3.269 \(\int \frac{x^2 (d^2-e^2 x^2)^p}{d+e x} \, dx\)
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*Hypergeom
etric2F1[3/2, 1 - p, 5/2, (e^2*x^2)/d^2])/(3*d*(1 - (e^2*x^2)/d^2)^p)
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Rubi [A] time = 0.0854197, 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, Rules used =
{850, 764, 365, 364, 266, 43} \[ \frac{x^3 \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac{3}{2},1-p;\frac{5}{2};\frac{e^2 x^2}{d^2}\right )}{3 d}+\frac{d^2 \left (d^2-e^2 x^2\right )^p}{2 e^3 p}-\frac{\left (d^2-e^2 x^2\right )^{p+1}}{2 e^3 (p+1)} \]
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*Hypergeom
etric2F1[3/2, 1 - p, 5/2, (e^2*x^2)/d^2])/(3*d*(1 - (e^2*x^2)/d^2)^p)
Rule 850
Int[((x_)^(n_.)*((a_) + (c_.)*(x_)^2)^(p_))/((d_) + (e_.)*(x_)), x_Symbol] :> Int[x^n*(a/d + (c*x)/e)*(a + c*x
^2)^(p - 1), x] /; FreeQ[{a, c, d, e, n, p}, x] && EqQ[c*d^2 + a*e^2, 0] && !IntegerQ[p] && ( !IntegerQ[n] ||
!IntegerQ[2*p] || IGtQ[n, 2] || (GtQ[p, 0] && NeQ[n, 2]))
Rule 764
Int[(x_)^(m_.)*((f_) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[f, Int[x^m*(a + c*x^2)^p, x]
, x] + Dist[g, Int[x^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, f, g, p}, x] && IntegerQ[m] && !IntegerQ[2
*p]
Rule 365
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a + b*x^n)^FracPart[p])
/(1 + (b*x^n)/a)^FracPart[p], Int[(c*x)^m*(1 + (b*x^n)/a)^p, x], x] /; FreeQ[{a, b, c, m, n, p}, x] && !IGtQ[
p, 0] && !(ILtQ[p, 0] || GtQ[a, 0])
Rule 364
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])
Rule 266
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
Rule 43
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
Rubi steps
\begin{align*} \int \frac{x^2 \left (d^2-e^2 x^2\right )^p}{d+e x} \, dx &=\int x^2 (d-e x) \left (d^2-e^2 x^2\right )^{-1+p} \, dx\\ &=d \int x^2 \left (d^2-e^2 x^2\right )^{-1+p} \, dx-e \int x^3 \left (d^2-e^2 x^2\right )^{-1+p} \, dx\\ &=-\left (\frac{1}{2} e \operatorname{Subst}\left (\int x \left (d^2-e^2 x\right )^{-1+p} \, dx,x,x^2\right )\right )+\frac{\left (\left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p}\right ) \int x^2 \left (1-\frac{e^2 x^2}{d^2}\right )^{-1+p} \, dx}{d}\\ &=\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{1}{2} e \operatorname{Subst}\left (\int \left (\frac{d^2 \left (d^2-e^2 x\right )^{-1+p}}{e^2}-\frac{\left (d^2-e^2 x\right )^p}{e^2}\right ) \, dx,x,x^2\right )\\ &=\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 )^{1+p}}{2 e^3 (1+p)}+\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}\\ \end{align*}
