3.954 \(\int \frac{\left (1-\frac{e^2 x^2}{d^2}\right )^p}{(d+e x)^2} \, dx\)

Optimal. Leaf size=57 \[ -\frac{2^{p-2} \left (\frac{d-e x}{d}\right )^{p+1} \, _2F_1\left (2-p,p+1;p+2;\frac{d-e x}{2 d}\right )}{d e (p+1)} \]

[Out]

-((2^(-2 + p)*((d - e*x)/d)^(1 + p)*Hypergeometric2F1[2 - p, 1 + p, 2 + p, (d -
e*x)/(2*d)])/(d*e*(1 + p)))

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Rubi [A]  time = 0.110468, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087 \[ -\frac{2^{p-2} \left (\frac{d-e x}{d}\right )^{p+1} \, _2F_1\left (2-p,p+1;p+2;\frac{d-e x}{2 d}\right )}{d e (p+1)} \]

Antiderivative was successfully verified.

[In]  Int[(1 - (e^2*x^2)/d^2)^p/(d + e*x)^2,x]

[Out]

-((2^(-2 + p)*((d - e*x)/d)^(1 + p)*Hypergeometric2F1[2 - p, 1 + p, 2 + p, (d -
e*x)/(2*d)])/(d*e*(1 + p)))

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Rubi in Sympy [A]  time = 30.5739, size = 75, normalized size = 1.32 \[ - \frac{\left (\frac{\frac{d}{2} + \frac{e x}{2}}{d}\right )^{- p} \left (1 - \frac{e^{2} x^{2}}{d^{2}}\right )^{p} \left (\frac{1}{d} - \frac{e x}{d^{2}}\right )^{- p} \left (\frac{1}{d} - \frac{e x}{d^{2}}\right )^{p + 1}{{}_{2}F_{1}\left (\begin{matrix} - p + 2, p + 1 \\ p + 2 \end{matrix}\middle |{\frac{1}{2} - \frac{e x}{2 d}} \right )}}{4 e \left (p + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1-e**2*x**2/d**2)**p/(e*x+d)**2,x)

[Out]

-((d/2 + e*x/2)/d)**(-p)*(1 - e**2*x**2/d**2)**p*(1/d - e*x/d**2)**(-p)*(1/d - e
*x/d**2)**(p + 1)*hyper((-p + 2, p + 1), (p + 2,), 1/2 - e*x/(2*d))/(4*e*(p + 1)
)

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Mathematica [A]  time = 0.106841, size = 76, normalized size = 1.33 \[ -\frac{2^{p-2} (d-e x) \left (\frac{e x}{d}+1\right )^{-p} \left (1-\frac{e^2 x^2}{d^2}\right )^p \, _2F_1\left (2-p,p+1;p+2;\frac{d-e x}{2 d}\right )}{d^2 e (p+1)} \]

Antiderivative was successfully verified.

[In]  Integrate[(1 - (e^2*x^2)/d^2)^p/(d + e*x)^2,x]

[Out]

-((2^(-2 + p)*(d - e*x)*(1 - (e^2*x^2)/d^2)^p*Hypergeometric2F1[2 - p, 1 + p, 2
+ p, (d - e*x)/(2*d)])/(d^2*e*(1 + p)*(1 + (e*x)/d)^p))

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Maple [F]  time = 0.093, size = 0, normalized size = 0. \[ \int{\frac{1}{ \left ( ex+d \right ) ^{2}} \left ( 1-{\frac{{e}^{2}{x}^{2}}{{d}^{2}}} \right ) ^{p}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-e^2*x^2/d^2)^p/(e*x+d)^2,x)

[Out]

int((1-e^2*x^2/d^2)^p/(e*x+d)^2,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-\frac{e^{2} x^{2}}{d^{2}} + 1\right )}^{p}}{{\left (e x + d\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e^2*x^2/d^2 + 1)^p/(e*x + d)^2,x, algorithm="maxima")

[Out]

integrate((-e^2*x^2/d^2 + 1)^p/(e*x + d)^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e^2*x^2/d^2 + 1)^p/(e*x + d)^2,x, algorithm="fricas")

[Out]

integral((-(e^2*x^2 - d^2)/d^2)^p/(e^2*x^2 + 2*d*e*x + d^2), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (- \left (-1 + \frac{e x}{d}\right ) \left (1 + \frac{e x}{d}\right )\right )^{p}}{\left (d + e x\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((1-e**2*x**2/d**2)**p/(e*x+d)**2,x)

[Out]

Integral((-(-1 + e*x/d)*(1 + e*x/d))**p/(d + e*x)**2, x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-\frac{e^{2} x^{2}}{d^{2}} + 1\right )}^{p}}{{\left (e x + d\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e^2*x^2/d^2 + 1)^p/(e*x + d)^2,x, algorithm="giac")

[Out]

integrate((-e^2*x^2/d^2 + 1)^p/(e*x + d)^2, x)