Integrand size = 25, antiderivative size = 285 \[ \int x^{-5-2 p} \left (d+e x^2\right )^p (a+b \arctan (c x)) \, dx=-\frac {b \left (e+c^2 d (1+p)\right ) x^{-3-2 p} \left (d+e x^2\right )^p \left (1+\frac {e x^2}{d}\right )^{-p} \operatorname {AppellF1}\left (\frac {1}{2} (-3-2 p),1,-1-p,\frac {1}{2} (-1-2 p),-c^2 x^2,-\frac {e x^2}{d}\right )}{2 c d (1+p) (2+p) (3+2 p)}+\frac {e x^{-2 (1+p)} \left (d+e x^2\right )^{1+p} (a+b \arctan (c x))}{2 d^2 (1+p) (2+p)}-\frac {x^{-2 (2+p)} \left (d+e x^2\right )^{1+p} (a+b \arctan (c x))}{2 d (2+p)}+\frac {b e x^{-3-2 p} \left (d+e x^2\right )^p \left (1+\frac {e x^2}{d}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (-3-2 p),-1-p,\frac {1}{2} (-1-2 p),-\frac {e x^2}{d}\right )}{2 c d \left (6+13 p+9 p^2+2 p^3\right )} \]
-1/2*b*(e+c^2*d*(p+1))*x^(-3-2*p)*(e*x^2+d)^p*AppellF1(-3/2-p,1,-1-p,-1/2- p,-c^2*x^2,-e*x^2/d)/c/d/(3+2*p)/(p^2+3*p+2)/((1+e*x^2/d)^p)+1/2*e*(e*x^2+ d)^(p+1)*(a+b*arctan(c*x))/d^2/(p+1)/(2+p)/(x^(2*p+2))-1/2*(e*x^2+d)^(p+1) *(a+b*arctan(c*x))/d/(2+p)/(x^(4+2*p))+1/2*b*e*x^(-3-2*p)*(e*x^2+d)^p*hype rgeom([-1-p, -3/2-p],[-1/2-p],-e*x^2/d)/c/d/(2*p^3+9*p^2+13*p+6)/((1+e*x^2 /d)^p)
\[ \int x^{-5-2 p} \left (d+e x^2\right )^p (a+b \arctan (c x)) \, dx=\int x^{-5-2 p} \left (d+e x^2\right )^p (a+b \arctan (c x)) \, dx \]
Time = 0.57 (sec) , antiderivative size = 270, normalized size of antiderivative = 0.95, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {5511, 27, 446, 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.
| \(\displaystyle \int x^{-2 p-5} \left (d+e x^2\right )^p (a+b \arctan (c x)) \, dx\) |
| \(\Big \downarrow \) 5511 |
| \(\displaystyle -b c \int -\frac {x^{-2 (p+2)} \left (d (p+1)-e x^2\right ) \left (e x^2+d\right )^{p+1}}{2 d^2 (p+1) (p+2) \left (c^2 x^2+1\right )}dx+\frac {e x^{-2 (p+1)} \left (d+e x^2\right )^{p+1} (a+b \arctan (c x))}{2 d^2 (p+1) (p+2)}-\frac {x^{-2 (p+2)} \left (d+e x^2\right )^{p+1} (a+b \arctan (c x))}{2 d (p+2)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b c \int \frac {x^{-2 (p+2)} \left (d (p+1)-e x^2\right ) \left (e x^2+d\right )^{p+1}}{c^2 x^2+1}dx}{2 d^2 (p+1) (p+2)}+\frac {e x^{-2 (p+1)} \left (d+e x^2\right )^{p+1} (a+b \arctan (c x))}{2 d^2 (p+1) (p+2)}-\frac {x^{-2 (p+2)} \left (d+e x^2\right )^{p+1} (a+b \arctan (c x))}{2 d (p+2)}\) |
| \(\Big \downarrow \) 446 |
| \(\displaystyle \frac {b c \int \left (\frac {\left (d (p+1) c^2+e\right ) x^{-2 (p+2)} \left (e x^2+d\right )^{p+1}}{c^2 \left (c^2 x^2+1\right )}-\frac {e x^{-2 (p+2)} \left (e x^2+d\right )^{p+1}}{c^2}\right )dx}{2 d^2 (p+1) (p+2)}+\frac {e x^{-2 (p+1)} \left (d+e x^2\right )^{p+1} (a+b \arctan (c x))}{2 d^2 (p+1) (p+2)}-\frac {x^{-2 (p+2)} \left (d+e x^2\right )^{p+1} (a+b \arctan (c x))}{2 d (p+2)}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {e x^{-2 (p+1)} \left (d+e x^2\right )^{p+1} (a+b \arctan (c x))}{2 d^2 (p+1) (p+2)}-\frac {x^{-2 (p+2)} \left (d+e x^2\right )^{p+1} (a+b \arctan (c x))}{2 d (p+2)}+\frac {b c \left (\frac {d e x^{-2 p-3} \left (d+e x^2\right )^p \left (\frac {e x^2}{d}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (-2 p-3),-p-1,\frac {1}{2} (-2 p-1),-\frac {e x^2}{d}\right )}{c^2 (2 p+3)}-\frac {d x^{-2 p-3} \left (c^2 d (p+1)+e\right ) \left (d+e x^2\right )^p \left (\frac {e x^2}{d}+1\right )^{-p} \operatorname {AppellF1}\left (-p-\frac {3}{2},-p-1,1,-p-\frac {1}{2},-\frac {e x^2}{d},-c^2 x^2\right )}{c^2 (2 p+3)}\right )}{2 d^2 (p+1) (p+2)}\) |
(e*(d + e*x^2)^(1 + p)*(a + b*ArcTan[c*x]))/(2*d^2*(1 + p)*(2 + p)*x^(2*(1 + p))) - ((d + e*x^2)^(1 + p)*(a + b*ArcTan[c*x]))/(2*d*(2 + p)*x^(2*(2 + p))) + (b*c*(-((d*(e + c^2*d*(1 + p))*x^(-3 - 2*p)*(d + e*x^2)^p*AppellF1 [-3/2 - p, -1 - p, 1, -1/2 - p, -((e*x^2)/d), -(c^2*x^2)])/(c^2*(3 + 2*p)* (1 + (e*x^2)/d)^p)) + (d*e*x^(-3 - 2*p)*(d + e*x^2)^p*Hypergeometric2F1[(- 3 - 2*p)/2, -1 - p, (-1 - 2*p)/2, -((e*x^2)/d)])/(c^2*(3 + 2*p)*(1 + (e*x^ 2)/d)^p)))/(2*d^2*(1 + p)*(2 + p))
3.13.43.3.1 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[(((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((e_) + (f_.)*(x_)^2))/( (c_) + (d_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(g*x)^m*(a + b*x^2)^ p*((e + f*x^2)/(c + d*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x ]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_.)*((d_.) + (e_.)*(x _)^2)^(q_.), x_Symbol] :> With[{u = IntHide[(f*x)^m*(d + e*x^2)^q, x]}, Sim p[(a + b*ArcTan[c*x]) u, x] - Simp[b*c Int[SimplifyIntegrand[u/(1 + c^2 *x^2), x], x], x]] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && ((IGtQ[q, 0] && !(ILtQ[(m - 1)/2, 0] && GtQ[m + 2*q + 3, 0])) || (IGtQ[(m + 1)/2, 0] && !(ILtQ[q, 0] && GtQ[m + 2*q + 3, 0])) || (ILtQ[(m + 2*q + 1)/2, 0] && !ILt Q[(m - 1)/2, 0]))
\[\int x^{-5-2 p} \left (e \,x^{2}+d \right )^{p} \left (a +b \arctan \left (c x \right )\right )d x\]
\[ \int x^{-5-2 p} \left (d+e x^2\right )^p (a+b \arctan (c x)) \, dx=\int { {\left (b \arctan \left (c x\right ) + a\right )} {\left (e x^{2} + d\right )}^{p} x^{-2 \, p - 5} \,d x } \]
Timed out. \[ \int x^{-5-2 p} \left (d+e x^2\right )^p (a+b \arctan (c x)) \, dx=\text {Timed out} \]
\[ \int x^{-5-2 p} \left (d+e x^2\right )^p (a+b \arctan (c x)) \, dx=\int { {\left (b \arctan \left (c x\right ) + a\right )} {\left (e x^{2} + d\right )}^{p} x^{-2 \, p - 5} \,d x } \]
b*integrate(arctan(c*x)*e^(p*log(e*x^2 + d) - 2*p*log(x))/x^5, x) + 1/2*(e ^2*x^4 - d*e*p*x^2 - d^2*(p + 1))*a*e^(p*log(e*x^2 + d) - 2*p*log(x))/((p^ 2 + 3*p + 2)*d^2*x^4)
\[ \int x^{-5-2 p} \left (d+e x^2\right )^p (a+b \arctan (c x)) \, dx=\int { {\left (b \arctan \left (c x\right ) + a\right )} {\left (e x^{2} + d\right )}^{p} x^{-2 \, p - 5} \,d x } \]
Timed out. \[ \int x^{-5-2 p} \left (d+e x^2\right )^p (a+b \arctan (c x)) \, dx=\int \frac {\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )\,{\left (e\,x^2+d\right )}^p}{x^{2\,p+5}} \,d x \]