3.269 \(\int \frac{x^2 \left (d^2-e^2 x^2\right )^p}{d+e x} \, dx\)
Optimal. Leaf size=119 \[ \frac{x^3 \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac{3}{2},1-p;\frac{5}{2};\frac{e^2 x^2}{d^2}\right )}{3 d}+\frac{d^2 \left (d^2-e^2 x^2\right )^p}{2 e^3 p}-\frac{\left (d^2-e^2 x^2\right )^{p+1}}{2 e^3 (p+1)} \]
[Out]
(d^2*(d^2 - e^2*x^2)^p)/(2*e^3*p) - (d^2 - e^2*x^2)^(1 + p)/(2*e^3*(1 + p)) + (x
^3*(d^2 - e^2*x^2)^p*Hypergeometric2F1[3/2, 1 - p, 5/2, (e^2*x^2)/d^2])/(3*d*(1
- (e^2*x^2)/d^2)^p)
_______________________________________________________________________________________
Rubi [A] time = 0.235812, antiderivative size = 119, 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
\[ \frac{x^3 \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac{3}{2},1-p;\frac{5}{2};\frac{e^2 x^2}{d^2}\right )}{3 d}+\frac{d^2 \left (d^2-e^2 x^2\right )^p}{2 e^3 p}-\frac{\left (d^2-e^2 x^2\right )^{p+1}}{2 e^3 (p+1)} \]
Antiderivative was successfully verified.
[In] Int[(x^2*(d^2 - e^2*x^2)^p)/(d + e*x),x]
[Out]
(d^2*(d^2 - e^2*x^2)^p)/(2*e^3*p) - (d^2 - e^2*x^2)^(1 + p)/(2*e^3*(1 + p)) + (x
^3*(d^2 - e^2*x^2)^p*Hypergeometric2F1[3/2, 1 - p, 5/2, (e^2*x^2)/d^2])/(3*d*(1
- (e^2*x^2)/d^2)^p)
_______________________________________________________________________________________
Rubi in Sympy [A] time = 44.7078, size = 92, normalized size = 0.77 \[ \frac{d^{2} \left (d^{2} - e^{2} x^{2}\right )^{p}}{2 e^{3} p} - \frac{\left (d^{2} - e^{2} x^{2}\right )^{p + 1}}{2 e^{3} \left (p + 1\right )} + \frac{x^{3} \left (1 - \frac{e^{2} x^{2}}{d^{2}}\right )^{- p} \left (d^{2} - e^{2} x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p + 1, \frac{3}{2} \\ \frac{5}{2} \end{matrix}\middle |{\frac{e^{2} x^{2}}{d^{2}}} \right )}}{3 d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2*(-e**2*x**2+d**2)**p/(e*x+d),x)
[Out]
d**2*(d**2 - e**2*x**2)**p/(2*e**3*p) - (d**2 - e**2*x**2)**(p + 1)/(2*e**3*(p +
1)) + x**3*(1 - e**2*x**2/d**2)**(-p)*(d**2 - e**2*x**2)**p*hyper((-p + 1, 3/2)
, (5/2,), e**2*x**2/d**2)/(3*d)
_______________________________________________________________________________________
Mathematica [A] time = 0.427147, size = 198, normalized size = 1.66 \[ -\frac{\left (\frac{e x}{d}+1\right )^{-p} \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \left (2 d e (p+1) x \left (\frac{e x}{d}+1\right )^p \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};\frac{e^2 x^2}{d^2}\right )+d (d-e x) \left (2-\frac{2 e^2 x^2}{d^2}\right )^p \, _2F_1\left (1-p,p+1;p+2;\frac{d-e x}{2 d}\right )+\left (d^2 \left (\left (1-\frac{e^2 x^2}{d^2}\right )^p-1\right )-e^2 x^2 \left (1-\frac{e^2 x^2}{d^2}\right )^p\right ) \left (\frac{e x}{d}+1\right )^p\right )}{2 e^3 (p+1)} \]
Antiderivative was successfully verified.
