3.242 \(\int x^3 (d+e x) \left (d^2-e^2 x^2\right )^p \, dx\)

Optimal. Leaf size=120 \[ \frac{1}{5} e x^5 \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac{5}{2},-p;\frac{7}{2};\frac{e^2 x^2}{d^2}\right )+\frac{d \left (d^2-e^2 x^2\right )^{p+2}}{2 e^4 (p+2)}-\frac{d^3 \left (d^2-e^2 x^2\right )^{p+1}}{2 e^4 (p+1)} \]

[Out]

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

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Rubi [A]  time = 0.171427, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217 \[ \frac{1}{5} e x^5 \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac{5}{2},-p;\frac{7}{2};\frac{e^2 x^2}{d^2}\right )+\frac{d \left (d^2-e^2 x^2\right )^{p+2}}{2 e^4 (p+2)}-\frac{d^3 \left (d^2-e^2 x^2\right )^{p+1}}{2 e^4 (p+1)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 33.398, size = 97, normalized size = 0.81 \[ - \frac{d^{3} \left (d^{2} - e^{2} x^{2}\right )^{p + 1}}{2 e^{4} \left (p + 1\right )} + \frac{d \left (d^{2} - e^{2} x^{2}\right )^{p + 2}}{2 e^{4} \left (p + 2\right )} + \frac{e x^{5} \left (1 - \frac{e^{2} x^{2}}{d^{2}}\right )^{- p} \left (d^{2} - e^{2} x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{5}{2} \\ \frac{7}{2} \end{matrix}\middle |{\frac{e^{2} x^{2}}{d^{2}}} \right )}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-d**3*(d**2 - e**2*x**2)**(p + 1)/(2*e**4*(p + 1)) + d*(d**2 - e**2*x**2)**(p +
2)/(2*e**4*(p + 2)) + e*x**5*(1 - e**2*x**2/d**2)**(-p)*(d**2 - e**2*x**2)**p*hy
per((-p, 5/2), (7/2,), e**2*x**2/d**2)/5

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Mathematica [A]  time = 0.150373, size = 161, normalized size = 1.34 \[ \frac{\left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \left (2 e^5 \left (p^2+3 p+2\right ) x^5 \, _2F_1\left (\frac{5}{2},-p;\frac{7}{2};\frac{e^2 x^2}{d^2}\right )+5 d e^4 (p+1) x^4 \left (1-\frac{e^2 x^2}{d^2}\right )^p-5 d^5 \left (\left (1-\frac{e^2 x^2}{d^2}\right )^p-1\right )-5 d^3 e^2 p x^2 \left (1-\frac{e^2 x^2}{d^2}\right )^p\right )}{10 e^4 (p+1) (p+2)} \]

Antiderivative was successfully verified.

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

[Out]

((d^2 - e^2*x^2)^p*(-5*d^3*e^2*p*x^2*(1 - (e^2*x^2)/d^2)^p + 5*d*e^4*(1 + p)*x^4
*(1 - (e^2*x^2)/d^2)^p - 5*d^5*(-1 + (1 - (e^2*x^2)/d^2)^p) + 2*e^5*(2 + 3*p + p
^2)*x^5*Hypergeometric2F1[5/2, -p, 7/2, (e^2*x^2)/d^2]))/(10*e^4*(1 + p)*(2 + p)
*(1 - (e^2*x^2)/d^2)^p)

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Maple [F]  time = 0.066, size = 0, normalized size = 0. \[ \int{x}^{3} \left ( ex+d \right ) \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ e \int x^{4} e^{\left (p \log \left (e x + d\right ) + p \log \left (-e x + d\right )\right )}\,{d x} + \frac{{\left (e^{4}{\left (p + 1\right )} x^{4} - d^{2} e^{2} p x^{2} - d^{4}\right )}{\left (-e^{2} x^{2} + d^{2}\right )}^{p} d}{2 \,{\left (p^{2} + 3 \, p + 2\right )} e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

