Integrand size = 25, antiderivative size = 185 \[ \int x^4 (d+e x)^2 \left (d^2-e^2 x^2\right )^p \, dx=-\frac {d^5 \left (d^2-e^2 x^2\right )^{1+p}}{e^5 (1+p)}-\frac {x^5 \left (d^2-e^2 x^2\right )^{1+p}}{7+2 p}+\frac {2 d^3 \left (d^2-e^2 x^2\right )^{2+p}}{e^5 (2+p)}-\frac {d \left (d^2-e^2 x^2\right )^{3+p}}{e^5 (3+p)}+\frac {2 d^2 (6+p) x^5 \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},-p,\frac {7}{2},\frac {e^2 x^2}{d^2}\right )}{5 (7+2 p)} \]
-d^5*(-e^2*x^2+d^2)^(p+1)/e^5/(p+1)-x^5*(-e^2*x^2+d^2)^(p+1)/(7+2*p)+2*d^3 *(-e^2*x^2+d^2)^(2+p)/e^5/(2+p)-d*(-e^2*x^2+d^2)^(3+p)/e^5/(3+p)+2/5*d^2*( 6+p)*x^5*(-e^2*x^2+d^2)^p*hypergeom([5/2, -p],[7/2],e^2*x^2/d^2)/(7+2*p)/( (1-e^2*x^2/d^2)^p)
Time = 0.29 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.01 \[ \int x^4 (d+e x)^2 \left (d^2-e^2 x^2\right )^p \, dx=\frac {1}{35} \left (d^2-e^2 x^2\right )^p \left (-\frac {35 d^5 \left (d^2-e^2 x^2\right )}{e^5 (1+p)}+\frac {70 d^3 \left (d^2-e^2 x^2\right )^2}{e^5 (2+p)}-\frac {35 d \left (d^2-e^2 x^2\right )^3}{e^5 (3+p)}+7 d^2 x^5 \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},-p,\frac {7}{2},\frac {e^2 x^2}{d^2}\right )+5 e^2 x^7 \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {7}{2},-p,\frac {9}{2},\frac {e^2 x^2}{d^2}\right )\right ) \]
((d^2 - e^2*x^2)^p*((-35*d^5*(d^2 - e^2*x^2))/(e^5*(1 + p)) + (70*d^3*(d^2 - e^2*x^2)^2)/(e^5*(2 + p)) - (35*d*(d^2 - e^2*x^2)^3)/(e^5*(3 + p)) + (7 *d^2*x^5*Hypergeometric2F1[5/2, -p, 7/2, (e^2*x^2)/d^2])/(1 - (e^2*x^2)/d^ 2)^p + (5*e^2*x^7*Hypergeometric2F1[7/2, -p, 9/2, (e^2*x^2)/d^2])/(1 - (e^ 2*x^2)/d^2)^p))/35
Time = 0.33 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.02, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {543, 27, 243, 53, 363, 279, 278, 2009}
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.
| \(\displaystyle \int x^4 (d+e x)^2 \left (d^2-e^2 x^2\right )^p \, dx\) |
| \(\Big \downarrow \) 543 |
| \(\displaystyle \int 2 d e x^5 \left (d^2-e^2 x^2\right )^pdx+\int x^4 \left (d^2-e^2 x^2\right )^p \left (d^2+e^2 x^2\right )dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 d e \int x^5 \left (d^2-e^2 x^2\right )^pdx+\int x^4 \left (d^2-e^2 x^2\right )^p \left (d^2+e^2 x^2\right )dx\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle d e \int x^4 \left (d^2-e^2 x^2\right )^pdx^2+\int x^4 \left (d^2-e^2 x^2\right )^p \left (d^2+e^2 x^2\right )dx\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle \int x^4 \left (d^2-e^2 x^2\right )^p \left (d^2+e^2 x^2\right )dx+d e \int \left (\frac {d^4 \left (d^2-e^2 x^2\right )^p}{e^4}-\frac {2 d^2 \left (d^2-e^2 x^2\right )^{p+1}}{e^4}+\frac {\left (d^2-e^2 x^2\right )^{p+2}}{e^4}\right )dx^2\) |
| \(\Big \downarrow \) 363 |
| \(\displaystyle \frac {2 d^2 (p+6) \int x^4 \left (d^2-e^2 x^2\right )^pdx}{2 p+7}+d e \int \left (\frac {d^4 \left (d^2-e^2 x^2\right )^p}{e^4}-\frac {2 d^2 \left (d^2-e^2 x^2\right )^{p+1}}{e^4}+\frac {\left (d^2-e^2 x^2\right )^{p+2}}{e^4}\right )dx^2-\frac {x^5 \left (d^2-e^2 x^2\right )^{p+1}}{2 p+7}\) |
| \(\Big \downarrow \) 279 |
| \(\displaystyle \frac {2 d^2 (p+6) \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \int x^4 \left (1-\frac {e^2 x^2}{d^2}\right )^pdx}{2 p+7}+d e \int \left (\frac {d^4 \left (d^2-e^2 x^2\right )^p}{e^4}-\frac {2 d^2 \left (d^2-e^2 x^2\right )^{p+1}}{e^4}+\frac {\left (d^2-e^2 x^2\right )^{p+2}}{e^4}\right )dx^2-\frac {x^5 \left (d^2-e^2 x^2\right )^{p+1}}{2 p+7}\) |
| \(\Big \downarrow \) 278 |
