Optimal. Leaf size=88 \[ -\frac{2^{p+\frac{1}{2}} \sqrt{a+b x} \left (\frac{b x}{a}+1\right )^{-p-\frac{3}{2}} \left (a^2-b^2 x^2\right )^{p+1} \, _2F_1\left (-p-\frac{1}{2},p+1;p+2;\frac{a-b x}{2 a}\right )}{a b (p+1)} \]
[Out]
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Rubi [A] time = 0.163843, 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 \[ -\frac{2^{p+\frac{1}{2}} \sqrt{a+b x} \left (\frac{b x}{a}+1\right )^{-p-\frac{3}{2}} \left (a^2-b^2 x^2\right )^{p+1} \, _2F_1\left (-p-\frac{1}{2},p+1;p+2;\frac{a-b x}{2 a}\right )}{a b (p+1)} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[a + b*x]*(a^2 - b^2*x^2)^p,x]
[Out]
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Rubi in Sympy [A] time = 27.2303, size = 87, normalized size = 0.99 \[ - \frac{\left (\frac{\frac{a}{2} + \frac{b x}{2}}{a}\right )^{- p - \frac{1}{2}} \left (a - b x\right )^{- p} \left (a - b x\right )^{p + 1} \left (a + b x\right )^{- p} \left (a + b x\right )^{p + \frac{1}{2}} \left (a^{2} - b^{2} x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p - \frac{1}{2}, p + 1 \\ p + 2 \end{matrix}\middle |{\frac{\frac{a}{2} - \frac{b x}{2}}{a}} \right )}}{b \left (p + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**(1/2)*(-b**2*x**2+a**2)**p,x)
[Out]
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Mathematica [A] time = 0.117504, size = 89, normalized size = 1.01 \[ \frac{2^{p+\frac{1}{2}} (b x-a) \sqrt{a+b x} \left (\frac{b x}{a}+1\right )^{-p-\frac{1}{2}} \left (a^2-b^2 x^2\right )^p \, _2F_1\left (-p-\frac{1}{2},p+1;p+2;\frac{a-b x}{2 a}\right )}{b (p+1)} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[a + b*x]*(a^2 - b^2*x^2)^p,x]
[Out]
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Maple [F] time = 0.049, size = 0, normalized size = 0. \[ \int \sqrt{bx+a} \left ( -{b}^{2}{x}^{2}+{a}^{2} \right ) ^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^(1/2)*(-b^2*x^2+a^2)^p,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{b x + a}{\left (-b^{2} x^{2} + a^{2}\right )}^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*x + a)*(-b^2*x^2 + a^2)^p,x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\sqrt{b x + a}{\left (-b^{2} x^{2} + a^{2}\right )}^{p}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*x + a)*(-b^2*x^2 + a^2)^p,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((b*x+a)**(1/2)*(-b**2*x**2+a**2)**p,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{b x + a}{\left (-b^{2} x^{2} + a^{2}\right )}^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*x + a)*(-b^2*x^2 + a^2)^p,x, algorithm="giac")
[Out]