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

Optimal. Leaf size=106 \[ \frac{e \left (d^2-e^2 x^2\right )^p \, _2F_1\left (1,p;p+1;1-\frac{e^2 x^2}{d^2}\right )}{2 d^2 p}-\frac{\left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (-\frac{1}{2},1-p;\frac{1}{2};\frac{e^2 x^2}{d^2}\right )}{d x} \]

[Out]

-(((d^2 - e^2*x^2)^p*Hypergeometric2F1[-1/2, 1 - p, 1/2, (e^2*x^2)/d^2])/(d*x*(1
- (e^2*x^2)/d^2)^p)) + (e*(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 = 0.212387, antiderivative size = 106, 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 \[ \frac{e \left (d^2-e^2 x^2\right )^p \, _2F_1\left (1,p;p+1;1-\frac{e^2 x^2}{d^2}\right )}{2 d^2 p}-\frac{\left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (-\frac{1}{2},1-p;\frac{1}{2};\frac{e^2 x^2}{d^2}\right )}{d x} \]

Antiderivative was successfully verified.

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

[Out]

-(((d^2 - e^2*x^2)^p*Hypergeometric2F1[-1/2, 1 - p, 1/2, (e^2*x^2)/d^2])/(d*x*(1
- (e^2*x^2)/d^2)^p)) + (e*(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 = 33.4714, size = 82, normalized size = 0.77 \[ - \frac{\left (1 - \frac{e^{2} x^{2}}{d^{2}}\right )^{- p} \left (d^{2} - e^{2} x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p + 1, - \frac{1}{2} \\ \frac{1}{2} \end{matrix}\middle |{\frac{e^{2} x^{2}}{d^{2}}} \right )}}{d x} + \frac{e \left (d^{2} - e^{2} x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} 1, p \\ p + 1 \end{matrix}\middle |{1 - \frac{e^{2} x^{2}}{d^{2}}} \right )}}{2 d^{2} p} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.2559, size = 167, normalized size = 1.58 \[ \frac{\left (d^2-e^2 x^2\right )^p \left (-\frac{d e \left (1-\frac{d^2}{e^2 x^2}\right )^{-p} \, _2F_1\left (-p,-p;1-p;\frac{d^2}{e^2 x^2}\right )}{p}-\frac{2 d^2 \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (-\frac{1}{2},-p;\frac{1}{2};\frac{e^2 x^2}{d^2}\right )}{x}+\frac{e 2^p (e x-d) \left (\frac{e x}{d}+1\right )^{-p} \, _2F_1\left (1-p,p+1;p+2;\frac{d-e x}{2 d}\right )}{p+1}\right )}{2 d^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.077, size = 0, normalized size = 0. \[ \int{\frac{ \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{p}}{{x}^{2} \left ( ex+d \right ) }}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 18.2182, size = 450, normalized size = 4.25 \[ \begin{cases} - \frac{0^{p} d^{2 p}}{d x} - \frac{0^{p} d^{2 p} e \log{\left (\frac{e^{2} x^{2}}{d^{2}} \right )}}{2 d^{2}} + \frac{0^{p} d^{2 p} e \log{\left (-1 + \frac{e^{2} x^{2}}{d^{2}} \right )}}{2 d^{2}} + \frac{0^{p} d^{2 p} e \operatorname{acoth}{\left (\frac{e x}{d} \right )}}{d^{2}} + \frac{d e^{2 p} p x^{2 p} e^{i \pi p} \Gamma \left (p\right ) \Gamma \left (- p + \frac{3}{2}\right ){{}_{2}F_{1}\left (\begin{matrix} - p + 1, - p + \frac{3}{2} \\ - p + \frac{5}{2} \end{matrix}\middle |{\frac{d^{2}}{e^{2} x^{2}}} \right )}}{2 e^{2} x^{3} \Gamma \left (- p + \frac{5}{2}\right ) \Gamma \left (p + 1\right )} - \frac{e^{2 p} p x^{2 p} e^{i \pi p} \Gamma \left (p\right ) \Gamma \left (- p + 1\right ){{}_{2}F_{1}\left (\begin{matrix} - p + 1, - p + 1 \\ - p + 2 \end{matrix}\middle |{\frac{d^{2}}{e^{2} x^{2}}} \right )}}{2 e x^{2} \Gamma \left (- p + 2\right ) \Gamma \left (p + 1\right )} & \text{for}\: \left |{\frac{e^{2} x^{2}}{d^{2}}}\right | > 1 \\- \frac{0^{p} d^{2 p}}{d x} - \frac{0^{p} d^{2 p} e \log{\left (\frac{e^{2} x^{2}}{d^{2}} \right )}}{2 d^{2}} + \frac{0^{p} d^{2 p} e \log{\left (1 - \frac{e^{2} x^{2}}{d^{2}} \right )}}{2 d^{2}} + \frac{0^{p} d^{2 p} e \operatorname{atanh}{\left (\frac{e x}{d} \right )}}{d^{2}} + \frac{d e^{2 p} p x^{2 p} e^{i \pi p} \Gamma \left (p\right ) \Gamma \left (- p + \frac{3}{2}\right ){{}_{2}F_{1}\left (\begin{matrix} - p + 1, - p + \frac{3}{2} \\ - p + \frac{5}{2} \end{matrix}\middle |{\frac{d^{2}}{e^{2} x^{2}}} \right )}}{2 e^{2} x^{3} \Gamma \left (- p + \frac{5}{2}\right ) \Gamma \left (p + 1\right )} - \frac{e^{2 p} p x^{2 p} e^{i \pi p} \Gamma \left (p\right ) \Gamma \left (- p + 1\right ){{}_{2}F_{1}\left (\begin{matrix} - p + 1, - p + 1 \\ - p + 2 \end{matrix}\middle |{\frac{d^{2}}{e^{2} x^{2}}} \right )}}{2 e x^{2} \Gamma \left (- p + 2\right ) \Gamma \left (p + 1\right )} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-0**p*d**(2*p)/(d*x) - 0**p*d**(2*p)*e*log(e**2*x**2/d**2)/(2*d**2) +
0**p*d**(2*p)*e*log(-1 + e**2*x**2/d**2)/(2*d**2) + 0**p*d**(2*p)*e*acoth(e*x/d
)/d**2 + d*e**(2*p)*p*x**(2*p)*exp(I*pi*p)*gamma(p)*gamma(-p + 3/2)*hyper((-p +
1, -p + 3/2), (-p + 5/2,), d**2/(e**2*x**2))/(2*e**2*x**3*gamma(-p + 5/2)*gamma(
p + 1)) - e**(2*p)*p*x**(2*p)*exp(I*pi*p)*gamma(p)*gamma(-p + 1)*hyper((-p + 1,
-p + 1), (-p + 2,), d**2/(e**2*x**2))/(2*e*x**2*gamma(-p + 2)*gamma(p + 1)), Abs
(e**2*x**2/d**2) > 1), (-0**p*d**(2*p)/(d*x) - 0**p*d**(2*p)*e*log(e**2*x**2/d**
2)/(2*d**2) + 0**p*d**(2*p)*e*log(1 - e**2*x**2/d**2)/(2*d**2) + 0**p*d**(2*p)*e
*atanh(e*x/d)/d**2 + d*e**(2*p)*p*x**(2*p)*exp(I*pi*p)*gamma(p)*gamma(-p + 3/2)*
hyper((-p + 1, -p + 3/2), (-p + 5/2,), d**2/(e**2*x**2))/(2*e**2*x**3*gamma(-p +
5/2)*gamma(p + 1)) - e**(2*p)*p*x**(2*p)*exp(I*pi*p)*gamma(p)*gamma(-p + 1)*hyp
er((-p + 1, -p + 1), (-p + 2,), d**2/(e**2*x**2))/(2*e*x**2*gamma(-p + 2)*gamma(
p + 1)), True))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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