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

Optimal. Leaf size=147 \[ \frac{1}{5} d x^5 \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac{5}{2},-p;\frac{7}{2};\frac{e^2 x^2}{d^2}\right )-\frac{d^4 \left (d^2-e^2 x^2\right )^{p+1}}{2 e^5 (p+1)}+\frac{d^2 \left (d^2-e^2 x^2\right )^{p+2}}{e^5 (p+2)}-\frac{\left (d^2-e^2 x^2\right )^{p+3}}{2 e^5 (p+3)} \]

[Out]

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

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Rubi [A]  time = 0.0909236, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {764, 365, 364, 266, 43} \[ \frac{1}{5} d x^5 \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac{5}{2},-p;\frac{7}{2};\frac{e^2 x^2}{d^2}\right )-\frac{d^4 \left (d^2-e^2 x^2\right )^{p+1}}{2 e^5 (p+1)}+\frac{d^2 \left (d^2-e^2 x^2\right )^{p+2}}{e^5 (p+2)}-\frac{\left (d^2-e^2 x^2\right )^{p+3}}{2 e^5 (p+3)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 764

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 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 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 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 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])

Rubi steps

\begin{align*} \int x^4 (d+e x) \left (d^2-e^2 x^2\right )^p \, dx &=d \int x^4 \left (d^2-e^2 x^2\right )^p \, dx+e \int x^5 \left (d^2-e^2 x^2\right )^p \, dx\\ &=\frac{1}{2} e \operatorname{Subst}\left (\int x^2 \left (d^2-e^2 x\right )^p \, dx,x,x^2\right )+\left (d \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\\ &=\frac{1}{5} d 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 )+\frac{1}{2} 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^4 \left (d^2-e^2 x^2\right )^{1+p}}{2 e^5 (1+p)}+\frac{d^2 \left (d^2-e^2 x^2\right )^{2+p}}{e^5 (2+p)}-\frac{\left (d^2-e^2 x^2\right )^{3+p}}{2 e^5 (3+p)}+\frac{1}{5} d 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 )\\ \end{align*}

Mathematica [A]  time = 0.091418, size = 129, normalized size = 0.88 \[ \frac{1}{10} \left (d^2-e^2 x^2\right )^p \left (2 d x^5 \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac{5}{2},-p;\frac{7}{2};\frac{e^2 x^2}{d^2}\right )-\frac{5 \left (d^2-e^2 x^2\right ) \left (2 d^2 e^2 (p+1) x^2+2 d^4+e^4 \left (p^2+3 p+2\right ) x^4\right )}{e^5 (p+1) (p+2) (p+3)}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

((d^2 - e^2*x^2)^p*((-5*(d^2 - e^2*x^2)*(2*d^4 + 2*d^2*e^2*(1 + p)*x^2 + e^4*(2 + 3*p + p^2)*x^4))/(e^5*(1 + p
)*(2 + p)*(3 + p)) + (2*d*x^5*Hypergeometric2F1[5/2, -p, 7/2, (e^2*x^2)/d^2])/(1 - (e^2*x^2)/d^2)^p))/10

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Maple [F]  time = 0.458, size = 0, normalized size = 0. \begin{align*} \int{x}^{4} \left ( ex+d \right ) \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{p}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (e x + d\right )}{\left (-e^{2} x^{2} + d^{2}\right )}^{p} x^{4}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (e x^{5} + d x^{4}\right )}{\left (-e^{2} x^{2} + d^{2}\right )}^{p}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 7.92963, size = 967, normalized size = 6.58 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d*d**(2*p)*x**5*hyper((5/2, -p), (7/2,), e**2*x**2*exp_polar(2*I*pi)/d**2)/5 + 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) - 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) - 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) + 2*e**4*x**4/(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))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (e x + d\right )}{\left (-e^{2} x^{2} + d^{2}\right )}^{p} x^{4}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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