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

Optimal. Leaf size=220 \[ -\frac{3 d x^5 \left (d^2-e^2 x^2\right )^{p-2}}{2 p+1}+\frac{3 d^2 \left (d^2-e^2 x^2\right )^p}{e^5 p}-\frac{\left (d^2-e^2 x^2\right )^{p+1}}{2 e^5 (p+1)}-\frac{2 d^6 \left (d^2-e^2 x^2\right )^{p-2}}{e^5 (2-p)}+\frac{9 d^4 \left (d^2-e^2 x^2\right )^{p-1}}{2 e^5 (1-p)}+\frac{2 (p+8) x^5 \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \left (d^2-e^2 x^2\right )^p \, _2F_1\left (\frac{5}{2},3-p;\frac{7}{2};\frac{e^2 x^2}{d^2}\right )}{5 d^3 (2 p+1)} \]

[Out]

(-2*d^6*(d^2 - e^2*x^2)^(-2 + p))/(e^5*(2 - p)) - (3*d*x^5*(d^2 - e^2*x^2)^(-2 +
p))/(1 + 2*p) + (9*d^4*(d^2 - e^2*x^2)^(-1 + p))/(2*e^5*(1 - p)) + (3*d^2*(d^2
- e^2*x^2)^p)/(e^5*p) - (d^2 - e^2*x^2)^(1 + p)/(2*e^5*(1 + p)) + (2*(8 + p)*x^5
*(d^2 - e^2*x^2)^p*Hypergeometric2F1[5/2, 3 - p, 7/2, (e^2*x^2)/d^2])/(5*d^3*(1
+ 2*p)*(1 - (e^2*x^2)/d^2)^p)

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Rubi [A]  time = 0.531737, antiderivative size = 220, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.28 \[ -\frac{3 d x^5 \left (d^2-e^2 x^2\right )^{p-2}}{2 p+1}+\frac{3 d^2 \left (d^2-e^2 x^2\right )^p}{e^5 p}-\frac{\left (d^2-e^2 x^2\right )^{p+1}}{2 e^5 (p+1)}-\frac{2 d^6 \left (d^2-e^2 x^2\right )^{p-2}}{e^5 (2-p)}+\frac{9 d^4 \left (d^2-e^2 x^2\right )^{p-1}}{2 e^5 (1-p)}+\frac{2 (p+8) x^5 \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \left (d^2-e^2 x^2\right )^p \, _2F_1\left (\frac{5}{2},3-p;\frac{7}{2};\frac{e^2 x^2}{d^2}\right )}{5 d^3 (2 p+1)} \]

Antiderivative was successfully verified.

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

[Out]

(-2*d^6*(d^2 - e^2*x^2)^(-2 + p))/(e^5*(2 - p)) - (3*d*x^5*(d^2 - e^2*x^2)^(-2 +
p))/(1 + 2*p) + (9*d^4*(d^2 - e^2*x^2)^(-1 + p))/(2*e^5*(1 - p)) + (3*d^2*(d^2
- e^2*x^2)^p)/(e^5*p) - (d^2 - e^2*x^2)^(1 + p)/(2*e^5*(1 + p)) + (2*(8 + p)*x^5
*(d^2 - e^2*x^2)^p*Hypergeometric2F1[5/2, 3 - p, 7/2, (e^2*x^2)/d^2])/(5*d^3*(1
+ 2*p)*(1 - (e^2*x^2)/d^2)^p)

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Rubi in Sympy [A]  time = 132.337, size = 202, normalized size = 0.92 \[ - \frac{2 d^{6} \left (d^{2} - e^{2} x^{2}\right )^{p - 2}}{e^{5} \left (- p + 2\right )} + \frac{9 d^{4} \left (d^{2} - e^{2} x^{2}\right )^{p - 1}}{2 e^{5} \left (- p + 1\right )} + \frac{3 d^{2} \left (d^{2} - e^{2} x^{2}\right )^{p}}{e^{5} p} - \frac{\left (d^{2} - e^{2} x^{2}\right )^{p + 1}}{2 e^{5} \left (p + 1\right )} + \frac{x^{5} \left (1 - \frac{e^{2} x^{2}}{d^{2}}\right )^{- p} \left (d^{2} - e^{2} x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p + 3, \frac{5}{2} \\ \frac{7}{2} \end{matrix}\middle |{\frac{e^{2} x^{2}}{d^{2}}} \right )}}{5 d^{3}} + \frac{3 e^{2} x^{7} \left (1 - \frac{e^{2} x^{2}}{d^{2}}\right )^{- p} \left (d^{2} - e^{2} x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p + 3, \frac{7}{2} \\ \frac{9}{2} \end{matrix}\middle |{\frac{e^{2} x^{2}}{d^{2}}} \right )}}{7 d^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.436041, size = 140, normalized size = 0.64 \[ -\frac{6 d x^5 (d-e x)^p (d+e x)^{p-3} F_1\left (5;-p,3-p;6;\frac{e x}{d},-\frac{e x}{d}\right )}{5 \left (e x \left (p F_1\left (6;1-p,3-p;7;\frac{e x}{d},-\frac{e x}{d}\right )-(p-3) F_1\left (6;-p,4-p;7;\frac{e x}{d},-\frac{e x}{d}\right )\right )-6 d F_1\left (5;-p,3-p;6;\frac{e x}{d},-\frac{e x}{d}\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(-6*d*x^5*(d - e*x)^p*(d + e*x)^(-3 + p)*AppellF1[5, -p, 3 - p, 6, (e*x)/d, -((e
*x)/d)])/(5*(-6*d*AppellF1[5, -p, 3 - p, 6, (e*x)/d, -((e*x)/d)] + e*x*(p*Appell
F1[6, 1 - p, 3 - p, 7, (e*x)/d, -((e*x)/d)] - (-3 + p)*AppellF1[6, -p, 4 - p, 7,
(e*x)/d, -((e*x)/d)])))

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Maple [F]  time = 0.153, size = 0, normalized size = 0. \[ \int{\frac{{x}^{4} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{p}}{ \left ( ex+d \right ) ^{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x**4*(-(-d + e*x)*(d + e*x))**p/(d + e*x)**3, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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