3.3.75 \(\int \frac {x^5 (d^2-e^2 x^2)^p}{(d+e x)^2} \, dx\) [275]

Optimal. Leaf size=179 \[ \frac {d^6 \left (d^2-e^2 x^2\right )^{-1+p}}{e^6 (1-p)}+\frac {5 d^4 \left (d^2-e^2 x^2\right )^p}{2 e^6 p}-\frac {2 d^2 \left (d^2-e^2 x^2\right )^{1+p}}{e^6 (1+p)}+\frac {\left (d^2-e^2 x^2\right )^{2+p}}{2 e^6 (2+p)}-\frac {2 e x^7 \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac {7}{2},2-p;\frac {9}{2};\frac {e^2 x^2}{d^2}\right )}{7 d^3} \]

[Out]

d^6*(-e^2*x^2+d^2)^(-1+p)/e^6/(1-p)+5/2*d^4*(-e^2*x^2+d^2)^p/e^6/p-2*d^2*(-e^2*x^2+d^2)^(1+p)/e^6/(1+p)+1/2*(-
e^2*x^2+d^2)^(2+p)/e^6/(2+p)-2/7*e*x^7*(-e^2*x^2+d^2)^p*hypergeom([7/2, 2-p],[9/2],e^2*x^2/d^2)/d^3/((1-e^2*x^
2/d^2)^p)

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Rubi [A]
time = 0.12, antiderivative size = 179, 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 = {866, 1666, 457, 78, 12, 372, 371} \begin {gather*} -\frac {2 d^2 \left (d^2-e^2 x^2\right )^{p+1}}{e^6 (p+1)}+\frac {\left (d^2-e^2 x^2\right )^{p+2}}{2 e^6 (p+2)}+\frac {d^6 \left (d^2-e^2 x^2\right )^{p-1}}{e^6 (1-p)}+\frac {5 d^4 \left (d^2-e^2 x^2\right )^p}{2 e^6 p}-\frac {2 e x^7 \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \left (d^2-e^2 x^2\right )^p \, _2F_1\left (\frac {7}{2},2-p;\frac {9}{2};\frac {e^2 x^2}{d^2}\right )}{7 d^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^5*(d^2 - e^2*x^2)^p)/(d + e*x)^2,x]

[Out]

(d^6*(d^2 - e^2*x^2)^(-1 + p))/(e^6*(1 - p)) + (5*d^4*(d^2 - e^2*x^2)^p)/(2*e^6*p) - (2*d^2*(d^2 - e^2*x^2)^(1
+ p))/(e^6*(1 + p)) + (d^2 - e^2*x^2)^(2 + p)/(2*e^6*(2 + p)) - (2*e*x^7*(d^2 - e^2*x^2)^p*Hypergeometric2F1[
7/2, 2 - p, 9/2, (e^2*x^2)/d^2])/(7*d^3*(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 78

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 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 372

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 457

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 866

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[d^(2*m)/a
^m, Int[(f + g*x)^n*((a + c*x^2)^(m + p)/(d - e*x)^m), x], x] /; FreeQ[{a, c, d, e, f, g, n, p}, x] && NeQ[e*f
- d*g, 0] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && EqQ[f, 0] && ILtQ[m, -1] &&  !(IGtQ[n, 0] && ILtQ[m +
n, 0] &&  !GtQ[p, 1])

Rule 1666

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 \frac {x^5 \left (d^2-e^2 x^2\right )^p}{(d+e x)^2} \, dx &=\int x^5 (d-e x)^2 \left (d^2-e^2 x^2\right )^{-2+p} \, dx\\ &=\int -2 d e x^6 \left (d^2-e^2 x^2\right )^{-2+p} \, dx+\int x^5 \left (d^2-e^2 x^2\right )^{-2+p} \left (d^2+e^2 x^2\right ) \, dx\\ &=\frac {1}{2} \text {Subst}\left (\int x^2 \left (d^2-e^2 x\right )^{-2+p} \left (d^2+e^2 x\right ) \, dx,x,x^2\right )-(2 d e) \int x^6 \left (d^2-e^2 x^2\right )^{-2+p} \, dx\\ &=\frac {1}{2} \text {Subst}\left (\int \left (\frac {2 d^6 \left (d^2-e^2 x\right )^{-2+p}}{e^4}-\frac {5 d^4 \left (d^2-e^2 x\right )^{-1+p}}{e^4}+\frac {4 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 {\left (2 e \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p}\right ) \int x^6 \left (1-\frac {e^2 x^2}{d^2}\right )^{-2+p} \, dx}{d^3}\\ &=\frac {d^6 \left (d^2-e^2 x^2\right )^{-1+p}}{e^6 (1-p)}+\frac {5 d^4 \left (d^2-e^2 x^2\right )^p}{2 e^6 p}-\frac {2 d^2 \left (d^2-e^2 x^2\right )^{1+p}}{e^6 (1+p)}+\frac {\left (d^2-e^2 x^2\right )^{2+p}}{2 e^6 (2+p)}-\frac {2 e x^7 \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac {7}{2},2-p;\frac {9}{2};\frac {e^2 x^2}{d^2}\right )}{7 d^3}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.
time = 0.34, size = 66, normalized size = 0.37 \begin {gather*} \frac {x^6 (d-e x)^p (d+e x)^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} F_1\left (6;-p,2-p;7;\frac {e x}{d},-\frac {e x}{d}\right )}{6 d^2} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[(x^5*(d^2 - e^2*x^2)^p)/(d + e*x)^2,x]

[Out]

(x^6*(d - e*x)^p*(d + e*x)^p*AppellF1[6, -p, 2 - p, 7, (e*x)/d, -((e*x)/d)])/(6*d^2*(1 - (e^2*x^2)/d^2)^p)

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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {x^{5} \left (-e^{2} x^{2}+d^{2}\right )^{p}}{\left (e x +d \right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(-e^2*x^2+d^2)^p/(e*x+d)^2,x)

[Out]

int(x^5*(-e^2*x^2+d^2)^p/(e*x+d)^2,x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(-e^2*x^2+d^2)^p/(e*x+d)^2,x, algorithm="maxima")

[Out]

integrate((-x^2*e^2 + d^2)^p*x^5/(x*e + d)^2, x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(-e^2*x^2+d^2)^p/(e*x+d)^2,x, algorithm="fricas")

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{5} \left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{p}}{\left (d + e x\right )^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*(-e**2*x**2+d**2)**p/(e*x+d)**2,x)

[Out]

Integral(x**5*(-(-d + e*x)*(d + e*x))**p/(d + e*x)**2, x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(-e^2*x^2+d^2)^p/(e*x+d)^2,x, algorithm="giac")

[Out]

integrate((-x^2*e^2 + d^2)^p*x^5/(x*e + d)^2, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^5\,{\left (d^2-e^2\,x^2\right )}^p}{{\left (d+e\,x\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^5*(d^2 - e^2*x^2)^p)/(d + e*x)^2,x)

[Out]

int((x^5*(d^2 - e^2*x^2)^p)/(d + e*x)^2, x)

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