3.297 \(\int \frac{x^3 \left (d^2-e^2 x^2\right )^p}{(d+e x)^4} \, dx\)

Optimal. Leaf size=211 \[ \frac{3\ 2^{p-2} (p+2) \left (d^2-e^2 x^2\right )^{p+1} \left (\frac{e x}{d}+1\right )^{-p-1} \, _2F_1\left (3-p,p+1;p+2;\frac{d-e x}{2 d}\right )}{d^2 e^4 (1-2 p) (3-p) p (p+1)}-\frac{\left (d^2-e^2 x^2\right )^{p+1}}{2 e^4 p (d+e x)^2}-\frac{d (2 p+1) \left (d^2-e^2 x^2\right )^{p+1}}{e^4 (1-2 p) p (d+e x)^3}+\frac{d^2 \left (d^2-e^2 x^2\right )^{p+1}}{2 e^4 (3-p) (d+e x)^4} \]

[Out]

(d^2*(d^2 - e^2*x^2)^(1 + p))/(2*e^4*(3 - p)*(d + e*x)^4) - (d*(1 + 2*p)*(d^2 -
e^2*x^2)^(1 + p))/(e^4*(1 - 2*p)*p*(d + e*x)^3) - (d^2 - e^2*x^2)^(1 + p)/(2*e^4
*p*(d + e*x)^2) + (3*2^(-2 + p)*(2 + p)*(1 + (e*x)/d)^(-1 - p)*(d^2 - e^2*x^2)^(
1 + p)*Hypergeometric2F1[3 - p, 1 + p, 2 + p, (d - e*x)/(2*d)])/(d^2*e^4*(1 - 2*
p)*(3 - p)*p*(1 + p))

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Rubi [A]  time = 0.678008, antiderivative size = 211, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16 \[ \frac{3\ 2^{p-2} (p+2) \left (d^2-e^2 x^2\right )^{p+1} \left (\frac{e x}{d}+1\right )^{-p-1} \, _2F_1\left (3-p,p+1;p+2;\frac{d-e x}{2 d}\right )}{d^2 e^4 (1-2 p) (3-p) p (p+1)}-\frac{\left (d^2-e^2 x^2\right )^{p+1}}{2 e^4 p (d+e x)^2}-\frac{d (2 p+1) \left (d^2-e^2 x^2\right )^{p+1}}{e^4 (1-2 p) p (d+e x)^3}+\frac{d^2 \left (d^2-e^2 x^2\right )^{p+1}}{2 e^4 (3-p) (d+e x)^4} \]

Antiderivative was successfully verified.

[In]  Int[(x^3*(d^2 - e^2*x^2)^p)/(d + e*x)^4,x]

[Out]

(d^2*(d^2 - e^2*x^2)^(1 + p))/(2*e^4*(3 - p)*(d + e*x)^4) - (d*(1 + 2*p)*(d^2 -
e^2*x^2)^(1 + p))/(e^4*(1 - 2*p)*p*(d + e*x)^3) - (d^2 - e^2*x^2)^(1 + p)/(2*e^4
*p*(d + e*x)^2) + (3*2^(-2 + p)*(2 + p)*(1 + (e*x)/d)^(-1 - p)*(d^2 - e^2*x^2)^(
1 + p)*Hypergeometric2F1[3 - p, 1 + p, 2 + p, (d - e*x)/(2*d)])/(d^2*e^4*(1 - 2*
p)*(3 - p)*p*(1 + p))

