\(\int x^{-7-2 p} (d+e x^2)^p (a+b \arctan (c x)) \, dx\) [1243]

Optimal result

Integrand size = 25, antiderivative size = 466 \[ \int x^{-7-2 p} \left (d+e x^2\right )^p (a+b \arctan (c x)) \, dx=-\frac {b \left (2 e^2+2 c^2 d e (1+p)+c^4 d^2 \left (2+3 p+p^2\right )\right ) x^{-5-2 p} \left (d+e x^2\right )^p \left (1+\frac {e x^2}{d}\right )^{-p} \operatorname {AppellF1}\left (\frac {1}{2} (-5-2 p),1,-1-p,\frac {1}{2} (-3-2 p),-c^2 x^2,-\frac {e x^2}{d}\right )}{2 c^3 d^2 (1+p) (2+p) (3+p) (5+2 p)}-\frac {e^2 x^{-2 (1+p)} \left (d+e x^2\right )^{1+p} (a+b \arctan (c x))}{d^3 (1+p) (2+p) (3+p)}+\frac {e x^{-2 (2+p)} \left (d+e x^2\right )^{1+p} (a+b \arctan (c x))}{d^2 (2+p) (3+p)}-\frac {x^{-2 (3+p)} \left (d+e x^2\right )^{1+p} (a+b \arctan (c x))}{2 d (3+p)}+\frac {b e \left (e+c^2 d (1+p)\right ) x^{-5-2 p} \left (d+e x^2\right )^p \left (1+\frac {e x^2}{d}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (-5-2 p),-1-p,\frac {1}{2} (-3-2 p),-\frac {e x^2}{d}\right )}{c^3 d^2 (1+p) (2+p) (3+p) (5+2 p)}-\frac {b e^2 x^{-3-2 p} \left (d+e x^2\right )^p \left (1+\frac {e x^2}{d}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (-3-2 p),-1-p,\frac {1}{2} (-1-2 p),-\frac {e x^2}{d}\right )}{c d^2 (1+p) (2+p) (3+p) (3+2 p)} \] Output:

-1/2*b*(2*e^2+2*c^2*d*e*(p+1)+c^4*d^2*(p^2+3*p+2))*x^(-5-2*p)*(e*x^2+d)^p* 
AppellF1(-5/2-p,1,-1-p,-3/2-p,-c^2*x^2,-e*x^2/d)/c^3/d^2/(p+1)/(2+p)/(3+p) 
/(5+2*p)/((1+e*x^2/d)^p)-e^2*(e*x^2+d)^(p+1)*(a+b*arctan(c*x))/d^3/(p+1)/( 
2+p)/(3+p)/(x^(2*p+2))+e*(e*x^2+d)^(p+1)*(a+b*arctan(c*x))/d^2/(2+p)/(3+p) 
/(x^(4+2*p))-1/2*(e*x^2+d)^(p+1)*(a+b*arctan(c*x))/d/(3+p)/(x^(6+2*p))+b*e 
*(e+c^2*d*(p+1))*x^(-5-2*p)*(e*x^2+d)^p*hypergeom([-1-p, -5/2-p],[-3/2-p], 
-e*x^2/d)/c^3/d^2/(p+1)/(2+p)/(3+p)/(5+2*p)/((1+e*x^2/d)^p)-b*e^2*x^(-3-2* 
p)*(e*x^2+d)^p*hypergeom([-1-p, -3/2-p],[-1/2-p],-e*x^2/d)/c/d^2/(p+1)/(2+ 
p)/(3+p)/(3+2*p)/((1+e*x^2/d)^p)
 

Mathematica [F]

\[ \int x^{-7-2 p} \left (d+e x^2\right )^p (a+b \arctan (c x)) \, dx=\int x^{-7-2 p} \left (d+e x^2\right )^p (a+b \arctan (c x)) \, dx \] Input:

Integrate[x^(-7 - 2*p)*(d + e*x^2)^p*(a + b*ArcTan[c*x]),x]
 

Output:

Integrate[x^(-7 - 2*p)*(d + e*x^2)^p*(a + b*ArcTan[c*x]), x]
 

Rubi [A] (warning: unable to verify)

Time = 1.71 (sec) , antiderivative size = 432, normalized size of antiderivative = 0.93, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {5511, 27, 7276, 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.

