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

Optimal. Leaf size=139 \[ -\frac{2 d e \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (-\frac{1}{2},-p;\frac{1}{2};\frac{e^2 x^2}{d^2}\right )}{x}-\frac{e^2 (1-p) \left (d^2-e^2 x^2\right )^{p+1} \, _2F_1\left (1,p+1;p+2;1-\frac{e^2 x^2}{d^2}\right )}{2 d^2 (p+1)}-\frac{\left (d^2-e^2 x^2\right )^{p+1}}{2 x^2} \]

[Out]

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

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Rubi [A]  time = 0.248563, antiderivative size = 139, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24 \[ -\frac{2 d e \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (-\frac{1}{2},-p;\frac{1}{2};\frac{e^2 x^2}{d^2}\right )}{x}-\frac{e^2 (1-p) \left (d^2-e^2 x^2\right )^{p+1} \, _2F_1\left (1,p+1;p+2;1-\frac{e^2 x^2}{d^2}\right )}{2 d^2 (p+1)}-\frac{\left (d^2-e^2 x^2\right )^{p+1}}{2 x^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 35.0233, size = 136, normalized size = 0.98 \[ - \frac{2 d e \left (1 - \frac{e^{2} x^{2}}{d^{2}}\right )^{- p} \left (d^{2} - e^{2} x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, - \frac{1}{2} \\ \frac{1}{2} \end{matrix}\middle |{\frac{e^{2} x^{2}}{d^{2}}} \right )}}{x} - \frac{e^{2} \left (d^{2} - e^{2} x^{2}\right )^{p + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, p + 1 \\ p + 2 \end{matrix}\middle |{1 - \frac{e^{2} x^{2}}{d^{2}}} \right )}}{2 d^{2} \left (p + 1\right )} - \frac{e^{2} \left (d^{2} - e^{2} x^{2}\right )^{p + 1}{{}_{2}F_{1}\left (\begin{matrix} 2, p + 1 \\ p + 2 \end{matrix}\middle |{1 - \frac{e^{2} x^{2}}{d^{2}}} \right )}}{2 d^{2} \left (p + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.233056, size = 152, normalized size = 1.09 \[ \frac{\left (d^2-e^2 x^2\right )^p \left (\frac{\left (1-\frac{d^2}{e^2 x^2}\right )^{-p} \left (d^2 p \, _2F_1\left (1-p,-p;2-p;\frac{d^2}{e^2 x^2}\right )+e^2 (p-1) x^2 \, _2F_1\left (-p,-p;1-p;\frac{d^2}{e^2 x^2}\right )\right )}{(p-1) p}-4 d e x \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (-\frac{1}{2},-p;\frac{1}{2};\frac{e^2 x^2}{d^2}\right )\right )}{2 x^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.057, size = 0, normalized size = 0. \[ \int{\frac{ \left ( ex+d \right ) ^{2} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{p}}{{x}^{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 15.2524, size = 139, normalized size = 1. \[ - \frac{d^{2} e^{2 p} x^{2 p} e^{i \pi p} \Gamma \left (- p + 1\right ){{}_{2}F_{1}\left (\begin{matrix} - p, - p + 1 \\ - p + 2 \end{matrix}\middle |{\frac{d^{2}}{e^{2} x^{2}}} \right )}}{2 x^{2} \Gamma \left (- p + 2\right )} - \frac{2 d d^{2 p} e{{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, - p \\ \frac{1}{2} \end{matrix}\middle |{\frac{e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{x} - \frac{e^{2} e^{2 p} x^{2 p} e^{i \pi p} \Gamma \left (- p\right ){{}_{2}F_{1}\left (\begin{matrix} - p, - p \\ - p + 1 \end{matrix}\middle |{\frac{d^{2}}{e^{2} x^{2}}} \right )}}{2 \Gamma \left (- p + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-d**2*e**(2*p)*x**(2*p)*exp(I*pi*p)*gamma(-p + 1)*hyper((-p, -p + 1), (-p + 2,),
d**2/(e**2*x**2))/(2*x**2*gamma(-p + 2)) - 2*d*d**(2*p)*e*hyper((-1/2, -p), (1/
2,), e**2*x**2*exp_polar(2*I*pi)/d**2)/x - e**2*e**(2*p)*x**(2*p)*exp(I*pi*p)*ga
mma(-p)*hyper((-p, -p), (-p + 1,), d**2/(e**2*x**2))/(2*gamma(-p + 1))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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