3.302 \(\int \frac{\left (d^2-e^2 x^2\right )^p}{x^2 (d+e x)^4} \, dx\)

Optimal. Leaf size=207 \[ -\frac{2 e \left (d^2-e^2 x^2\right )^{p-2} \, _2F_1\left (1,p-2;p-1;1-\frac{e^2 x^2}{d^2}\right )}{d (2-p)}+\frac{e^2 x \left (d^2-e^2 x^2\right )^{p-3}}{5-2 p}-\frac{4 d e \left (d^2-e^2 x^2\right )^{p-3}}{3-p}-\frac{d^2 \left (d^2-e^2 x^2\right )^{p-3}}{x}+\frac{4 e^2 \left (p^2-9 p+16\right ) x \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \left (d^2-e^2 x^2\right )^p \, _2F_1\left (\frac{1}{2},4-p;\frac{3}{2};\frac{e^2 x^2}{d^2}\right )}{d^6 (5-2 p)} \]

[Out]

(-4*d*e*(d^2 - e^2*x^2)^(-3 + p))/(3 - p) - (d^2*(d^2 - e^2*x^2)^(-3 + p))/x + (
e^2*x*(d^2 - e^2*x^2)^(-3 + p))/(5 - 2*p) + (4*e^2*(16 - 9*p + p^2)*x*(d^2 - e^2
*x^2)^p*Hypergeometric2F1[1/2, 4 - p, 3/2, (e^2*x^2)/d^2])/(d^6*(5 - 2*p)*(1 - (
e^2*x^2)/d^2)^p) - (2*e*(d^2 - e^2*x^2)^(-2 + p)*Hypergeometric2F1[1, -2 + p, -1
+ p, 1 - (e^2*x^2)/d^2])/(d*(2 - p))

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Rubi [A]  time = 0.527922, antiderivative size = 207, 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 \[ -\frac{2 e \left (d^2-e^2 x^2\right )^{p-2} \, _2F_1\left (1,p-2;p-1;1-\frac{e^2 x^2}{d^2}\right )}{d (2-p)}+\frac{e^2 x \left (d^2-e^2 x^2\right )^{p-3}}{5-2 p}-\frac{4 d e \left (d^2-e^2 x^2\right )^{p-3}}{3-p}-\frac{d^2 \left (d^2-e^2 x^2\right )^{p-3}}{x}+\frac{4 e^2 \left (p^2-9 p+16\right ) x \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \left (d^2-e^2 x^2\right )^p \, _2F_1\left (\frac{1}{2},4-p;\frac{3}{2};\frac{e^2 x^2}{d^2}\right )}{d^6 (5-2 p)} \]

Antiderivative was successfully verified.

[In]  Int[(d^2 - e^2*x^2)^p/(x^2*(d + e*x)^4),x]

[Out]

(-4*d*e*(d^2 - e^2*x^2)^(-3 + p))/(3 - p) - (d^2*(d^2 - e^2*x^2)^(-3 + p))/x + (
e^2*x*(d^2 - e^2*x^2)^(-3 + p))/(5 - 2*p) + (4*e^2*(16 - 9*p + p^2)*x*(d^2 - e^2
*x^2)^p*Hypergeometric2F1[1/2, 4 - p, 3/2, (e^2*x^2)/d^2])/(d^6*(5 - 2*p)*(1 - (
e^2*x^2)/d^2)^p) - (2*e*(d^2 - e^2*x^2)^(-2 + p)*Hypergeometric2F1[1, -2 + p, -1
+ p, 1 - (e^2*x^2)/d^2])/(d*(2 - p))

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Rubi in Sympy [A]  time = 73.6857, size = 216, normalized size = 1.04 \[ - \frac{2 d e \left (d^{2} - e^{2} x^{2}\right )^{p - 3}{{}_{2}F_{1}\left (\begin{matrix} 1, p - 3 \\ p - 2 \end{matrix}\middle |{1 - \frac{e^{2} x^{2}}{d^{2}}} \right )}}{- p + 3} - \frac{2 d e \left (d^{2} - e^{2} x^{2}\right )^{p - 3}}{- p + 3} - \frac{\left (1 - \frac{e^{2} x^{2}}{d^{2}}\right )^{- p} \left (d^{2} - e^{2} x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p + 4, - \frac{1}{2} \\ \frac{1}{2} \end{matrix}\middle |{\frac{e^{2} x^{2}}{d^{2}}} \right )}}{d^{4} x} + \frac{6 e^{2} x \left (1 - \frac{e^{2} x^{2}}{d^{2}}\right )^{- p} \left (d^{2} - e^{2} x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p + 4, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{\frac{e^{2} x^{2}}{d^{2}}} \right )}}{d^{6}} + \frac{e^{4} x^{3} \left (1 - \frac{e^{2} x^{2}}{d^{2}}\right )^{- p} \left (d^{2} - e^{2} x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p + 4, \frac{3}{2} \\ \frac{5}{2} \end{matrix}\middle |{\frac{e^{2} x^{2}}{d^{2}}} \right )}}{3 d^{8}} \]

