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

Optimal. Leaf size=58 \[ -\frac{\left (a^2-b^2 x^2\right )^{p+1} \, _2F_1\left (1,2 p;p;\frac{a+b x}{2 a}\right )}{2 a b (1-p) (a+b x)^2} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0911718, antiderivative size = 73, normalized size of antiderivative = 1.26, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ -\frac{2^{p-2} \left (\frac{b x}{a}+1\right )^{-p-1} \left (a^2-b^2 x^2\right )^{p+1} \, _2F_1\left (2-p,p+1;p+2;\frac{a-b x}{2 a}\right )}{a^3 b (p+1)} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 26.2866, size = 66, normalized size = 1.14 \[ - \frac{\left (\frac{\frac{a}{2} + \frac{b x}{2}}{a}\right )^{- p} \left (a - b x\right )^{- p} \left (a - b x\right )^{p + 1} \left (a^{2} - b^{2} x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p + 2, p + 1 \\ p + 2 \end{matrix}\middle |{\frac{\frac{a}{2} - \frac{b x}{2}}{a}} \right )}}{4 a^{2} b \left (p + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-b**2*x**2+a**2)**p/(b*x+a)**2,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 + 2, p + 1), (p + 2,), (a/2 - b*x/2)/a)/(4*a**2*b*(p + 1))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0696443, size = 75, normalized size = 1.29 \[ -\frac{2^{p-2} (a-b x) \left (\frac{b x}{a}+1\right )^{-p} \left (a^2-b^2 x^2\right )^p \, _2F_1\left (2-p,p+1;p+2;\frac{a-b x}{2 a}\right )}{a^2 b (p+1)} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [F]  time = 0.087, size = 0, normalized size = 0. \[ \int{\frac{ \left ( -{b}^{2}{x}^{2}+{a}^{2} \right ) ^{p}}{ \left ( bx+a \right ) ^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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