[In] Integrate[(x^2*(d^2 - e^2*x^2)^p)/(d + e*x),x]
[Out]
-((d^2 - e^2*x^2)^p*((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)) + 2*d*e*(1 + p)*x*(1 + (e*x)/d)^p*Hypergeometric2F1[1
/2, -p, 3/2, (e^2*x^2)/d^2] + d*(d - e*x)*(2 - (2*e^2*x^2)/d^2)^p*Hypergeometric
2F1[1 - p, 1 + p, 2 + p, (d - e*x)/(2*d)]))/(2*e^3*(1 + p)*(1 + (e*x)/d)^p*(1 -
(e^2*x^2)/d^2)^p)
_______________________________________________________________________________________
Maple [F] time = 0.065, size = 0, normalized size = 0. \[ \int{\frac{{x}^{2} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{p}}{ex+d}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2*(-e^2*x^2+d^2)^p/(e*x+d),x)
[Out]
int(x^2*(-e^2*x^2+d^2)^p/(e*x+d),x)
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{p} x^{2}}{e x + d}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-e^2*x^2 + d^2)^p*x^2/(e*x + d),x, algorithm="maxima")
[Out]
integrate((-e^2*x^2 + d^2)^p*x^2/(e*x + d), x)
_______________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{p} x^{2}}{e x + d}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-e^2*x^2 + d^2)^p*x^2/(e*x + d),x, algorithm="fricas")
[Out]
integral((-e^2*x^2 + d^2)^p*x^2/(e*x + d), x)
_______________________________________________________________________________________
Sympy [A] time = 23.2794, size = 4124, normalized size = 34.66 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**2*(-e**2*x**2+d**2)**p/(e*x+d),x)
[Out]
Piecewise((-0**p*d**2*d**(2*p)*p*log(d**2/(e**2*x**2))*gamma(-p + 1/2)*gamma(p +
1)/(2*e**3*p*gamma(-p + 1/2)*gamma(p + 1) + 2*e**3*gamma(-p + 1/2)*gamma(p + 1)
) + 0**p*d**2*d**(2*p)*p*log(d**2/(e**2*x**2) - 1)*gamma(-p + 1/2)*gamma(p + 1)/
(2*e**3*p*gamma(-p + 1/2)*gamma(p + 1) + 2*e**3*gamma(-p + 1/2)*gamma(p + 1)) +
2*0**p*d**2*d**(2*p)*p*acoth(d/(e*x))*gamma(-p + 1/2)*gamma(p + 1)/(2*e**3*p*gam
ma(-p + 1/2)*gamma(p + 1) + 2*e**3*gamma(-p + 1/2)*gamma(p + 1)) - 0**p*d**2*d**
(2*p)*log(d**2/(e**2*x**2))*gamma(-p + 1/2)*gamma(p + 1)/(2*e**3*p*gamma(-p + 1/
2)*gamma(p + 1) + 2*e**3*gamma(-p + 1/2)*gamma(p + 1)) + 0**p*d**2*d**(2*p)*log(
