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

Optimal. Leaf size=137 \[ -\frac{e \left (d^2-e^2 x^2\right )^{p-1} \, _2F_1\left (1,p-1;p;1-\frac{e^2 x^2}{d^2}\right )}{d (1-p)}-\frac{\left (d^2-e^2 x^2\right )^{p-1}}{x}+\frac{2 e^2 (2-p) x \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \left (d^2-e^2 x^2\right )^p \, _2F_1\left (\frac{1}{2},2-p;\frac{3}{2};\frac{e^2 x^2}{d^2}\right )}{d^4} \]

[Out]

-((d^2 - e^2*x^2)^(-1 + p)/x) + (2*e^2*(2 - p)*x*(d^2 - e^2*x^2)^p*Hypergeometri
c2F1[1/2, 2 - p, 3/2, (e^2*x^2)/d^2])/(d^4*(1 - (e^2*x^2)/d^2)^p) - (e*(d^2 - e^
2*x^2)^(-1 + p)*Hypergeometric2F1[1, -1 + p, p, 1 - (e^2*x^2)/d^2])/(d*(1 - p))

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Rubi [A]  time = 0.340382, antiderivative size = 137, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.28 \[ -\frac{e \left (d^2-e^2 x^2\right )^{p-1} \, _2F_1\left (1,p-1;p;1-\frac{e^2 x^2}{d^2}\right )}{d (1-p)}-\frac{\left (d^2-e^2 x^2\right )^{p-1}}{x}+\frac{2 e^2 (2-p) x \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \left (d^2-e^2 x^2\right )^p \, _2F_1\left (\frac{1}{2},2-p;\frac{3}{2};\frac{e^2 x^2}{d^2}\right )}{d^4} \]

Antiderivative was successfully verified.

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

[Out]

-((d^2 - e^2*x^2)^(-1 + p)/x) + (2*e^2*(2 - p)*x*(d^2 - e^2*x^2)^p*Hypergeometri
c2F1[1/2, 2 - p, 3/2, (e^2*x^2)/d^2])/(d^4*(1 - (e^2*x^2)/d^2)^p) - (e*(d^2 - e^
2*x^2)^(-1 + p)*Hypergeometric2F1[1, -1 + p, p, 1 - (e^2*x^2)/d^2])/(d*(1 - p))

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Rubi in Sympy [A]  time = 68.7371, size = 134, normalized size = 0.98 \[ - \frac{e \left (d^{2} - e^{2} x^{2}\right )^{p - 1}{{}_{2}F_{1}\left (\begin{matrix} 1, p - 1 \\ p \end{matrix}\middle |{1 - \frac{e^{2} x^{2}}{d^{2}}} \right )}}{d \left (- p + 1\right )} - \frac{\left (1 - \frac{e^{2} x^{2}}{d^{2}}\right )^{- p} \left (d^{2} - e^{2} x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p + 2, - \frac{1}{2} \\ \frac{1}{2} \end{matrix}\middle |{\frac{e^{2} x^{2}}{d^{2}}} \right )}}{d^{2} x} + \frac{e^{2} x \left (1 - \frac{e^{2} x^{2}}{d^{2}}\right )^{- p} \left (d^{2} - e^{2} x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p + 2, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{\frac{e^{2} x^{2}}{d^{2}}} \right )}}{d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-e*(d**2 - e**2*x**2)**(p - 1)*hyper((1, p - 1), (p,), 1 - e**2*x**2/d**2)/(d*(-
p + 1)) - (1 - e**2*x**2/d**2)**(-p)*(d**2 - e**2*x**2)**p*hyper((-p + 2, -1/2),
(1/2,), e**2*x**2/d**2)/(d**2*x) + e**2*x*(1 - e**2*x**2/d**2)**(-p)*(d**2 - e*
*2*x**2)**p*hyper((-p + 2, 1/2), (3/2,), e**2*x**2/d**2)/d**4

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Mathematica [C]  time = 0.475811, size = 195, normalized size = 1.42 \[ \frac{2 e (p-2) (d-e x)^p (d+e x)^{p-2} F_1\left (3-2 p;-p,2-p;4-2 p;\frac{d}{e x},-\frac{d}{e x}\right )}{(2 p-3) \left (2 e (p-2) x F_1\left (3-2 p;-p,2-p;4-2 p;\frac{d}{e x},-\frac{d}{e x}\right )+d p F_1\left (4-2 p;1-p,2-p;5-2 p;\frac{d}{e x},-\frac{d}{e x}\right )-d (p-2) F_1\left (4-2 p;-p,3-p;5-2 p;\frac{d}{e x},-\frac{d}{e x}\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(2*e*(-2 + p)*(d - e*x)^p*(d + e*x)^(-2 + p)*AppellF1[3 - 2*p, -p, 2 - p, 4 - 2*
p, d/(e*x), -(d/(e*x))])/((-3 + 2*p)*(2*e*(-2 + p)*x*AppellF1[3 - 2*p, -p, 2 - p
, 4 - 2*p, d/(e*x), -(d/(e*x))] + d*p*AppellF1[4 - 2*p, 1 - p, 2 - p, 5 - 2*p, d
/(e*x), -(d/(e*x))] - d*(-2 + p)*AppellF1[4 - 2*p, -p, 3 - p, 5 - 2*p, d/(e*x),
-(d/(e*x))]))

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Maple [F]  time = 0.08, size = 0, normalized size = 0. \[ \int{\frac{ \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{p}}{{x}^{2} \left ( ex+d \right ) ^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{p}}{e^{2} x^{4} + 2 \, d e x^{3} + d^{2} x^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((-e^2*x^2 + d^2)^p/(e^2*x^4 + 2*d*e*x^3 + d^2*x^2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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