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

Optimal. Leaf size=115 \[ \frac{\left (d^2-e^2 x^2\right )^{p+1}}{2 e^2 (1-p) (d+e x)^2}-\frac{2^{p-1} \left (\frac{e x}{d}+1\right )^{-p-1} \left (d^2-e^2 x^2\right )^{p+1} \, _2F_1\left (1-p,p+1;p+2;\frac{d-e x}{2 d}\right )}{d^2 e^2 \left (1-p^2\right )} \]

[Out]

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

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Rubi [A]  time = 0.154419, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13 \[ \frac{\left (d^2-e^2 x^2\right )^{p+1}}{2 e^2 (1-p) (d+e x)^2}-\frac{2^{p-1} \left (\frac{e x}{d}+1\right )^{-p-1} \left (d^2-e^2 x^2\right )^{p+1} \, _2F_1\left (1-p,p+1;p+2;\frac{d-e x}{2 d}\right )}{d^2 e^2 \left (1-p^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 48.4315, size = 75, normalized size = 0.65 \[ \frac{\left (d - e x\right )^{2} \left (d^{2} - e^{2} x^{2}\right )^{p - 1}}{2 e^{2} \left (- p + 1\right )} + \frac{\left (\frac{\frac{d}{2} - \frac{e x}{2}}{d}\right )^{- p} \left (d^{2} - e^{2} x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, p \\ p + 1 \end{matrix}\middle |{\frac{\frac{d}{2} + \frac{e x}{2}}{d}} \right )}}{e^{2} p \left (- p + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0878113, size = 102, normalized size = 0.89 \[ \frac{2^{p-2} (d-e x) \left (\frac{e x}{d}+1\right )^{-p} \left (d^2-e^2 x^2\right )^p \left (\, _2F_1\left (2-p,p+1;p+2;\frac{d-e x}{2 d}\right )-2 \, _2F_1\left (1-p,p+1;p+2;\frac{d-e x}{2 d}\right )\right )}{d e^2 (p+1)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.088, size = 0, normalized size = 0. \[ \int{\frac{x \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{p}}{ \left ( ex+d \right ) ^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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