d**2/(e**2*x**2) - 1)*gamma(-p + 1/2)*gamma(p + 1)/(2*e**3*p*gamma(-p + 1/2)*gam
ma(p + 1) + 2*e**3*gamma(-p + 1/2)*gamma(p + 1)) + 2*0**p*d**2*d**(2*p)*acoth(d/
(e*x))*gamma(-p + 1/2)*gamma(p + 1)/(2*e**3*p*gamma(-p + 1/2)*gamma(p + 1) + 2*e
**3*gamma(-p + 1/2)*gamma(p + 1)) - 2*0**p*d*d**(2*p)*e*p*x*gamma(-p + 1/2)*gamm
a(p + 1)/(2*e**3*p*gamma(-p + 1/2)*gamma(p + 1) + 2*e**3*gamma(-p + 1/2)*gamma(p
+ 1)) - 2*0**p*d*d**(2*p)*e*x*gamma(-p + 1/2)*gamma(p + 1)/(2*e**3*p*gamma(-p +
1/2)*gamma(p + 1) + 2*e**3*gamma(-p + 1/2)*gamma(p + 1)) + 0**p*d**(2*p)*e**2*p
*x**2*gamma(-p + 1/2)*gamma(p + 1)/(2*e**3*p*gamma(-p + 1/2)*gamma(p + 1) + 2*e*
*3*gamma(-p + 1/2)*gamma(p + 1)) + 0**p*d**(2*p)*e**2*x**2*gamma(-p + 1/2)*gamma
(p + 1)/(2*e**3*p*gamma(-p + 1/2)*gamma(p + 1) + 2*e**3*gamma(-p + 1/2)*gamma(p
+ 1)) + d**2*d**(2*p)*(-1 + e**2*x**2/d**2)**p*exp(I*pi*p)*gamma(p)*gamma(-p + 1
/2)/(2*e**3*p*gamma(-p + 1/2)*gamma(p + 1) + 2*e**3*gamma(-p + 1/2)*gamma(p + 1)
) + d*e*e**(2*p)*p**2*x*x**(2*p)*exp(I*pi*p)*gamma(p)*gamma(-p - 1/2)*hyper((-p
+ 1, -p - 1/2), (-p + 1/2,), d**2/(e**2*x**2))/(2*e**3*p*gamma(-p + 1/2)*gamma(p
+ 1) + 2*e**3*gamma(-p + 1/2)*gamma(p + 1)) + d*e*e**(2*p)*p*x*x**(2*p)*exp(I*p
i*p)*gamma(p)*gamma(-p - 1/2)*hyper((-p + 1, -p - 1/2), (-p + 1/2,), d**2/(e**2*
x**2))/(2*e**3*p*gamma(-p + 1/2)*gamma(p + 1) + 2*e**3*gamma(-p + 1/2)*gamma(p +
1)) + 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)/(2*e**3*p*gamma(-p + 1/2)*gamma(p + 1) + 2*e**3*gamma(-p + 1/2)*gamma(
p + 1)), (Abs(e**2*x**2/d**2) > 1) & (Abs(d**2/(e**2*x**2)) > 1)), (-0**p*d**2*d
**(2*p)*p*log(d**2/(e**2*x**2))*gamma(-p + 1/2)*gamma(p + 1)/(2*e**3*p*gamma(-p
+ 1/2)*gamma(p + 1) + 2*e**3*gamma(-p + 1/2)*gamma(p + 1)) + 0**p*d**2*d**(2*p)*
p*log(d**2/(e**2*x**2) - 1)*gamma(-p + 1/2)*gamma(p + 1)/(2*e**3*p*gamma(-p + 1/
2)*gamma(p + 1) + 2*e**3*gamma(-p + 1/2)*gamma(p + 1)) + 2*0**p*d**2*d**(2*p)*p*
acoth(d/(e*x))*gamma(-p + 1/2)*gamma(p + 1)/(2*e**3*p*gamma(-p + 1/2)*gamma(p +
1) + 2*e**3*gamma(-p + 1/2)*gamma(p + 1)) - 0**p*d**2*d**(2*p)*log(d**2/(e**2*x*
