3.961 \(\int \frac{\left (a^2-b^2 x^2\right )^p}{(a+b x)^3} \, dx\)

Optimal. Leaf size=62 \[ -\frac{\left (a^2-b^2 x^2\right )^{p+1} \, _2F_1\left (1,2 p-1;p-1;\frac{a+b x}{2 a}\right )}{2 a b (2-p) (a+b x)^3} \]

[Out]

-((a^2 - b^2*x^2)^(1 + p)*Hypergeometric2F1[1, -1 + 2*p, -1 + p, (a + b*x)/(2*a)
])/(2*a*b*(2 - p)*(a + b*x)^3)

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Rubi [A]  time = 0.0909334, antiderivative size = 73, normalized size of antiderivative = 1.18, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ -\frac{2^{p-3} \left (\frac{b x}{a}+1\right )^{-p-1} \left (a^2-b^2 x^2\right )^{p+1} \, _2F_1\left (3-p,p+1;p+2;\frac{a-b x}{2 a}\right )}{a^4 b (p+1)} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 - b^2*x^2)^p/(a + b*x)^3,x]

[Out]

-((2^(-3 + p)*(1 + (b*x)/a)^(-1 - p)*(a^2 - b^2*x^2)^(1 + p)*Hypergeometric2F1[3
- p, 1 + p, 2 + p, (a - b*x)/(2*a)])/(a^4*b*(1 + p)))

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Rubi in Sympy [A]  time = 29.7058, size = 66, normalized size = 1.06 \[ - \frac{\left (\frac{\frac{a}{2} + \frac{b x}{2}}{a}\right )^{- p} \left (a - b x\right )^{- p} \left (a - b x\right )^{p + 1} \left (a^{2} - b^{2} x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p + 3, p + 1 \\ p + 2 \end{matrix}\middle |{\frac{\frac{a}{2} - \frac{b x}{2}}{a}} \right )}}{8 a^{3} b \left (p + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-b**2*x**2+a**2)**p/(b*x+a)**3,x)

[Out]

-((a/2 + b*x/2)/a)**(-p)*(a - b*x)**(-p)*(a - b*x)**(p + 1)*(a**2 - b**2*x**2)**
p*hyper((-p + 3, p + 1), (p + 2,), (a/2 - b*x/2)/a)/(8*a**3*b*(p + 1))

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Mathematica [A]  time = 0.0701105, size = 75, normalized size = 1.21 \[ -\frac{2^{p-3} (a-b x) \left (\frac{b x}{a}+1\right )^{-p} \left (a^2-b^2 x^2\right )^p \, _2F_1\left (3-p,p+1;p+2;\frac{a-b x}{2 a}\right )}{a^3 b (p+1)} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 - b^2*x^2)^p/(a + b*x)^3,x]

[Out]

-((2^(-3 + p)*(a - b*x)*(a^2 - b^2*x^2)^p*Hypergeometric2F1[3 - p, 1 + p, 2 + p,
(a - b*x)/(2*a)])/(a^3*b*(1 + p)*(1 + (b*x)/a)^p))

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Maple [F]  time = 0.119, size = 0, normalized size = 0. \[ \int{\frac{ \left ( -{b}^{2}{x}^{2}+{a}^{2} \right ) ^{p}}{ \left ( bx+a \right ) ^{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-b^2*x^2+a^2)^p/(b*x+a)^3,x)

[Out]

int((-b^2*x^2+a^2)^p/(b*x+a)^3,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-b^{2} x^{2} + a^{2}\right )}^{p}}{{\left (b x + a\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b^2*x^2 + a^2)^p/(b*x + a)^3,x, algorithm="maxima")

[Out]

integrate((-b^2*x^2 + a^2)^p/(b*x + a)^3, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (-b^{2} x^{2} + a^{2}\right )}^{p}}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b^2*x^2 + a^2)^p/(b*x + a)^3,x, algorithm="fricas")

[Out]

integral((-b^2*x^2 + a^2)^p/(b^3*x^3 + 3*a*b^2*x^2 + 3*a^2*b*x + a^3), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (- \left (- a + b x\right ) \left (a + b x\right )\right )^{p}}{\left (a + b x\right )^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b**2*x**2+a**2)**p/(b*x+a)**3,x)

[Out]

Integral((-(-a + b*x)*(a + b*x))**p/(a + b*x)**3, x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-b^{2} x^{2} + a^{2}\right )}^{p}}{{\left (b x + a\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b^2*x^2 + a^2)^p/(b*x + a)^3,x, algorithm="giac")

[Out]

integrate((-b^2*x^2 + a^2)^p/(b*x + a)^3, x)