3.250 \(\int x^4 (d+e x)^2 (d^2-e^2 x^2)^p \, dx\)

Optimal. Leaf size=185 \[ \frac {2 d^2 (p+6) x^5 \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac {5}{2},-p;\frac {7}{2};\frac {e^2 x^2}{d^2}\right )}{5 (2 p+7)}-\frac {x^5 \left (d^2-e^2 x^2\right )^{p+1}}{2 p+7}-\frac {d \left (d^2-e^2 x^2\right )^{p+3}}{e^5 (p+3)}-\frac {d^5 \left (d^2-e^2 x^2\right )^{p+1}}{e^5 (p+1)}+\frac {2 d^3 \left (d^2-e^2 x^2\right )^{p+2}}{e^5 (p+2)} \]

[Out]

-d^5*(-e^2*x^2+d^2)^(1+p)/e^5/(1+p)-x^5*(-e^2*x^2+d^2)^(1+p)/(7+2*p)+2*d^3*(-e^2*x^2+d^2)^(2+p)/e^5/(2+p)-d*(-
e^2*x^2+d^2)^(3+p)/e^5/(3+p)+2/5*d^2*(6+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 = 0.17, antiderivative size = 185, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {1652, 459, 365, 364, 12, 266, 43} \[ \frac {2 d^2 (p+6) x^5 \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac {5}{2},-p;\frac {7}{2};\frac {e^2 x^2}{d^2}\right )}{5 (2 p+7)}-\frac {x^5 \left (d^2-e^2 x^2\right )^{p+1}}{2 p+7}-\frac {d^5 \left (d^2-e^2 x^2\right )^{p+1}}{e^5 (p+1)}+\frac {2 d^3 \left (d^2-e^2 x^2\right )^{p+2}}{e^5 (p+2)}-\frac {d \left (d^2-e^2 x^2\right )^{p+3}}{e^5 (p+3)} \]

Antiderivative was successfully verified.

[In]

Int[x^4*(d + e*x)^2*(d^2 - e^2*x^2)^p,x]

[Out]

-((d^5*(d^2 - e^2*x^2)^(1 + p))/(e^5*(1 + p))) - (x^5*(d^2 - e^2*x^2)^(1 + p))/(7 + 2*p) + (2*d^3*(d^2 - e^2*x
^2)^(2 + p))/(e^5*(2 + p)) - (d*(d^2 - e^2*x^2)^(3 + p))/(e^5*(3 + p)) + (2*d^2*(6 + 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 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 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 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 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^4 (d+e x)^2 \left (d^2-e^2 x^2\right )^p \, dx &=\int 2 d e x^5 \left (d^2-e^2 x^2\right )^p \, dx+\int x^4 \left (d^2-e^2 x^2\right )^p \left (d^2+e^2 x^2\right ) \, dx\\ &=-\frac {x^5 \left (d^2-e^2 x^2\right )^{1+p}}{7+2 p}+(2 d e) \int x^5 \left (d^2-e^2 x^2\right )^p \, dx+\frac {\left (2 d^2 (6+p)\right ) \int x^4 \left (d^2-e^2 x^2\right )^p \, dx}{7+2 p}\\ &=-\frac {x^5 \left (d^2-e^2 x^2\right )^{1+p}}{7+2 p}+(d e) \operatorname {Subst}\left (\int x^2 \left (d^2-e^2 x\right )^p \, dx,x,x^2\right )+\frac {\left (2 d^2 (6+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 {x^5 \left (d^2-e^2 x^2\right )^{1+p}}{7+2 p}+\frac {2 d^2 (6+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)}+(d e) \operatorname {Subst}\left (\int \left (\frac {d^4 \left (d^2-e^2 x\right )^p}{e^4}-\frac {2 d^2 \left (d^2-e^2 x\right )^{1+p}}{e^4}+\frac {\left (d^2-e^2 x\right )^{2+p}}{e^4}\right ) \, dx,x,x^2\right )\\ &=-\frac {d^5 \left (d^2-e^2 x^2\right )^{1+p}}{e^5 (1+p)}-\frac {x^5 \left (d^2-e^2 x^2\right )^{1+p}}{7+2 p}+\frac {2 d^3 \left (d^2-e^2 x^2\right )^{2+p}}{e^5 (2+p)}-\frac {d \left (d^2-e^2 x^2\right )^{3+p}}{e^5 (3+p)}+\frac {2 d^2 (6+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 = 0.12, size = 186, normalized size = 1.01 \[ \frac {1}{35} \left (d^2-e^2 x^2\right )^p \left (5 e^2 x^7 \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac {7}{2},-p;\frac {9}{2};\frac {e^2 x^2}{d^2}\right )+7 d^2 x^5 \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac {5}{2},-p;\frac {7}{2};\frac {e^2 x^2}{d^2}\right )-\frac {35 d \left (d^2-e^2 x^2\right )^3}{e^5 (p+3)}-\frac {35 d^5 \left (d^2-e^2 x^2\right )}{e^5 (p+1)}+\frac {70 d^3 \left (d^2-e^2 x^2\right )^2}{e^5 (p+2)}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^4*(d + e*x)^2*(d^2 - e^2*x^2)^p,x]

