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

Optimal. Leaf size=108 \[ -\frac{d \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (-\frac{1}{2},-p;\frac{1}{2};\frac{e^2 x^2}{d^2}\right )}{x}-\frac{e \left (d^2-e^2 x^2\right )^{p+1} \, _2F_1\left (1,p+1;p+2;1-\frac{e^2 x^2}{d^2}\right )}{2 d^2 (p+1)} \]

[Out]

-((d*(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*(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 = 0.139177, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217 \[ -\frac{d \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (-\frac{1}{2},-p;\frac{1}{2};\frac{e^2 x^2}{d^2}\right )}{x}-\frac{e \left (d^2-e^2 x^2\right )^{p+1} \, _2F_1\left (1,p+1;p+2;1-\frac{e^2 x^2}{d^2}\right )}{2 d^2 (p+1)} \]

Antiderivative was successfully verified.

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

[Out]

-((d*(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*(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 = 22.1809, size = 88, normalized size = 0.81 \[ - \frac{d \left (1 - \frac{e^{2} x^{2}}{d^{2}}\right )^{- p} \left (d^{2} - e^{2} x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, - \frac{1}{2} \\ \frac{1}{2} \end{matrix}\middle |{\frac{e^{2} x^{2}}{d^{2}}} \right )}}{x} - \frac{e \left (d^{2} - e^{2} x^{2}\right )^{p + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, p + 1 \\ p + 2 \end{matrix}\middle |{1 - \frac{e^{2} x^{2}}{d^{2}}} \right )}}{2 d^{2} \left (p + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-d*(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*(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))

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Mathematica [A]  time = 0.0699195, size = 106, normalized size = 0.98 \[ \frac{1}{2} \left (d^2-e^2 x^2\right )^p \left (\frac{e \left (1-\frac{d^2}{e^2 x^2}\right )^{-p} \, _2F_1\left (-p,-p;1-p;\frac{d^2}{e^2 x^2}\right )}{p}-\frac{2 d \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (-\frac{1}{2},-p;\frac{1}{2};\frac{e^2 x^2}{d^2}\right )}{x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.046, size = 0, normalized size = 0. \[ \int{\frac{ \left ( ex+d \right ) \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{p}}{{x}^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 10.0617, size = 82, normalized size = 0.76 \[ - \frac{d d^{2 p}{{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, - p \\ \frac{1}{2} \end{matrix}\middle |{\frac{e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{x} - \frac{e e^{2 p} x^{2 p} e^{i \pi p} \Gamma \left (- p\right ){{}_{2}F_{1}\left (\begin{matrix} - p, - p \\ - p + 1 \end{matrix}\middle |{\frac{d^{2}}{e^{2} x^{2}}} \right )}}{2 \Gamma \left (- p + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-d*d**(2*p)*hyper((-1/2, -p), (1/2,), e**2*x**2*exp_polar(2*I*pi)/d**2)/x - e*e*
*(2*p)*x**(2*p)*exp(I*pi*p)*gamma(-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. \[ \int \frac{{\left (e x + d\right )}{\left (-e^{2} x^{2} + d^{2}\right )}^{p}}{x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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