3.305 \(\int \frac{\left (d^2-e^2 x^2\right )^p}{x^5 (d+e x)^4} \, dx\)

Optimal. Leaf size=216 \[ -\frac{e^2 (17-p) \left (d^2-e^2 x^2\right )^{p-3}}{4 x^2}-\frac{d^2 \left (d^2-e^2 x^2\right )^{p-3}}{4 x^4}+\frac{4 d e \left (d^2-e^2 x^2\right )^{p-3}}{3 x^3}+\frac{e^4 \left (p^2-21 p+70\right ) \left (d^2-e^2 x^2\right )^{p-3} \, _2F_1\left (1,p-3;p-2;1-\frac{e^2 x^2}{d^2}\right )}{4 d^2 (3-p)}+\frac{8 e^3 (6-p) \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \left (d^2-e^2 x^2\right )^p \, _2F_1\left (-\frac{1}{2},4-p;\frac{1}{2};\frac{e^2 x^2}{d^2}\right )}{3 d^7 x} \]

[Out]

-(d^2*(d^2 - e^2*x^2)^(-3 + p))/(4*x^4) + (4*d*e*(d^2 - e^2*x^2)^(-3 + p))/(3*x^
3) - (e^2*(17 - p)*(d^2 - e^2*x^2)^(-3 + p))/(4*x^2) + (8*e^3*(6 - p)*(d^2 - e^2
*x^2)^p*Hypergeometric2F1[-1/2, 4 - p, 1/2, (e^2*x^2)/d^2])/(3*d^7*x*(1 - (e^2*x
^2)/d^2)^p) + (e^4*(70 - 21*p + p^2)*(d^2 - e^2*x^2)^(-3 + p)*Hypergeometric2F1[
1, -3 + p, -2 + p, 1 - (e^2*x^2)/d^2])/(4*d^2*(3 - p))

_______________________________________________________________________________________

Rubi [A]  time = 0.695903, antiderivative size = 216, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.28 \[ -\frac{e^2 (17-p) \left (d^2-e^2 x^2\right )^{p-3}}{4 x^2}-\frac{d^2 \left (d^2-e^2 x^2\right )^{p-3}}{4 x^4}+\frac{4 d e \left (d^2-e^2 x^2\right )^{p-3}}{3 x^3}+\frac{e^4 \left (p^2-21 p+70\right ) \left (d^2-e^2 x^2\right )^{p-3} \, _2F_1\left (1,p-3;p-2;1-\frac{e^2 x^2}{d^2}\right )}{4 d^2 (3-p)}+\frac{8 e^3 (6-p) \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \left (d^2-e^2 x^2\right )^p \, _2F_1\left (-\frac{1}{2},4-p;\frac{1}{2};\frac{e^2 x^2}{d^2}\right )}{3 d^7 x} \]

Antiderivative was successfully verified.

[In]  Int[(d^2 - e^2*x^2)^p/(x^5*(d + e*x)^4),x]

[Out]

-(d^2*(d^2 - e^2*x^2)^(-3 + p))/(4*x^4) + (4*d*e*(d^2 - e^2*x^2)^(-3 + p))/(3*x^
3) - (e^2*(17 - p)*(d^2 - e^2*x^2)^(-3 + p))/(4*x^2) + (8*e^3*(6 - p)*(d^2 - e^2
*x^2)^p*Hypergeometric2F1[-1/2, 4 - p, 1/2, (e^2*x^2)/d^2])/(3*d^7*x*(1 - (e^2*x
^2)/d^2)^p) + (e^4*(70 - 21*p + p^2)*(d^2 - e^2*x^2)^(-3 + p)*Hypergeometric2F1[
1, -3 + p, -2 + p, 1 - (e^2*x^2)/d^2])/(4*d^2*(3 - p))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 87.492, size = 238, normalized size = 1.1 \[ \frac{e^{4} \left (d^{2} - e^{2} x^{2}\right )^{p - 3}{{}_{2}F_{1}\left (\begin{matrix} 1, p - 3 \\ p - 2 \end{matrix}\middle |{1 - \frac{e^{2} x^{2}}{d^{2}}} \right )}}{2 d^{2} \left (- p + 3\right )} + \frac{3 e^{4} \left (d^{2} - e^{2} x^{2}\right )^{p - 3}{{}_{2}F_{1}\left (\begin{matrix} 2, p - 3 \\ p - 2 \end{matrix}\middle |{1 - \frac{e^{2} x^{2}}{d^{2}}} \right )}}{d^{2} \left (- p + 3\right )} + \frac{e^{4} \left (d^{2} - e^{2} x^{2}\right )^{p - 3}{{}_{2}F_{1}\left (\begin{matrix} 3, p - 3 \\ p - 2 \end{matrix}\middle |{1 - \frac{e^{2} x^{2}}{d^{2}}} \right )}}{2 d^{2} \left (- p + 3\right )} + \frac{4 e \left (1 - \frac{e^{2} x^{2}}{d^{2}}\right )^{- p} \left (d^{2} - e^{2} x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p + 4, - \frac{3}{2} \\ - \frac{1}{2} \end{matrix}\middle |{\frac{e^{2} x^{2}}{d^{2}}} \right )}}{3 d^{5} x^{3}} + \frac{4 e^{3} \left (1 - \frac{e^{2} x^{2}}{d^{2}}\right )^{- p} \left (d^{2} - e^{2} x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p + 4, - \frac{1}{2} \\ \frac{1}{2} \end{matrix}\middle |{\frac{e^{2} x^{2}}{d^{2}}} \right )}}{d^{7} x} \]

