3.267 \(\int \frac{x^4 \left (d^2-e^2 x^2\right )^p}{d+e x} \, dx\)
Optimal. Leaf size=148 \[ \frac{x^5 \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac{5}{2},1-p;\frac{7}{2};\frac{e^2 x^2}{d^2}\right )}{5 d}-\frac{d^2 \left (d^2-e^2 x^2\right )^{p+1}}{e^5 (p+1)}+\frac{\left (d^2-e^2 x^2\right )^{p+2}}{2 e^5 (p+2)}+\frac{d^4 \left (d^2-e^2 x^2\right )^p}{2 e^5 p} \]
[Out]
(d^4*(d^2 - e^2*x^2)^p)/(2*e^5*p) - (d^2*(d^2 - e^2*x^2)^(1 + p))/(e^5*(1 + p))
+ (d^2 - e^2*x^2)^(2 + p)/(2*e^5*(2 + p)) + (x^5*(d^2 - e^2*x^2)^p*Hypergeometri
c2F1[5/2, 1 - p, 7/2, (e^2*x^2)/d^2])/(5*d*(1 - (e^2*x^2)/d^2)^p)
_______________________________________________________________________________________
Rubi [A] time = 0.296313, antiderivative size = 148, normalized size of antiderivative = 1.,
number of steps used = 7, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24
\[ \frac{x^5 \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac{5}{2},1-p;\frac{7}{2};\frac{e^2 x^2}{d^2}\right )}{5 d}-\frac{d^2 \left (d^2-e^2 x^2\right )^{p+1}}{e^5 (p+1)}+\frac{\left (d^2-e^2 x^2\right )^{p+2}}{2 e^5 (p+2)}+\frac{d^4 \left (d^2-e^2 x^2\right )^p}{2 e^5 p} \]
Antiderivative was successfully verified.
[In] Int[(x^4*(d^2 - e^2*x^2)^p)/(d + e*x),x]
[Out]
(d^4*(d^2 - e^2*x^2)^p)/(2*e^5*p) - (d^2*(d^2 - e^2*x^2)^(1 + p))/(e^5*(1 + p))
+ (d^2 - e^2*x^2)^(2 + p)/(2*e^5*(2 + p)) + (x^5*(d^2 - e^2*x^2)^p*Hypergeometri
c2F1[5/2, 1 - p, 7/2, (e^2*x^2)/d^2])/(5*d*(1 - (e^2*x^2)/d^2)^p)
_______________________________________________________________________________________
Rubi in Sympy [A] time = 56.8357, size = 116, normalized size = 0.78 \[ \frac{d^{4} \left (d^{2} - e^{2} x^{2}\right )^{p}}{2 e^{5} p} - \frac{d^{2} \left (d^{2} - e^{2} x^{2}\right )^{p + 1}}{e^{5} \left (p + 1\right )} + \frac{\left (d^{2} - e^{2} x^{2}\right )^{p + 2}}{2 e^{5} \left (p + 2\right )} + \frac{x^{5} \left (1 - \frac{e^{2} x^{2}}{d^{2}}\right )^{- p} \left (d^{2} - e^{2} x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p + 1, \frac{5}{2} \\ \frac{7}{2} \end{matrix}\middle |{\frac{e^{2} x^{2}}{d^{2}}} \right )}}{5 d} \]
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),x)
[Out]
d**4*(d**2 - e**2*x**2)**p/(2*e**5*p) - d**2*(d**2 - e**2*x**2)**(p + 1)/(e**5*(
p + 1)) + (d**2 - e**2*x**2)**(p + 2)/(2*e**5*(p + 2)) + x**5*(1 - e**2*x**2/d**
2)**(-p)*(d**2 - e**2*x**2)**p*hyper((-p + 1, 5/2), (7/2,), e**2*x**2/d**2)/(5*d
)
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Mathematica [C] time = 0.738148, size = 225, normalized size = 1.52 \[ \frac{6 d e^5 (p+1) (p+2) x^{10} (d-e x)^p (d+e x)^{p-1} F_1\left (5;-p,1-p;6;\frac{e x}{d},-\frac{e x}{d}\right )}{5 \left (e^6 \left (p^3+2 p^2-p-2\right ) x^6 F_1\left (6;-p,2-p;7;\frac{e x}{d},-\frac{e x}{d}\right )+6 d e^5 \left (p^2+3 p+2\right ) x^5 F_1\left (5;-p,1-p;6;\frac{e x}{d},-\frac{e x}{d}\right )+3 d^2 e^4 p (p+1) x^4 \left (1-\frac{e^2 x^2}{d^2}\right )^p+6 d^6 \left (\left (1-\frac{e^2 x^2}{d^2}\right )^p-1\right )+6 d^4 e^2 p x^2 \left (1-\frac{e^2 x^2}{d^2}\right )^p\right )} \]
Warning: Unable to verify antiderivative.
