3.240 \(\int x^5 (d+e x) \left (d^2-e^2 x^2\right )^p \, dx\)

Optimal. Leaf size=148 \[ \frac{1}{7} e x^7 \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac{7}{2},-p;\frac{9}{2};\frac{e^2 x^2}{d^2}\right )-\frac{d \left (d^2-e^2 x^2\right )^{p+3}}{2 e^6 (p+3)}-\frac{d^5 \left (d^2-e^2 x^2\right )^{p+1}}{2 e^6 (p+1)}+\frac{d^3 \left (d^2-e^2 x^2\right )^{p+2}}{e^6 (p+2)} \]

[Out]

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

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Rubi [A]  time = 0.226445, antiderivative size = 148, 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 \[ \frac{1}{7} e x^7 \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac{7}{2},-p;\frac{9}{2};\frac{e^2 x^2}{d^2}\right )-\frac{d \left (d^2-e^2 x^2\right )^{p+3}}{2 e^6 (p+3)}-\frac{d^5 \left (d^2-e^2 x^2\right )^{p+1}}{2 e^6 (p+1)}+\frac{d^3 \left (d^2-e^2 x^2\right )^{p+2}}{e^6 (p+2)} \]

Antiderivative was successfully verified.

[In]  Int[x^5*(d + e*x)*(d^2 - e^2*x^2)^p,x]

[Out]

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

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Rubi in Sympy [A]  time = 38.1723, size = 121, normalized size = 0.82 \[ - \frac{d^{5} \left (d^{2} - e^{2} x^{2}\right )^{p + 1}}{2 e^{6} \left (p + 1\right )} + \frac{d^{3} \left (d^{2} - e^{2} x^{2}\right )^{p + 2}}{e^{6} \left (p + 2\right )} - \frac{d \left (d^{2} - e^{2} x^{2}\right )^{p + 3}}{2 e^{6} \left (p + 3\right )} + \frac{e x^{7} \left (1 - \frac{e^{2} x^{2}}{d^{2}}\right )^{- p} \left (d^{2} - e^{2} x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{7}{2} \\ \frac{9}{2} \end{matrix}\middle |{\frac{e^{2} x^{2}}{d^{2}}} \right )}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5*(e*x+d)*(-e**2*x**2+d**2)**p,x)

[Out]

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

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Mathematica [A]  time = 0.244916, size = 208, normalized size = 1.41 \[ \frac{\left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \left (2 e^7 \left (p^3+6 p^2+11 p+6\right ) x^7 \, _2F_1\left (\frac{7}{2},-p;\frac{9}{2};\frac{e^2 x^2}{d^2}\right )-7 d \left (-e^6 \left (p^2+3 p+2\right ) x^6 \left (1-\frac{e^2 x^2}{d^2}\right )^p+d^2 e^4 p (p+1) x^4 \left (1-\frac{e^2 x^2}{d^2}\right )^p+2 d^6 \left (\left (1-\frac{e^2 x^2}{d^2}\right )^p-1\right )+2 d^4 e^2 p x^2 \left (1-\frac{e^2 x^2}{d^2}\right )^p\right )\right )}{14 e^6 (p+1) (p+2) (p+3)} \]

Antiderivative was successfully verified.

[In]  Integrate[x^5*(d + e*x)*(d^2 - e^2*x^2)^p,x]

[Out]

((d^2 - e^2*x^2)^p*(-7*d*(2*d^4*e^2*p*x^2*(1 - (e^2*x^2)/d^2)^p + d^2*e^4*p*(1 +
p)*x^4*(1 - (e^2*x^2)/d^2)^p - e^6*(2 + 3*p + p^2)*x^6*(1 - (e^2*x^2)/d^2)^p +
2*d^6*(-1 + (1 - (e^2*x^2)/d^2)^p)) + 2*e^7*(6 + 11*p + 6*p^2 + p^3)*x^7*Hyperge
ometric2F1[7/2, -p, 9/2, (e^2*x^2)/d^2]))/(14*e^6*(1 + p)*(2 + p)*(3 + p)*(1 - (
e^2*x^2)/d^2)^p)

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Maple [F]  time = 0.127, size = 0, normalized size = 0. \[ \int{x}^{5} \left ( ex+d \right ) \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5*(e*x+d)*(-e^2*x^2+d^2)^p,x)

[Out]

int(x^5*(e*x+d)*(-e^2*x^2+d^2)^p,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ e \int x^{6} e^{\left (p \log \left (e x + d\right ) + p \log \left (-e x + d\right )\right )}\,{d x} + \frac{{\left ({\left (p^{2} + 3 \, p + 2\right )} e^{6} x^{6} -{\left (p^{2} + p\right )} d^{2} e^{4} x^{4} - 2 \, d^{4} e^{2} p x^{2} - 2 \, d^{6}\right )}{\left (-e^{2} x^{2} + d^{2}\right )}^{p} d}{2 \,{\left (p^{3} + 6 \, p^{2} + 11 \, p + 6\right )} e^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)*(-e^2*x^2 + d^2)^p*x^5,x, algorithm="maxima")

