3.295 \(\int \frac{\left (d^2-e^2 x^2\right )^p}{x^5 (d+e x)^3} \, dx\)

Optimal. Leaf size=174 \[ -\frac{d \left (d^2-e^2 x^2\right )^{p-2}}{4 x^4}+\frac{e \left (d^2-e^2 x^2\right )^{p-2}}{x^3}+\frac{2 e^3 (4-p) \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \left (d^2-e^2 x^2\right )^p \, _2F_1\left (-\frac{1}{2},3-p;\frac{1}{2};\frac{e^2 x^2}{d^2}\right )}{d^6 x}+\frac{e^4 (10-p) \left (d^2-e^2 x^2\right )^{p-2} \, _2F_1\left (2,p-2;p-1;1-\frac{e^2 x^2}{d^2}\right )}{4 d^3 (2-p)} \]

[Out]

-(d*(d^2 - e^2*x^2)^(-2 + p))/(4*x^4) + (e*(d^2 - e^2*x^2)^(-2 + p))/x^3 + (2*e^
3*(4 - p)*(d^2 - e^2*x^2)^p*Hypergeometric2F1[-1/2, 3 - p, 1/2, (e^2*x^2)/d^2])/
(d^6*x*(1 - (e^2*x^2)/d^2)^p) + (e^4*(10 - p)*(d^2 - e^2*x^2)^(-2 + p)*Hypergeom
etric2F1[2, -2 + p, -1 + p, 1 - (e^2*x^2)/d^2])/(4*d^3*(2 - p))

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Rubi [A]  time = 0.519324, antiderivative size = 174, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.28 \[ -\frac{d \left (d^2-e^2 x^2\right )^{p-2}}{4 x^4}+\frac{e \left (d^2-e^2 x^2\right )^{p-2}}{x^3}+\frac{2 e^3 (4-p) \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \left (d^2-e^2 x^2\right )^p \, _2F_1\left (-\frac{1}{2},3-p;\frac{1}{2};\frac{e^2 x^2}{d^2}\right )}{d^6 x}+\frac{e^4 (10-p) \left (d^2-e^2 x^2\right )^{p-2} \, _2F_1\left (2,p-2;p-1;1-\frac{e^2 x^2}{d^2}\right )}{4 d^3 (2-p)} \]

Antiderivative was successfully verified.

[In]  Int[(d^2 - e^2*x^2)^p/(x^5*(d + e*x)^3),x]

[Out]

-(d*(d^2 - e^2*x^2)^(-2 + p))/(4*x^4) + (e*(d^2 - e^2*x^2)^(-2 + p))/x^3 + (2*e^
3*(4 - p)*(d^2 - e^2*x^2)^p*Hypergeometric2F1[-1/2, 3 - p, 1/2, (e^2*x^2)/d^2])/
(d^6*x*(1 - (e^2*x^2)/d^2)^p) + (e^4*(10 - p)*(d^2 - e^2*x^2)^(-2 + p)*Hypergeom
etric2F1[2, -2 + p, -1 + p, 1 - (e^2*x^2)/d^2])/(4*d^3*(2 - p))

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Rubi in Sympy [A]  time = 74.2036, size = 192, normalized size = 1.1 \[ \frac{3 e^{4} \left (d^{2} - e^{2} x^{2}\right )^{p - 2}{{}_{2}F_{1}\left (\begin{matrix} 2, p - 2 \\ p - 1 \end{matrix}\middle |{1 - \frac{e^{2} x^{2}}{d^{2}}} \right )}}{2 d^{3} \left (- p + 2\right )} + \frac{e^{4} \left (d^{2} - e^{2} x^{2}\right )^{p - 2}{{}_{2}F_{1}\left (\begin{matrix} 3, p - 2 \\ p - 1 \end{matrix}\middle |{1 - \frac{e^{2} x^{2}}{d^{2}}} \right )}}{2 d^{3} \left (- p + 2\right )} + \frac{e \left (1 - \frac{e^{2} x^{2}}{d^{2}}\right )^{- p} \left (d^{2} - e^{2} x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p + 3, - \frac{3}{2} \\ - \frac{1}{2} \end{matrix}\middle |{\frac{e^{2} x^{2}}{d^{2}}} \right )}}{d^{4} x^{3}} + \frac{e^{3} \left (1 - \frac{e^{2} x^{2}}{d^{2}}\right )^{- p} \left (d^{2} - e^{2} x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p + 3, - \frac{1}{2} \\ \frac{1}{2} \end{matrix}\middle |{\frac{e^{2} x^{2}}{d^{2}}} \right )}}{d^{6} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-e**2*x**2+d**2)**p/x**5/(e*x+d)**3,x)

[Out]