Mathematica [A] time = 0.260615, 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*Hypergeometric2F1[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)
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Maple [F] time = 0.67, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{2} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{p}}{ex+d}}\, dx \end{align*}
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. \begin{align*} \int \frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{p} x^{2}}{e x + d}\,{d x} \end{align*}
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, 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. \begin{align*}{\rm integral}\left (\frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{p} x^{2}}{e x + d}, x\right ) \end{align*}
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, algorithm="fricas")
[Out]
integral((-e^2*x^2 + d^2)^p*x^2/(e*x + d), x)
________________________________________________________________________________________
Sympy [C] time = 10.6224, size = 4134, normalized size = 34.74 \begin{align*} \text{result too large to display} \end{align*}
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(1/2 - p)*gamma(p + 1)/(2*e**3*p*gamma(1/2 - p)*ga
mma(p + 1) + 2*e**3*gamma(1/2 - p)*gamma(p + 1)) + 0**p*d**2*d**(2*p)*p*log(d**2/(e**2*x**2) - 1)*gamma(1/2 -
p)*gamma(p + 1)/(2*e**3*p*gamma(1/2 - p)*gamma(p + 1) + 2*e**3*gamma(1/2 - p)*gamma(p + 1)) + 2*0**p*d**2*d**(
2*p)*p*acoth(d/(e*x))*gamma(1/2 - p)*gamma(p + 1)/(2*e**3*p*gamma(1/2 - p)*gamma(p + 1) + 2*e**3*gamma(1/2 - p
)*gamma(p + 1)) - 0**p*d**2*d**(2*p)*log(d**2/(e**2*x**2))*gamma(1/2 - p)*gamma(p + 1)/(2*e**3*p*gamma(1/2 - p
)*gamma(p + 1) + 2*e**3*gamma(1/2 - p)*gamma(p + 1)) + 0**p*d**2*d**(2*p)*log(d**2/(e**2*x**2) - 1)*gamma(1/2
- p)*gamma(p + 1)/(2*e**3*p*gamma(1/2 - p)*gamma(p + 1) + 2*e**3*gamma(1/2 - p)*gamma(p + 1)) + 2*0**p*d**2*d*
*(2*p)*acoth(d/(e*x))*gamma(1/2 - p)*gamma(p + 1)/(2*e**3*p*gamma(1/2 - p)*gamma(p + 1) + 2*e**3*gamma(1/2 - p
)*gamma(p + 1)) - 2*0**p*d*d**(2*p)*e*p*x*gamma(1/2 - p)*gamma(p + 1)/(2*e**3*p*gamma(1/2 - p)*gamma(p + 1) +
2*e**3*gamma(1/2 - p)*gamma(p + 1)) - 2*0**p*d*d**(2*p)*e*x*gamma(1/2 - p)*gamma(p + 1)/(2*e**3*p*gamma(1/2 -
p)*gamma(p + 1) + 2*e**3*gamma(1/2 - p)*gamma(p + 1)) + 0**p*d**(2*p)*e**2*p*x**2*gamma(1/2 - p)*gamma(p + 1)/