\(\displaystyle \int x^{-2 p-7} \left (d+e x^2\right )^p (a+b \arctan (c x)) \, dx\)

\(\Big \downarrow \) 5511

\(\displaystyle -b c \int -\frac {x^{-2 (p+3)} \left (e x^2+d\right )^{p+1} \left (2 e^2 x^4-2 d e (p+1) x^2+d^2 (p+1) (p+2)\right )}{2 d^3 (p+1) (p+2) (p+3) \left (c^2 x^2+1\right )}dx-\frac {e^2 x^{-2 (p+1)} \left (d+e x^2\right )^{p+1} (a+b \arctan (c x))}{d^3 (p+1) (p+2) (p+3)}+\frac {e x^{-2 (p+2)} \left (d+e x^2\right )^{p+1} (a+b \arctan (c x))}{d^2 (p+2) (p+3)}-\frac {x^{-2 (p+3)} \left (d+e x^2\right )^{p+1} (a+b \arctan (c x))}{2 d (p+3)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {b c \int \frac {x^{-2 (p+3)} \left (e x^2+d\right )^{p+1} \left (2 e^2 x^4-2 d e (p+1) x^2+d^2 (p+1) (p+2)\right )}{c^2 x^2+1}dx}{2 d^3 (p+1) (p+2) (p+3)}-\frac {e^2 x^{-2 (p+1)} \left (d+e x^2\right )^{p+1} (a+b \arctan (c x))}{d^3 (p+1) (p+2) (p+3)}+\frac {e x^{-2 (p+2)} \left (d+e x^2\right )^{p+1} (a+b \arctan (c x))}{d^2 (p+2) (p+3)}-\frac {x^{-2 (p+3)} \left (d+e x^2\right )^{p+1} (a+b \arctan (c x))}{2 d (p+3)}\)

\(\Big \downarrow \) 7276

\(\displaystyle \frac {b c \int \left (-\frac {2 e \left (d (p+1) c^2+e\right ) x^{-2 (p+3)} \left (e x^2+d\right )^{p+1}}{c^4}+\frac {2 e^2 x^{2-2 (p+3)} \left (e x^2+d\right )^{p+1}}{c^2}+\frac {\left (2 d^2 c^4+d^2 p^2 c^4+3 d^2 p c^4+2 d e c^2+2 d e p c^2+2 e^2\right ) x^{-2 (p+3)} \left (e x^2+d\right )^{p+1}}{c^4 \left (c^2 x^2+1\right )}\right )dx}{2 d^3 (p+1) (p+2) (p+3)}-\frac {e^2 x^{-2 (p+1)} \left (d+e x^2\right )^{p+1} (a+b \arctan (c x))}{d^3 (p+1) (p+2) (p+3)}+\frac {e x^{-2 (p+2)} \left (d+e x^2\right )^{p+1} (a+b \arctan (c x))}{d^2 (p+2) (p+3)}-\frac {x^{-2 (p+3)} \left (d+e x^2\right )^{p+1} (a+b \arctan (c x))}{2 d (p+3)}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {e^2 x^{-2 (p+1)} \left (d+e x^2\right )^{p+1} (a+b \arctan (c x))}{d^3 (p+1) (p+2) (p+3)}+\frac {e x^{-2 (p+2)} \left (d+e x^2\right )^{p+1} (a+b \arctan (c x))}{d^2 (p+2) (p+3)}-\frac {x^{-2 (p+3)} \left (d+e x^2\right )^{p+1} (a+b \arctan (c x))}{2 d (p+3)}+\frac {b c \left (-\frac {d x^{-2 p-5} \left (c^4 d^2 \left (p^2+3 p+2\right )+2 c^2 d e (p+1)+2 e^2\right ) \left (d+e x^2\right )^p \left (\frac {e x^2}{d}+1\right )^{-p} \operatorname {AppellF1}\left (-p-\frac {5}{2},-p-1,1,-p-\frac {3}{2},-\frac {e x^2}{d},-c^2 x^2\right )}{c^4 (2 p+5)}-\frac {2 d e^2 x^{-2 p-3} \left (d+e x^2\right )^p \left (\frac {e x^2}{d}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (-2 p-3),-p-1,\frac {1}{2} (-2 p-1),-\frac {e x^2}{d}\right )}{c^2 (2 p+3)}+\frac {2 d e x^{-2 p-5} \left (c^2 d (p+1)+e\right ) \left (d+e x^2\right )^p \left (\frac {e x^2}{d}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (-2 p-5),-p-1,\frac {1}{2} (-2 p-3),-\frac {e x^2}{d}\right )}{c^4 (2 p+5)}\right )}{2 d^3 (p+1) (p+2) (p+3)}\)