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

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Mathematica [B]  time = 1.01305, size = 446, normalized size = 2.56 \[ \frac{\left (d^2-e^2 x^2\right )^p \left (\frac{60 d e^4 \left (1-\frac{d^2}{e^2 x^2}\right )^{-p} \, _2F_1\left (-p,-p;1-p;\frac{d^2}{e^2 x^2}\right )}{p}+\frac{80 d^2 e^3 \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (-\frac{1}{2},-p;\frac{1}{2};\frac{e^2 x^2}{d^2}\right )}{x}+\frac{4 d^5 \left (1-\frac{d^2}{e^2 x^2}\right )^{-p} \, _2F_1\left (2-p,-p;3-p;\frac{d^2}{e^2 x^2}\right )}{(p-2) x^4}+\frac{8 d^4 e \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (-\frac{3}{2},-p;-\frac{1}{2};\frac{e^2 x^2}{d^2}\right )}{x^3}+\frac{24 d^3 e^2 \left (1-\frac{d^2}{e^2 x^2}\right )^{-p} \, _2F_1\left (1-p,-p;2-p;\frac{d^2}{e^2 x^2}\right )}{(p-1) x^2}+\frac{15 e^4 2^{p+2} (d-e x) \left (\frac{e x}{d}+1\right )^{-p} \, _2F_1\left (1-p,p+1;p+2;\frac{d-e x}{2 d}\right )}{p+1}+\frac{5 e^4 2^{p+1} (d-e x) \left (\frac{e x}{d}+1\right )^{-p} \, _2F_1\left (2-p,p+1;p+2;\frac{d-e x}{2 d}\right )}{p+1}+\frac{e^4 2^p (d-e x) \left (\frac{e x}{d}+1\right )^{-p} \, _2F_1\left (3-p,p+1;p+2;\frac{d-e x}{2 d}\right )}{p+1}\right )}{8 d^8} \]

Antiderivative was successfully verified.

[In]  Integrate[(d^2 - e^2*x^2)^p/(x^5*(d + e*x)^3),x]

[Out]

((d^2 - e^2*x^2)^p*((8*d^4*e*Hypergeometric2F1[-3/2, -p, -1/2, (e^2*x^2)/d^2])/(
x^3*(1 - (e^2*x^2)/d^2)^p) + (80*d^2*e^3*Hypergeometric2F1[-1/2, -p, 1/2, (e^2*x
^2)/d^2])/(x*(1 - (e^2*x^2)/d^2)^p) + (24*d^3*e^2*Hypergeometric2F1[1 - p, -p, 2
- p, d^2/(e^2*x^2)])/((-1 + p)*(1 - d^2/(e^2*x^2))^p*x^2) + (15*2^(2 + p)*e^4*(
d - e*x)*Hypergeometric2F1[1 - p, 1 + p, 2 + p, (d - e*x)/(2*d)])/((1 + p)*(1 +
(e*x)/d)^p) + (4*d^5*Hypergeometric2F1[2 - p, -p, 3 - p, d^2/(e^2*x^2)])/((-2 +
p)*(1 - d^2/(e^2*x^2))^p*x^4) + (5*2^(1 + p)*e^4*(d - e*x)*Hypergeometric2F1[2 -
p, 1 + p, 2 + p, (d - e*x)/(2*d)])/((1 + p)*(1 + (e*x)/d)^p) + (2^p*e^4*(d - e*
x)*Hypergeometric2F1[3 - p, 1 + p, 2 + p, (d - e*x)/(2*d)])/((1 + p)*(1 + (e*x)/
d)^p) + (60*d*e^4*Hypergeometric2F1[-p, -p, 1 - p, d^2/(e^2*x^2)])/(p*(1 - d^2/(
e^2*x^2))^p)))/(8*d^8)

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Maple [F]  time = 0.109, size = 0, normalized size = 0. \[ \int{\frac{ \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{p}}{{x}^{5} \left ( ex+d \right ) ^{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-e^2*x^2+d^2)^p/x^5/(e*x+d)^3,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e^2*x^2 + d^2)^p/((e*x + d)^3*x^5),x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{p}}{e^{3} x^{8} + 3 \, d e^{2} x^{7} + 3 \, d^{2} e x^{6} + d^{3} x^{5}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e^2*x^2 + d^2)^p/((e*x + d)^3*x^5),x, algorithm="fricas")

[Out]

integral((-e^2*x^2 + d^2)^p/(e^3*x^8 + 3*d*e^2*x^7 + 3*d^2*e*x^6 + d^3*x^5), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{p}}{x^{5} \left (d + e x\right )^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e**2*x**2+d**2)**p/x**5/(e*x+d)**3,x)

[Out]

Integral((-(-d + e*x)*(d + e*x))**p/(x**5*(d + e*x)**3), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{p}}{{\left (e x + d\right )}^{3} x^{5}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e^2*x^2 + d^2)^p/((e*x + d)^3*x^5),x, algorithm="giac")

[Out]

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