e*integrate(x^4*e^(p*log(e*x + d) + p*log(-e*x + d)), x) + 1/2*(e^4*(p + 1)*x^4
- d^2*e^2*p*x^2 - d^4)*(-e^2*x^2 + d^2)^p*d/((p^2 + 3*p + 2)*e^4)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 12.0498, size = 382, normalized size = 3.18 \[ d \left (\begin{cases} \frac{x^{4} \left (d^{2}\right )^{p}}{4} & \text{for}\: e = 0 \\- \frac{d^{2} \log{\left (- \frac{d}{e} + x \right )}}{- 2 d^{2} e^{4} + 2 e^{6} x^{2}} - \frac{d^{2} \log{\left (\frac{d}{e} + x \right )}}{- 2 d^{2} e^{4} + 2 e^{6} x^{2}} - \frac{d^{2}}{- 2 d^{2} e^{4} + 2 e^{6} x^{2}} + \frac{e^{2} x^{2} \log{\left (- \frac{d}{e} + x \right )}}{- 2 d^{2} e^{4} + 2 e^{6} x^{2}} + \frac{e^{2} x^{2} \log{\left (\frac{d}{e} + x \right )}}{- 2 d^{2} e^{4} + 2 e^{6} x^{2}} & \text{for}\: p = -2 \\- \frac{d^{2} \log{\left (- \frac{d}{e} + x \right )}}{2 e^{4}} - \frac{d^{2} \log{\left (\frac{d}{e} + x \right )}}{2 e^{4}} - \frac{x^{2}}{2 e^{2}} & \text{for}\: p = -1 \\- \frac{d^{4} \left (d^{2} - e^{2} x^{2}\right )^{p}}{2 e^{4} p^{2} + 6 e^{4} p + 4 e^{4}} - \frac{d^{2} e^{2} p x^{2} \left (d^{2} - e^{2} x^{2}\right )^{p}}{2 e^{4} p^{2} + 6 e^{4} p + 4 e^{4}} + \frac{e^{4} p x^{4} \left (d^{2} - e^{2} x^{2}\right )^{p}}{2 e^{4} p^{2} + 6 e^{4} p + 4 e^{4}} + \frac{e^{4} x^{4} \left (d^{2} - e^{2} x^{2}\right )^{p}}{2 e^{4} p^{2} + 6 e^{4} p + 4 e^{4}} & \text{otherwise} \end{cases}\right ) + \frac{d^{2 p} e x^{5}{{}_{2}F_{1}\left (\begin{matrix} \frac{5}{2}, - p \\ \frac{7}{2} \end{matrix}\middle |{\frac{e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d*Piecewise((x**4*(d**2)**p/4, Eq(e, 0)), (-d**2*log(-d/e + x)/(-2*d**2*e**4 + 2
*e**6*x**2) - d**2*log(d/e + x)/(-2*d**2*e**4 + 2*e**6*x**2) - d**2/(-2*d**2*e**
4 + 2*e**6*x**2) + e**2*x**2*log(-d/e + x)/(-2*d**2*e**4 + 2*e**6*x**2) + e**2*x
**2*log(d/e + x)/(-2*d**2*e**4 + 2*e**6*x**2), Eq(p, -2)), (-d**2*log(-d/e + x)/
(2*e**4) - d**2*log(d/e + x)/(2*e**4) - x**2/(2*e**2), Eq(p, -1)), (-d**4*(d**2
- e**2*x**2)**p/(2*e**4*p**2 + 6*e**4*p + 4*e**4) - d**2*e**2*p*x**2*(d**2 - e**
2*x**2)**p/(2*e**4*p**2 + 6*e**4*p + 4*e**4) + e**4*p*x**4*(d**2 - e**2*x**2)**p
/(2*e**4*p**2 + 6*e**4*p + 4*e**4) + e**4*x**4*(d**2 - e**2*x**2)**p/(2*e**4*p**
2 + 6*e**4*p + 4*e**4), True)) + d**(2*p)*e*x**5*hyper((5/2, -p), (7/2,), e**2*x
**2*exp_polar(2*I*pi)/d**2)/5

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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