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

Optimal. Leaf size=175 \[ \frac{\left (d^2-e^2 x^2\right )^{p-1} \, _2F_1\left (1,p-1;p;1-\frac{e^2 x^2}{d^2}\right )}{2 d (1-p)}-\frac{e x \left (d^2-e^2 x^2\right )^{p-2}}{3-2 p}+\frac{2 d \left (d^2-e^2 x^2\right )^{p-2}}{2-p}-\frac{2 e (4-3 p) x \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \left (d^2-e^2 x^2\right )^p \, _2F_1\left (\frac{1}{2},3-p;\frac{3}{2};\frac{e^2 x^2}{d^2}\right )}{d^4 (3-2 p)} \]

[Out]

(2*d*(d^2 - e^2*x^2)^(-2 + p))/(2 - p) - (e*x*(d^2 - e^2*x^2)^(-2 + p))/(3 - 2*p
) - (2*e*(4 - 3*p)*x*(d^2 - e^2*x^2)^p*Hypergeometric2F1[1/2, 3 - p, 3/2, (e^2*x
^2)/d^2])/(d^4*(3 - 2*p)*(1 - (e^2*x^2)/d^2)^p) + ((d^2 - e^2*x^2)^(-1 + p)*Hype
rgeometric2F1[1, -1 + p, p, 1 - (e^2*x^2)/d^2])/(2*d*(1 - p))

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Rubi [A]  time = 0.359227, antiderivative size = 175, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.32 \[ \frac{\left (d^2-e^2 x^2\right )^{p-1} \, _2F_1\left (1,p-1;p;1-\frac{e^2 x^2}{d^2}\right )}{2 d (1-p)}-\frac{e x \left (d^2-e^2 x^2\right )^{p-2}}{3-2 p}+\frac{2 d \left (d^2-e^2 x^2\right )^{p-2}}{2-p}-\frac{2 e (4-3 p) x \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \left (d^2-e^2 x^2\right )^p \, _2F_1\left (\frac{1}{2},3-p;\frac{3}{2};\frac{e^2 x^2}{d^2}\right )}{d^4 (3-2 p)} \]

Antiderivative was successfully verified.

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

[Out]

(2*d*(d^2 - e^2*x^2)^(-2 + p))/(2 - p) - (e*x*(d^2 - e^2*x^2)^(-2 + p))/(3 - 2*p
) - (2*e*(4 - 3*p)*x*(d^2 - e^2*x^2)^p*Hypergeometric2F1[1/2, 3 - p, 3/2, (e^2*x
^2)/d^2])/(d^4*(3 - 2*p)*(1 - (e^2*x^2)/d^2)^p) + ((d^2 - e^2*x^2)^(-1 + p)*Hype
rgeometric2F1[1, -1 + p, p, 1 - (e^2*x^2)/d^2])/(2*d*(1 - p))

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Rubi in Sympy [A]  time = 75.9342, size = 163, normalized size = 0.93 \[ \frac{d \left (d^{2} - e^{2} x^{2}\right )^{p - 2}{{}_{2}F_{1}\left (\begin{matrix} 1, p - 2 \\ p - 1 \end{matrix}\middle |{1 - \frac{e^{2} x^{2}}{d^{2}}} \right )}}{2 \left (- p + 2\right )} + \frac{3 d \left (d^{2} - e^{2} x^{2}\right )^{p - 2}}{2 \left (- p + 2\right )} - \frac{3 e x \left (1 - \frac{e^{2} x^{2}}{d^{2}}\right )^{- p} \left (d^{2} - e^{2} x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p + 3, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{\frac{e^{2} x^{2}}{d^{2}}} \right )}}{d^{4}} - \frac{e^{3} x^{3} \left (1 - \frac{e^{2} x^{2}}{d^{2}}\right )^{- p} \left (d^{2} - e^{2} x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p + 3, \frac{3}{2} \\ \frac{5}{2} \end{matrix}\middle |{\frac{e^{2} x^{2}}{d^{2}}} \right )}}{3 d^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.474111, size = 196, normalized size = 1.12 \[ \frac{2 e (p-2) x (d-e x)^p (d+e x)^{p-3} F_1\left (3-2 p;-p,3-p;4-2 p;\frac{d}{e x},-\frac{d}{e x}\right )}{(2 p-3) \left (2 e (p-2) x F_1\left (3-2 p;-p,3-p;4-2 p;\frac{d}{e x},-\frac{d}{e x}\right )+d p F_1\left (4-2 p;1-p,3-p;5-2 p;\frac{d}{e x},-\frac{d}{e x}\right )-d (p-3) F_1\left (4-2 p;-p,4-p;5-2 p;\frac{d}{e x},-\frac{d}{e x}\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

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

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Maple [F]  time = 0.072, size = 0, normalized size = 0. \[ \int{\frac{ \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{p}}{x \left ( ex+d \right ) ^{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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