3.267 \(\int \frac{x^4 (d^2-e^2 x^2)^p}{d+e x} \, dx\)
Optimal. Leaf size=148 \[ \frac{x^5 \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac{5}{2},1-p;\frac{7}{2};\frac{e^2 x^2}{d^2}\right )}{5 d}+\frac{d^4 \left (d^2-e^2 x^2\right )^p}{2 e^5 p}-\frac{d^2 \left (d^2-e^2 x^2\right )^{p+1}}{e^5 (p+1)}+\frac{\left (d^2-e^2 x^2\right )^{p+2}}{2 e^5 (p+2)} \]
[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*Hypergeometric2F1[5/2, 1 - p, 7/2, (e^2*x^2)/d^2])/(5*d*(1 - (e^2*x^2)/d^
2)^p)
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Rubi [A] time = 0.108949, 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, Rules used =
{850, 764, 365, 364, 266, 43} \[ \frac{x^5 \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac{5}{2},1-p;\frac{7}{2};\frac{e^2 x^2}{d^2}\right )}{5 d}+\frac{d^4 \left (d^2-e^2 x^2\right )^p}{2 e^5 p}-\frac{d^2 \left (d^2-e^2 x^2\right )^{p+1}}{e^5 (p+1)}+\frac{\left (d^2-e^2 x^2\right )^{p+2}}{2 e^5 (p+2)} \]
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*Hypergeometric2F1[5/2, 1 - p, 7/2, (e^2*x^2)/d^2])/(5*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^4 \left (d^2-e^2 x^2\right )^p}{d+e x} \, dx &=\int x^4 (d-e x) \left (d^2-e^2 x^2\right )^{-1+p} \, dx\\ &=d \int x^4 \left (d^2-e^2 x^2\right )^{-1+p} \, dx-e \int x^5 \left (d^2-e^2 x^2\right )^{-1+p} \, dx\\ &=-\left (\frac{1}{2} e \operatorname{Subst}\left (\int x^2 \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^4 \left (1-\frac{e^2 x^2}{d^2}\right )^{-1+p} \, dx}{d}\\ &=\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{1}{2} e \operatorname{Subst}\left (\int \left (\frac{d^4 \left (d^2-e^2 x\right )^{-1+p}}{e^4}-\frac{2 d^2 \left (d^2-e^2 x\right )^p}{e^4}+\frac{\left (d^2-e^2 x\right )^{1+p}}{e^4}\right ) \, dx,x,x^2\right )\\ &=\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 )^{1+p}}{e^5 (1+p)}+\frac{\left (d^2-e^2 x^2\right )^{2+p}}{2 e^5 (2+p)}+\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}\\ \end{align*}
Mathematica [C] time = 0.113559, size = 66, normalized size = 0.45 \[ \frac{x^5 (d-e x)^p (d+e x)^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} F_1\left (5;-p,1-p;6;\frac{e x}{d},-\frac{e x}{d}\right )}{5 d} \]
Warning: Unable to verify antiderivative.
[In]
Integrate[(x^4*(d^2 - e^2*x^2)^p)/(d + e*x),x]
[Out]
(x^5*(d - e*x)^p*(d + e*x)^p*AppellF1[5, -p, 1 - p, 6, (e*x)/d, -((e*x)/d)])/(5*d*(1 - (e^2*x^2)/d^2)^p)
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Maple [F] time = 0.656, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{4} \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^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)
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{p} x^{4}}{e x + d}\,{d x} \end{align*}
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, algorithm="maxima")
[Out]
integrate((-e^2*x^2 + d^2)^p*x^4/(e*x + d), x)
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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^{4}}{e x + d}, x\right ) \end{align*}
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, algorithm="fricas")
[Out]
integral((-e^2*x^2 + d^2)^p*x^4/(e*x + d), x)
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Sympy [C] time = 22.9988, size = 4446, normalized size = 30.04 \begin{align*} \text{result too large to display} \end{align*}
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)*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*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)*gam
ma(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)*gamm
a(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)*gam
ma(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)*gamma(p + 3)) + 12*0**p*d**4*d**(2*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)) - 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*g
amma(-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)*g
amma(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) + 12*e**5*gam
ma(-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((1 - p, -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((1 - p, -p - 3/2), (-p - 1/2,), d**2/(e**2*x**2))/(12*e**5*p*gamma(-p)*gamma(-p - 1/2)*ga
mma(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*gamma(-p)*gamma(p)*gamma(-p - 1/2)*gamma(p + 2)/(12*e**5*p*gamma(-p)*g
amma(-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)*gamma(-p - 1/2)*gamma(p + 1)*gamma(
p + 3)), Abs(d**2)/(Abs(e**2)*Abs(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*g
amma(-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*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*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))*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)) + 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)*gamm
a(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*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)*g
amma(-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)*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)) - 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)*ga
mma(-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) + 12*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*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))
+ 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)*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((1 - p, -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((1 - p, -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)*g
amma(-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)*gamm
a(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)*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)), 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^{4}}{e x + d}\,{d x} \end{align*}
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, algorithm="giac")
[Out]
integrate((-e^2*x^2 + d^2)^p*x^4/(e*x + d), x)