3.268 \(\int \frac{x^3 (d^2-e^2 x^2)^p}{d+e x} \, dx\)

Optimal. Leaf size=121 \[ -\frac{e x^5 \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac{5}{2},1-p;\frac{7}{2};\frac{e^2 x^2}{d^2}\right )}{5 d^2}-\frac{d^3 \left (d^2-e^2 x^2\right )^p}{2 e^4 p}+\frac{d \left (d^2-e^2 x^2\right )^{p+1}}{2 e^4 (p+1)} \]

[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*Hy
pergeometric2F1[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 = 0.0930484, 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, Rules used = {850, 764, 266, 43, 365, 364} \[ -\frac{e x^5 \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac{5}{2},1-p;\frac{7}{2};\frac{e^2 x^2}{d^2}\right )}{5 d^2}-\frac{d^3 \left (d^2-e^2 x^2\right )^p}{2 e^4 p}+\frac{d \left (d^2-e^2 x^2\right )^{p+1}}{2 e^4 (p+1)} \]

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*Hy
pergeometric2F1[5/2, 1 - p, 7/2, (e^2*x^2)/d^2])/(5*d^2*(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 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])

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])

Rubi steps

\begin{align*} \int \frac{x^3 \left (d^2-e^2 x^2\right )^p}{d+e x} \, dx &=\int x^3 (d-e x) \left (d^2-e^2 x^2\right )^{-1+p} \, dx\\ &=d \int x^3 \left (d^2-e^2 x^2\right )^{-1+p} \, dx-e \int x^4 \left (d^2-e^2 x^2\right )^{-1+p} \, dx\\ &=\frac{1}{2} d \operatorname{Subst}\left (\int x \left (d^2-e^2 x\right )^{-1+p} \, dx,x,x^2\right )-\frac{\left (e \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^2}\\ &=-\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{1}{2} d \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^3 \left (d^2-e^2 x^2\right )^p}{2 e^4 p}+\frac{d \left (d^2-e^2 x^2\right )^{1+p}}{2 e^4 (1+p)}-\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}\\ \end{align*}

Mathematica [B]  time = 0.317096, size = 245, normalized size = 2.02 \[ \frac{\left (\frac{e x}{d}+1\right )^{-p} \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \left (6 d^2 e (p+1) x \left (\frac{e x}{d}+1\right )^p \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};\frac{e^2 x^2}{d^2}\right )+2 e^3 (p+1) x^3 \left (\frac{e x}{d}+1\right )^p \, _2F_1\left (\frac{3}{2},-p;\frac{5}{2};\frac{e^2 x^2}{d^2}\right )+3 d \left (d (d-e x) \left (2-\frac{2 e^2 x^2}{d^2}\right )^p \, _2F_1\left (1-p,p+1;p+2;\frac{d-e x}{2 d}\right )+\left (d^2 \left (\left (1-\frac{e^2 x^2}{d^2}\right )^p-1\right )-e^2 x^2 \left (1-\frac{e^2 x^2}{d^2}\right )^p\right ) \left (\frac{e x}{d}+1\right )^p\right )\right )}{6 e^4 (p+1)} \]

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*Hypergeometric2
F1[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 = 0.648, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{3} \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^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. \begin{align*} \int \frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{p} x^{3}}{e x + d}\,{d x} \end{align*}

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, 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. \begin{align*}{\rm integral}\left (\frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{p} x^{3}}{e x + d}, x\right ) \end{align*}

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, algorithm="fricas")

[Out]

integral((-e^2*x^2 + d^2)^p*x^3/(e*x + d), x)

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Sympy [C]  time = 15.9521, size = 5100, normalized size = 42.15 \begin{align*} \text{result too large to display} \end{align*}

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*ga
mma(-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*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*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)*gamm
a(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*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)*ga
mma(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*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*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*e**3*e**(2*p)*p**2*x**
3*x**(2*p)*exp(I*pi*p)*gamma(p)*gamma(-p - 3/2)*hyper((1 - p, -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((1 - p, -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)/Abs(d**2) > 1) & (Abs(d**2)/(Abs(e**
2)*Abs(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*gamm
a(-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*gam
ma(-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)*ga
mma(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*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*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((1 - p, -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((1 - p, -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)/Abs(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*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*ac
oth(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)*gamm
a(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*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)*gam
ma(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*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((1 - p, -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*ga
mma(-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((1 -
p, -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)*g
amma(p + 1)), Abs(d**2)/(Abs(e**2)*Abs(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)*ga
mma(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)*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*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*g
amma(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((1 - p, -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((1 - p, -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))

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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^{3}}{e x + d}\,{d x} \end{align*}

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, algorithm="giac")

[Out]

integrate((-e^2*x^2 + d^2)^p*x^3/(e*x + d), x)