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

Optimal. Leaf size=145 \[ -\frac{\left (d^2-e^2 x^2\right )^{p-1}}{3 x^3}-\frac{2 e^2 (4-p) \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \left (d^2-e^2 x^2\right )^p \, _2F_1\left (-\frac{1}{2},2-p;\frac{1}{2};\frac{e^2 x^2}{d^2}\right )}{3 d^4 x}-\frac{e^3 \left (d^2-e^2 x^2\right )^{p-1} \, _2F_1\left (2,p-1;p;1-\frac{e^2 x^2}{d^2}\right )}{d^3 (1-p)} \]

[Out]

-(d^2 - e^2*x^2)^(-1 + p)/(3*x^3) - (2*e^2*(4 - p)*(d^2 - e^2*x^2)^p*Hypergeomet
ric2F1[-1/2, 2 - p, 1/2, (e^2*x^2)/d^2])/(3*d^4*x*(1 - (e^2*x^2)/d^2)^p) - (e^3*
(d^2 - e^2*x^2)^(-1 + p)*Hypergeometric2F1[2, -1 + p, p, 1 - (e^2*x^2)/d^2])/(d^
3*(1 - p))

_______________________________________________________________________________________

Rubi [A]  time = 0.380129, antiderivative size = 145, 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 \[ -\frac{\left (d^2-e^2 x^2\right )^{p-1}}{3 x^3}-\frac{2 e^2 (4-p) \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \left (d^2-e^2 x^2\right )^p \, _2F_1\left (-\frac{1}{2},2-p;\frac{1}{2};\frac{e^2 x^2}{d^2}\right )}{3 d^4 x}-\frac{e^3 \left (d^2-e^2 x^2\right )^{p-1} \, _2F_1\left (2,p-1;p;1-\frac{e^2 x^2}{d^2}\right )}{d^3 (1-p)} \]

Antiderivative was successfully verified.

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

[Out]

-(d^2 - e^2*x^2)^(-1 + p)/(3*x^3) - (2*e^2*(4 - p)*(d^2 - e^2*x^2)^p*Hypergeomet
ric2F1[-1/2, 2 - p, 1/2, (e^2*x^2)/d^2])/(3*d^4*x*(1 - (e^2*x^2)/d^2)^p) - (e^3*
(d^2 - e^2*x^2)^(-1 + p)*Hypergeometric2F1[2, -1 + p, p, 1 - (e^2*x^2)/d^2])/(d^
3*(1 - p))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 62.504, size = 146, normalized size = 1.01 \[ - \frac{\left (1 - \frac{e^{2} x^{2}}{d^{2}}\right )^{- p} \left (d^{2} - e^{2} x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p + 2, - \frac{3}{2} \\ - \frac{1}{2} \end{matrix}\middle |{\frac{e^{2} x^{2}}{d^{2}}} \right )}}{3 d^{2} x^{3}} - \frac{e^{3} \left (d^{2} - e^{2} x^{2}\right )^{p - 1}{{}_{2}F_{1}\left (\begin{matrix} 2, p - 1 \\ p \end{matrix}\middle |{1 - \frac{e^{2} x^{2}}{d^{2}}} \right )}}{d^{3} \left (- p + 1\right )} - \frac{e^{2} \left (1 - \frac{e^{2} x^{2}}{d^{2}}\right )^{- p} \left (d^{2} - e^{2} x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p + 2, - \frac{1}{2} \\ \frac{1}{2} \end{matrix}\middle |{\frac{e^{2} x^{2}}{d^{2}}} \right )}}{d^{4} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [B]  time = 0.602752, size = 334, normalized size = 2.3 \[ \frac{\left (d^2-e^2 x^2\right )^p \left (-\frac{36 d^2 e^2 \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (-\frac{1}{2},-p;\frac{1}{2};\frac{e^2 x^2}{d^2}\right )}{x}-\frac{24 d e^3 \left (1-\frac{d^2}{e^2 x^2}\right )^{-p} \, _2F_1\left (-p,-p;1-p;\frac{d^2}{e^2 x^2}\right )}{p}-\frac{4 d^4 \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (-\frac{3}{2},-p;-\frac{1}{2};\frac{e^2 x^2}{d^2}\right )}{x^3}-\frac{12 d^3 e \left (1-\frac{d^2}{e^2 x^2}\right )^{-p} \, _2F_1\left (1-p,-p;2-p;\frac{d^2}{e^2 x^2}\right )}{(p-1) x^2}+\frac{3 e^3 2^{p+3} (e x-d) \left (\frac{e x}{d}+1\right )^{-p} \, _2F_1\left (1-p,p+1;p+2;\frac{d-e x}{2 d}\right )}{p+1}+\frac{3 e^3 2^p (e x-d) \left (\frac{e x}{d}+1\right )^{-p} \, _2F_1\left (2-p,p+1;p+2;\frac{d-e x}{2 d}\right )}{p+1}\right )}{12 d^6} \]

Antiderivative was successfully verified.

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

[Out]

((d^2 - e^2*x^2)^p*((-4*d^4*Hypergeometric2F1[-3/2, -p, -1/2, (e^2*x^2)/d^2])/(x
^3*(1 - (e^2*x^2)/d^2)^p) - (36*d^2*e^2*Hypergeometric2F1[-1/2, -p, 1/2, (e^2*x^
2)/d^2])/(x*(1 - (e^2*x^2)/d^2)^p) - (12*d^3*e*Hypergeometric2F1[1 - p, -p, 2 -
p, d^2/(e^2*x^2)])/((-1 + p)*(1 - d^2/(e^2*x^2))^p*x^2) + (3*2^(3 + p)*e^3*(-d +
e*x)*Hypergeometric2F1[1 - p, 1 + p, 2 + p, (d - e*x)/(2*d)])/((1 + p)*(1 + (e*
x)/d)^p) + (3*2^p*e^3*(-d + e*x)*Hypergeometric2F1[2 - p, 1 + p, 2 + p, (d - e*x
)/(2*d)])/((1 + p)*(1 + (e*x)/d)^p) - (24*d*e^3*Hypergeometric2F1[-p, -p, 1 - p,
d^2/(e^2*x^2)])/(p*(1 - d^2/(e^2*x^2))^p)))/(12*d^6)

_______________________________________________________________________________________

Maple [F]  time = 0.109, size = 0, normalized size = 0. \[ \int{\frac{ \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{p}}{{x}^{4} \left ( ex+d \right ) ^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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