3.268 \(\int \frac{x^3 \left (d^2-e^2 x^2\right )^p}{d+e x} \, dx\)
Optimal. Leaf size=121 \[ -\frac{e x^5 \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac{5}{2},1-p;\frac{7}{2};\frac{e^2 x^2}{d^2}\right )}{5 d^2}+\frac{d \left (d^2-e^2 x^2\right )^{p+1}}{2 e^4 (p+1)}-\frac{d^3 \left (d^2-e^2 x^2\right )^p}{2 e^4 p} \]
[Out]
-(d^3*(d^2 - e^2*x^2)^p)/(2*e^4*p) + (d*(d^2 - e^2*x^2)^(1 + p))/(2*e^4*(1 + p))
- (e*x^5*(d^2 - e^2*x^2)^p*Hypergeometric2F1[5/2, 1 - p, 7/2, (e^2*x^2)/d^2])/(
5*d^2*(1 - (e^2*x^2)/d^2)^p)
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Rubi [A] time = 0.241402, antiderivative size = 121, 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
\[ -\frac{e x^5 \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac{5}{2},1-p;\frac{7}{2};\frac{e^2 x^2}{d^2}\right )}{5 d^2}+\frac{d \left (d^2-e^2 x^2\right )^{p+1}}{2 e^4 (p+1)}-\frac{d^3 \left (d^2-e^2 x^2\right )^p}{2 e^4 p} \]
Antiderivative was successfully verified.
[In] Int[(x^3*(d^2 - e^2*x^2)^p)/(d + e*x),x]
[Out]
-(d^3*(d^2 - e^2*x^2)^p)/(2*e^4*p) + (d*(d^2 - e^2*x^2)^(1 + p))/(2*e^4*(1 + p))
- (e*x^5*(d^2 - e^2*x^2)^p*Hypergeometric2F1[5/2, 1 - p, 7/2, (e^2*x^2)/d^2])/(
5*d^2*(1 - (e^2*x^2)/d^2)^p)
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Rubi in Sympy [A] time = 44.9319, size = 97, normalized size = 0.8 \[ - \frac{d^{3} \left (d^{2} - e^{2} x^{2}\right )^{p}}{2 e^{4} p} + \frac{d \left (d^{2} - e^{2} x^{2}\right )^{p + 1}}{2 e^{4} \left (p + 1\right )} - \frac{e x^{5} \left (1 - \frac{e^{2} x^{2}}{d^{2}}\right )^{- p} \left (d^{2} - e^{2} x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p + 1, \frac{5}{2} \\ \frac{7}{2} \end{matrix}\middle |{\frac{e^{2} x^{2}}{d^{2}}} \right )}}{5 d^{2}} \]
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),x)
[Out]
-d**3*(d**2 - e**2*x**2)**p/(2*e**4*p) + d*(d**2 - e**2*x**2)**(p + 1)/(2*e**4*(
p + 1)) - e*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**2)
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Mathematica [B] time = 0.463687, size = 245, normalized size = 2.02 \[ \frac{\left (\frac{e x}{d}+1\right )^{-p} \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \left (6 d^2 e (p+1) x \left (\frac{e x}{d}+1\right )^p \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};\frac{e^2 x^2}{d^2}\right )+3 d \left (d (d-e x) \left (2-\frac{2 e^2 x^2}{d^2}\right )^p \, _2F_1\left (1-p,p+1;p+2;\frac{d-e x}{2 d}\right )+\left (d^2 \left (\left (1-\frac{e^2 x^2}{d^2}\right )^p-1\right )-e^2 x^2 \left (1-\frac{e^2 x^2}{d^2}\right )^p\right ) \left (\frac{e x}{d}+1\right )^p\right )+2 e^3 (p+1) x^3 \left (\frac{e x}{d}+1\right )^p \, _2F_1\left (\frac{3}{2},-p;\frac{5}{2};\frac{e^2 x^2}{d^2}\right )\right )}{6 e^4 (p+1)} \]
Antiderivative was successfully verified.