(2*e**3*p*gamma(1/2 - p)*gamma(p + 1) + 2*e**3*gamma(1/2 - p)*gamma(p + 1)) + 0**p*d**(2*p)*e**2*x**2*gamma(1/
2 - p)*gamma(p + 1)/(2*e**3*p*gamma(1/2 - p)*gamma(p + 1) + 2*e**3*gamma(1/2 - p)*gamma(p + 1)) + d**2*d**(2*p
)*(-1 + e**2*x**2/d**2)**p*exp(I*pi*p)*gamma(p)*gamma(1/2 - p)/(2*e**3*p*gamma(1/2 - p)*gamma(p + 1) + 2*e**3*
gamma(1/2 - p)*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((1 - p,
-p - 1/2), (1/2 - p,), d**2/(e**2*x**2))/(2*e**3*p*gamma(1/2 - p)*gamma(p + 1) + 2*e**3*gamma(1/2 - p)*gamma(
p + 1)) + d*e*e**(2*p)*p*x*x**(2*p)*exp(I*pi*p)*gamma(p)*gamma(-p - 1/2)*hyper((1 - p, -p - 1/2), (1/2 - p,),
d**2/(e**2*x**2))/(2*e**3*p*gamma(1/2 - p)*gamma(p + 1) + 2*e**3*gamma(1/2 - p)*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(1/2 - p)/(2*e**3*p*gamma(1/2 - p)*gamma(p + 1) + 2*
e**3*gamma(1/2 - p)*gamma(p + 1)), (Abs(e**2*x**2)/Abs(d**2) > 1) & (Abs(d**2)/(Abs(e**2)*Abs(x**2)) > 1)), (-
0**p*d**2*d**(2*p)*p*log(d**2/(e**2*x**2))*gamma(1/2 - p)*gamma(p + 1)/(2*e**3*p*gamma(1/2 - p)*gamma(p + 1) +
2*e**3*gamma(1/2 - p)*gamma(p + 1)) + 0**p*d**2*d**(2*p)*p*log(d**2/(e**2*x**2) - 1)*gamma(1/2 - p)*gamma(p +
1)/(2*e**3*p*gamma(1/2 - p)*gamma(p + 1) + 2*e**3*gamma(1/2 - p)*gamma(p + 1)) + 2*0**p*d**2*d**(2*p)*p*acoth
(d/(e*x))*gamma(1/2 - p)*gamma(p + 1)/(2*e**3*p*gamma(1/2 - p)*gamma(p + 1) + 2*e**3*gamma(1/2 - p)*gamma(p +
1)) - 0**p*d**2*d**(2*p)*log(d**2/(e**2*x**2))*gamma(1/2 - p)*gamma(p + 1)/(2*e**3*p*gamma(1/2 - p)*gamma(p +
1) + 2*e**3*gamma(1/2 - p)*gamma(p + 1)) + 0**p*d**2*d**(2*p)*log(d**2/(e**2*x**2) - 1)*gamma(1/2 - p)*gamma(p
+ 1)/(2*e**3*p*gamma(1/2 - p)*gamma(p + 1) + 2*e**3*gamma(1/2 - p)*gamma(p + 1)) + 2*0**p*d**2*d**(2*p)*acoth
(d/(e*x))*gamma(1/2 - p)*gamma(p + 1)/(2*e**3*p*gamma(1/2 - p)*gamma(p + 1) + 2*e**3*gamma(1/2 - p)*gamma(p +
1)) - 2*0**p*d*d**(2*p)*e*p*x*gamma(1/2 - p)*gamma(p + 1)/(2*e**3*p*gamma(1/2 - p)*gamma(p + 1) + 2*e**3*gamma
(1/2 - p)*gamma(p + 1)) - 2*0**p*d*d**(2*p)*e*x*gamma(1/2 - p)*gamma(p + 1)/(2*e**3*p*gamma(1/2 - p)*gamma(p +
1) + 2*e**3*gamma(1/2 - p)*gamma(p + 1)) + 0**p*d**(2*p)*e**2*p*x**2*gamma(1/2 - p)*gamma(p + 1)/(2*e**3*p*ga
mma(1/2 - p)*gamma(p + 1) + 2*e**3*gamma(1/2 - p)*gamma(p + 1)) + 0**p*d**(2*p)*e**2*x**2*gamma(1/2 - p)*gamma
(p + 1)/(2*e**3*p*gamma(1/2 - p)*gamma(p + 1) + 2*e**3*gamma(1/2 - p)*gamma(p + 1)) + d**2*d**(2*p)*(1 - e**2*
x**2/d**2)**p*gamma(p)*gamma(1/2 - p)/(2*e**3*p*gamma(1/2 - p)*gamma(p + 1) + 2*e**3*gamma(1/2 - p)*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((1 - p, -p - 1/2), (1/2 - p,), d