[In] Integrate[(x^4*(d^2 - e^2*x^2)^p)/(d + e*x),x]
[Out]
(6*d*e^5*(1 + p)*(2 + p)*x^10*(d - e*x)^p*(d + e*x)^(-1 + p)*AppellF1[5, -p, 1 -
p, 6, (e*x)/d, -((e*x)/d)])/(5*(6*d^4*e^2*p*x^2*(1 - (e^2*x^2)/d^2)^p + 3*d^2*e
^4*p*(1 + p)*x^4*(1 - (e^2*x^2)/d^2)^p + 6*d^6*(-1 + (1 - (e^2*x^2)/d^2)^p) + 6*
d*e^5*(2 + 3*p + p^2)*x^5*AppellF1[5, -p, 1 - p, 6, (e*x)/d, -((e*x)/d)] + e^6*(
-2 - p + 2*p^2 + p^3)*x^6*AppellF1[6, -p, 2 - p, 7, (e*x)/d, -((e*x)/d)]))
_______________________________________________________________________________________
Maple [F] time = 0.085, size = 0, normalized size = 0. \[ \int{\frac{{x}^{4} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{p}}{ex+d}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^4*(-e^2*x^2+d^2)^p/(e*x+d),x)
[Out]
int(x^4*(-e^2*x^2+d^2)^p/(e*x+d),x)
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{p} x^{4}}{e x + d}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-e^2*x^2 + d^2)^p*x^4/(e*x + d),x, algorithm="maxima")
[Out]
integrate((-e^2*x^2 + d^2)^p*x^4/(e*x + d), x)
_______________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{p} x^{4}}{e x + d}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-e^2*x^2 + d^2)^p*x^4/(e*x + d),x, algorithm="fricas")
[Out]
integral((-e^2*x^2 + d^2)^p*x^4/(e*x + d), x)
_______________________________________________________________________________________
Sympy [A] time = 57.2155, size = 4442, normalized size = 30.01 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**4*(-e**2*x**2+d**2)**p/(e*x+d),x)
[Out]
Piecewise((-6*0**p*d**4*d**(2*p)*p*log(d**2/(e**2*x**2))*gamma(-p)*gamma(-p - 1/
2)*gamma(p + 1)*gamma(p + 3)/(12*e**5*p*gamma(-p)*gamma(-p - 1/2)*gamma(p + 1)*g
amma(p + 3) + 12*e**5*gamma(-p)*gamma(-p - 1/2)*gamma(p + 1)*gamma(p + 3)) + 6*0
**p*d**4*d**(2*p)*p*log(d**2/(e**2*x**2) - 1)*gamma(-p)*gamma(-p - 1/2)*gamma(p
+ 1)*gamma(p + 3)/(12*e**5*p*gamma(-p)*gamma(-p - 1/2)*gamma(p + 1)*gamma(p + 3)
+ 12*e**5*gamma(-p)*gamma(-p - 1/2)*gamma(p + 1)*gamma(p + 3)) + 12*0**p*d**4*d
**(2*p)*p*acoth(d/(e*x))*gamma(-p)*gamma(-p - 1/2)*gamma(p + 1)*gamma(p + 3)/(12
*e**5*p*gamma(-p)*gamma(-p - 1/2)*gamma(p + 1)*gamma(p + 3) + 12*e**5*gamma(-p)*
