3.3.41 \(\int x^4 (d+e x) (d^2-e^2 x^2)^p \, dx\) [241]

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

[Out]

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

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Rubi [A]
time = 0.06, antiderivative size = 147, normalized size of antiderivative = 1.00, 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 = {778, 372, 371, 272, 45} \begin {gather*} \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^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)}-\frac {d^4 \left (d^2-e^2 x^2\right )^{p+1}}{2 e^5 (p+1)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 \text {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 \text {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*}

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Mathematica [A]
time = 0.20, size = 129, normalized size = 0.88 \begin {gather*} \frac {1}{10} \left (d^2-e^2 x^2\right )^p \left (-\frac {5 \left (d^2-e^2 x^2\right ) \left (2 d^4+2 d^2 e^2 (1+p) x^2+e^4 \left (2+3 p+p^2\right ) x^4\right )}{e^5 (1+p) (2+p) (3+p)}+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 )\right ) \end {gather*}

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 = 0.02, size = 0, normalized size = 0.00 \[\int x^{4} \left (e x +d \right ) \left (-e^{2} x^{2}+d^{2}\right )^{p}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(e*x+d)*(-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 = 0.00, 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*x+d)*(-e^2*x^2+d^2)^p,x, algorithm="maxima")

[Out]

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

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 937 vs. \(2 (119) = 238\).
time = 2.17, size = 972, normalized size = 6.61 \begin {gather*} \frac {d 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} + 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 ) \end {gather*}

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) - 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*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 = 0.00, 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*x+d)*(-e^2*x^2+d^2)^p,x, algorithm="giac")

[Out]

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

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