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

Optimal. Leaf size=128 \[ -\frac{\left (d^2-e^2 x^2\right )^p \, _2F_1\left (1,p;p+1;1-\frac{e^2 x^2}{d^2}\right )}{2 d^2 p}+\frac{\left (d^2-e^2 x^2\right )^{p-1}}{1-p}-\frac{2 e x \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \left (d^2-e^2 x^2\right )^p \, _2F_1\left (\frac{1}{2},2-p;\frac{3}{2};\frac{e^2 x^2}{d^2}\right )}{d^3} \]

[Out]

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

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Rubi [A]  time = 0.274992, antiderivative size = 128, 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 \[ -\frac{\left (d^2-e^2 x^2\right )^p \, _2F_1\left (1,p;p+1;1-\frac{e^2 x^2}{d^2}\right )}{2 d^2 p}+\frac{\left (d^2-e^2 x^2\right )^{p-1}}{1-p}-\frac{2 e x \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \left (d^2-e^2 x^2\right )^p \, _2F_1\left (\frac{1}{2},2-p;\frac{3}{2};\frac{e^2 x^2}{d^2}\right )}{d^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 60.8306, size = 102, normalized size = 0.8 \[ \frac{\left (d^{2} - e^{2} x^{2}\right )^{p - 1}{{}_{2}F_{1}\left (\begin{matrix} 1, p - 1 \\ p \end{matrix}\middle |{1 - \frac{e^{2} x^{2}}{d^{2}}} \right )}}{2 \left (- p + 1\right )} + \frac{\left (d^{2} - e^{2} x^{2}\right )^{p - 1}}{2 \left (- p + 1\right )} - \frac{2 e x \left (1 - \frac{e^{2} x^{2}}{d^{2}}\right )^{- p} \left (d^{2} - e^{2} x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p + 2, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{\frac{e^{2} x^{2}}{d^{2}}} \right )}}{d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.223476, size = 201, normalized size = 1.57 \[ \frac{2^{p-2} \left (1-\frac{d^2}{e^2 x^2}\right )^{-p} \left (\frac{e x}{d}+1\right )^{-p} \left (d^2-e^2 x^2\right )^p \left (2 p (d-e x) \left (1-\frac{d^2}{e^2 x^2}\right )^p \, _2F_1\left (1-p,p+1;p+2;\frac{d-e x}{2 d}\right )+p (d-e x) \left (1-\frac{d^2}{e^2 x^2}\right )^p \, _2F_1\left (2-p,p+1;p+2;\frac{d-e x}{2 d}\right )+2 d (p+1) \left (\frac{e x}{2 d}+\frac{1}{2}\right )^p \, _2F_1\left (-p,-p;1-p;\frac{d^2}{e^2 x^2}\right )\right )}{d^3 p (p+1)} \]

Antiderivative was successfully verified.

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

[Out]

(2^(-2 + p)*(d^2 - e^2*x^2)^p*(2*p*(1 - d^2/(e^2*x^2))^p*(d - e*x)*Hypergeometri
c2F1[1 - p, 1 + p, 2 + p, (d - e*x)/(2*d)] + p*(1 - d^2/(e^2*x^2))^p*(d - e*x)*H
ypergeometric2F1[2 - p, 1 + p, 2 + p, (d - e*x)/(2*d)] + 2*d*(1 + p)*(1/2 + (e*x
)/(2*d))^p*Hypergeometric2F1[-p, -p, 1 - p, d^2/(e^2*x^2)]))/(d^3*p*(1 + p)*(1 -
d^2/(e^2*x^2))^p*(1 + (e*x)/d)^p)

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Maple [F]  time = 0.095, size = 0, normalized size = 0. \[ \int{\frac{ \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{p}}{x \left ( ex+d \right ) ^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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