Integrand size = 23, antiderivative size = 118 \[ \int x (d+e x)^2 \left (d^2-e^2 x^2\right )^p \, dx=-\frac {d^2 \left (d^2-e^2 x^2\right )^{1+p}}{e^2 (1+p)}+\frac {\left (d^2-e^2 x^2\right )^{2+p}}{2 e^2 (2+p)}+\frac {2}{3} d e x^3 \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},-p,\frac {5}{2},\frac {e^2 x^2}{d^2}\right ) \]
-d^2*(-e^2*x^2+d^2)^(p+1)/e^2/(p+1)+1/2*(-e^2*x^2+d^2)^(2+p)/e^2/(2+p)+2/3 *d*e*x^3*(-e^2*x^2+d^2)^p*hypergeom([3/2, -p],[5/2],e^2*x^2/d^2)/((1-e^2*x ^2/d^2)^p)
Time = 0.18 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.93 \[ \int x (d+e x)^2 \left (d^2-e^2 x^2\right )^p \, dx=\frac {\left (d^2-e^2 x^2\right )^p \left (-\frac {3 \left (d^2-e^2 x^2\right ) \left (d^2 (3+p)+e^2 (1+p) x^2\right )}{(1+p) (2+p)}+4 d e^3 x^3 \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},-p,\frac {5}{2},\frac {e^2 x^2}{d^2}\right )\right )}{6 e^2} \]
((d^2 - e^2*x^2)^p*((-3*(d^2 - e^2*x^2)*(d^2*(3 + p) + e^2*(1 + p)*x^2))/( (1 + p)*(2 + p)) + (4*d*e^3*x^3*Hypergeometric2F1[3/2, -p, 5/2, (e^2*x^2)/ d^2])/(1 - (e^2*x^2)/d^2)^p))/(6*e^2)
Time = 0.23 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.97, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {572, 473, 79}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int x (d+e x)^2 \left (d^2-e^2 x^2\right )^p \, dx\) |
| \(\Big \downarrow \) 572 |
| \(\displaystyle \frac {d \int (d+e x)^2 \left (d^2-e^2 x^2\right )^pdx}{e (p+2)}-\frac {(d+e x)^2 \left (d^2-e^2 x^2\right )^{p+1}}{2 e^2 (p+2)}\) |
| \(\Big \downarrow \) 473 |
| \(\displaystyle \frac {d^2 (d-e x)^{-p-1} \left (\frac {e x}{d}+1\right )^{-p-1} \left (d^2-e^2 x^2\right )^{p+1} \int (d-e x)^p \left (\frac {e x}{d}+1\right )^{p+2}dx}{e (p+2)}-\frac {(d+e x)^2 \left (d^2-e^2 x^2\right )^{p+1}}{2 e^2 (p+2)}\) |
| \(\Big \downarrow \) 79 |
| \(\displaystyle -\frac {d^2 2^{p+2} \left (d^2-e^2 x^2\right )^{p+1} \left (\frac {e x}{d}+1\right )^{-p-1} \operatorname {Hypergeometric2F1}\left (-p-2,p+1,p+2,\frac {d-e x}{2 d}\right )}{e^2 (p+1) (p+2)}-\frac {(d+e x)^2 \left (d^2-e^2 x^2\right )^{p+1}}{2 e^2 (p+2)}\) |
-1/2*((d + e*x)^2*(d^2 - e^2*x^2)^(1 + p))/(e^2*(2 + p)) - (2^(2 + p)*d^2* (1 + (e*x)/d)^(-1 - p)*(d^2 - e^2*x^2)^(1 + p)*Hypergeometric2F1[-2 - p, 1 + p, 2 + p, (d - e*x)/(2*d)])/(e^2*(1 + p)*(2 + p))
3.3.53.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(( a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c - a*d))^n))*Hypergeometric2F1[-n, m + 1 , m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m, n}, x] && !IntegerQ[m] && !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] || !(RationalQ[n] && GtQ[-d/(b*c - a*d), 0]))
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ c^(n - 1)*((a + b*x^2)^(p + 1)/((1 + d*(x/c))^(p + 1)*(a/c + (b*x)/d)^(p + 1))) Int[(1 + d*(x/c))^(n + p)*(a/c + (b/d)*x)^p, x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c^2 + a*d^2, 0] && (IntegerQ[n] || GtQ[c, 0]) && !Gt Q[a, 0] && !(IntegerQ[n] && (IntegerQ[3*p] || IntegerQ[4*p]))
Int[(x_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c + d*x)^n*((a + b*x^2)^(p + 1)/(b*(n + 2*p + 2))), x] + Simp[c*(n/(d *(n + 2*p + 2))) Int[(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, n, p}, x] && EqQ[b*c^2 + a*d^2, 0] && NeQ[n + 2*p + 2, 0]
\[\int x \left (e x +d \right )^{2} \left (-e^{2} x^{2}+d^{2}\right )^{p}d x\]