[In] Integrate[(x^3*(d^2 - e^2*x^2)^p)/(d + e*x),x]
[Out]
((d^2 - e^2*x^2)^p*(6*d^2*e*(1 + p)*x*(1 + (e*x)/d)^p*Hypergeometric2F1[1/2, -p,
3/2, (e^2*x^2)/d^2] + 2*e^3*(1 + p)*x^3*(1 + (e*x)/d)^p*Hypergeometric2F1[3/2,
-p, 5/2, (e^2*x^2)/d^2] + 3*d*((1 + (e*x)/d)^p*(-(e^2*x^2*(1 - (e^2*x^2)/d^2)^p)
+ d^2*(-1 + (1 - (e^2*x^2)/d^2)^p)) + d*(d - e*x)*(2 - (2*e^2*x^2)/d^2)^p*Hyper
geometric2F1[1 - p, 1 + p, 2 + p, (d - e*x)/(2*d)])))/(6*e^4*(1 + p)*(1 + (e*x)/
d)^p*(1 - (e^2*x^2)/d^2)^p)
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Maple [F] time = 0.074, size = 0, normalized size = 0. \[ \int{\frac{{x}^{3} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{p}}{ex+d}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^3*(-e^2*x^2+d^2)^p/(e*x+d),x)
[Out]
int(x^3*(-e^2*x^2+d^2)^p/(e*x+d),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{p} x^{3}}{e x + d}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-e^2*x^2 + d^2)^p*x^3/(e*x + d),x, algorithm="maxima")
[Out]
integrate((-e^2*x^2 + d^2)^p*x^3/(e*x + d), x)
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{p} x^{3}}{e x + d}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-e^2*x^2 + d^2)^p*x^3/(e*x + d),x, algorithm="fricas")
[Out]
integral((-e^2*x^2 + d^2)^p*x^3/(e*x + d), x)
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Sympy [A] time = 36.6861, size = 5090, normalized size = 42.07 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**3*(-e**2*x**2+d**2)**p/(e*x+d),x)
[Out]
Piecewise((3*0**p*d**3*d**(2*p)*p*log(d**2/(e**2*x**2))*gamma(-p - 1/2)*gamma(p
+ 1)/(6*e**4*p*gamma(-p - 1/2)*gamma(p + 1) + 6*e**4*gamma(-p - 1/2)*gamma(p + 1
)) - 3*0**p*d**3*d**(2*p)*p*log(d**2/(e**2*x**2) - 1)*gamma(-p - 1/2)*gamma(p +
1)/(6*e**4*p*gamma(-p - 1/2)*gamma(p + 1) + 6*e**4*gamma(-p - 1/2)*gamma(p + 1))
- 6*0**p*d**3*d**(2*p)*p*acoth(d/(e*x))*gamma(-p - 1/2)*gamma(p + 1)/(6*e**4*p*
gamma(-p - 1/2)*gamma(p + 1) + 6*e**4*gamma(-p - 1/2)*gamma(p + 1)) + 3*0**p*d**
3*d**(2*p)*log(d**2/(e**2*x**2))*gamma(-p - 1/2)*gamma(p + 1)/(6*e**4*p*gamma(-p
- 1/2)*gamma(p + 1) + 6*e**4*gamma(-p - 1/2)*gamma(p + 1)) - 3*0**p*d**3*d**(2*
p)*log(d**2/(e**2*x**2) - 1)*gamma(-p - 1/2)*gamma(p + 1)/(6*e**4*p*gamma(-p - 1
/2)*gamma(p + 1) + 6*e**4*gamma(-p - 1/2)*gamma(p + 1)) - 6*0**p*d**3*d**(2*p)*a
coth(d/(e*x))*gamma(-p - 1/2)*gamma(p + 1)/(6*e**4*p*gamma(-p - 1/2)*gamma(p + 1
