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

Optimal. Leaf size=104 \[ -\frac{e x \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac{1}{2},1-p;\frac{3}{2};\frac{e^2 x^2}{d^2}\right )}{d^2}-\frac{\left (d^2-e^2 x^2\right )^p \, _2F_1\left (1,p;p+1;1-\frac{e^2 x^2}{d^2}\right )}{2 d p} \]

[Out]

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

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Rubi [A]  time = 0.1861, antiderivative size = 104, 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 \[ -\frac{e x \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac{1}{2},1-p;\frac{3}{2};\frac{e^2 x^2}{d^2}\right )}{d^2}-\frac{\left (d^2-e^2 x^2\right )^p \, _2F_1\left (1,p;p+1;1-\frac{e^2 x^2}{d^2}\right )}{2 d p} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 41.5426, size = 82, normalized size = 0.79 \[ - \frac{\left (d^{2} - e^{2} x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} 1, p \\ p + 1 \end{matrix}\middle |{1 - \frac{e^{2} x^{2}}{d^{2}}} \right )}}{2 d p} - \frac{e x \left (1 - \frac{e^{2} x^{2}}{d^{2}}\right )^{- p} \left (d^{2} - e^{2} x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p + 1, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{\frac{e^{2} x^{2}}{d^{2}}} \right )}}{d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.145778, size = 151, normalized size = 1.45 \[ \frac{2^{p-1} \left (1-\frac{d^2}{e^2 x^2}\right )^{-p} \left (\frac{e x}{d}+1\right )^{-p} \left (d^2-e^2 x^2\right )^p \left (p (d-e x) \left (1-\frac{d^2}{e^2 x^2}\right )^p \, _2F_1\left (1-p,p+1;p+2;\frac{d-e x}{2 d}\right )+d (p+1) \left (\frac{e x}{2 d}+\frac{1}{2}\right )^p \, _2F_1\left (-p,-p;1-p;\frac{d^2}{e^2 x^2}\right )\right )}{d^2 p (p+1)} \]

Antiderivative was successfully verified.

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

[Out]

(2^(-1 + p)*(d^2 - e^2*x^2)^p*(p*(1 - d^2/(e^2*x^2))^p*(d - e*x)*Hypergeometric2
F1[1 - p, 1 + p, 2 + p, (d - e*x)/(2*d)] + d*(1 + p)*(1/2 + (e*x)/(2*d))^p*Hyper
geometric2F1[-p, -p, 1 - p, d^2/(e^2*x^2)]))/(d^2*p*(1 + p)*(1 - d^2/(e^2*x^2))^
p*(1 + (e*x)/d)^p)

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Maple [F]  time = 0.065, size = 0, normalized size = 0. \[ \int{\frac{ \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{p}}{x \left ( ex+d \right ) }}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 14.8412, size = 355, normalized size = 3.41 \[ \begin{cases} - \frac{0^{p} d^{2 p} \log{\left (\frac{d^{2}}{e^{2} x^{2}} - 1 \right )}}{2 d} - \frac{0^{p} d^{2 p} \operatorname{acoth}{\left (\frac{d}{e x} \right )}}{d} + \frac{d e^{2 p} p x^{2 p} e^{i \pi p} \Gamma \left (p\right ) \Gamma \left (- p + 1\right ){{}_{2}F_{1}\left (\begin{matrix} - p + 1, - p + 1 \\ - p + 2 \end{matrix}\middle |{\frac{d^{2}}{e^{2} x^{2}}} \right )}}{2 e^{2} x^{2} \Gamma \left (- p + 2\right ) \Gamma \left (p + 1\right )} - \frac{e^{2 p} p x^{2 p} e^{i \pi p} \Gamma \left (p\right ) \Gamma \left (- p + \frac{1}{2}\right ){{}_{2}F_{1}\left (\begin{matrix} - p + 1, - p + \frac{1}{2} \\ - p + \frac{3}{2} \end{matrix}\middle |{\frac{d^{2}}{e^{2} x^{2}}} \right )}}{2 e x \Gamma \left (- p + \frac{3}{2}\right ) \Gamma \left (p + 1\right )} & \text{for}\: \left |{\frac{d^{2}}{e^{2} x^{2}}}\right | > 1 \\- \frac{0^{p} d^{2 p} \log{\left (- \frac{d^{2}}{e^{2} x^{2}} + 1 \right )}}{2 d} - \frac{0^{p} d^{2 p} \operatorname{atanh}{\left (\frac{d}{e x} \right )}}{d} + \frac{d e^{2 p} p x^{2 p} e^{i \pi p} \Gamma \left (p\right ) \Gamma \left (- p + 1\right ){{}_{2}F_{1}\left (\begin{matrix} - p + 1, - p + 1 \\ - p + 2 \end{matrix}\middle |{\frac{d^{2}}{e^{2} x^{2}}} \right )}}{2 e^{2} x^{2} \Gamma \left (- p + 2\right ) \Gamma \left (p + 1\right )} - \frac{e^{2 p} p x^{2 p} e^{i \pi p} \Gamma \left (p\right ) \Gamma \left (- p + \frac{1}{2}\right ){{}_{2}F_{1}\left (\begin{matrix} - p + 1, - p + \frac{1}{2} \\ - p + \frac{3}{2} \end{matrix}\middle |{\frac{d^{2}}{e^{2} x^{2}}} \right )}}{2 e x \Gamma \left (- p + \frac{3}{2}\right ) \Gamma \left (p + 1\right )} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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