3.260 \(\int x^3 (d+e x)^3 (d^2-e^2 x^2)^p \, dx\)
Optimal. Leaf size=193 \[ \frac {2 d^2 e (3 p+13) x^5 \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac {5}{2},-p;\frac {7}{2};\frac {e^2 x^2}{d^2}\right )}{5 (2 p+7)}-\frac {e x^5 \left (d^2-e^2 x^2\right )^{p+1}}{2 p+7}-\frac {3 d \left (d^2-e^2 x^2\right )^{p+3}}{2 e^4 (p+3)}-\frac {2 d^5 \left (d^2-e^2 x^2\right )^{p+1}}{e^4 (p+1)}+\frac {7 d^3 \left (d^2-e^2 x^2\right )^{p+2}}{2 e^4 (p+2)} \]
[Out]
-2*d^5*(-e^2*x^2+d^2)^(1+p)/e^4/(1+p)-e*x^5*(-e^2*x^2+d^2)^(1+p)/(7+2*p)+7/2*d^3*(-e^2*x^2+d^2)^(2+p)/e^4/(2+p
)-3/2*d*(-e^2*x^2+d^2)^(3+p)/e^4/(3+p)+2/5*d^2*e*(13+3*p)*x^5*(-e^2*x^2+d^2)^p*hypergeom([5/2, -p],[7/2],e^2*x
^2/d^2)/(7+2*p)/((1-e^2*x^2/d^2)^p)
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Rubi [A] time = 0.18, antiderivative size = 193, normalized size of antiderivative = 1.00,
number of steps used = 7, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used =
{1652, 446, 77, 459, 365, 364} \[ \frac {2 d^2 e (3 p+13) x^5 \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac {5}{2},-p;\frac {7}{2};\frac {e^2 x^2}{d^2}\right )}{5 (2 p+7)}-\frac {e x^5 \left (d^2-e^2 x^2\right )^{p+1}}{2 p+7}-\frac {2 d^5 \left (d^2-e^2 x^2\right )^{p+1}}{e^4 (p+1)}+\frac {7 d^3 \left (d^2-e^2 x^2\right )^{p+2}}{2 e^4 (p+2)}-\frac {3 d \left (d^2-e^2 x^2\right )^{p+3}}{2 e^4 (p+3)} \]
Antiderivative was successfully verified.
[In]
Int[x^3*(d + e*x)^3*(d^2 - e^2*x^2)^p,x]
[Out]
(-2*d^5*(d^2 - e^2*x^2)^(1 + p))/(e^4*(1 + p)) - (e*x^5*(d^2 - e^2*x^2)^(1 + p))/(7 + 2*p) + (7*d^3*(d^2 - e^2
*x^2)^(2 + p))/(2*e^4*(2 + p)) - (3*d*(d^2 - e^2*x^2)^(3 + p))/(2*e^4*(3 + p)) + (2*d^2*e*(13 + 3*p)*x^5*(d^2
- e^2*x^2)^p*Hypergeometric2F1[5/2, -p, 7/2, (e^2*x^2)/d^2])/(5*(7 + 2*p)*(1 - (e^2*x^2)/d^2)^p)
Rule 77
Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
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 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 446
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]
Rule 459
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(d*(e*x)^(m
+ 1)*(a + b*x^n)^(p + 1))/(b*e*(m + n*(p + 1) + 1)), x] - Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(b*(m +
n*(p + 1) + 1)), Int[(e*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && NeQ[b*c - a*d, 0]
&& NeQ[m + n*(p + 1) + 1, 0]
Rule 1652
Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Module[{q = Expon[Pq, x], k}, Int[x^m*Sum[Coeff[
Pq, x, 2*k]*x^(2*k), {k, 0, q/2}]*(a + b*x^2)^p, x] + Int[x^(m + 1)*Sum[Coeff[Pq, x, 2*k + 1]*x^(2*k), {k, 0,
(q - 1)/2}]*(a + b*x^2)^p, x]] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x] && !PolyQ[Pq, x^2] && IGtQ[m, -2] && !