Input:

Int[x^(-7 - 2*p)*(d + e*x^2)^p*(a + b*ArcTan[c*x]),x]
 

Output:

-((e^2*(d + e*x^2)^(1 + p)*(a + b*ArcTan[c*x]))/(d^3*(1 + p)*(2 + p)*(3 + 
p)*x^(2*(1 + p)))) + (e*(d + e*x^2)^(1 + p)*(a + b*ArcTan[c*x]))/(d^2*(2 + 
 p)*(3 + p)*x^(2*(2 + p))) - ((d + e*x^2)^(1 + p)*(a + b*ArcTan[c*x]))/(2* 
d*(3 + p)*x^(2*(3 + p))) + (b*c*(-((d*(2*e^2 + 2*c^2*d*e*(1 + p) + c^4*d^2 
*(2 + 3*p + p^2))*x^(-5 - 2*p)*(d + e*x^2)^p*AppellF1[-5/2 - p, -1 - p, 1, 
 -3/2 - p, -((e*x^2)/d), -(c^2*x^2)])/(c^4*(5 + 2*p)*(1 + (e*x^2)/d)^p)) + 
 (2*d*e*(e + c^2*d*(1 + p))*x^(-5 - 2*p)*(d + e*x^2)^p*Hypergeometric2F1[( 
-5 - 2*p)/2, -1 - p, (-3 - 2*p)/2, -((e*x^2)/d)])/(c^4*(5 + 2*p)*(1 + (e*x 
^2)/d)^p) - (2*d*e^2*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^3*(1 + p)*(2 + p)*(3 + p))
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 5511

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]))
 

rule 7276

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE 
xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ 
[n, 0]
 

Maple [F]

\[\int x^{-7-2 p} \left (e \,x^{2}+d \right )^{p} \left (a +b \arctan \left (c x \right )\right )d x\]

Input:

int(x^(-7-2*p)*(e*x^2+d)^p*(a+b*arctan(c*x)),x)
 

Output:

int(x^(-7-2*p)*(e*x^2+d)^p*(a+b*arctan(c*x)),x)
 

Fricas [F]

\[ \int x^{-7-2 p} \left (d+e x^2\right )^p (a+b \arctan (c x)) \, dx=\int { {\left (b \arctan \left (c x\right ) + a\right )} {\left (e x^{2} + d\right )}^{p} x^{-2 \, p - 7} \,d x } \] Input:

integrate(x^(-7-2*p)*(e*x^2+d)^p*(a+b*arctan(c*x)),x, algorithm="fricas")
 

Output:

integral((b*arctan(c*x) + a)*(e*x^2 + d)^p*x^(-2*p - 7), x)
 

Sympy [F(-1)]

Timed out. \[ \int x^{-7-2 p} \left (d+e x^2\right )^p (a+b \arctan (c x)) \, dx=\text {Timed out} \] Input:

integrate(x**(-7-2*p)*(e*x**2+d)**p*(a+b*atan(c*x)),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int x^{-7-2 p} \left (d+e x^2\right )^p (a+b \arctan (c x)) \, dx=\int { {\left (b \arctan \left (c x\right ) + a\right )} {\left (e x^{2} + d\right )}^{p} x^{-2 \, p - 7} \,d x } \] Input:

integrate(x^(-7-2*p)*(e*x^2+d)^p*(a+b*arctan(c*x)),x, algorithm="maxima")
 

Output:

b*integrate(arctan(c*x)*e^(p*log(e*x^2 + d) - 2*p*log(x))/x^7, x) - 1/2*(2 
*e^3*x^6 - 2*d*e^2*p*x^4 + (p^2 + p)*d^2*e*x^2 + (p^2 + 3*p + 2)*d^3)*a*e^ 
(p*log(e*x^2 + d) - 2*p*log(x))/((p^3 + 6*p^2 + 11*p + 6)*d^3*x^6)
 

