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

Optimal. Leaf size=73 \[ -\frac{2^{p-1} \left (\frac{e x}{d}+1\right )^{-p-1} \left (d^2-e^2 x^2\right )^{p+1} \, _2F_1\left (1-p,p+1;p+2;\frac{d-e x}{2 d}\right )}{d^2 e (p+1)} \]

[Out]

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

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Rubi [A]  time = 0.0808427, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ -\frac{2^{p-1} \left (\frac{e x}{d}+1\right )^{-p-1} \left (d^2-e^2 x^2\right )^{p+1} \, _2F_1\left (1-p,p+1;p+2;\frac{d-e x}{2 d}\right )}{d^2 e (p+1)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 24.4089, size = 41, normalized size = 0.56 \[ \frac{\left (\frac{\frac{d}{2} - \frac{e x}{2}}{d}\right )^{- p} \left (d^{2} - e^{2} x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, p \\ p + 1 \end{matrix}\middle |{\frac{\frac{d}{2} + \frac{e x}{2}}{d}} \right )}}{e p} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0460123, size = 75, normalized size = 1.03 \[ -\frac{2^{p-1} (d-e x) \left (\frac{e x}{d}+1\right )^{-p} \left (d^2-e^2 x^2\right )^p \, _2F_1\left (1-p,p+1;p+2;\frac{d-e x}{2 d}\right )}{d e (p+1)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.063, size = 0, normalized size = 0. \[ \int{\frac{ \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{p}}{ex+d}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-e^2*x^2+d^2)^p/(e*x+d),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 13.389, size = 321, normalized size = 4.4 \[ \begin{cases} \frac{0^{p} \log{\left (-1 + \frac{e^{2} x^{2}}{d^{2}} \right )}}{2 e} + \frac{0^{p} \operatorname{acoth}{\left (\frac{e x}{d} \right )}}{e} + \frac{d e^{2 p} p x^{2 p} e^{i \pi p} \Gamma \left (p\right ) \Gamma \left (- p + \frac{1}{2}\right ){{}_{2}F_{1}\left (\begin{matrix} - p + 1, - p + \frac{1}{2} \\ - p + \frac{3}{2} \end{matrix}\middle |{\frac{d^{2}}{e^{2} x^{2}}} \right )}}{2 e^{2} x \Gamma \left (- p + \frac{3}{2}\right ) \Gamma \left (p + 1\right )} + \frac{d^{2 p} e x^{2} \Gamma \left (p\right ) \Gamma \left (- p + 1\right ){{}_{3}F_{2}\left (\begin{matrix} 2, 1, - p + 1 \\ 2, 2 \end{matrix}\middle |{\frac{e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{2 d^{2} \Gamma \left (- p\right ) \Gamma \left (p + 1\right )} & \text{for}\: \left |{\frac{e^{2} x^{2}}{d^{2}}}\right | > 1 \\\frac{0^{p} \log{\left (1 - \frac{e^{2} x^{2}}{d^{2}} \right )}}{2 e} + \frac{0^{p} \operatorname{atanh}{\left (\frac{e x}{d} \right )}}{e} + \frac{d e^{2 p} p x^{2 p} e^{i \pi p} \Gamma \left (p\right ) \Gamma \left (- p + \frac{1}{2}\right ){{}_{2}F_{1}\left (\begin{matrix} - p + 1, - p + \frac{1}{2} \\ - p + \frac{3}{2} \end{matrix}\middle |{\frac{d^{2}}{e^{2} x^{2}}} \right )}}{2 e^{2} x \Gamma \left (- p + \frac{3}{2}\right ) \Gamma \left (p + 1\right )} + \frac{d^{2 p} e x^{2} \Gamma \left (p\right ) \Gamma \left (- p + 1\right ){{}_{3}F_{2}\left (\begin{matrix} 2, 1, - p + 1 \\ 2, 2 \end{matrix}\middle |{\frac{e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{2 d^{2} \Gamma \left (- p\right ) \Gamma \left (p + 1\right )} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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