3.300 \(\int \frac{\left (d^2-e^2 x^2\right )^p}{(d+e x)^4} \, dx\)

Optimal. Leaf size=73 \[ -\frac{2^{p-4} \left (\frac{e x}{d}+1\right )^{-p-1} \left (d^2-e^2 x^2\right )^{p+1} \, _2F_1\left (4-p,p+1;p+2;\frac{d-e x}{2 d}\right )}{d^5 e (p+1)} \]

[Out]

-((2^(-4 + p)*(1 + (e*x)/d)^(-1 - p)*(d^2 - e^2*x^2)^(1 + p)*Hypergeometric2F1[4
- p, 1 + p, 2 + p, (d - e*x)/(2*d)])/(d^5*e*(1 + p)))

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Rubi [A]  time = 0.0763143, 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 \[ -\frac{2^{p-4} \left (\frac{e x}{d}+1\right )^{-p-1} \left (d^2-e^2 x^2\right )^{p+1} \, _2F_1\left (4-p,p+1;p+2;\frac{d-e x}{2 d}\right )}{d^5 e (p+1)} \]

Antiderivative was successfully verified.

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

[Out]

-((2^(-4 + p)*(1 + (e*x)/d)^(-1 - p)*(d^2 - e^2*x^2)^(1 + p)*Hypergeometric2F1[4
- p, 1 + p, 2 + p, (d - e*x)/(2*d)])/(d^5*e*(1 + p)))

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Rubi in Sympy [A]  time = 11.8307, size = 66, normalized size = 0.9 \[ - \frac{\left (\frac{\frac{d}{2} + \frac{e x}{2}}{d}\right )^{- p} \left (d - e x\right )^{- p} \left (d - e x\right )^{p + 1} \left (d^{2} - e^{2} x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p + 4, p + 1 \\ p + 2 \end{matrix}\middle |{\frac{\frac{d}{2} - \frac{e x}{2}}{d}} \right )}}{16 d^{4} e \left (p + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-e**2*x**2+d**2)**p/(e*x+d)**4,x)

[Out]

-((d/2 + e*x/2)/d)**(-p)*(d - e*x)**(-p)*(d - e*x)**(p + 1)*(d**2 - e**2*x**2)**
p*hyper((-p + 4, p + 1), (p + 2,), (d/2 - e*x/2)/d)/(16*d**4*e*(p + 1))

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Mathematica [A]  time = 0.0516472, size = 75, normalized size = 1.03 \[ -\frac{2^{p-4} (d-e x) \left (\frac{e x}{d}+1\right )^{-p} \left (d^2-e^2 x^2\right )^p \, _2F_1\left (4-p,p+1;p+2;\frac{d-e x}{2 d}\right )}{d^4 e (p+1)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.14, size = 0, normalized size = 0. \[ \int{\frac{ \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{p}}{ \left ( ex+d \right ) ^{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int((-e^2*x^2+d^2)^p/(e*x+d)^4,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{p}}{{\left (e x + d\right )}^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e^2*x^2 + d^2)^p/(e*x + d)^4,x, algorithm="maxima")

[Out]

integrate((-e^2*x^2 + d^2)^p/(e*x + d)^4, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{p}}{e^{4} x^{4} + 4 \, d e^{3} x^{3} + 6 \, d^{2} e^{2} x^{2} + 4 \, d^{3} e x + d^{4}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e^2*x^2 + d^2)^p/(e*x + d)^4,x, algorithm="fricas")

[Out]

integral((-e^2*x^2 + d^2)^p/(e^4*x^4 + 4*d*e^3*x^3 + 6*d^2*e^2*x^2 + 4*d^3*e*x +
d^4), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{p}}{\left (d + e x\right )^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e**2*x**2+d**2)**p/(e*x+d)**4,x)

[Out]

Integral((-(-d + e*x)*(d + e*x))**p/(d + e*x)**4, x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{p}}{{\left (e x + d\right )}^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e^2*x^2 + d^2)^p/(e*x + d)^4,x, algorithm="giac")

[Out]

integrate((-e^2*x^2 + d^2)^p/(e*x + d)^4, x)