3.964 \(\int \frac{\left (a^2-b^2 x^2\right )^p}{\sqrt{a+b x}} \, dx\)

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]

-((2^(-1/2 + p)*(1 + (b*x)/a)^(-1/2 - p)*(a^2 - b^2*x^2)^(1 + p)*Hypergeometric2
F1[1/2 - p, 1 + p, 2 + p, (a - b*x)/(2*a)])/(a*b*(1 + p)*Sqrt[a + b*x]))

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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]

-((2^(-1/2 + p)*(1 + (b*x)/a)^(-1/2 - p)*(a^2 - b^2*x^2)^(1 + p)*Hypergeometric2
F1[1/2 - p, 1 + p, 2 + p, (a - b*x)/(2*a)])/(a*b*(1 + p)*Sqrt[a + b*x]))

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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]

-((a/2 + b*x/2)/a)**(-p - 1/2)*(a - b*x)**(-p)*(a - b*x)**(p + 1)*(a + b*x)**(-p
)*(a + b*x)**(p + 1/2)*(a**2 - b**2*x**2)**p*hyper((-p + 1/2, p + 1), (p + 2,),
(a/2 - b*x/2)/a)/(2*a*b*(p + 1))

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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]

(2^(-1/2 + p)*(-a + b*x)*(1 + (b*x)/a)^(1/2 - p)*(a^2 - b^2*x^2)^p*Hypergeometri
c2F1[1/2 - p, 1 + p, 2 + p, (a - b*x)/(2*a)])/(b*(1 + p)*Sqrt[a + b*x])

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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]

int((-b^2*x^2+a^2)^p/(b*x+a)^(1/2),x)

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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]

integrate((-b^2*x^2 + a^2)^p/sqrt(b*x + a), x)

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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]

integral((-b^2*x^2 + a^2)^p/sqrt(b*x + a), x)

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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]

Integral((-(-a + b*x)*(a + b*x))**p/sqrt(a + b*x), x)

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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]

integrate((-b^2*x^2 + a^2)^p/sqrt(b*x + a), x)