*2))*gamma(-p + 1/2)*gamma(p + 1)/(2*e**3*p*gamma(-p + 1/2)*gamma(p + 1) + 2*e**
3*gamma(-p + 1/2)*gamma(p + 1)) + 0**p*d**2*d**(2*p)*log(d**2/(e**2*x**2) - 1)*g
amma(-p + 1/2)*gamma(p + 1)/(2*e**3*p*gamma(-p + 1/2)*gamma(p + 1) + 2*e**3*gamm
a(-p + 1/2)*gamma(p + 1)) + 2*0**p*d**2*d**(2*p)*acoth(d/(e*x))*gamma(-p + 1/2)*
gamma(p + 1)/(2*e**3*p*gamma(-p + 1/2)*gamma(p + 1) + 2*e**3*gamma(-p + 1/2)*gam
ma(p + 1)) - 2*0**p*d*d**(2*p)*e*p*x*gamma(-p + 1/2)*gamma(p + 1)/(2*e**3*p*gamm
a(-p + 1/2)*gamma(p + 1) + 2*e**3*gamma(-p + 1/2)*gamma(p + 1)) - 2*0**p*d*d**(2
*p)*e*x*gamma(-p + 1/2)*gamma(p + 1)/(2*e**3*p*gamma(-p + 1/2)*gamma(p + 1) + 2*
e**3*gamma(-p + 1/2)*gamma(p + 1)) + 0**p*d**(2*p)*e**2*p*x**2*gamma(-p + 1/2)*g
amma(p + 1)/(2*e**3*p*gamma(-p + 1/2)*gamma(p + 1) + 2*e**3*gamma(-p + 1/2)*gamm
a(p + 1)) + 0**p*d**(2*p)*e**2*x**2*gamma(-p + 1/2)*gamma(p + 1)/(2*e**3*p*gamma
(-p + 1/2)*gamma(p + 1) + 2*e**3*gamma(-p + 1/2)*gamma(p + 1)) + d**2*d**(2*p)*(
1 - e**2*x**2/d**2)**p*gamma(p)*gamma(-p + 1/2)/(2*e**3*p*gamma(-p + 1/2)*gamma(
p + 1) + 2*e**3*gamma(-p + 1/2)*gamma(p + 1)) + d*e*e**(2*p)*p**2*x*x**(2*p)*exp
(I*pi*p)*gamma(p)*gamma(-p - 1/2)*hyper((-p + 1, -p - 1/2), (-p + 1/2,), d**2/(e
**2*x**2))/(2*e**3*p*gamma(-p + 1/2)*gamma(p + 1) + 2*e**3*gamma(-p + 1/2)*gamma
(p + 1)) + d*e*e**(2*p)*p*x*x**(2*p)*exp(I*pi*p)*gamma(p)*gamma(-p - 1/2)*hyper(
(-p + 1, -p - 1/2), (-p + 1/2,), d**2/(e**2*x**2))/(2*e**3*p*gamma(-p + 1/2)*gam
ma(p + 1) + 2*e**3*gamma(-p + 1/2)*gamma(p + 1)) + d**(2*p)*e**2*p*x**2*(1 - e**
2*x**2/d**2)**p*gamma(p)*gamma(-p + 1/2)/(2*e**3*p*gamma(-p + 1/2)*gamma(p + 1)
+ 2*e**3*gamma(-p + 1/2)*gamma(p + 1)), Abs(d**2/(e**2*x**2)) > 1), (-0**p*d**2*
d**(2*p)*p*log(d**2/(e**2*x**2))*gamma(-p + 1/2)*gamma(p + 1)/(2*e**3*p*gamma(-p
+ 1/2)*gamma(p + 1) + 2*e**3*gamma(-p + 1/2)*gamma(p + 1)) + 0**p*d**2*d**(2*p)
*p*log(-d**2/(e**2*x**2) + 1)*gamma(-p + 1/2)*gamma(p + 1)/(2*e**3*p*gamma(-p +
1/2)*gamma(p + 1) + 2*e**3*gamma(-p + 1/2)*gamma(p + 1)) + 2*0**p*d**2*d**(2*p)*
p*atanh(d/(e*x))*gamma(-p + 1/2)*gamma(p + 1)/(2*e**3*p*gamma(-p + 1/2)*gamma(p