) + 6*e**4*gamma(-p - 1/2)*gamma(p + 1)) + 6*0**p*d**2*d**(2*p)*e*p*x*gamma(-p -
1/2)*gamma(p + 1)/(6*e**4*p*gamma(-p - 1/2)*gamma(p + 1) + 6*e**4*gamma(-p - 1/
2)*gamma(p + 1)) + 6*0**p*d**2*d**(2*p)*e*x*gamma(-p - 1/2)*gamma(p + 1)/(6*e**4
*p*gamma(-p - 1/2)*gamma(p + 1) + 6*e**4*gamma(-p - 1/2)*gamma(p + 1)) - 3*0**p*
d*d**(2*p)*e**2*p*x**2*gamma(-p - 1/2)*gamma(p + 1)/(6*e**4*p*gamma(-p - 1/2)*ga
mma(p + 1) + 6*e**4*gamma(-p - 1/2)*gamma(p + 1)) - 3*0**p*d*d**(2*p)*e**2*x**2*
gamma(-p - 1/2)*gamma(p + 1)/(6*e**4*p*gamma(-p - 1/2)*gamma(p + 1) + 6*e**4*gam
ma(-p - 1/2)*gamma(p + 1)) + 2*0**p*d**(2*p)*e**3*p*x**3*gamma(-p - 1/2)*gamma(p
+ 1)/(6*e**4*p*gamma(-p - 1/2)*gamma(p + 1) + 6*e**4*gamma(-p - 1/2)*gamma(p +
1)) + 2*0**p*d**(2*p)*e**3*x**3*gamma(-p - 1/2)*gamma(p + 1)/(6*e**4*p*gamma(-p
- 1/2)*gamma(p + 1) + 6*e**4*gamma(-p - 1/2)*gamma(p + 1)) - 3*d**3*d**(2*p)*(-1
+ e**2*x**2/d**2)**p*exp(I*pi*p)*gamma(p)*gamma(-p - 1/2)/(6*e**4*p*gamma(-p -
1/2)*gamma(p + 1) + 6*e**4*gamma(-p - 1/2)*gamma(p + 1)) - 3*d*d**(2*p)*e**2*p*x
**2*(-1 + e**2*x**2/d**2)**p*exp(I*pi*p)*gamma(p)*gamma(-p - 1/2)/(6*e**4*p*gamm
a(-p - 1/2)*gamma(p + 1) + 6*e**4*gamma(-p - 1/2)*gamma(p + 1)) - 3*e**3*e**(2*p
)*p**2*x**3*x**(2*p)*exp(I*pi*p)*gamma(p)*gamma(-p - 3/2)*hyper((-p + 1, -p - 3/
2), (-p - 1/2,), d**2/(e**2*x**2))/(6*e**4*p*gamma(-p - 1/2)*gamma(p + 1) + 6*e*
*4*gamma(-p - 1/2)*gamma(p + 1)) - 3*e**3*e**(2*p)*p*x**3*x**(2*p)*exp(I*pi*p)*g
amma(p)*gamma(-p - 3/2)*hyper((-p + 1, -p - 3/2), (-p - 1/2,), d**2/(e**2*x**2))
/(6*e**4*p*gamma(-p - 1/2)*gamma(p + 1) + 6*e**4*gamma(-p - 1/2)*gamma(p + 1)),
(Abs(e**2*x**2/d**2) > 1) & (Abs(d**2/(e**2*x**2)) > 1)), (3*0**p*d**3*d**(2*p)*
p*log(d**2/(e**2*x**2))*gamma(-p - 1/2)*gamma(p + 1)/(6*e**4*p*gamma(-p - 1/2)*g
amma(p + 1) + 6*e**4*gamma(-p - 1/2)*gamma(p + 1)) - 3*0**p*d**3*d**(2*p)*p*log(
-d**2/(e**2*x**2) + 1)*gamma(-p - 1/2)*gamma(p + 1)/(6*e**4*p*gamma(-p - 1/2)*ga
mma(p + 1) + 6*e**4*gamma(-p - 1/2)*gamma(p + 1)) - 6*0**p*d**3*d**(2*p)*p*atanh
(d/(e*x))*gamma(-p - 1/2)*gamma(p + 1)/(6*e**4*p*gamma(-p - 1/2)*gamma(p + 1) +
6*e**4*gamma(-p - 1/2)*gamma(p + 1)) + 3*0**p*d**3*d**(2*p)*log(d**2/(e**2*x**2)
)*gamma(-p - 1/2)*gamma(p + 1)/(6*e**4*p*gamma(-p - 1/2)*gamma(p + 1) + 6*e**4*g
amma(-p - 1/2)*gamma(p + 1)) - 3*0**p*d**3*d**(2*p)*log(-d**2/(e**2*x**2) + 1)*g
