Integrand size = 23, antiderivative size = 115 \[ \int \frac {x \left (d^2-e^2 x^2\right )^p}{(d+e x)^2} \, dx=\frac {\left (d^2-e^2 x^2\right )^{1+p}}{2 e^2 (1-p) (d+e x)^2}-\frac {2^{-1+p} \left (1+\frac {e x}{d}\right )^{-1-p} \left (d^2-e^2 x^2\right )^{1+p} \operatorname {Hypergeometric2F1}\left (1-p,1+p,2+p,\frac {d-e x}{2 d}\right )}{d^2 e^2 \left (1-p^2\right )} \]
1/2*(-e^2*x^2+d^2)^(p+1)/e^2/(1-p)/(e*x+d)^2-2^(-1+p)*(1+e*x/d)^(-1-p)*(-e ^2*x^2+d^2)^(p+1)*hypergeom([1-p, p+1],[2+p],1/2*(-e*x+d)/d)/d^2/e^2/(-p^2 +1)
Time = 0.21 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.89 \[ \int \frac {x \left (d^2-e^2 x^2\right )^p}{(d+e x)^2} \, dx=\frac {2^{-2+p} (d-e x) \left (1+\frac {e x}{d}\right )^{-p} \left (d^2-e^2 x^2\right )^p \left (-2 \operatorname {Hypergeometric2F1}\left (1-p,1+p,2+p,\frac {d-e x}{2 d}\right )+\operatorname {Hypergeometric2F1}\left (2-p,1+p,2+p,\frac {d-e x}{2 d}\right )\right )}{d e^2 (1+p)} \]
(2^(-2 + p)*(d - e*x)*(d^2 - e^2*x^2)^p*(-2*Hypergeometric2F1[1 - p, 1 + p , 2 + p, (d - e*x)/(2*d)] + Hypergeometric2F1[2 - p, 1 + p, 2 + p, (d - e* x)/(2*d)]))/(d*e^2*(1 + p)*(1 + (e*x)/d)^p)
Time = 0.23 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.03, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {571, 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.
\(\displaystyle \int \frac {x \left (d^2-e^2 x^2\right )^p}{(d+e x)^2} \, dx\) |
\(\Big \downarrow \) 571 |
\(\displaystyle \frac {\int \frac {\left (d^2-e^2 x^2\right )^p}{d+e x}dx}{e (1-p)}+\frac {\left (d^2-e^2 x^2\right )^{p+1}}{2 e^2 (1-p) (d+e x)^2}\) |
\(\Big \downarrow \) 473 |
\(\displaystyle \frac {(d-e x)^{-p-1} \left (d^2-e^2 x^2\right )^{p+1} \left (\frac {e x}{d}+1\right )^{-p-1} \int (d-e x)^p \left (\frac {e x}{d}+1\right )^{p-1}dx}{d^2 e (1-p)}+\frac {\left (d^2-e^2 x^2\right )^{p+1}}{2 e^2 (1-p) (d+e x)^2}\) |
\(\Big \downarrow \) 79 |
\(\displaystyle \frac {\left (d^2-e^2 x^2\right )^{p+1}}{2 e^2 (1-p) (d+e x)^2}-\frac {2^{p-1} \left (\frac {e x}{d}+1\right )^{-p-1} \left (d^2-e^2 x^2\right )^{p+1} \operatorname {Hypergeometric2F1}\left (1-p,p+1,p+2,\frac {d-e x}{2 d}\right )}{d^2 e^2 (1-p) (p+1)}\) |
(d^2 - e^2*x^2)^(1 + p)/(2*e^2*(1 - p)*(d + e*x)^2) - (2^(-1 + p)*(1 + (e* x)/d)^(-1 - p)*(d^2 - e^2*x^2)^(1 + p)*Hypergeometric2F1[1 - p, 1 + p, 2 + p, (d - e*x)/(2*d)])/(d^2*e^2*(1 - p)*(1 + p))
3.3.79.3.1 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)/(2*b*(n + p + 1))), x] + Simp[n/(2*d* (n + p + 1)) Int[(c + d*x)^(n + 1)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, n, p}, x] && EqQ[b*c^2 + a*d^2, 0] && ((LtQ[n, -1] && !IGtQ[n + p + 1, 0]) || (LtQ[n, 0] && LtQ[p, -1]) || EqQ[n + 2*p + 2, 0]) && NeQ[n + p + 1, 0]
\[\int \frac {x \left (-e^{2} x^{2}+d^{2}\right )^{p}}{\left (e x +d \right )^{2}}d x\]
\[ \int \frac {x \left (d^2-e^2 x^2\right )^p}{(d+e x)^2} \, dx=\int { \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{p} x}{{\left (e x + d\right )}^{2}} \,d x } \]
\[ \int \frac {x \left (d^2-e^2 x^2\right )^p}{(d+e x)^2} \, dx=\int \frac {x \left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{p}}{\left (d + e x\right )^{2}}\, dx \]
\[ \int \frac {x \left (d^2-e^2 x^2\right )^p}{(d+e x)^2} \, dx=\int { \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{p} x}{{\left (e x + d\right )}^{2}} \,d x } \]
\[ \int \frac {x \left (d^2-e^2 x^2\right )^p}{(d+e x)^2} \, dx=\int { \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{p} x}{{\left (e x + d\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {x \left (d^2-e^2 x^2\right )^p}{(d+e x)^2} \, dx=\int \frac {x\,{\left (d^2-e^2\,x^2\right )}^p}{{\left (d+e\,x\right )}^2} \,d x \]