[Out]

((d^2 - e^2*x^2)^p*((-35*d^5*(d^2 - e^2*x^2))/(e^5*(1 + p)) + (70*d^3*(d^2 - e^2*x^2)^2)/(e^5*(2 + p)) - (35*d
*(d^2 - e^2*x^2)^3)/(e^5*(3 + p)) + (7*d^2*x^5*Hypergeometric2F1[5/2, -p, 7/2, (e^2*x^2)/d^2])/(1 - (e^2*x^2)/
d^2)^p + (5*e^2*x^7*Hypergeometric2F1[7/2, -p, 9/2, (e^2*x^2)/d^2])/(1 - (e^2*x^2)/d^2)^p))/35

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fricas [F]  time = 0.82, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (e^{2} x^{6} + 2 \, d e x^{5} + d^{2} x^{4}\right )} {\left (-e^{2} x^{2} + d^{2}\right )}^{p}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(e*x+d)^2*(-e^2*x^2+d^2)^p,x, algorithm="fricas")

[Out]

integral((e^2*x^6 + 2*d*e*x^5 + d^2*x^4)*(-e^2*x^2 + d^2)^p, x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (e x + d\right )}^{2} {\left (-e^{2} x^{2} + d^{2}\right )}^{p} x^{4}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(e*x+d)^2*(-e^2*x^2+d^2)^p,x, algorithm="giac")

[Out]

integrate((e*x + d)^2*(-e^2*x^2 + d^2)^p*x^4, x)

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maple [F]  time = 0.06, size = 0, normalized size = 0.00 \[ \int \left (e x +d \right )^{2} x^{4} \left (-e^{2} x^{2}+d^{2}\right )^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(e*x+d)^2*(-e^2*x^2+d^2)^p,x)

[Out]

int(x^4*(e*x+d)^2*(-e^2*x^2+d^2)^p,x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (e x + d\right )}^{2} {\left (-e^{2} x^{2} + d^{2}\right )}^{p} x^{4}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(e*x+d)^2*(-e^2*x^2+d^2)^p,x, algorithm="maxima")

[Out]

integrate((e*x + d)^2*(-e^2*x^2 + d^2)^p*x^4, x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int x^4\,{\left (d^2-e^2\,x^2\right )}^p\,{\left (d+e\,x\right )}^2 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(d^2 - e^2*x^2)^p*(d + e*x)^2,x)

[Out]

int(x^4*(d^2 - e^2*x^2)^p*(d + e*x)^2, x)