amma(-p - 1/2)*gamma(p + 1)/(6*e**4*p*gamma(-p - 1/2)*gamma(p + 1) + 6*e**4*gamm
a(-p - 1/2)*gamma(p + 1)) - 6*0**p*d**3*d**(2*p)*atanh(d/(e*x))*gamma(-p - 1/2)*
gamma(p + 1)/(6*e**4*p*gamma(-p - 1/2)*gamma(p + 1) + 6*e**4*gamma(-p - 1/2)*gam
ma(p + 1)) + 6*0**p*d**2*d**(2*p)*e*p*x*gamma(-p - 1/2)*gamma(p + 1)/(6*e**4*p*g
amma(-p - 1/2)*gamma(p + 1) + 6*e**4*gamma(-p - 1/2)*gamma(p + 1)) + 6*0**p*d**2
*d**(2*p)*e*x*gamma(-p - 1/2)*gamma(p + 1)/(6*e**4*p*gamma(-p - 1/2)*gamma(p + 1
) + 6*e**4*gamma(-p - 1/2)*gamma(p + 1)) - 3*0**p*d*d**(2*p)*e**2*p*x**2*gamma(-
p - 1/2)*gamma(p + 1)/(6*e**4*p*gamma(-p - 1/2)*gamma(p + 1) + 6*e**4*gamma(-p -
1/2)*gamma(p + 1)) - 3*0**p*d*d**(2*p)*e**2*x**2*gamma(-p - 1/2)*gamma(p + 1)/(
6*e**4*p*gamma(-p - 1/2)*gamma(p + 1) + 6*e**4*gamma(-p - 1/2)*gamma(p + 1)) + 2
*0**p*d**(2*p)*e**3*p*x**3*gamma(-p - 1/2)*gamma(p + 1)/(6*e**4*p*gamma(-p - 1/2
)*gamma(p + 1) + 6*e**4*gamma(-p - 1/2)*gamma(p + 1)) + 2*0**p*d**(2*p)*e**3*x**
3*gamma(-p - 1/2)*gamma(p + 1)/(6*e**4*p*gamma(-p - 1/2)*gamma(p + 1) + 6*e**4*g
amma(-p - 1/2)*gamma(p + 1)) - 3*d**3*d**(2*p)*(-1 + e**2*x**2/d**2)**p*exp(I*pi
*p)*gamma(p)*gamma(-p - 1/2)/(6*e**4*p*gamma(-p - 1/2)*gamma(p + 1) + 6*e**4*gam
ma(-p - 1/2)*gamma(p + 1)) - 3*d*d**(2*p)*e**2*p*x**2*(-1 + e**2*x**2/d**2)**p*e
xp(I*pi*p)*gamma(p)*gamma(-p - 1/2)/(6*e**4*p*gamma(-p - 1/2)*gamma(p + 1) + 6*e
**4*gamma(-p - 1/2)*gamma(p + 1)) - 3*e**3*e**(2*p)*p**2*x**3*x**(2*p)*exp(I*pi*
p)*gamma(p)*gamma(-p - 3/2)*hyper((-p + 1, -p - 3/2), (-p - 1/2,), d**2/(e**2*x*
*2))/(6*e**4*p*gamma(-p - 1/2)*gamma(p + 1) + 6*e**4*gamma(-p - 1/2)*gamma(p + 1
)) - 3*e**3*e**(2*p)*p*x**3*x**(2*p)*exp(I*pi*p)*gamma(p)*gamma(-p - 3/2)*hyper(
(-p + 1, -p - 3/2), (-p - 1/2,), d**2/(e**2*x**2))/(6*e**4*p*gamma(-p - 1/2)*gam
ma(p + 1) + 6*e**4*gamma(-p - 1/2)*gamma(p + 1)), Abs(e**2*x**2/d**2) > 1), (3*0
**p*d**3*d**(2*p)*p*log(d**2/(e**2*x**2))*gamma(-p - 1/2)*gamma(p + 1)/(6*e**4*p
*gamma(-p - 1/2)*gamma(p + 1) + 6*e**4*gamma(-p - 1/2)*gamma(p + 1)) - 3*0**p*d*
*3*d**(2*p)*p*log(d**2/(e**2*x**2) - 1)*gamma(-p - 1/2)*gamma(p + 1)/(6*e**4*p*g
amma(-p - 1/2)*gamma(p + 1) + 6*e**4*gamma(-p - 1/2)*gamma(p + 1)) - 6*0**p*d**3
*d**(2*p)*p*acoth(d/(e*x))*gamma(-p - 1/2)*gamma(p + 1)/(6*e**4*p*gamma(-p - 1/2
)*gamma(p + 1) + 6*e**4*gamma(-p - 1/2)*gamma(p + 1)) + 3*0**p*d**3*d**(2*p)*log