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Rubi in Sympy [A]  time = 141.427, size = 206, normalized size = 0.98 \[ \frac{4 d^{6} \left (d^{2} - e^{2} x^{2}\right )^{p - 3}}{e^{4} \left (- p + 3\right )} - \frac{8 d^{4} \left (d^{2} - e^{2} x^{2}\right )^{p - 2}}{e^{4} \left (- p + 2\right )} + \frac{9 d^{2} \left (d^{2} - e^{2} x^{2}\right )^{p - 1}}{2 e^{4} \left (- p + 1\right )} + \frac{\left (d^{2} - e^{2} x^{2}\right )^{p}}{2 e^{4} p} - \frac{4 e x^{5} \left (1 - \frac{e^{2} x^{2}}{d^{2}}\right )^{- p} \left (d^{2} - e^{2} x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p + 4, \frac{5}{2} \\ \frac{7}{2} \end{matrix}\middle |{\frac{e^{2} x^{2}}{d^{2}}} \right )}}{5 d^{5}} - \frac{4 e^{3} x^{7} \left (1 - \frac{e^{2} x^{2}}{d^{2}}\right )^{- p} \left (d^{2} - e^{2} x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p + 4, \frac{7}{2} \\ \frac{9}{2} \end{matrix}\middle |{\frac{e^{2} x^{2}}{d^{2}}} \right )}}{7 d^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*(-e**2*x**2+d**2)**p/(e*x+d)**4,x)

[Out]

4*d**6*(d**2 - e**2*x**2)**(p - 3)/(e**4*(-p + 3)) - 8*d**4*(d**2 - e**2*x**2)**
(p - 2)/(e**4*(-p + 2)) + 9*d**2*(d**2 - e**2*x**2)**(p - 1)/(2*e**4*(-p + 1)) +
(d**2 - e**2*x**2)**p/(2*e**4*p) - 4*e*x**5*(1 - e**2*x**2/d**2)**(-p)*(d**2 -
e**2*x**2)**p*hyper((-p + 4, 5/2), (7/2,), e**2*x**2/d**2)/(5*d**5) - 4*e**3*x**
7*(1 - e**2*x**2/d**2)**(-p)*(d**2 - e**2*x**2)**p*hyper((-p + 4, 7/2), (9/2,),
e**2*x**2/d**2)/(7*d**7)

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Mathematica [C]  time = 0.457616, size = 140, normalized size = 0.66 \[ -\frac{5 d x^4 (d-e x)^p (d+e x)^{p-4} F_1\left (4;-p,4-p;5;\frac{e x}{d},-\frac{e x}{d}\right )}{4 \left (e x \left (p F_1\left (5;1-p,4-p;6;\frac{e x}{d},-\frac{e x}{d}\right )-(p-4) F_1\left (5;-p,5-p;6;\frac{e x}{d},-\frac{e x}{d}\right )\right )-5 d F_1\left (4;-p,4-p;5;\frac{e x}{d},-\frac{e x}{d}\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(x^3*(d^2 - e^2*x^2)^p)/(d + e*x)^4,x]

[Out]

(-5*d*x^4*(d - e*x)^p*(d + e*x)^(-4 + p)*AppellF1[4, -p, 4 - p, 5, (e*x)/d, -((e
*x)/d)])/(4*(-5*d*AppellF1[4, -p, 4 - p, 5, (e*x)/d, -((e*x)/d)] + e*x*(p*Appell
F1[5, 1 - p, 4 - p, 6, (e*x)/d, -((e*x)/d)] - (-4 + p)*AppellF1[5, -p, 5 - p, 6,
(e*x)/d, -((e*x)/d)])))

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Maple [F]  time = 0.163, size = 0, normalized size = 0. \[ \int{\frac{{x}^{3} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{p}}{ \left ( ex+d \right ) ^{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(-e^2*x^2+d^2)^p/(e*x+d)^4,x)

[Out]

int(x^3*(-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. \[ \int \frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{p} x^{3}}{{\left (e x + d\right )}^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e^2*x^2 + d^2)^p*x^3/(e*x + d)^4,x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e^2*x^2 + d^2)^p*x^3/(e*x + d)^4,x, algorithm="fricas")

[Out]

integral((-e^2*x^2 + d^2)^p*x^3/(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. \[ \int \frac{x^{3} \left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{p}}{\left (d + e x\right )^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3*(-e**2*x**2+d**2)**p/(e*x+d)**4,x)

[Out]

Integral(x**3*(-(-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. \[ \int \frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{p} x^{3}}{{\left (e x + d\right )}^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e^2*x^2 + d^2)^p*x^3/(e*x + d)^4,x, algorithm="giac")

[Out]

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