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sympy [B]  time = 8.90, size = 1015, normalized size = 5.49 \[ \frac {d^{2} d^{2 p} x^{5} {{}_{2}F_{1}\left (\begin {matrix} \frac {5}{2}, - p \\ \frac {7}{2} \end {matrix}\middle | {\frac {e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{5} + 2 d e \left (\begin {cases} \frac {x^{6} \left (d^{2}\right )^{p}}{6} & \text {for}\: e = 0 \\- \frac {2 d^{4} \log {\left (- \frac {d}{e} + x \right )}}{4 d^{4} e^{6} - 8 d^{2} e^{8} x^{2} + 4 e^{10} x^{4}} - \frac {2 d^{4} \log {\left (\frac {d}{e} + x \right )}}{4 d^{4} e^{6} - 8 d^{2} e^{8} x^{2} + 4 e^{10} x^{4}} - \frac {3 d^{4}}{4 d^{4} e^{6} - 8 d^{2} e^{8} x^{2} + 4 e^{10} x^{4}} + \frac {4 d^{2} e^{2} x^{2} \log {\left (- \frac {d}{e} + x \right )}}{4 d^{4} e^{6} - 8 d^{2} e^{8} x^{2} + 4 e^{10} x^{4}} + \frac {4 d^{2} e^{2} x^{2} \log {\left (\frac {d}{e} + x \right )}}{4 d^{4} e^{6} - 8 d^{2} e^{8} x^{2} + 4 e^{10} x^{4}} + \frac {4 d^{2} e^{2} x^{2}}{4 d^{4} e^{6} - 8 d^{2} e^{8} x^{2} + 4 e^{10} x^{4}} - \frac {2 e^{4} x^{4} \log {\left (- \frac {d}{e} + x \right )}}{4 d^{4} e^{6} - 8 d^{2} e^{8} x^{2} + 4 e^{10} x^{4}} - \frac {2 e^{4} x^{4} \log {\left (\frac {d}{e} + x \right )}}{4 d^{4} e^{6} - 8 d^{2} e^{8} x^{2} + 4 e^{10} x^{4}} & \text {for}\: p = -3 \\- \frac {2 d^{4} \log {\left (- \frac {d}{e} + x \right )}}{- 2 d^{2} e^{6} + 2 e^{8} x^{2}} - \frac {2 d^{4} \log {\left (\frac {d}{e} + x \right )}}{- 2 d^{2} e^{6} + 2 e^{8} x^{2}} - \frac {2 d^{4}}{- 2 d^{2} e^{6} + 2 e^{8} x^{2}} + \frac {2 d^{2} e^{2} x^{2} \log {\left (- \frac {d}{e} + x \right )}}{- 2 d^{2} e^{6} + 2 e^{8} x^{2}} + \frac {2 d^{2} e^{2} x^{2} \log {\left (\frac {d}{e} + x \right )}}{- 2 d^{2} e^{6} + 2 e^{8} x^{2}} + \frac {e^{4} x^{4}}{- 2 d^{2} e^{6} + 2 e^{8} x^{2}} & \text {for}\: p = -2 \\- \frac {d^{4} \log {\left (- \frac {d}{e} + x \right )}}{2 e^{6}} - \frac {d^{4} \log {\left (\frac {d}{e} + x \right )}}{2 e^{6}} - \frac {d^{2} x^{2}}{2 e^{4}} - \frac {x^{4}}{4 e^{2}} & \text {for}\: p = -1 \\- \frac {2 d^{6} \left (d^{2} - e^{2} x^{2}\right )^{p}}{2 e^{6} p^{3} + 12 e^{6} p^{2} + 22 e^{6} p + 12 e^{6}} - \frac {2 d^{4} e^{2} p x^{2} \left (d^{2} - e^{2} x^{2}\right )^{p}}{2 e^{6} p^{3} + 12 e^{6} p^{2} + 22 e^{6} p + 12 e^{6}} - \frac {d^{2} e^{4} p^{2} x^{4} \left (d^{2} - e^{2} x^{2}\right )^{p}}{2 e^{6} p^{3} + 12 e^{6} p^{2} + 22 e^{6} p + 12 e^{6}} - \frac {d^{2} e^{4} p x^{4} \left (d^{2} - e^{2} x^{2}\right )^{p}}{2 e^{6} p^{3} + 12 e^{6} p^{2} + 22 e^{6} p + 12 e^{6}} + \frac {e^{6} p^{2} x^{6} \left (d^{2} - e^{2} x^{2}\right )^{p}}{2 e^{6} p^{3} + 12 e^{6} p^{2} + 22 e^{6} p + 12 e^{6}} + \frac {3 e^{6} p x^{6} \left (d^{2} - e^{2} x^{2}\right )^{p}}{2 e^{6} p^{3} + 12 e^{6} p^{2} + 22 e^{6} p + 12 e^{6}} + \frac {2 e^{6} x^{6} \left (d^{2} - e^{2} x^{2}\right )^{p}}{2 e^{6} p^{3} + 12 e^{6} p^{2} + 22 e^{6} p + 12 e^{6}} & \text {otherwise} \end {cases}\right ) + \frac {d^{2 p} e^{2} x^{7} {{}_{2}F_{1}\left (\begin {matrix} \frac {7}{2}, - p \\ \frac {9}{2} \end {matrix}\middle | {\frac {e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(e*x+d)**2*(-e**2*x**2+d**2)**p,x)

[Out]

d**2*d**(2*p)*x**5*hyper((5/2, -p), (7/2,), e**2*x**2*exp_polar(2*I*pi)/d**2)/5 + 2*d*e*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*lo
g(-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**2*x**7*hyper((7/2, -p), (9/2,), e**2*x**2*exp_polar(2*I*pi)/d**2)/7

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