Integrand size = 25, antiderivative size = 148 \[ \int \frac {x^4 \left (d^2-e^2 x^2\right )^p}{d+e x} \, dx=\frac {d^4 \left (d^2-e^2 x^2\right )^p}{2 e^5 p}-\frac {d^2 \left (d^2-e^2 x^2\right )^{1+p}}{e^5 (1+p)}+\frac {\left (d^2-e^2 x^2\right )^{2+p}}{2 e^5 (2+p)}+\frac {x^5 \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},1-p,\frac {7}{2},\frac {e^2 x^2}{d^2}\right )}{5 d} \]
1/2*d^4*(-e^2*x^2+d^2)^p/e^5/p-d^2*(-e^2*x^2+d^2)^(p+1)/e^5/(p+1)+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)
Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.
Time = 0.33 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.45 \[ \int \frac {x^4 \left (d^2-e^2 x^2\right )^p}{d+e x} \, dx=\frac {x^5 (d-e x)^p (d+e x)^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {AppellF1}\left (5,-p,1-p,6,\frac {e x}{d},-\frac {e x}{d}\right )}{5 d} \]
(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)
Time = 0.30 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.01, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {583, 542, 243, 53, 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.
\(\displaystyle \int \frac {x^4 \left (d^2-e^2 x^2\right )^p}{d+e x} \, dx\) |
\(\Big \downarrow \) 583 |
\(\displaystyle \int x^4 (d-e x) \left (d^2-e^2 x^2\right )^{p-1}dx\) |
\(\Big \downarrow \) 542 |
\(\displaystyle d \int x^4 \left (d^2-e^2 x^2\right )^{p-1}dx-e \int x^5 \left (d^2-e^2 x^2\right )^{p-1}dx\) |
\(\Big \downarrow \) 243 |
\(\displaystyle d \int x^4 \left (d^2-e^2 x^2\right )^{p-1}dx-\frac {1}{2} e \int x^4 \left (d^2-e^2 x^2\right )^{p-1}dx^2\) |
\(\Big \downarrow \) 53 |
\(\displaystyle d \int x^4 \left (d^2-e^2 x^2\right )^{p-1}dx-\frac {1}{2} e \int \left (\frac {d^4 \left (d^2-e^2 x^2\right )^{p-1}}{e^4}-\frac {2 d^2 \left (d^2-e^2 x^2\right )^p}{e^4}+\frac {\left (d^2-e^2 x^2\right )^{p+1}}{e^4}\right )dx^2\) |
\(\Big \downarrow \) 279 |
\(\displaystyle \frac {\left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \int x^4 \left (1-\frac {e^2 x^2}{d^2}\right )^{p-1}dx}{d}-\frac {1}{2} e \int \left (\frac {d^4 \left (d^2-e^2 x^2\right )^{p-1}}{e^4}-\frac {2 d^2 \left (d^2-e^2 x^2\right )^p}{e^4}+\frac {\left (d^2-e^2 x^2\right )^{p+1}}{e^4}\right )dx^2\) |
\(\Big \downarrow \) 278 |
\(\displaystyle \frac {x^5 \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},1-p,\frac {7}{2},\frac {e^2 x^2}{d^2}\right )}{5 d}-\frac {1}{2} e \int \left (\frac {d^4 \left (d^2-e^2 x^2\right )^{p-1}}{e^4}-\frac {2 d^2 \left (d^2-e^2 x^2\right )^p}{e^4}+\frac {\left (d^2-e^2 x^2\right )^{p+1}}{e^4}\right )dx^2\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {x^5 \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},1-p,\frac {7}{2},\frac {e^2 x^2}{d^2}\right )}{5 d}-\frac {1}{2} e \left (\frac {2 d^2 \left (d^2-e^2 x^2\right )^{p+1}}{e^6 (p+1)}-\frac {\left (d^2-e^2 x^2\right )^{p+2}}{e^6 (p+2)}-\frac {d^4 \left (d^2-e^2 x^2\right )^p}{e^6 p}\right )\) |
-1/2*(e*(-((d^4*(d^2 - e^2*x^2)^p)/(e^6*p)) + (2*d^2*(d^2 - e^2*x^2)^(1 + p))/(e^6*(1 + p)) - (d^2 - e^2*x^2)^(2 + p)/(e^6*(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)
3.3.67.3.1 Defintions of rubi rules used
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[(x_)^(m_.)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c Int[x^m*(a + b*x^2)^p, x], x] + Simp[d Int[x^(m + 1)*(a + b*x^2 )^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && IntegerQ[m] && !IntegerQ[2*p]
Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c^(2*n)/a^n Int[(e*x)^m*((a + b*x^2)^(n + p)/(c - d*x)^ n), x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[b*c^2 + a*d^2, 0] && I LtQ[n, 0]
\[\int \frac {x^{4} \left (-e^{2} x^{2}+d^{2}\right )^{p}}{e x +d}d x\]
\[ \int \frac {x^4 \left (d^2-e^2 x^2\right )^p}{d+e x} \, dx=\int { \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{p} x^{4}}{e x + d} \,d x } \]
Timed out. \[ \int \frac {x^4 \left (d^2-e^2 x^2\right )^p}{d+e x} \, dx=\text {Timed out} \]
\[ \int \frac {x^4 \left (d^2-e^2 x^2\right )^p}{d+e x} \, dx=\int { \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{p} x^{4}}{e x + d} \,d x } \]
\[ \int \frac {x^4 \left (d^2-e^2 x^2\right )^p}{d+e x} \, dx=\int { \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{p} x^{4}}{e x + d} \,d x } \]
Timed out. \[ \int \frac {x^4 \left (d^2-e^2 x^2\right )^p}{d+e x} \, dx=\int \frac {x^4\,{\left (d^2-e^2\,x^2\right )}^p}{d+e\,x} \,d x \]