Optimal. Leaf size=88 \[ -\frac{2^{p-\frac{1}{2}} \left (\frac{b x}{a}+1\right )^{-p-\frac{1}{2}} \left (a^2-b^2 x^2\right )^{p+1} \, _2F_1\left (\frac{1}{2}-p,p+1;p+2;\frac{a-b x}{2 a}\right )}{a b (p+1) \sqrt{a+b x}} \]
[Out]
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Rubi [A] time = 0.182121, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{2^{p-\frac{1}{2}} \left (\frac{b x}{a}+1\right )^{-p-\frac{1}{2}} \left (a^2-b^2 x^2\right )^{p+1} \, _2F_1\left (\frac{1}{2}-p,p+1;p+2;\frac{a-b x}{2 a}\right )}{a b (p+1) \sqrt{a+b x}} \]
Antiderivative was successfully verified.
[In] Int[(a^2 - b^2*x^2)^p/Sqrt[a + b*x],x]
[Out]
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Rubi in Sympy [A] time = 27.9202, size = 88, normalized size = 1. \[ - \frac{\left (\frac{\frac{a}{2} + \frac{b x}{2}}{a}\right )^{- p - \frac{1}{2}} \left (a - b x\right )^{- p} \left (a - b x\right )^{p + 1} \left (a + b x\right )^{- p} \left (a + b x\right )^{p + \frac{1}{2}} \left (a^{2} - b^{2} x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p + \frac{1}{2}, p + 1 \\ p + 2 \end{matrix}\middle |{\frac{\frac{a}{2} - \frac{b x}{2}}{a}} \right )}}{2 a b \left (p + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((-b**2*x**2+a**2)**p/(b*x+a)**(1/2),x)
[Out]
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Mathematica [A] time = 0.136037, size = 89, normalized size = 1.01 \[ \frac{2^{p-\frac{1}{2}} (b x-a) \left (\frac{b x}{a}+1\right )^{\frac{1}{2}-p} \left (a^2-b^2 x^2\right )^p \, _2F_1\left (\frac{1}{2}-p,p+1;p+2;\frac{a-b x}{2 a}\right )}{b (p+1) \sqrt{a+b x}} \]
Antiderivative was successfully verified.
[In] Integrate[(a^2 - b^2*x^2)^p/Sqrt[a + b*x],x]
[Out]
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Maple [F] time = 0.052, size = 0, normalized size = 0. \[ \int{ \left ( -{b}^{2}{x}^{2}+{a}^{2} \right ) ^{p}{\frac{1}{\sqrt{bx+a}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((-b^2*x^2+a^2)^p/(b*x+a)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-b^{2} x^{2} + a^{2}\right )}^{p}}{\sqrt{b x + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-b^2*x^2 + a^2)^p/sqrt(b*x + a),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (-b^{2} x^{2} + a^{2}\right )}^{p}}{\sqrt{b x + a}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-b^2*x^2 + a^2)^p/sqrt(b*x + a),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (- \left (- a + b x\right ) \left (a + b x\right )\right )^{p}}{\sqrt{a + b x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-b**2*x**2+a**2)**p/(b*x+a)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-b^{2} x^{2} + a^{2}\right )}^{p}}{\sqrt{b x + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-b^2*x^2 + a^2)^p/sqrt(b*x + a),x, algorithm="giac")
[Out]