| \(\displaystyle d e \int \left (\frac {d^4 \left (d^2-e^2 x^2\right )^p}{e^4}-\frac {2 d^2 \left (d^2-e^2 x^2\right )^{p+1}}{e^4}+\frac {\left (d^2-e^2 x^2\right )^{p+2}}{e^4}\right )dx^2+\frac {2 d^2 (p+6) x^5 \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},-p,\frac {7}{2},\frac {e^2 x^2}{d^2}\right )}{5 (2 p+7)}-\frac {x^5 \left (d^2-e^2 x^2\right )^{p+1}}{2 p+7}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 d^2 (p+6) x^5 \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},-p,\frac {7}{2},\frac {e^2 x^2}{d^2}\right )}{5 (2 p+7)}-\frac {x^5 \left (d^2-e^2 x^2\right )^{p+1}}{2 p+7}+d e \left (\frac {2 d^2 \left (d^2-e^2 x^2\right )^{p+2}}{e^6 (p+2)}-\frac {\left (d^2-e^2 x^2\right )^{p+3}}{e^6 (p+3)}-\frac {d^4 \left (d^2-e^2 x^2\right )^{p+1}}{e^6 (p+1)}\right )\) |
-((x^5*(d^2 - e^2*x^2)^(1 + p))/(7 + 2*p)) + d*e*(-((d^4*(d^2 - e^2*x^2)^( 1 + p))/(e^6*(1 + p))) + (2*d^2*(d^2 - e^2*x^2)^(2 + p))/(e^6*(2 + p)) - ( d^2 - e^2*x^2)^(3 + p)/(e^6*(3 + p))) + (2*d^2*(6 + p)*x^5*(d^2 - e^2*x^2) ^p*Hypergeometric2F1[5/2, -p, 7/2, (e^2*x^2)/d^2])/(5*(7 + 2*p)*(1 - (e^2* x^2)/d^2)^p)
3.3.50.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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] && 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])
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^p*(( c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/2, (m + 1)/2 + 1, ( -b)*(x^2/a)], x] /; FreeQ[{a, b, c, m, p}, x] && !IGtQ[p, 0] && (ILtQ[p, 0 ] || GtQ[a, 0])
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^IntP art[p]*((a + b*x^2)^FracPart[p]/(1 + b*(x^2/a))^FracPart[p]) Int[(c*x)^m* (1 + b*(x^2/a))^p, x], x] /; FreeQ[{a, b, c, m, p}, x] && !IGtQ[p, 0] && !(ILtQ[p, 0] || GtQ[a, 0])
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x _Symbol] :> Simp[d*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(b*e*(m + 2*p + 3))), x] - Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(b*(m + 2*p + 3)) Int[(e*x)^ m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b*c - a*d , 0] && NeQ[m + 2*p + 3, 0]
Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo l] :> Module[{k}, Int[x^m*Sum[Binomial[n, 2*k]*c^(n - 2*k)*d^(2*k)*x^(2*k), {k, 0, n/2}]*(a + b*x^2)^p, x] + Int[x^(m + 1)*Sum[Binomial[n, 2*k + 1]*c^ (n - 2*k - 1)*d^(2*k + 1)*x^(2*k), {k, 0, (n - 1)/2}]*(a + b*x^2)^p, x]] /; FreeQ[{a, b, c, d, p}, x] && IGtQ[n, 1] && IntegerQ[m] && !IntegerQ[2*p] && !(EqQ[m, 1] && EqQ[b*c^2 + a*d^2, 0])
\[\int x^{4} \left (e x +d \right )^{2} \left (-e^{2} x^{2}+d^{2}\right )^{p}d x\]
\[ \int x^4 (d+e x)^2 \left (d^2-e^2 x^2\right )^p \, dx=\int { {\left (e x + d\right )}^{2} {\left (-e^{2} x^{2} + d^{2}\right )}^{p} x^{4} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 940 vs. \(2 (153) = 306\).
Time = 2.39 (sec) , antiderivative size = 1015, normalized size of antiderivative = 5.49 \[ \int x^4 (d+e x)^2 \left (d^2-e^2 x^2\right )^p \, dx=\frac {d^{2} 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} + 2 d 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 ) + \frac {d^{2 p} e^{2} x^{7} {{}_{2}F_{1}\left (\begin {matrix} \frac {7}{2}, - p \\ \frac {9}{2} \end {matrix}\middle | {\frac {e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{7} \]
d**2*d**(2*p)*x**5*hyper((5/2, -p), (7/2,), e**2*x**2*exp_polar(2*I*pi)/d* *2)/5 + 2*d*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*...
\[ \int x^4 (d+e x)^2 \left (d^2-e^2 x^2\right )^p \, dx=\int { {\left (e x + d\right )}^{2} {\left (-e^{2} x^{2} + d^{2}\right )}^{p} x^{4} \,d x } \]
\[ \int x^4 (d+e x)^2 \left (d^2-e^2 x^2\right )^p \, dx=\int { {\left (e x + d\right )}^{2} {\left (-e^{2} x^{2} + d^{2}\right )}^{p} x^{4} \,d x } \]
Timed out. \[ \int x^4 (d+e x)^2 \left (d^2-e^2 x^2\right )^p \, dx=\int x^4\,{\left (d^2-e^2\,x^2\right )}^p\,{\left (d+e\,x\right )}^2 \,d x \]