IntegerQ[2*p]
Rubi steps
\begin {align*} \int x^3 (d+e x)^3 \left (d^2-e^2 x^2\right )^p \, dx &=\int x^3 \left (d^2-e^2 x^2\right )^p \left (d^3+3 d e^2 x^2\right ) \, dx+\int x^4 \left (d^2-e^2 x^2\right )^p \left (3 d^2 e+e^3 x^2\right ) \, dx\\ &=-\frac {e x^5 \left (d^2-e^2 x^2\right )^{1+p}}{7+2 p}+\frac {1}{2} \operatorname {Subst}\left (\int x \left (d^2-e^2 x\right )^p \left (d^3+3 d e^2 x\right ) \, dx,x,x^2\right )+\frac {\left (2 d^2 e (13+3 p)\right ) \int x^4 \left (d^2-e^2 x^2\right )^p \, dx}{7+2 p}\\ &=-\frac {e x^5 \left (d^2-e^2 x^2\right )^{1+p}}{7+2 p}+\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {4 d^5 \left (d^2-e^2 x\right )^p}{e^2}-\frac {7 d^3 \left (d^2-e^2 x\right )^{1+p}}{e^2}+\frac {3 d \left (d^2-e^2 x\right )^{2+p}}{e^2}\right ) \, dx,x,x^2\right )+\frac {\left (2 d^2 e (13+3 p) \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 )^p \, dx}{7+2 p}\\ &=-\frac {2 d^5 \left (d^2-e^2 x^2\right )^{1+p}}{e^4 (1+p)}-\frac {e x^5 \left (d^2-e^2 x^2\right )^{1+p}}{7+2 p}+\frac {7 d^3 \left (d^2-e^2 x^2\right )^{2+p}}{2 e^4 (2+p)}-\frac {3 d \left (d^2-e^2 x^2\right )^{3+p}}{2 e^4 (3+p)}+\frac {2 d^2 e (13+3 p) 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},-p;\frac {7}{2};\frac {e^2 x^2}{d^2}\right )}{5 (7+2 p)}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 187, normalized size = 0.97 \[ \frac {\left (d^2-e^2 x^2\right )^p \left (10 e^7 x^7 \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac {7}{2},-p;\frac {9}{2};\frac {e^2 x^2}{d^2}\right )+42 d^2 e^5 x^5 \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac {5}{2},-p;\frac {7}{2};\frac {e^2 x^2}{d^2}\right )-\frac {35 d \left (d^2-e^2 x^2\right ) \left (d^4 (p+9)+d^2 e^2 \left (p^2+10 p+9\right ) x^2+3 e^4 \left (p^2+3 p+2\right ) x^4\right )}{(p+1) (p+2) (p+3)}\right )}{70 e^4} \]
Antiderivative was successfully verified.