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)**4,x)

[Out]

-2*d*e*(d**2 - e**2*x**2)**(p - 3)*hyper((1, p - 3), (p - 2,), 1 - e**2*x**2/d**
2)/(-p + 3) - 2*d*e*(d**2 - e**2*x**2)**(p - 3)/(-p + 3) - (1 - e**2*x**2/d**2)*
*(-p)*(d**2 - e**2*x**2)**p*hyper((-p + 4, -1/2), (1/2,), e**2*x**2/d**2)/(d**4*
x) + 6*e**2*x*(1 - e**2*x**2/d**2)**(-p)*(d**2 - e**2*x**2)**p*hyper((-p + 4, 1/
2), (3/2,), e**2*x**2/d**2)/d**6 + e**4*x**3*(1 - e**2*x**2/d**2)**(-p)*(d**2 -
e**2*x**2)**p*hyper((-p + 4, 3/2), (5/2,), e**2*x**2/d**2)/(3*d**8)

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Mathematica [C]  time = 0.53431, size = 195, normalized size = 0.94 \[ \frac{2 e (p-3) (d-e x)^p (d+e x)^{p-4} F_1\left (5-2 p;-p,4-p;6-2 p;\frac{d}{e x},-\frac{d}{e x}\right )}{(2 p-5) \left (2 e (p-3) x F_1\left (5-2 p;-p,4-p;6-2 p;\frac{d}{e x},-\frac{d}{e x}\right )+d p F_1\left (6-2 p;1-p,4-p;7-2 p;\frac{d}{e x},-\frac{d}{e x}\right )-d (p-4) F_1\left (6-2 p;-p,5-p;7-2 p;\frac{d}{e x},-\frac{d}{e x}\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(d^2 - e^2*x^2)^p/(x^2*(d + e*x)^4),x]

[Out]

(2*e*(-3 + p)*(d - e*x)^p*(d + e*x)^(-4 + p)*AppellF1[5 - 2*p, -p, 4 - p, 6 - 2*
p, d/(e*x), -(d/(e*x))])/((-5 + 2*p)*(2*e*(-3 + p)*x*AppellF1[5 - 2*p, -p, 4 - p
, 6 - 2*p, d/(e*x), -(d/(e*x))] + d*p*AppellF1[6 - 2*p, 1 - p, 4 - p, 7 - 2*p, d
/(e*x), -(d/(e*x))] - d*(-4 + p)*AppellF1[6 - 2*p, -p, 5 - p, 7 - 2*p, d/(e*x),
-(d/(e*x))]))

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Maple [F]  time = 0.108, size = 0, normalized size = 0. \[ \int{\frac{ \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{p}}{{x}^{2} \left ( ex+d \right ) ^{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-e^2*x^2+d^2)^p/x^2/(e*x+d)^4,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e^2*x^2 + d^2)^p/((e*x + d)^4*x^2),x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{p}}{e^{4} x^{6} + 4 \, d e^{3} x^{5} + 6 \, d^{2} e^{2} x^{4} + 4 \, d^{3} e x^{3} + d^{4} x^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e^2*x^2 + d^2)^p/((e*x + d)^4*x^2),x, algorithm="fricas")

[Out]

integral((-e^2*x^2 + d^2)^p/(e^4*x^6 + 4*d*e^3*x^5 + 6*d^2*e^2*x^4 + 4*d^3*e*x^3
+ d^4*x^2), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{p}}{x^{2} \left (d + e x\right )^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e**2*x**2+d**2)**p/x**2/(e*x+d)**4,x)

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{p}}{{\left (e x + d\right )}^{4} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e^2*x^2 + d^2)^p/((e*x + d)^4*x^2),x, algorithm="giac")

[Out]

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