\(\int \frac {x^4 (d^2-e^2 x^2)^p}{d+e x} \, dx\) [267]

Optimal result

Integrand size = 25, antiderivative size = 148 \[ \int \frac {x^4 \left (d^2-e^2 x^2\right )^p}{d+e x} \, dx=\frac {d^4 \left (d^2-e^2 x^2\right )^p}{2 e^5 p}-\frac {d^2 \left (d^2-e^2 x^2\right )^{1+p}}{e^5 (1+p)}+\frac {\left (d^2-e^2 x^2\right )^{2+p}}{2 e^5 (2+p)}+\frac {x^5 \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},1-p,\frac {7}{2},\frac {e^2 x^2}{d^2}\right )}{5 d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {864, 778, 372, 371, 272, 45} \[ \int \frac {x^4 \left (d^2-e^2 x^2\right )^p}{d+e x} \, dx=\frac {x^5 \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},1-p,\frac {7}{2},\frac {e^2 x^2}{d^2}\right )}{5 d}-\frac {d^2 \left (d^2-e^2 x^2\right )^{p+1}}{e^5 (p+1)}+\frac {\left (d^2-e^2 x^2\right )^{p+2}}{2 e^5 (p+2)}+\frac {d^4 \left (d^2-e^2 x^2\right )^p}{2 e^5 p} \]

[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 45

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 272

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 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 778

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 864

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]))

Rubi steps \begin{align*} \text {integral}& = \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 \text {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 \text {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] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.

Time = 0.33 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.45 \[ \int \frac {x^4 \left (d^2-e^2 x^2\right )^p}{d+e x} \, dx=\frac {x^5 (d-e x)^p (d+e x)^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {AppellF1}\left (5,-p,1-p,6,\frac {e x}{d},-\frac {e x}{d}\right )}{5 d} \]

[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)

Maple [F]

\[\int \frac {x^{4} \left (-e^{2} x^{2}+d^{2}\right )^{p}}{e x +d}d x\]

[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)

Fricas [F]

\[ \int \frac {x^4 \left (d^2-e^2 x^2\right )^p}{d+e x} \, dx=\int { \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{p} x^{4}}{e x + d} \,d x } \]

[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)

Sympy [F(-1)]

Timed out. \[ \int \frac {x^4 \left (d^2-e^2 x^2\right )^p}{d+e x} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int \frac {x^4 \left (d^2-e^2 x^2\right )^p}{d+e x} \, dx=\int { \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{p} x^{4}}{e x + d} \,d x } \]

[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)

Giac [F]

\[ \int \frac {x^4 \left (d^2-e^2 x^2\right )^p}{d+e x} \, dx=\int { \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{p} x^{4}}{e x + d} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^4 \left (d^2-e^2 x^2\right )^p}{d+e x} \, dx=\int \frac {x^4\,{\left (d^2-e^2\,x^2\right )}^p}{d+e\,x} \,d x \]

[In]

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

[Out]

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