Optimal. Leaf size=46 \[ -\frac{2 x^m \left (-\frac{b x}{a}\right )^{-m} \, _2F_1\left (-\frac{1}{2},-m;\frac{1}{2};\frac{b x}{a}+1\right )}{b \sqrt{a+b x}} \]
[Out]
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Rubi [A] time = 0.0392248, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\frac{2 x^m \left (-\frac{b x}{a}\right )^{-m} \, _2F_1\left (-\frac{1}{2},-m;\frac{1}{2};\frac{b x}{a}+1\right )}{b \sqrt{a+b x}} \]
Antiderivative was successfully verified.
[In] Int[x^m/(a + b*x)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 5.97762, size = 39, normalized size = 0.85 \[ - \frac{2 x^{m} \left (- \frac{b x}{a}\right )^{- m}{{}_{2}F_{1}\left (\begin{matrix} - m, - \frac{1}{2} \\ \frac{1}{2} \end{matrix}\middle |{1 + \frac{b x}{a}} \right )}}{b \sqrt{a + b x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**m/(b*x+a)**(3/2),x)
[Out]
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Mathematica [A] time = 0.034299, size = 53, normalized size = 1.15 \[ \frac{x^{m+1} \sqrt{\frac{a+b x}{a}} \, _2F_1\left (\frac{3}{2},m+1;m+2;-\frac{b x}{a}\right )}{a (m+1) \sqrt{a+b x}} \]
Antiderivative was successfully verified.
[In] Integrate[x^m/(a + b*x)^(3/2),x]
[Out]
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Maple [F] time = 0.026, size = 0, normalized size = 0. \[ \int{{x}^{m} \left ( bx+a \right ) ^{-{\frac{3}{2}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^m/(b*x+a)^(3/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{m}}{{\left (b x + a\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^m/(b*x + a)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{m}}{{\left (b x + a\right )}^{\frac{3}{2}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^m/(b*x + a)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 6.79872, size = 36, normalized size = 0.78 \[ \frac{x x^{m} \Gamma \left (m + 1\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{2}, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{b x e^{i \pi }}{a}} \right )}}{a^{\frac{3}{2}} \Gamma \left (m + 2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**m/(b*x+a)**(3/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{m}}{{\left (b x + a\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^m/(b*x + a)^(3/2),x, algorithm="giac")
[Out]