac{1}{2} \end{matrix}\middle |{1 + \frac{b x}{a}} \right )}}{\sqrt{a + b x}} - \frac{2 m x^{m} \left (- \frac{b x}{a}\right )^{- m} \sqrt{a + b x}{{}_{2}F_{1}\left (\begin{matrix} - m + 1, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{1 + \frac{b x}{a}} \right )}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(-1/2*b*x**m/(b*x+a)**(3/2)+m*x**(-1+m)/(b*x+a)**(1/2),x)
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Mathematica [C] time = 0.0474768, size = 100, normalized size = 7.69 \[ \frac{x^m \sqrt{a+b x} \left (2 a (m+1) \, _2F_1\left (-\frac{1}{2},m;m+1;-\frac{b x}{a}\right )-b x \left (2 m \, _2F_1\left (\frac{1}{2},m+1;m+2;-\frac{b x}{a}\right )+\, _2F_1\left (\frac{3}{2},m+1;m+2;-\frac{b x}{a}\right )\right )\right )}{2 a^2 (m+1) \sqrt{\frac{b x}{a}+1}} \]
Antiderivative was successfully verified.
[In] Integrate[-(b*x^m)/(2*(a + b*x)^(3/2)) + (m*x^(-1 + m))/Sqrt[a + b*x],x]
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Maple [F] time = 0.15, size = 0, normalized size = 0. \[ \int -{\frac{b{x}^{m}}{2} \left ( bx+a \right ) ^{-{\frac{3}{2}}}}+{m{x}^{-1+m}{\frac{1}{\sqrt{bx+a}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(-1/2*b*x^m/(b*x+a)^(3/2)+m*x^(-1+m)/(b*x+a)^(1/2),x)
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Maxima [A] time = 1.53198, size = 15, normalized size = 1.15 \[ \frac{x^{m}}{\sqrt{b x + a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(m*x^(m - 1)/sqrt(b*x + a) - 1/2*b*x^m/(b*x + a)^(3/2),x, algorithm="maxima")
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Fricas [A] time = 0.227745, size = 15, normalized size = 1.15 \[ \frac{x^{m}}{\sqrt{b x + a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(m*x^(m - 1)/sqrt(b*x + a) - 1/2*b*x^m/(b*x + a)^(3/2),x, algorithm="fricas")
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Sympy [A] time = 28.0152, size = 73, normalized size = 5.62 \[ \frac{m x^{m} \Gamma \left (m\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, m \\ m + 1 \end{matrix}\middle |{\frac{b x e^{i \pi }}{a}} \right )}}{\sqrt{a} \Gamma \left (m + 1\right )} - \frac{b x x^{m} \Gamma \left (m + 1\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{2}, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{b x e^{i \pi }}{a}} \right )}}{2 a^{\frac{3}{2}} \Gamma \left (m + 2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-1/2*b*x**m/(b*x+a)**(3/2)+m*x**(-1+m)/(b*x+a)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{m x^{m - 1}}{\sqrt{b x + a}} - \frac{b x^{m}}{2 \,{\left (b x + a\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(m*x^(m - 1)/sqrt(b*x + a) - 1/2*b*x^m/(b*x + a)^(3/2),x, algorithm="giac")
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