[Out]

e*integrate(x^6*e^(p*log(e*x + d) + p*log(-e*x + d)), x) + 1/2*((p^2 + 3*p + 2)*
e^6*x^6 - (p^2 + p)*d^2*e^4*x^4 - 2*d^4*e^2*p*x^2 - 2*d^6)*(-e^2*x^2 + d^2)^p*d/
((p^3 + 6*p^2 + 11*p + 6)*e^6)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (e x^{6} + d x^{5}\right )}{\left (-e^{2} x^{2} + d^{2}\right )}^{p}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)*(-e^2*x^2 + d^2)^p*x^5,x, algorithm="fricas")

[Out]

integral((e*x^6 + d*x^5)*(-e^2*x^2 + d^2)^p, x)

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Sympy [A]  time = 19.0671, size = 972, normalized size = 6.57 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5*(e*x+d)*(-e**2*x**2+d**2)**p,x)

[Out]

d*Piecewise((x**6*(d**2)**p/6, Eq(e, 0)), (-2*d**4*log(-d/e + x)/(4*d**4*e**6 -
8*d**2*e**8*x**2 + 4*e**10*x**4) - 2*d**4*log(d/e + x)/(4*d**4*e**6 - 8*d**2*e**
8*x**2 + 4*e**10*x**4) - 3*d**4/(4*d**4*e**6 - 8*d**2*e**8*x**2 + 4*e**10*x**4)
+ 4*d**2*e**2*x**2*log(-d/e + x)/(4*d**4*e**6 - 8*d**2*e**8*x**2 + 4*e**10*x**4)
+ 4*d**2*e**2*x**2*log(d/e + x)/(4*d**4*e**6 - 8*d**2*e**8*x**2 + 4*e**10*x**4)
+ 4*d**2*e**2*x**2/(4*d**4*e**6 - 8*d**2*e**8*x**2 + 4*e**10*x**4) - 2*e**4*x**
4*log(-d/e + x)/(4*d**4*e**6 - 8*d**2*e**8*x**2 + 4*e**10*x**4) - 2*e**4*x**4*lo
g(d/e + x)/(4*d**4*e**6 - 8*d**2*e**8*x**2 + 4*e**10*x**4), Eq(p, -3)), (-2*d**4
*log(-d/e + x)/(-2*d**2*e**6 + 2*e**8*x**2) - 2*d**4*log(d/e + x)/(-2*d**2*e**6
+ 2*e**8*x**2) - 2*d**4/(-2*d**2*e**6 + 2*e**8*x**2) + 2*d**2*e**2*x**2*log(-d/e
+ x)/(-2*d**2*e**6 + 2*e**8*x**2) + 2*d**2*e**2*x**2*log(d/e + x)/(-2*d**2*e**6
+ 2*e**8*x**2) + e**4*x**4/(-2*d**2*e**6 + 2*e**8*x**2), Eq(p, -2)), (-d**4*log
(-d/e + x)/(2*e**6) - d**4*log(d/e + x)/(2*e**6) - d**2*x**2/(2*e**4) - x**4/(4*
e**2), Eq(p, -1)), (-2*d**6*(d**2 - e**2*x**2)**p/(2*e**6*p**3 + 12*e**6*p**2 +
22*e**6*p + 12*e**6) - 2*d**4*e**2*p*x**2*(d**2 - e**2*x**2)**p/(2*e**6*p**3 + 1
2*e**6*p**2 + 22*e**6*p + 12*e**6) - d**2*e**4*p**2*x**4*(d**2 - e**2*x**2)**p/(
2*e**6*p**3 + 12*e**6*p**2 + 22*e**6*p + 12*e**6) - d**2*e**4*p*x**4*(d**2 - e**
2*x**2)**p/(2*e**6*p**3 + 12*e**6*p**2 + 22*e**6*p + 12*e**6) + e**6*p**2*x**6*(
d**2 - e**2*x**2)**p/(2*e**6*p**3 + 12*e**6*p**2 + 22*e**6*p + 12*e**6) + 3*e**6
*p*x**6*(d**2 - e**2*x**2)**p/(2*e**6*p**3 + 12*e**6*p**2 + 22*e**6*p + 12*e**6)
+ 2*e**6*x**6*(d**2 - e**2*x**2)**p/(2*e**6*p**3 + 12*e**6*p**2 + 22*e**6*p + 1
2*e**6), True)) + d**(2*p)*e*x**7*hyper((7/2, -p), (9/2,), e**2*x**2*exp_polar(2
*I*pi)/d**2)/7

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (e x + d\right )}{\left (-e^{2} x^{2} + d^{2}\right )}^{p} x^{5}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)*(-e^2*x^2 + d^2)^p*x^5,x, algorithm="giac")

[Out]

integrate((e*x + d)*(-e^2*x^2 + d^2)^p*x^5, x)