Giac [F]

\[ \int x^{-7-2 p} \left (d+e x^2\right )^p (a+b \arctan (c x)) \, dx=\int { {\left (b \arctan \left (c x\right ) + a\right )} {\left (e x^{2} + d\right )}^{p} x^{-2 \, p - 7} \,d x } \] Input:

integrate(x^(-7-2*p)*(e*x^2+d)^p*(a+b*arctan(c*x)),x, algorithm="giac")
 

Output:

integrate((b*arctan(c*x) + a)*(e*x^2 + d)^p*x^(-2*p - 7), x)
 

Mupad [F(-1)]

Timed out. \[ \int x^{-7-2 p} \left (d+e x^2\right )^p (a+b \arctan (c x)) \, dx=\int \frac {\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )\,{\left (e\,x^2+d\right )}^p}{x^{2\,p+7}} \,d x \] Input:

int(((a + b*atan(c*x))*(d + e*x^2)^p)/x^(2*p + 7),x)
 

Output:

int(((a + b*atan(c*x))*(d + e*x^2)^p)/x^(2*p + 7), x)
 

Reduce [F]

\[ \int x^{-7-2 p} \left (d+e x^2\right )^p (a+b \arctan (c x)) \, dx=\text {too large to display} \] Input:

int(x^(-7-2*p)*(e*x^2+d)^p*(a+b*atan(c*x)),x)
 

Output:

( - 6*(d + e*x**2)**p*atan(c*x)*b*c**3*d**3*p**3 - 33*(d + e*x**2)**p*atan 
(c*x)*b*c**3*d**3*p**2 - 57*(d + e*x**2)**p*atan(c*x)*b*c**3*d**3*p - 30*( 
d + e*x**2)**p*atan(c*x)*b*c**3*d**3 - 6*(d + e*x**2)**p*atan(c*x)*b*c**3* 
d**2*e*p**3*x**2 - 21*(d + e*x**2)**p*atan(c*x)*b*c**3*d**2*e*p**2*x**2 - 
15*(d + e*x**2)**p*atan(c*x)*b*c**3*d**2*e*p*x**2 + 12*(d + e*x**2)**p*ata 
n(c*x)*b*c**3*d*e**2*p**2*x**4 + 30*(d + e*x**2)**p*atan(c*x)*b*c**3*d*e** 
2*p*x**4 - 12*(d + e*x**2)**p*atan(c*x)*b*c**3*e**3*p*x**6 - 30*(d + e*x** 
2)**p*atan(c*x)*b*c**3*e**3*x**6 - 6*(d + e*x**2)**p*a*c**3*d**3*p**3 - 33 
*(d + e*x**2)**p*a*c**3*d**3*p**2 - 57*(d + e*x**2)**p*a*c**3*d**3*p - 30* 
(d + e*x**2)**p*a*c**3*d**3 - 6*(d + e*x**2)**p*a*c**3*d**2*e*p**3*x**2 - 
21*(d + e*x**2)**p*a*c**3*d**2*e*p**2*x**2 - 15*(d + e*x**2)**p*a*c**3*d** 
2*e*p*x**2 + 12*(d + e*x**2)**p*a*c**3*d*e**2*p**2*x**4 + 30*(d + e*x**2)* 
*p*a*c**3*d*e**2*p*x**4 - 12*(d + e*x**2)**p*a*c**3*e**3*p*x**6 - 30*(d + 
e*x**2)**p*a*c**3*e**3*x**6 - 3*(d + e*x**2)**p*b*c**4*d**3*p**2*x - 9*(d 
+ e*x**2)**p*b*c**4*d**3*p*x - 6*(d + e*x**2)**p*b*c**4*d**3*x + 12*(d + e 
*x**2)**p*b*c**2*d*e**2*p**2*x**3 + 30*(d + e*x**2)**p*b*c**2*d*e**2*p*x** 
3 - 12*(d + e*x**2)**p*b*c**2*e**3*p*x**5 - 30*(d + e*x**2)**p*b*c**2*e**3 
*x**5 + 4*(d + e*x**2)**p*b*e**3*p*x**3 + 10*(d + e*x**2)**p*b*e**3*x**3 - 
 24*x**(2*p)*int((d + e*x**2)**p/(4*x**(2*p)*c**4*d**2*p**5*x**6 + 44*x**( 
2*p)*c**4*d**2*p**4*x**6 + 189*x**(2*p)*c**4*d**2*p**3*x**6 + 394*x**(2...