**2/(e**2*x**2))/(2*e**3*p*gamma(1/2 - p)*gamma(p + 1) + 2*e**3*gamma(1/2 - p)*gamma(p + 1)) + d*e*e**(2*p)*p*
x*x**(2*p)*exp(I*pi*p)*gamma(p)*gamma(-p - 1/2)*hyper((1 - p, -p - 1/2), (1/2 - p,), d**2/(e**2*x**2))/(2*e**3
*p*gamma(1/2 - p)*gamma(p + 1) + 2*e**3*gamma(1/2 - p)*gamma(p + 1)) + d**(2*p)*e**2*p*x**2*(1 - e**2*x**2/d**
2)**p*gamma(p)*gamma(1/2 - p)/(2*e**3*p*gamma(1/2 - p)*gamma(p + 1) + 2*e**3*gamma(1/2 - p)*gamma(p + 1)), Abs
(d**2)/(Abs(e**2)*Abs(x**2)) > 1), (-0**p*d**2*d**(2*p)*p*log(d**2/(e**2*x**2))*gamma(1/2 - p)*gamma(p + 1)/(2
*e**3*p*gamma(1/2 - p)*gamma(p + 1) + 2*e**3*gamma(1/2 - p)*gamma(p + 1)) + 0**p*d**2*d**(2*p)*p*log(-d**2/(e*
*2*x**2) + 1)*gamma(1/2 - p)*gamma(p + 1)/(2*e**3*p*gamma(1/2 - p)*gamma(p + 1) + 2*e**3*gamma(1/2 - p)*gamma(
p + 1)) + 2*0**p*d**2*d**(2*p)*p*atanh(d/(e*x))*gamma(1/2 - p)*gamma(p + 1)/(2*e**3*p*gamma(1/2 - p)*gamma(p +
1) + 2*e**3*gamma(1/2 - p)*gamma(p + 1)) - 0**p*d**2*d**(2*p)*log(d**2/(e**2*x**2))*gamma(1/2 - p)*gamma(p +
1)/(2*e**3*p*gamma(1/2 - p)*gamma(p + 1) + 2*e**3*gamma(1/2 - p)*gamma(p + 1)) + 0**p*d**2*d**(2*p)*log(-d**2/
(e**2*x**2) + 1)*gamma(1/2 - p)*gamma(p + 1)/(2*e**3*p*gamma(1/2 - p)*gamma(p + 1) + 2*e**3*gamma(1/2 - p)*gam
ma(p + 1)) + 2*0**p*d**2*d**(2*p)*atanh(d/(e*x))*gamma(1/2 - p)*gamma(p + 1)/(2*e**3*p*gamma(1/2 - p)*gamma(p
+ 1) + 2*e**3*gamma(1/2 - p)*gamma(p + 1)) - 2*0**p*d*d**(2*p)*e*p*x*gamma(1/2 - p)*gamma(p + 1)/(2*e**3*p*gam
ma(1/2 - p)*gamma(p + 1) + 2*e**3*gamma(1/2 - p)*gamma(p + 1)) - 2*0**p*d*d**(2*p)*e*x*gamma(1/2 - p)*gamma(p
+ 1)/(2*e**3*p*gamma(1/2 - p)*gamma(p + 1) + 2*e**3*gamma(1/2 - p)*gamma(p + 1)) + 0**p*d**(2*p)*e**2*p*x**2*g
amma(1/2 - p)*gamma(p + 1)/(2*e**3*p*gamma(1/2 - p)*gamma(p + 1) + 2*e**3*gamma(1/2 - p)*gamma(p + 1)) + 0**p*
d**(2*p)*e**2*x**2*gamma(1/2 - p)*gamma(p + 1)/(2*e**3*p*gamma(1/2 - p)*gamma(p + 1) + 2*e**3*gamma(1/2 - p)*g
amma(p + 1)) + d**2*d**(2*p)*(-1 + e**2*x**2/d**2)**p*exp(I*pi*p)*gamma(p)*gamma(1/2 - p)/(2*e**3*p*gamma(1/2
- p)*gamma(p + 1) + 2*e**3*gamma(1/2 - p)*gamma(p + 1)) + d*e*e**(2*p)*p**2*x*x**(2*p)*exp(I*pi*p)*gamma(p)*ga
mma(-p - 1/2)*hyper((1 - p, -p - 1/2), (1/2 - p,), d**2/(e**2*x**2))/(2*e**3*p*gamma(1/2 - p)*gamma(p + 1) + 2
*e**3*gamma(1/2 - p)*gamma(p + 1)) + d*e*e**(2*p)*p*x*x**(2*p)*exp(I*pi*p)*gamma(p)*gamma(-p - 1/2)*hyper((1 -
p, -p - 1/2), (1/2 - p,), d**2/(e**2*x**2))/(2*e**3*p*gamma(1/2 - p)*gamma(p + 1) + 2*e**3*gamma(1/2 - p)*gam