\[ \int x (d+e x)^2 \left (d^2-e^2 x^2\right )^p \, dx=\int { {\left (e x + d\right )}^{2} {\left (-e^{2} x^{2} + d^{2}\right )}^{p} x \,d x } \]
Time = 1.58 (sec) , antiderivative size = 440, normalized size of antiderivative = 3.73 \[ \int x (d+e x)^2 \left (d^2-e^2 x^2\right )^p \, dx=d^{2} \left (\begin {cases} \frac {x^{2} \left (d^{2}\right )^{p}}{2} & \text {for}\: e^{2} = 0 \\- \frac {\begin {cases} \frac {\left (d^{2} - e^{2} x^{2}\right )^{p + 1}}{p + 1} & \text {for}\: p \neq -1 \\\log {\left (d^{2} - e^{2} x^{2} \right )} & \text {otherwise} \end {cases}}{2 e^{2}} & \text {otherwise} \end {cases}\right ) + \frac {2 d d^{2 p} e x^{3} {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, - p \\ \frac {5}{2} \end {matrix}\middle | {\frac {e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{3} + e^{2} \left (\begin {cases} \frac {x^{4} \left (d^{2}\right )^{p}}{4} & \text {for}\: e = 0 \\- \frac {d^{2} \log {\left (- \frac {d}{e} + x \right )}}{- 2 d^{2} e^{4} + 2 e^{6} x^{2}} - \frac {d^{2} \log {\left (\frac {d}{e} + x \right )}}{- 2 d^{2} e^{4} + 2 e^{6} x^{2}} - \frac {d^{2}}{- 2 d^{2} e^{4} + 2 e^{6} x^{2}} + \frac {e^{2} x^{2} \log {\left (- \frac {d}{e} + x \right )}}{- 2 d^{2} e^{4} + 2 e^{6} x^{2}} + \frac {e^{2} x^{2} \log {\left (\frac {d}{e} + x \right )}}{- 2 d^{2} e^{4} + 2 e^{6} x^{2}} & \text {for}\: p = -2 \\- \frac {d^{2} \log {\left (- \frac {d}{e} + x \right )}}{2 e^{4}} - \frac {d^{2} \log {\left (\frac {d}{e} + x \right )}}{2 e^{4}} - \frac {x^{2}}{2 e^{2}} & \text {for}\: p = -1 \\- \frac {d^{4} \left (d^{2} - e^{2} x^{2}\right )^{p}}{2 e^{4} p^{2} + 6 e^{4} p + 4 e^{4}} - \frac {d^{2} e^{2} p x^{2} \left (d^{2} - e^{2} x^{2}\right )^{p}}{2 e^{4} p^{2} + 6 e^{4} p + 4 e^{4}} + \frac {e^{4} p x^{4} \left (d^{2} - e^{2} x^{2}\right )^{p}}{2 e^{4} p^{2} + 6 e^{4} p + 4 e^{4}} + \frac {e^{4} x^{4} \left (d^{2} - e^{2} x^{2}\right )^{p}}{2 e^{4} p^{2} + 6 e^{4} p + 4 e^{4}} & \text {otherwise} \end {cases}\right ) \]
d**2*Piecewise((x**2*(d**2)**p/2, Eq(e**2, 0)), (-Piecewise(((d**2 - e**2* x**2)**(p + 1)/(p + 1), Ne(p, -1)), (log(d**2 - e**2*x**2), True))/(2*e**2 ), True)) + 2*d*d**(2*p)*e*x**3*hyper((3/2, -p), (5/2,), e**2*x**2*exp_pol ar(2*I*pi)/d**2)/3 + e**2*Piecewise((x**4*(d**2)**p/4, Eq(e, 0)), (-d**2*l og(-d/e + x)/(-2*d**2*e**4 + 2*e**6*x**2) - d**2*log(d/e + x)/(-2*d**2*e** 4 + 2*e**6*x**2) - d**2/(-2*d**2*e**4 + 2*e**6*x**2) + e**2*x**2*log(-d/e + x)/(-2*d**2*e**4 + 2*e**6*x**2) + e**2*x**2*log(d/e + x)/(-2*d**2*e**4 + 2*e**6*x**2), Eq(p, -2)), (-d**2*log(-d/e + x)/(2*e**4) - d**2*log(d/e + x)/(2*e**4) - x**2/(2*e**2), Eq(p, -1)), (-d**4*(d**2 - e**2*x**2)**p/(2*e **4*p**2 + 6*e**4*p + 4*e**4) - d**2*e**2*p*x**2*(d**2 - e**2*x**2)**p/(2* e**4*p**2 + 6*e**4*p + 4*e**4) + e**4*p*x**4*(d**2 - e**2*x**2)**p/(2*e**4 *p**2 + 6*e**4*p + 4*e**4) + e**4*x**4*(d**2 - e**2*x**2)**p/(2*e**4*p**2 + 6*e**4*p + 4*e**4), True))
\[ \int x (d+e x)^2 \left (d^2-e^2 x^2\right )^p \, dx=\int { {\left (e x + d\right )}^{2} {\left (-e^{2} x^{2} + d^{2}\right )}^{p} x \,d x } \]
-1/2*(-e^2*x^2 + d^2)^(p + 1)*d^2/(e^2*(p + 1)) + integrate((e^2*x^3 + 2*d *e*x^2)*e^(p*log(e*x + d) + p*log(-e*x + d)), x)
\[ \int x (d+e x)^2 \left (d^2-e^2 x^2\right )^p \, dx=\int { {\left (e x + d\right )}^{2} {\left (-e^{2} x^{2} + d^{2}\right )}^{p} x \,d x } \]
Timed out. \[ \int x (d+e x)^2 \left (d^2-e^2 x^2\right )^p \, dx=\int x\,{\left (d^2-e^2\,x^2\right )}^p\,{\left (d+e\,x\right )}^2 \,d x \]