gamma(-p - 1/2)*gamma(p + 1)*gamma(p + 3)) - 6*0**p*d**4*d**(2*p)*log(d**2/(e**2
*x**2))*gamma(-p)*gamma(-p - 1/2)*gamma(p + 1)*gamma(p + 3)/(12*e**5*p*gamma(-p)
*gamma(-p - 1/2)*gamma(p + 1)*gamma(p + 3) + 12*e**5*gamma(-p)*gamma(-p - 1/2)*g
amma(p + 1)*gamma(p + 3)) + 6*0**p*d**4*d**(2*p)*log(d**2/(e**2*x**2) - 1)*gamma
(-p)*gamma(-p - 1/2)*gamma(p + 1)*gamma(p + 3)/(12*e**5*p*gamma(-p)*gamma(-p - 1
/2)*gamma(p + 1)*gamma(p + 3) + 12*e**5*gamma(-p)*gamma(-p - 1/2)*gamma(p + 1)*g
amma(p + 3)) + 12*0**p*d**4*d**(2*p)*acoth(d/(e*x))*gamma(-p)*gamma(-p - 1/2)*ga
mma(p + 1)*gamma(p + 3)/(12*e**5*p*gamma(-p)*gamma(-p - 1/2)*gamma(p + 1)*gamma(
p + 3) + 12*e**5*gamma(-p)*gamma(-p - 1/2)*gamma(p + 1)*gamma(p + 3)) - 12*0**p*
d**3*d**(2*p)*e*p*x*gamma(-p)*gamma(-p - 1/2)*gamma(p + 1)*gamma(p + 3)/(12*e**5
*p*gamma(-p)*gamma(-p - 1/2)*gamma(p + 1)*gamma(p + 3) + 12*e**5*gamma(-p)*gamma
(-p - 1/2)*gamma(p + 1)*gamma(p + 3)) - 12*0**p*d**3*d**(2*p)*e*x*gamma(-p)*gamm
a(-p - 1/2)*gamma(p + 1)*gamma(p + 3)/(12*e**5*p*gamma(-p)*gamma(-p - 1/2)*gamma
(p + 1)*gamma(p + 3) + 12*e**5*gamma(-p)*gamma(-p - 1/2)*gamma(p + 1)*gamma(p +
3)) + 6*0**p*d**2*d**(2*p)*e**2*p*x**2*gamma(-p)*gamma(-p - 1/2)*gamma(p + 1)*ga
mma(p + 3)/(12*e**5*p*gamma(-p)*gamma(-p - 1/2)*gamma(p + 1)*gamma(p + 3) + 12*e
**5*gamma(-p)*gamma(-p - 1/2)*gamma(p + 1)*gamma(p + 3)) + 6*0**p*d**2*d**(2*p)*
e**2*x**2*gamma(-p)*gamma(-p - 1/2)*gamma(p + 1)*gamma(p + 3)/(12*e**5*p*gamma(-
p)*gamma(-p - 1/2)*gamma(p + 1)*gamma(p + 3) + 12*e**5*gamma(-p)*gamma(-p - 1/2)
*gamma(p + 1)*gamma(p + 3)) - 4*0**p*d*d**(2*p)*e**3*p*x**3*gamma(-p)*gamma(-p -
1/2)*gamma(p + 1)*gamma(p + 3)/(12*e**5*p*gamma(-p)*gamma(-p - 1/2)*gamma(p + 1
)*gamma(p + 3) + 12*e**5*gamma(-p)*gamma(-p - 1/2)*gamma(p + 1)*gamma(p + 3)) -
4*0**p*d*d**(2*p)*e**3*x**3*gamma(-p)*gamma(-p - 1/2)*gamma(p + 1)*gamma(p + 3)/
(12*e**5*p*gamma(-p)*gamma(-p - 1/2)*gamma(p + 1)*gamma(p + 3) + 12*e**5*gamma(-
p)*gamma(-p - 1/2)*gamma(p + 1)*gamma(p + 3)) + 3*0**p*d**(2*p)*e**4*p*x**4*gamm
a(-p)*gamma(-p - 1/2)*gamma(p + 1)*gamma(p + 3)/(12*e**5*p*gamma(-p)*gamma(-p -
1/2)*gamma(p + 1)*gamma(p + 3) + 12*e**5*gamma(-p)*gamma(-p - 1/2)*gamma(p + 1)*