[In]
Integrate[x^3*(d + e*x)^3*(d^2 - e^2*x^2)^p,x]
[Out]
((d^2 - e^2*x^2)^p*((-35*d*(d^2 - e^2*x^2)*(d^4*(9 + p) + d^2*e^2*(9 + 10*p + p^2)*x^2 + 3*e^4*(2 + 3*p + p^2)
*x^4))/((1 + p)*(2 + p)*(3 + p)) + (42*d^2*e^5*x^5*Hypergeometric2F1[5/2, -p, 7/2, (e^2*x^2)/d^2])/(1 - (e^2*x
^2)/d^2)^p + (10*e^7*x^7*Hypergeometric2F1[7/2, -p, 9/2, (e^2*x^2)/d^2])/(1 - (e^2*x^2)/d^2)^p))/(70*e^4)
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fricas [F] time = 0.70, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (e^{3} x^{6} + 3 \, d e^{2} x^{5} + 3 \, d^{2} e x^{4} + d^{3} x^{3}\right )} {\left (-e^{2} x^{2} + d^{2}\right )}^{p}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^3*(e*x+d)^3*(-e^2*x^2+d^2)^p,x, algorithm="fricas")
[Out]
integral((e^3*x^6 + 3*d*e^2*x^5 + 3*d^2*e*x^4 + d^3*x^3)*(-e^2*x^2 + d^2)^p, x)
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (e x + d\right )}^{3} {\left (-e^{2} x^{2} + d^{2}\right )}^{p} x^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^3*(e*x+d)^3*(-e^2*x^2+d^2)^p,x, algorithm="giac")
[Out]
integrate((e*x + d)^3*(-e^2*x^2 + d^2)^p*x^3, x)
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maple [F] time = 0.07, size = 0, normalized size = 0.00 \[ \int \left (e x +d \right )^{3} x^{3} \left (-e^{2} x^{2}+d^{2}\right )^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^3*(e*x+d)^3*(-e^2*x^2+d^2)^p,x)
[Out]
int(x^3*(e*x+d)^3*(-e^2*x^2+d^2)^p,x)
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {{\left (e^{4} {\left (p + 1\right )} x^{4} - d^{2} e^{2} p x^{2} - d^{4}\right )} {\left (-e^{2} x^{2} + d^{2}\right )}^{p} d^{3}}{2 \, {\left (p^{2} + 3 \, p + 2\right )} e^{4}} + \int {\left (e^{3} x^{6} + 3 \, d e^{2} x^{5} + 3 \, d^{2} e x^{4}\right )} e^{\left (p \log \left (e x + d\right ) + p \log \left (-e x + d\right )\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^3*(e*x+d)^3*(-e^2*x^2+d^2)^p,x, algorithm="maxima")
[Out]
1/2*(e^4*(p + 1)*x^4 - d^2*e^2*p*x^2 - d^4)*(-e^2*x^2 + d^2)^p*d^3/((p^2 + 3*p + 2)*e^4) + integrate((e^3*x^6
+ 3*d*e^2*x^5 + 3*d^2*e*x^4)*e^(p*log(e*x + d) + p*log(-e*x + d)), x)
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^3\,{\left (d^2-e^2\,x^2\right )}^p\,{\left (d+e\,x\right )}^3 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^3*(d^2 - e^2*x^2)^p*(d + e*x)^3,x)
[Out]
int(x^3*(d^2 - e^2*x^2)^p*(d + e*x)^3, x)
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sympy [B] time = 11.26, size = 1370, normalized size = 7.10 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x**3*(e*x+d)**3*(-e**2*x**2+d**2)**p,x)
[Out]