ma(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(1/2 - p)/(2*e**3*p*gamma
(1/2 - p)*gamma(p + 1) + 2*e**3*gamma(1/2 - p)*gamma(p + 1)), Abs(e**2*x**2)/Abs(d**2) > 1), (-0**p*d**2*d**(2
*p)*p*log(d**2/(e**2*x**2))*gamma(1/2 - p)*gamma(p + 1)/(2*e**3*p*gamma(1/2 - p)*gamma(p + 1) + 2*e**3*gamma(1
/2 - p)*gamma(p + 1)) + 0**p*d**2*d**(2*p)*p*log(-d**2/(e**2*x**2) + 1)*gamma(1/2 - p)*gamma(p + 1)/(2*e**3*p*
gamma(1/2 - p)*gamma(p + 1) + 2*e**3*gamma(1/2 - p)*gamma(p + 1)) + 2*0**p*d**2*d**(2*p)*p*atanh(d/(e*x))*gamm
a(1/2 - p)*gamma(p + 1)/(2*e**3*p*gamma(1/2 - p)*gamma(p + 1) + 2*e**3*gamma(1/2 - p)*gamma(p + 1)) - 0**p*d**
2*d**(2*p)*log(d**2/(e**2*x**2))*gamma(1/2 - p)*gamma(p + 1)/(2*e**3*p*gamma(1/2 - p)*gamma(p + 1) + 2*e**3*ga
mma(1/2 - p)*gamma(p + 1)) + 0**p*d**2*d**(2*p)*log(-d**2/(e**2*x**2) + 1)*gamma(1/2 - p)*gamma(p + 1)/(2*e**3
*p*gamma(1/2 - p)*gamma(p + 1) + 2*e**3*gamma(1/2 - p)*gamma(p + 1)) + 2*0**p*d**2*d**(2*p)*atanh(d/(e*x))*gam
ma(1/2 - p)*gamma(p + 1)/(2*e**3*p*gamma(1/2 - p)*gamma(p + 1) + 2*e**3*gamma(1/2 - p)*gamma(p + 1)) - 2*0**p*
d*d**(2*p)*e*p*x*gamma(1/2 - p)*gamma(p + 1)/(2*e**3*p*gamma(1/2 - p)*gamma(p + 1) + 2*e**3*gamma(1/2 - p)*gam
ma(p + 1)) - 2*0**p*d*d**(2*p)*e*x*gamma(1/2 - p)*gamma(p + 1)/(2*e**3*p*gamma(1/2 - p)*gamma(p + 1) + 2*e**3*
gamma(1/2 - p)*gamma(p + 1)) + 0**p*d**(2*p)*e**2*p*x**2*gamma(1/2 - p)*gamma(p + 1)/(2*e**3*p*gamma(1/2 - p)*
gamma(p + 1) + 2*e**3*gamma(1/2 - p)*gamma(p + 1)) + 0**p*d**(2*p)*e**2*x**2*gamma(1/2 - p)*gamma(p + 1)/(2*e*
*3*p*gamma(1/2 - p)*gamma(p + 1) + 2*e**3*gamma(1/2 - p)*gamma(p + 1)) + d**2*d**(2*p)*(1 - e**2*x**2/d**2)**p
*gamma(p)*gamma(1/2 - p)/(2*e**3*p*gamma(1/2 - p)*gamma(p + 1) + 2*e**3*gamma(1/2 - p)*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((1 - p, -p - 1/2), (1/2 - p,), d**2/(e**2*x**
2))/(2*e**3*p*gamma(1/2 - p)*gamma(p + 1) + 2*e**3*gamma(1/2 - p)*gamma(p + 1)) + d*e*e**(2*p)*p*x*x**(2*p)*ex
p(I*pi*p)*gamma(p)*gamma(-p - 1/2)*hyper((1 - p, -p - 1/2), (1/2 - p,), d**2/(e**2*x**2))/(2*e**3*p*gamma(1/2
- p)*gamma(p + 1) + 2*e**3*gamma(1/2 - p)*gamma(p + 1)) + d**(2*p)*e**2*p*x**2*(1 - e**2*x**2/d**2)**p*gamma(p
)*gamma(1/2 - p)/(2*e**3*p*gamma(1/2 - p)*gamma(p + 1) + 2*e**3*gamma(1/2 - p)*gamma(p + 1)), True))
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{p} x^{2}}{e x + d}\,{d x} \end{align*}
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, algorithm="giac")
[Out]
integrate((-e^2*x^2 + d^2)^p*x^2/(e*x + d), x)