gamma(p + 3)) + 3*0**p*d**(2*p)*e**4*x**4*gamma(-p)*gamma(-p - 1/2)*gamma(p + 1)
*gamma(p + 3)/(12*e**5*p*gamma(-p)*gamma(-p - 1/2)*gamma(p + 1)*gamma(p + 3) + 1
2*e**5*gamma(-p)*gamma(-p - 1/2)*gamma(p + 1)*gamma(p + 3)) + 12*d**4*e**(2*p)*x
**(2*p)*(d**2/(e**2*x**2) - 1)**p*gamma(-p)*gamma(p)*gamma(-p - 1/2)*gamma(p + 2
)/(12*e**5*p*gamma(-p)*gamma(-p - 1/2)*gamma(p + 1)*gamma(p + 3) + 12*e**5*gamma
(-p)*gamma(-p - 1/2)*gamma(p + 1)*gamma(p + 3)) + 12*d**2*e**2*e**(2*p)*p*x**2*x
**(2*p)*(d**2/(e**2*x**2) - 1)**p*gamma(-p)*gamma(p)*gamma(-p - 1/2)*gamma(p + 2
)/(12*e**5*p*gamma(-p)*gamma(-p - 1/2)*gamma(p + 1)*gamma(p + 3) + 12*e**5*gamma
(-p)*gamma(-p - 1/2)*gamma(p + 1)*gamma(p + 3)) + 6*d*e**3*e**(2*p)*p**2*x**3*x*
*(2*p)*exp(I*pi*p)*gamma(-p)*gamma(p)*gamma(-p - 3/2)*gamma(p + 3)*hyper((-p + 1
, -p - 3/2), (-p - 1/2,), d**2/(e**2*x**2))/(12*e**5*p*gamma(-p)*gamma(-p - 1/2)
*gamma(p + 1)*gamma(p + 3) + 12*e**5*gamma(-p)*gamma(-p - 1/2)*gamma(p + 1)*gamm
a(p + 3)) + 6*d*e**3*e**(2*p)*p*x**3*x**(2*p)*exp(I*pi*p)*gamma(-p)*gamma(p)*gam
ma(-p - 3/2)*gamma(p + 3)*hyper((-p + 1, -p - 3/2), (-p - 1/2,), d**2/(e**2*x**2
))/(12*e**5*p*gamma(-p)*gamma(-p - 1/2)*gamma(p + 1)*gamma(p + 3) + 12*e**5*gamm
a(-p)*gamma(-p - 1/2)*gamma(p + 1)*gamma(p + 3)) + 6*e**4*e**(2*p)*p**2*x**4*x**
(2*p)*(d**2/(e**2*x**2) - 1)**p*gamma(-p)*gamma(p)*gamma(-p - 1/2)*gamma(p + 2)/
(12*e**5*p*gamma(-p)*gamma(-p - 1/2)*gamma(p + 1)*gamma(p + 3) + 12*e**5*gamma(-
p)*gamma(-p - 1/2)*gamma(p + 1)*gamma(p + 3)) + 6*e**4*e**(2*p)*p*x**4*x**(2*p)*
(d**2/(e**2*x**2) - 1)**p*gamma(-p)*gamma(p)*gamma(-p - 1/2)*gamma(p + 2)/(12*e*
*5*p*gamma(-p)*gamma(-p - 1/2)*gamma(p + 1)*gamma(p + 3) + 12*e**5*gamma(-p)*gam
ma(-p - 1/2)*gamma(p + 1)*gamma(p + 3)), Abs(d**2/(e**2*x**2)) > 1), (-6*0**p*d*
*4*d**(2*p)*p*log(d**2/(e**2*x**2))*gamma(-p)*gamma(-p - 1/2)*gamma(p + 1)*gamma
(p + 3)/(12*e**5*p*gamma(-p)*gamma(-p - 1/2)*gamma(p + 1)*gamma(p + 3) + 12*e**5
*gamma(-p)*gamma(-p - 1/2)*gamma(p + 1)*gamma(p + 3)) + 6*0**p*d**4*d**(2*p)*p*l
og(-d**2/(e**2*x**2) + 1)*gamma(-p)*gamma(-p - 1/2)*gamma(p + 1)*gamma(p + 3)/(1
2*e**5*p*gamma(-p)*gamma(-p - 1/2)*gamma(p + 1)*gamma(p + 3) + 12*e**5*gamma(-p)
