3.292 \(\int \frac{\left (d^2-e^2 x^2\right )^p}{x^2 (d+e x)^3} \, dx\)

Optimal. Leaf size=166 \[ -\frac{3 e \left (d^2-e^2 x^2\right )^{p-1} \, _2F_1\left (1,p-1;p;1-\frac{e^2 x^2}{d^2}\right )}{2 d^2 (1-p)}-\frac{2 e \left (d^2-e^2 x^2\right )^{p-2}}{2-p}-\frac{d \left (d^2-e^2 x^2\right )^{p-2}}{x}+\frac{2 e^2 (4-p) x \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \left (d^2-e^2 x^2\right )^p \, _2F_1\left (\frac{1}{2},3-p;\frac{3}{2};\frac{e^2 x^2}{d^2}\right )}{d^5} \]

[Out]

(-2*e*(d^2 - e^2*x^2)^(-2 + p))/(2 - p) - (d*(d^2 - e^2*x^2)^(-2 + p))/x + (2*e^
2*(4 - p)*x*(d^2 - e^2*x^2)^p*Hypergeometric2F1[1/2, 3 - p, 3/2, (e^2*x^2)/d^2])
/(d^5*(1 - (e^2*x^2)/d^2)^p) - (3*e*(d^2 - e^2*x^2)^(-1 + p)*Hypergeometric2F1[1
, -1 + p, p, 1 - (e^2*x^2)/d^2])/(2*d^2*(1 - p))

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Rubi [A]  time = 0.430773, antiderivative size = 166, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.36 \[ -\frac{3 e \left (d^2-e^2 x^2\right )^{p-1} \, _2F_1\left (1,p-1;p;1-\frac{e^2 x^2}{d^2}\right )}{2 d^2 (1-p)}-\frac{2 e \left (d^2-e^2 x^2\right )^{p-2}}{2-p}-\frac{d \left (d^2-e^2 x^2\right )^{p-2}}{x}+\frac{2 e^2 (4-p) x \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \left (d^2-e^2 x^2\right )^p \, _2F_1\left (\frac{1}{2},3-p;\frac{3}{2};\frac{e^2 x^2}{d^2}\right )}{d^5} \]

Antiderivative was successfully verified.

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

[Out]

(-2*e*(d^2 - e^2*x^2)^(-2 + p))/(2 - p) - (d*(d^2 - e^2*x^2)^(-2 + p))/x + (2*e^
2*(4 - p)*x*(d^2 - e^2*x^2)^p*Hypergeometric2F1[1/2, 3 - p, 3/2, (e^2*x^2)/d^2])
/(d^5*(1 - (e^2*x^2)/d^2)^p) - (3*e*(d^2 - e^2*x^2)^(-1 + p)*Hypergeometric2F1[1
, -1 + p, p, 1 - (e^2*x^2)/d^2])/(2*d^2*(1 - p))

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Rubi in Sympy [A]  time = 74.2349, size = 160, normalized size = 0.96 \[ - \frac{3 e \left (d^{2} - e^{2} x^{2}\right )^{p - 2}{{}_{2}F_{1}\left (\begin{matrix} 1, p - 2 \\ p - 1 \end{matrix}\middle |{1 - \frac{e^{2} x^{2}}{d^{2}}} \right )}}{2 \left (- p + 2\right )} - \frac{e \left (d^{2} - e^{2} x^{2}\right )^{p - 2}}{2 \left (- p + 2\right )} - \frac{\left (1 - \frac{e^{2} x^{2}}{d^{2}}\right )^{- p} \left (d^{2} - e^{2} x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p + 3, - \frac{1}{2} \\ \frac{1}{2} \end{matrix}\middle |{\frac{e^{2} x^{2}}{d^{2}}} \right )}}{d^{3} x} + \frac{3 e^{2} x \left (1 - \frac{e^{2} x^{2}}{d^{2}}\right )^{- p} \left (d^{2} - e^{2} x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p + 3, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{\frac{e^{2} x^{2}}{d^{2}}} \right )}}{d^{5}} \]

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)**3,x)

[Out]

-3*e*(d**2 - e**2*x**2)**(p - 2)*hyper((1, p - 2), (p - 1,), 1 - e**2*x**2/d**2)
/(2*(-p + 2)) - e*(d**2 - e**2*x**2)**(p - 2)/(2*(-p + 2)) - (1 - e**2*x**2/d**2
)**(-p)*(d**2 - e**2*x**2)**p*hyper((-p + 3, -1/2), (1/2,), e**2*x**2/d**2)/(d**
3*x) + 3*e**2*x*(1 - e**2*x**2/d**2)**(-p)*(d**2 - e**2*x**2)**p*hyper((-p + 3,
1/2), (3/2,), e**2*x**2/d**2)/d**5

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Mathematica [C]  time = 0.481672, size = 198, normalized size = 1.19 \[ \frac{e (2 p-5) (d-e x)^p (d+e x)^{p-3} F_1\left (4-2 p;-p,3-p;5-2 p;\frac{d}{e x},-\frac{d}{e x}\right )}{2 (p-2) \left (e (2 p-5) x F_1\left (4-2 p;-p,3-p;5-2 p;\frac{d}{e x},-\frac{d}{e x}\right )+d p F_1\left (5-2 p;1-p,3-p;6-2 p;\frac{d}{e x},-\frac{d}{e x}\right )-d (p-3) F_1\left (5-2 p;-p,4-p;6-2 p;\frac{d}{e x},-\frac{d}{e x}\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(e*(-5 + 2*p)*(d - e*x)^p*(d + e*x)^(-3 + p)*AppellF1[4 - 2*p, -p, 3 - p, 5 - 2*
p, d/(e*x), -(d/(e*x))])/(2*(-2 + p)*(e*(-5 + 2*p)*x*AppellF1[4 - 2*p, -p, 3 - p
, 5 - 2*p, d/(e*x), -(d/(e*x))] + d*p*AppellF1[5 - 2*p, 1 - p, 3 - p, 6 - 2*p, d
/(e*x), -(d/(e*x))] - d*(-3 + p)*AppellF1[5 - 2*p, -p, 4 - p, 6 - 2*p, d/(e*x),
-(d/(e*x))]))

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Maple [F]  time = 0.105, size = 0, normalized size = 0. \[ \int{\frac{ \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{p}}{{x}^{2} \left ( ex+d \right ) ^{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{p}}{e^{3} x^{5} + 3 \, d e^{2} x^{4} + 3 \, d^{2} e x^{3} + d^{3} x^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((-e^2*x^2 + d^2)^p/(e^3*x^5 + 3*d*e^2*x^4 + 3*d^2*e*x^3 + d^3*x^2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e**2*x**2+d**2)**p/x**2/(e*x+d)**3,x)

[Out]

Integral((-(-d + e*x)*(d + e*x))**p/(x**2*(d + e*x)**3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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