(d**2/(e**2*x**2))*gamma(-p - 1/2)*gamma(p + 1)/(6*e**4*p*gamma(-p - 1/2)*gamma(
p + 1) + 6*e**4*gamma(-p - 1/2)*gamma(p + 1)) - 3*0**p*d**3*d**(2*p)*log(d**2/(e
**2*x**2) - 1)*gamma(-p - 1/2)*gamma(p + 1)/(6*e**4*p*gamma(-p - 1/2)*gamma(p +
1) + 6*e**4*gamma(-p - 1/2)*gamma(p + 1)) - 6*0**p*d**3*d**(2*p)*acoth(d/(e*x))*
gamma(-p - 1/2)*gamma(p + 1)/(6*e**4*p*gamma(-p - 1/2)*gamma(p + 1) + 6*e**4*gam
ma(-p - 1/2)*gamma(p + 1)) + 6*0**p*d**2*d**(2*p)*e*p*x*gamma(-p - 1/2)*gamma(p
+ 1)/(6*e**4*p*gamma(-p - 1/2)*gamma(p + 1) + 6*e**4*gamma(-p - 1/2)*gamma(p + 1
)) + 6*0**p*d**2*d**(2*p)*e*x*gamma(-p - 1/2)*gamma(p + 1)/(6*e**4*p*gamma(-p -
1/2)*gamma(p + 1) + 6*e**4*gamma(-p - 1/2)*gamma(p + 1)) - 3*0**p*d*d**(2*p)*e**
2*p*x**2*gamma(-p - 1/2)*gamma(p + 1)/(6*e**4*p*gamma(-p - 1/2)*gamma(p + 1) + 6
*e**4*gamma(-p - 1/2)*gamma(p + 1)) - 3*0**p*d*d**(2*p)*e**2*x**2*gamma(-p - 1/2
)*gamma(p + 1)/(6*e**4*p*gamma(-p - 1/2)*gamma(p + 1) + 6*e**4*gamma(-p - 1/2)*g
amma(p + 1)) + 2*0**p*d**(2*p)*e**3*p*x**3*gamma(-p - 1/2)*gamma(p + 1)/(6*e**4*
p*gamma(-p - 1/2)*gamma(p + 1) + 6*e**4*gamma(-p - 1/2)*gamma(p + 1)) + 2*0**p*d
**(2*p)*e**3*x**3*gamma(-p - 1/2)*gamma(p + 1)/(6*e**4*p*gamma(-p - 1/2)*gamma(p
+ 1) + 6*e**4*gamma(-p - 1/2)*gamma(p + 1)) - 3*d**3*d**(2*p)*(1 - e**2*x**2/d*
*2)**p*gamma(p)*gamma(-p - 1/2)/(6*e**4*p*gamma(-p - 1/2)*gamma(p + 1) + 6*e**4*
gamma(-p - 1/2)*gamma(p + 1)) - 3*d*d**(2*p)*e**2*p*x**2*(1 - e**2*x**2/d**2)**p
*gamma(p)*gamma(-p - 1/2)/(6*e**4*p*gamma(-p - 1/2)*gamma(p + 1) + 6*e**4*gamma(
-p - 1/2)*gamma(p + 1)) - 3*e**3*e**(2*p)*p**2*x**3*x**(2*p)*exp(I*pi*p)*gamma(p
)*gamma(-p - 3/2)*hyper((-p + 1, -p - 3/2), (-p - 1/2,), d**2/(e**2*x**2))/(6*e*
*4*p*gamma(-p - 1/2)*gamma(p + 1) + 6*e**4*gamma(-p - 1/2)*gamma(p + 1)) - 3*e**
3*e**(2*p)*p*x**3*x**(2*p)*exp(I*pi*p)*gamma(p)*gamma(-p - 3/2)*hyper((-p + 1, -
p - 3/2), (-p - 1/2,), d**2/(e**2*x**2))/(6*e**4*p*gamma(-p - 1/2)*gamma(p + 1)
+ 6*e**4*gamma(-p - 1/2)*gamma(p + 1)), Abs(d**2/(e**2*x**2)) > 1), (3*0**p*d**3
*d**(2*p)*p*log(d**2/(e**2*x**2))*gamma(-p - 1/2)*gamma(p + 1)/(6*e**4*p*gamma(-
p - 1/2)*gamma(p + 1) + 6*e**4*gamma(-p - 1/2)*gamma(p + 1)) - 3*0**p*d**3*d**(2
*p)*p*log(-d**2/(e**2*x**2) + 1)*gamma(-p - 1/2)*gamma(p + 1)/(6*e**4*p*gamma(-p
- 1/2)*gamma(p + 1) + 6*e**4*gamma(-p - 1/2)*gamma(p + 1)) - 6*0**p*d**3*d**(2*