+ 1) + 2*e**3*gamma(-p + 1/2)*gamma(p + 1)) - 0**p*d**2*d**(2*p)*log(d**2/(e**2*
x**2))*gamma(-p + 1/2)*gamma(p + 1)/(2*e**3*p*gamma(-p + 1/2)*gamma(p + 1) + 2*e
**3*gamma(-p + 1/2)*gamma(p + 1)) + 0**p*d**2*d**(2*p)*log(-d**2/(e**2*x**2) + 1
)*gamma(-p + 1/2)*gamma(p + 1)/(2*e**3*p*gamma(-p + 1/2)*gamma(p + 1) + 2*e**3*g
amma(-p + 1/2)*gamma(p + 1)) + 2*0**p*d**2*d**(2*p)*atanh(d/(e*x))*gamma(-p + 1/
2)*gamma(p + 1)/(2*e**3*p*gamma(-p + 1/2)*gamma(p + 1) + 2*e**3*gamma(-p + 1/2)*
gamma(p + 1)) - 2*0**p*d*d**(2*p)*e*p*x*gamma(-p + 1/2)*gamma(p + 1)/(2*e**3*p*g
amma(-p + 1/2)*gamma(p + 1) + 2*e**3*gamma(-p + 1/2)*gamma(p + 1)) - 2*0**p*d*d*
*(2*p)*e*x*gamma(-p + 1/2)*gamma(p + 1)/(2*e**3*p*gamma(-p + 1/2)*gamma(p + 1) +
2*e**3*gamma(-p + 1/2)*gamma(p + 1)) + 0**p*d**(2*p)*e**2*p*x**2*gamma(-p + 1/2
)*gamma(p + 1)/(2*e**3*p*gamma(-p + 1/2)*gamma(p + 1) + 2*e**3*gamma(-p + 1/2)*g
amma(p + 1)) + 0**p*d**(2*p)*e**2*x**2*gamma(-p + 1/2)*gamma(p + 1)/(2*e**3*p*ga
mma(-p + 1/2)*gamma(p + 1) + 2*e**3*gamma(-p + 1/2)*gamma(p + 1)) + d**2*d**(2*p
)*(-1 + e**2*x**2/d**2)**p*exp(I*pi*p)*gamma(p)*gamma(-p + 1/2)/(2*e**3*p*gamma(
-p + 1/2)*gamma(p + 1) + 2*e**3*gamma(-p + 1/2)*gamma(p + 1)) + d*e*e**(2*p)*p**
2*x*x**(2*p)*exp(I*pi*p)*gamma(p)*gamma(-p - 1/2)*hyper((-p + 1, -p - 1/2), (-p
+ 1/2,), d**2/(e**2*x**2))/(2*e**3*p*gamma(-p + 1/2)*gamma(p + 1) + 2*e**3*gamma
(-p + 1/2)*gamma(p + 1)) + d*e*e**(2*p)*p*x*x**(2*p)*exp(I*pi*p)*gamma(p)*gamma(
-p - 1/2)*hyper((-p + 1, -p - 1/2), (-p + 1/2,), d**2/(e**2*x**2))/(2*e**3*p*gam
ma(-p + 1/2)*gamma(p + 1) + 2*e**3*gamma(-p + 1/2)*gamma(p + 1)) + 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)/(2*e**3*p*
gamma(-p + 1/2)*gamma(p + 1) + 2*e**3*gamma(-p + 1/2)*gamma(p + 1)), Abs(e**2*x*
*2/d**2) > 1), (-0**p*d**2*d**(2*p)*p*log(d**2/(e**2*x**2))*gamma(-p + 1/2)*gamm
a(p + 1)/(2*e**3*p*gamma(-p + 1/2)*gamma(p + 1) + 2*e**3*gamma(-p + 1/2)*gamma(p
+ 1)) + 0**p*d**2*d**(2*p)*p*log(-d**2/(e**2*x**2) + 1)*gamma(-p + 1/2)*gamma(p
+ 1)/(2*e**3*p*gamma(-p + 1/2)*gamma(p + 1) + 2*e**3*gamma(-p + 1/2)*gamma(p +