d**3*Piecewise((x**4*(d**2)**p/4, Eq(e, 0)), (-d**2*log(-d/e + x)/(-2*d**2*e**4 + 2*e**6*x**2) - d**2*log(d/e
+ x)/(-2*d**2*e**4 + 2*e**6*x**2) - d**2/(-2*d**2*e**4 + 2*e**6*x**2) + e**2*x**2*log(-d/e + x)/(-2*d**2*e**4
+ 2*e**6*x**2) + e**2*x**2*log(d/e + x)/(-2*d**2*e**4 + 2*e**6*x**2), Eq(p, -2)), (-d**2*log(-d/e + x)/(2*e**4
) - d**2*log(d/e + x)/(2*e**4) - x**2/(2*e**2), Eq(p, -1)), (-d**4*(d**2 - e**2*x**2)**p/(2*e**4*p**2 + 6*e**4
*p + 4*e**4) - d**2*e**2*p*x**2*(d**2 - e**2*x**2)**p/(2*e**4*p**2 + 6*e**4*p + 4*e**4) + e**4*p*x**4*(d**2 -
e**2*x**2)**p/(2*e**4*p**2 + 6*e**4*p + 4*e**4) + e**4*x**4*(d**2 - e**2*x**2)**p/(2*e**4*p**2 + 6*e**4*p + 4*
e**4), True)) + 3*d**2*d**(2*p)*e*x**5*hyper((5/2, -p), (7/2,), e**2*x**2*exp_polar(2*I*pi)/d**2)/5 + 3*d*e**2
*Piecewise((x**6*(d**2)**p/6, Eq(e, 0)), (-2*d**4*log(-d/e + x)/(4*d**4*e**6 - 8*d**2*e**8*x**2 + 4*e**10*x**4
) - 2*d**4*log(d/e + x)/(4*d**4*e**6 - 8*d**2*e**8*x**2 + 4*e**10*x**4) - 3*d**4/(4*d**4*e**6 - 8*d**2*e**8*x*
*2 + 4*e**10*x**4) + 4*d**2*e**2*x**2*log(-d/e + x)/(4*d**4*e**6 - 8*d**2*e**8*x**2 + 4*e**10*x**4) + 4*d**2*e
**2*x**2*log(d/e + x)/(4*d**4*e**6 - 8*d**2*e**8*x**2 + 4*e**10*x**4) + 4*d**2*e**2*x**2/(4*d**4*e**6 - 8*d**2
*e**8*x**2 + 4*e**10*x**4) - 2*e**4*x**4*log(-d/e + x)/(4*d**4*e**6 - 8*d**2*e**8*x**2 + 4*e**10*x**4) - 2*e**
4*x**4*log(d/e + x)/(4*d**4*e**6 - 8*d**2*e**8*x**2 + 4*e**10*x**4), Eq(p, -3)), (-2*d**4*log(-d/e + x)/(-2*d*
*2*e**6 + 2*e**8*x**2) - 2*d**4*log(d/e + x)/(-2*d**2*e**6 + 2*e**8*x**2) - 2*d**4/(-2*d**2*e**6 + 2*e**8*x**2
) + 2*d**2*e**2*x**2*log(-d/e + x)/(-2*d**2*e**6 + 2*e**8*x**2) + 2*d**2*e**2*x**2*log(d/e + x)/(-2*d**2*e**6
+ 2*e**8*x**2) + e**4*x**4/(-2*d**2*e**6 + 2*e**8*x**2), Eq(p, -2)), (-d**4*log(-d/e + x)/(2*e**6) - d**4*log(
d/e + x)/(2*e**6) - d**2*x**2/(2*e**4) - x**4/(4*e**2), Eq(p, -1)), (-2*d**6*(d**2 - e**2*x**2)**p/(2*e**6*p**
3 + 12*e**6*p**2 + 22*e**6*p + 12*e**6) - 2*d**4*e**2*p*x**2*(d**2 - e**2*x**2)**p/(2*e**6*p**3 + 12*e**6*p**2
+ 22*e**6*p + 12*e**6) - d**2*e**4*p**2*x**4*(d**2 - e**2*x**2)**p/(2*e**6*p**3 + 12*e**6*p**2 + 22*e**6*p +
12*e**6) - d**2*e**4*p*x**4*(d**2 - e**2*x**2)**p/(2*e**6*p**3 + 12*e**6*p**2 + 22*e**6*p + 12*e**6) + e**6*p*
*2*x**6*(d**2 - e**2*x**2)**p/(2*e**6*p**3 + 12*e**6*p**2 + 22*e**6*p + 12*e**6) + 3*e**6*p*x**6*(d**2 - e**2*
x**2)**p/(2*e**6*p**3 + 12*e**6*p**2 + 22*e**6*p + 12*e**6) + 2*e**6*x**6*(d**2 - e**2*x**2)**p/(2*e**6*p**3 +
12*e**6*p**2 + 22*e**6*p + 12*e**6), True)) + d**(2*p)*e**3*x**7*hyper((7/2, -p), (9/2,), e**2*x**2*exp_polar
(2*I*pi)/d**2)/7
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