p)*p*atanh(d/(e*x))*gamma(-p - 1/2)*gamma(p + 1)/(6*e**4*p*gamma(-p - 1/2)*gamma
(p + 1) + 6*e**4*gamma(-p - 1/2)*gamma(p + 1)) + 3*0**p*d**3*d**(2*p)*log(d**2/(
e**2*x**2))*gamma(-p - 1/2)*gamma(p + 1)/(6*e**4*p*gamma(-p - 1/2)*gamma(p + 1)
+ 6*e**4*gamma(-p - 1/2)*gamma(p + 1)) - 3*0**p*d**3*d**(2*p)*log(-d**2/(e**2*x*
*2) + 1)*gamma(-p - 1/2)*gamma(p + 1)/(6*e**4*p*gamma(-p - 1/2)*gamma(p + 1) + 6
*e**4*gamma(-p - 1/2)*gamma(p + 1)) - 6*0**p*d**3*d**(2*p)*atanh(d/(e*x))*gamma(
-p - 1/2)*gamma(p + 1)/(6*e**4*p*gamma(-p - 1/2)*gamma(p + 1) + 6*e**4*gamma(-p
- 1/2)*gamma(p + 1)) + 6*0**p*d**2*d**(2*p)*e*p*x*gamma(-p - 1/2)*gamma(p + 1)/(
6*e**4*p*gamma(-p - 1/2)*gamma(p + 1) + 6*e**4*gamma(-p - 1/2)*gamma(p + 1)) + 6
*0**p*d**2*d**(2*p)*e*x*gamma(-p - 1/2)*gamma(p + 1)/(6*e**4*p*gamma(-p - 1/2)*g
amma(p + 1) + 6*e**4*gamma(-p - 1/2)*gamma(p + 1)) - 3*0**p*d*d**(2*p)*e**2*p*x*
*2*gamma(-p - 1/2)*gamma(p + 1)/(6*e**4*p*gamma(-p - 1/2)*gamma(p + 1) + 6*e**4*
gamma(-p - 1/2)*gamma(p + 1)) - 3*0**p*d*d**(2*p)*e**2*x**2*gamma(-p - 1/2)*gamm
a(p + 1)/(6*e**4*p*gamma(-p - 1/2)*gamma(p + 1) + 6*e**4*gamma(-p - 1/2)*gamma(p
+ 1)) + 2*0**p*d**(2*p)*e**3*p*x**3*gamma(-p - 1/2)*gamma(p + 1)/(6*e**4*p*gamm
a(-p - 1/2)*gamma(p + 1) + 6*e**4*gamma(-p - 1/2)*gamma(p + 1)) + 2*0**p*d**(2*p
)*e**3*x**3*gamma(-p - 1/2)*gamma(p + 1)/(6*e**4*p*gamma(-p - 1/2)*gamma(p + 1)
+ 6*e**4*gamma(-p - 1/2)*gamma(p + 1)) - 3*d**3*d**(2*p)*(1 - e**2*x**2/d**2)**p
*gamma(p)*gamma(-p - 1/2)/(6*e**4*p*gamma(-p - 1/2)*gamma(p + 1) + 6*e**4*gamma(
-p - 1/2)*gamma(p + 1)) - 3*d*d**(2*p)*e**2*p*x**2*(1 - e**2*x**2/d**2)**p*gamma
(p)*gamma(-p - 1/2)/(6*e**4*p*gamma(-p - 1/2)*gamma(p + 1) + 6*e**4*gamma(-p - 1
/2)*gamma(p + 1)) - 3*e**3*e**(2*p)*p**2*x**3*x**(2*p)*exp(I*pi*p)*gamma(p)*gamm
a(-p - 3/2)*hyper((-p + 1, -p - 3/2), (-p - 1/2,), d**2/(e**2*x**2))/(6*e**4*p*g
amma(-p - 1/2)*gamma(p + 1) + 6*e**4*gamma(-p - 1/2)*gamma(p + 1)) - 3*e**3*e**(
2*p)*p*x**3*x**(2*p)*exp(I*pi*p)*gamma(p)*gamma(-p - 3/2)*hyper((-p + 1, -p - 3/
2), (-p - 1/2,), d**2/(e**2*x**2))/(6*e**4*p*gamma(-p - 1/2)*gamma(p + 1) + 6*e*
*4*gamma(-p - 1/2)*gamma(p + 1)), True))
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{p} x^{3}}{e x + d}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-e^2*x^2 + d^2)^p*x^3/(e*x + d),x, algorithm="giac")
[Out]
integrate((-e^2*x^2 + d^2)^p*x^3/(e*x + d), x)