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

Optimal. Leaf size=148 \[ \frac {d^4 \left (d^2-e^2 x^2\right )^p}{2 e^5 p}-\frac {d^2 \left (d^2-e^2 x^2\right )^{1+p}}{e^5 (1+p)}+\frac {\left (d^2-e^2 x^2\right )^{2+p}}{2 e^5 (2+p)}+\frac {x^5 \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac {5}{2},1-p;\frac {7}{2};\frac {e^2 x^2}{d^2}\right )}{5 d} \]

[Out]

1/2*d^4*(-e^2*x^2+d^2)^p/e^5/p-d^2*(-e^2*x^2+d^2)^(1+p)/e^5/(1+p)+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)

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Rubi [A]
time = 0.07, antiderivative size = 148, 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 = {864, 778, 372, 371, 272, 45} \begin {gather*} \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 {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} \end {gather*}

Antiderivative was successfully verified.

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

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Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.
time = 0.27, size = 66, normalized size = 0.45 \begin {gather*} \frac {x^5 (d-e x)^p (d+e x)^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} F_1\left (5;-p,1-p;6;\frac {e x}{d},-\frac {e x}{d}\right )}{5 d} \end {gather*}

Warning: Unable to verify antiderivative.

[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)

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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {x^{4} \left (-e^{2} x^{2}+d^{2}\right )^{p}}{e x +d}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

[Out]

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

[Out]

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

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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