1)) + 2*0**p*d**2*d**(2*p)*p*atanh(d/(e*x))*gamma(-p + 1/2)*gamma(p + 1)/(2*e**3
*p*gamma(-p + 1/2)*gamma(p + 1) + 2*e**3*gamma(-p + 1/2)*gamma(p + 1)) - 0**p*d*
*2*d**(2*p)*log(d**2/(e**2*x**2))*gamma(-p + 1/2)*gamma(p + 1)/(2*e**3*p*gamma(-
p + 1/2)*gamma(p + 1) + 2*e**3*gamma(-p + 1/2)*gamma(p + 1)) + 0**p*d**2*d**(2*p
)*log(-d**2/(e**2*x**2) + 1)*gamma(-p + 1/2)*gamma(p + 1)/(2*e**3*p*gamma(-p + 1
/2)*gamma(p + 1) + 2*e**3*gamma(-p + 1/2)*gamma(p + 1)) + 2*0**p*d**2*d**(2*p)*a
tanh(d/(e*x))*gamma(-p + 1/2)*gamma(p + 1)/(2*e**3*p*gamma(-p + 1/2)*gamma(p + 1
) + 2*e**3*gamma(-p + 1/2)*gamma(p + 1)) - 2*0**p*d*d**(2*p)*e*p*x*gamma(-p + 1/
2)*gamma(p + 1)/(2*e**3*p*gamma(-p + 1/2)*gamma(p + 1) + 2*e**3*gamma(-p + 1/2)*
gamma(p + 1)) - 2*0**p*d*d**(2*p)*e*x*gamma(-p + 1/2)*gamma(p + 1)/(2*e**3*p*gam
ma(-p + 1/2)*gamma(p + 1) + 2*e**3*gamma(-p + 1/2)*gamma(p + 1)) + 0**p*d**(2*p)
*e**2*p*x**2*gamma(-p + 1/2)*gamma(p + 1)/(2*e**3*p*gamma(-p + 1/2)*gamma(p + 1)
+ 2*e**3*gamma(-p + 1/2)*gamma(p + 1)) + 0**p*d**(2*p)*e**2*x**2*gamma(-p + 1/2
)*gamma(p + 1)/(2*e**3*p*gamma(-p + 1/2)*gamma(p + 1) + 2*e**3*gamma(-p + 1/2)*g
amma(p + 1)) + d**2*d**(2*p)*(1 - e**2*x**2/d**2)**p*gamma(p)*gamma(-p + 1/2)/(2
*e**3*p*gamma(-p + 1/2)*gamma(p + 1) + 2*e**3*gamma(-p + 1/2)*gamma(p + 1)) + d*
e*e**(2*p)*p**2*x*x**(2*p)*exp(I*pi*p)*gamma(p)*gamma(-p - 1/2)*hyper((-p + 1, -
p - 1/2), (-p + 1/2,), d**2/(e**2*x**2))/(2*e**3*p*gamma(-p + 1/2)*gamma(p + 1)
+ 2*e**3*gamma(-p + 1/2)*gamma(p + 1)) + d*e*e**(2*p)*p*x*x**(2*p)*exp(I*pi*p)*g
amma(p)*gamma(-p - 1/2)*hyper((-p + 1, -p - 1/2), (-p + 1/2,), d**2/(e**2*x**2))
/(2*e**3*p*gamma(-p + 1/2)*gamma(p + 1) + 2*e**3*gamma(-p + 1/2)*gamma(p + 1)) +
d**(2*p)*e**2*p*x**2*(1 - e**2*x**2/d**2)**p*gamma(p)*gamma(-p + 1/2)/(2*e**3*p
*gamma(-p + 1/2)*gamma(p + 1) + 2*e**3*gamma(-p + 1/2)*gamma(p + 1)), True))
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{p} x^{2}}{e x + d}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-e^2*x^2 + d^2)^p*x^2/(e*x + d),x, algorithm="giac")
[Out]
integrate((-e^2*x^2 + d^2)^p*x^2/(e*x + d), x)