*gamma(-p - 1/2)*gamma(p + 1)*gamma(p + 3)) + 12*0**p*d**4*d**(2*p)*p*atanh(d/(e
*x))*gamma(-p)*gamma(-p - 1/2)*gamma(p + 1)*gamma(p + 3)/(12*e**5*p*gamma(-p)*ga
mma(-p - 1/2)*gamma(p + 1)*gamma(p + 3) + 12*e**5*gamma(-p)*gamma(-p - 1/2)*gamm
a(p + 1)*gamma(p + 3)) - 6*0**p*d**4*d**(2*p)*log(d**2/(e**2*x**2))*gamma(-p)*ga
mma(-p - 1/2)*gamma(p + 1)*gamma(p + 3)/(12*e**5*p*gamma(-p)*gamma(-p - 1/2)*gam
ma(p + 1)*gamma(p + 3) + 12*e**5*gamma(-p)*gamma(-p - 1/2)*gamma(p + 1)*gamma(p
+ 3)) + 6*0**p*d**4*d**(2*p)*log(-d**2/(e**2*x**2) + 1)*gamma(-p)*gamma(-p - 1/2
)*gamma(p + 1)*gamma(p + 3)/(12*e**5*p*gamma(-p)*gamma(-p - 1/2)*gamma(p + 1)*ga
mma(p + 3) + 12*e**5*gamma(-p)*gamma(-p - 1/2)*gamma(p + 1)*gamma(p + 3)) + 12*0
**p*d**4*d**(2*p)*atanh(d/(e*x))*gamma(-p)*gamma(-p - 1/2)*gamma(p + 1)*gamma(p
+ 3)/(12*e**5*p*gamma(-p)*gamma(-p - 1/2)*gamma(p + 1)*gamma(p + 3) + 12*e**5*ga
mma(-p)*gamma(-p - 1/2)*gamma(p + 1)*gamma(p + 3)) - 12*0**p*d**3*d**(2*p)*e*p*x
*gamma(-p)*gamma(-p - 1/2)*gamma(p + 1)*gamma(p + 3)/(12*e**5*p*gamma(-p)*gamma(
-p - 1/2)*gamma(p + 1)*gamma(p + 3) + 12*e**5*gamma(-p)*gamma(-p - 1/2)*gamma(p
+ 1)*gamma(p + 3)) - 12*0**p*d**3*d**(2*p)*e*x*gamma(-p)*gamma(-p - 1/2)*gamma(p
+ 1)*gamma(p + 3)/(12*e**5*p*gamma(-p)*gamma(-p - 1/2)*gamma(p + 1)*gamma(p + 3
) + 12*e**5*gamma(-p)*gamma(-p - 1/2)*gamma(p + 1)*gamma(p + 3)) + 6*0**p*d**2*d
**(2*p)*e**2*p*x**2*gamma(-p)*gamma(-p - 1/2)*gamma(p + 1)*gamma(p + 3)/(12*e**5
*p*gamma(-p)*gamma(-p - 1/2)*gamma(p + 1)*gamma(p + 3) + 12*e**5*gamma(-p)*gamma
(-p - 1/2)*gamma(p + 1)*gamma(p + 3)) + 6*0**p*d**2*d**(2*p)*e**2*x**2*gamma(-p)
*gamma(-p - 1/2)*gamma(p + 1)*gamma(p + 3)/(12*e**5*p*gamma(-p)*gamma(-p - 1/2)*
gamma(p + 1)*gamma(p + 3) + 12*e**5*gamma(-p)*gamma(-p - 1/2)*gamma(p + 1)*gamma
(p + 3)) - 4*0**p*d*d**(2*p)*e**3*p*x**3*gamma(-p)*gamma(-p - 1/2)*gamma(p + 1)*
gamma(p + 3)/(12*e**5*p*gamma(-p)*gamma(-p - 1/2)*gamma(p + 1)*gamma(p + 3) + 12
*e**5*gamma(-p)*gamma(-p - 1/2)*gamma(p + 1)*gamma(p + 3)) - 4*0**p*d*d**(2*p)*e
**3*x**3*gamma(-p)*gamma(-p - 1/2)*gamma(p + 1)*gamma(p + 3)/(12*e**5*p*gamma(-p
)*gamma(-p - 1/2)*gamma(p + 1)*gamma(p + 3) + 12*e**5*gamma(-p)*gamma(-p - 1/2)*
gamma(p + 1)*gamma(p + 3)) + 3*0**p*d**(2*p)*e**4*p*x**4*gamma(-p)*gamma(-p - 1/