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)**4,x)

[Out]

e**4*(d**2 - e**2*x**2)**(p - 3)*hyper((1, p - 3), (p - 2,), 1 - e**2*x**2/d**2)
/(2*d**2*(-p + 3)) + 3*e**4*(d**2 - e**2*x**2)**(p - 3)*hyper((2, p - 3), (p - 2
,), 1 - e**2*x**2/d**2)/(d**2*(-p + 3)) + e**4*(d**2 - e**2*x**2)**(p - 3)*hyper
((3, p - 3), (p - 2,), 1 - e**2*x**2/d**2)/(2*d**2*(-p + 3)) + 4*e*(1 - e**2*x**
2/d**2)**(-p)*(d**2 - e**2*x**2)**p*hyper((-p + 4, -3/2), (-1/2,), e**2*x**2/d**
2)/(3*d**5*x**3) + 4*e**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**7*x)

_______________________________________________________________________________________

Mathematica [B]  time = 1.42239, size = 505, normalized size = 2.34 \[ \frac{\left (d^2-e^2 x^2\right )^p \left (\frac{840 d e^4 \left (1-\frac{d^2}{e^2 x^2}\right )^{-p} \, _2F_1\left (-p,-p;1-p;\frac{d^2}{e^2 x^2}\right )}{p}+\frac{960 d^2 e^3 \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (-\frac{1}{2},-p;\frac{1}{2};\frac{e^2 x^2}{d^2}\right )}{x}+\frac{24 d^5 \left (1-\frac{d^2}{e^2 x^2}\right )^{-p} \, _2F_1\left (2-p,-p;3-p;\frac{d^2}{e^2 x^2}\right )}{(p-2) x^4}+\frac{64 d^4 e \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (-\frac{3}{2},-p;-\frac{1}{2};\frac{e^2 x^2}{d^2}\right )}{x^3}+\frac{240 d^3 e^2 \left (1-\frac{d^2}{e^2 x^2}\right )^{-p} \, _2F_1\left (1-p,-p;2-p;\frac{d^2}{e^2 x^2}\right )}{(p-1) x^2}+\frac{105 e^4 2^{p+3} (d-e x) \left (\frac{e x}{d}+1\right )^{-p} \, _2F_1\left (1-p,p+1;p+2;\frac{d-e x}{2 d}\right )}{p+1}+\frac{45 e^4 2^{p+2} (d-e x) \left (\frac{e x}{d}+1\right )^{-p} \, _2F_1\left (2-p,p+1;p+2;\frac{d-e x}{2 d}\right )}{p+1}+\frac{15 e^4 2^{p+1} (d-e x) \left (\frac{e x}{d}+1\right )^{-p} \, _2F_1\left (3-p,p+1;p+2;\frac{d-e x}{2 d}\right )}{p+1}+\frac{3 e^4 2^p (d-e x) \left (\frac{e x}{d}+1\right )^{-p} \, _2F_1\left (4-p,p+1;p+2;\frac{d-e x}{2 d}\right )}{p+1}\right )}{48 d^9} \]

Antiderivative was successfully verified.

[In]  Integrate[(d^2 - e^2*x^2)^p/(x^5*(d + e*x)^4),x]

[Out]

((d^2 - e^2*x^2)^p*((64*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) + (960*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) + (240*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) + (105*2^(3 + 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) + (24*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) + (45*2^(2 + p)*e^4*(d - e*x)*Hypergeometric2
F1[2 - p, 1 + p, 2 + p, (d - e*x)/(2*d)])/((1 + p)*(1 + (e*x)/d)^p) + (15*2^(1 +
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) + (3*2^p*e^4*(d - e*x)*Hypergeometric2F1[4 - p, 1 + p, 2 +
p, (d - e*x)/(2*d)])/((1 + p)*(1 + (e*x)/d)^p) + (840*d*e^4*Hypergeometric2F1[-p
, -p, 1 - p, d^2/(e^2*x^2)])/(p*(1 - d^2/(e^2*x^2))^p)))/(48*d^9)

_______________________________________________________________________________________

Maple [F]  time = 0.104, size = 0, normalized size = 0. \[ \int{\frac{ \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{p}}{{x}^{5} \left ( ex+d \right ) ^{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-e^2*x^2+d^2)^p/x^5/(e*x+d)^4,x)

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e^2*x^2 + d^2)^p/((e*x + d)^4*x^5),x, algorithm="maxima")

[Out]

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

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{p}}{e^{4} x^{9} + 4 \, d e^{3} x^{8} + 6 \, d^{2} e^{2} x^{7} + 4 \, d^{3} e x^{6} + d^{4} x^{5}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e^2*x^2 + d^2)^p/((e*x + d)^4*x^5),x, algorithm="fricas")

[Out]

integral((-e^2*x^2 + d^2)^p/(e^4*x^9 + 4*d*e^3*x^8 + 6*d^2*e^2*x^7 + 4*d^3*e*x^6
+ d^4*x^5), x)

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{p}}{x^{5} \left (d + e x\right )^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e**2*x**2+d**2)**p/x**5/(e*x+d)**4,x)

[Out]

Integral((-(-d + e*x)*(d + e*x))**p/(x**5*(d + e*x)**4), x)

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{p}}{{\left (e x + d\right )}^{4} x^{5}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e^2*x^2 + d^2)^p/((e*x + d)^4*x^5),x, algorithm="giac")

[Out]

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