2)*gamma(p + 1)*gamma(p + 3)/(12*e**5*p*gamma(-p)*gamma(-p - 1/2)*gamma(p + 1)*g
amma(p + 3) + 12*e**5*gamma(-p)*gamma(-p - 1/2)*gamma(p + 1)*gamma(p + 3)) + 3*0
**p*d**(2*p)*e**4*x**4*gamma(-p)*gamma(-p - 1/2)*gamma(p + 1)*gamma(p + 3)/(12*e
**5*p*gamma(-p)*gamma(-p - 1/2)*gamma(p + 1)*gamma(p + 3) + 12*e**5*gamma(-p)*ga
mma(-p - 1/2)*gamma(p + 1)*gamma(p + 3)) + 12*d**4*e**(2*p)*x**(2*p)*(-d**2/(e**
2*x**2) + 1)**p*exp(I*pi*p)*gamma(-p)*gamma(p)*gamma(-p - 1/2)*gamma(p + 2)/(12*
e**5*p*gamma(-p)*gamma(-p - 1/2)*gamma(p + 1)*gamma(p + 3) + 12*e**5*gamma(-p)*g
amma(-p - 1/2)*gamma(p + 1)*gamma(p + 3)) + 12*d**2*e**2*e**(2*p)*p*x**2*x**(2*p
)*(-d**2/(e**2*x**2) + 1)**p*exp(I*pi*p)*gamma(-p)*gamma(p)*gamma(-p - 1/2)*gamm
a(p + 2)/(12*e**5*p*gamma(-p)*gamma(-p - 1/2)*gamma(p + 1)*gamma(p + 3) + 12*e**
5*gamma(-p)*gamma(-p - 1/2)*gamma(p + 1)*gamma(p + 3)) + 6*d*e**3*e**(2*p)*p**2*
x**3*x**(2*p)*exp(I*pi*p)*gamma(-p)*gamma(p)*gamma(-p - 3/2)*gamma(p + 3)*hyper(
(-p + 1, -p - 3/2), (-p - 1/2,), d**2/(e**2*x**2))/(12*e**5*p*gamma(-p)*gamma(-p
- 1/2)*gamma(p + 1)*gamma(p + 3) + 12*e**5*gamma(-p)*gamma(-p - 1/2)*gamma(p +
1)*gamma(p + 3)) + 6*d*e**3*e**(2*p)*p*x**3*x**(2*p)*exp(I*pi*p)*gamma(-p)*gamma
(p)*gamma(-p - 3/2)*gamma(p + 3)*hyper((-p + 1, -p - 3/2), (-p - 1/2,), d**2/(e*
*2*x**2))/(12*e**5*p*gamma(-p)*gamma(-p - 1/2)*gamma(p + 1)*gamma(p + 3) + 12*e*
*5*gamma(-p)*gamma(-p - 1/2)*gamma(p + 1)*gamma(p + 3)) + 6*e**4*e**(2*p)*p**2*x
**4*x**(2*p)*(-d**2/(e**2*x**2) + 1)**p*exp(I*pi*p)*gamma(-p)*gamma(p)*gamma(-p
- 1/2)*gamma(p + 2)/(12*e**5*p*gamma(-p)*gamma(-p - 1/2)*gamma(p + 1)*gamma(p +
3) + 12*e**5*gamma(-p)*gamma(-p - 1/2)*gamma(p + 1)*gamma(p + 3)) + 6*e**4*e**(2
*p)*p*x**4*x**(2*p)*(-d**2/(e**2*x**2) + 1)**p*exp(I*pi*p)*gamma(-p)*gamma(p)*ga
mma(-p - 1/2)*gamma(p + 2)/(12*e**5*p*gamma(-p)*gamma(-p - 1/2)*gamma(p + 1)*gam
ma(p + 3) + 12*e**5*gamma(-p)*gamma(-p - 1/2)*gamma(p + 1)*gamma(p + 3)), True))
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{p} x^{4}}{e x + d}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-e^2*x^2 + d^2)^p*x^4/(e*x + d),x, algorithm="giac")
[Out]
integrate((-e^2*x^2 + d^2)^p*x^4/(e*x + d), x)