Integrand size = 20, antiderivative size = 270 \[ \int x^m \left (c+a^2 c x^2\right )^3 \arctan (a x) \, dx=\frac {c^3 x^{1+m} \arctan (a x)}{1+m}+\frac {3 a^2 c^3 x^{3+m} \arctan (a x)}{3+m}+\frac {3 a^4 c^3 x^{5+m} \arctan (a x)}{5+m}+\frac {a^6 c^3 x^{7+m} \arctan (a x)}{7+m}-\frac {a c^3 x^{2+m} \operatorname {Hypergeometric2F1}\left (1,\frac {2+m}{2},\frac {4+m}{2},-a^2 x^2\right )}{2+3 m+m^2}-\frac {3 a^3 c^3 x^{4+m} \operatorname {Hypergeometric2F1}\left (1,\frac {4+m}{2},\frac {6+m}{2},-a^2 x^2\right )}{12+7 m+m^2}-\frac {3 a^5 c^3 x^{6+m} \operatorname {Hypergeometric2F1}\left (1,\frac {6+m}{2},\frac {8+m}{2},-a^2 x^2\right )}{(5+m) (6+m)}-\frac {a^7 c^3 x^{8+m} \operatorname {Hypergeometric2F1}\left (1,\frac {8+m}{2},\frac {10+m}{2},-a^2 x^2\right )}{(7+m) (8+m)} \] Output:
c^3*x^(1+m)*arctan(a*x)/(1+m)+3*a^2*c^3*x^(3+m)*arctan(a*x)/(3+m)+3*a^4*c^ 3*x^(5+m)*arctan(a*x)/(5+m)+a^6*c^3*x^(7+m)*arctan(a*x)/(7+m)-a*c^3*x^(2+m )*hypergeom([1, 1+1/2*m],[2+1/2*m],-a^2*x^2)/(m^2+3*m+2)-3*a^3*c^3*x^(4+m) *hypergeom([1, 2+1/2*m],[3+1/2*m],-a^2*x^2)/(m^2+7*m+12)-3*a^5*c^3*x^(6+m) *hypergeom([1, 3+1/2*m],[4+1/2*m],-a^2*x^2)/(5+m)/(6+m)-a^7*c^3*x^(8+m)*hy pergeom([1, 4+1/2*m],[5+1/2*m],-a^2*x^2)/(7+m)/(8+m)
Time = 0.21 (sec) , antiderivative size = 234, normalized size of antiderivative = 0.87 \[ \int x^m \left (c+a^2 c x^2\right )^3 \arctan (a x) \, dx=c^3 x^{1+m} \left (\frac {\arctan (a x)}{1+m}+\frac {3 a^2 x^2 \arctan (a x)}{3+m}+\frac {3 a^4 x^4 \arctan (a x)}{5+m}+\frac {a^6 x^6 \arctan (a x)}{7+m}-\frac {a^7 x^7 \operatorname {Hypergeometric2F1}\left (1,4+\frac {m}{2},5+\frac {m}{2},-a^2 x^2\right )}{(7+m) (8+m)}-\frac {a x \operatorname {Hypergeometric2F1}\left (1,\frac {2+m}{2},\frac {4+m}{2},-a^2 x^2\right )}{2+3 m+m^2}-\frac {3 a^3 x^3 \operatorname {Hypergeometric2F1}\left (1,\frac {4+m}{2},\frac {6+m}{2},-a^2 x^2\right )}{12+7 m+m^2}-\frac {3 a^5 x^5 \operatorname {Hypergeometric2F1}\left (1,\frac {6+m}{2},\frac {8+m}{2},-a^2 x^2\right )}{(5+m) (6+m)}\right ) \] Input:
Integrate[x^m*(c + a^2*c*x^2)^3*ArcTan[a*x],x]
Output:
c^3*x^(1 + m)*(ArcTan[a*x]/(1 + m) + (3*a^2*x^2*ArcTan[a*x])/(3 + m) + (3* a^4*x^4*ArcTan[a*x])/(5 + m) + (a^6*x^6*ArcTan[a*x])/(7 + m) - (a^7*x^7*Hy pergeometric2F1[1, 4 + m/2, 5 + m/2, -(a^2*x^2)])/((7 + m)*(8 + m)) - (a*x *Hypergeometric2F1[1, (2 + m)/2, (4 + m)/2, -(a^2*x^2)])/(2 + 3*m + m^2) - (3*a^3*x^3*Hypergeometric2F1[1, (4 + m)/2, (6 + m)/2, -(a^2*x^2)])/(12 + 7*m + m^2) - (3*a^5*x^5*Hypergeometric2F1[1, (6 + m)/2, (8 + m)/2, -(a^2*x ^2)])/((5 + m)*(6 + m)))
Time = 0.50 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {5483, 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.
| \(\displaystyle \int x^m \arctan (a x) \left (a^2 c x^2+c\right )^3 \, dx\) |
| \(\Big \downarrow \) 5483 |
| \(\displaystyle \int \left (a^6 c^3 x^{m+6} \arctan (a x)+3 a^4 c^3 x^{m+4} \arctan (a x)+3 a^2 c^3 x^{m+2} \arctan (a x)+c^3 x^m \arctan (a x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a^6 c^3 x^{m+7} \arctan (a x)}{m+7}+\frac {3 a^4 c^3 x^{m+5} \arctan (a x)}{m+5}+\frac {3 a^2 c^3 x^{m+3} \arctan (a x)}{m+3}-\frac {a c^3 x^{m+2} \operatorname {Hypergeometric2F1}\left (1,\frac {m+2}{2},\frac {m+4}{2},-a^2 x^2\right )}{m^2+3 m+2}-\frac {a^7 c^3 x^{m+8} \operatorname {Hypergeometric2F1}\left (1,\frac {m+8}{2},\frac {m+10}{2},-a^2 x^2\right )}{(m+7) (m+8)}-\frac {3 a^5 c^3 x^{m+6} \operatorname {Hypergeometric2F1}\left (1,\frac {m+6}{2},\frac {m+8}{2},-a^2 x^2\right )}{(m+5) (m+6)}-\frac {3 a^3 c^3 x^{m+4} \operatorname {Hypergeometric2F1}\left (1,\frac {m+4}{2},\frac {m+6}{2},-a^2 x^2\right )}{m^2+7 m+12}+\frac {c^3 x^{m+1} \arctan (a x)}{m+1}\) |
Input:
Int[x^m*(c + a^2*c*x^2)^3*ArcTan[a*x],x]
Output:
(c^3*x^(1 + m)*ArcTan[a*x])/(1 + m) + (3*a^2*c^3*x^(3 + m)*ArcTan[a*x])/(3 + m) + (3*a^4*c^3*x^(5 + m)*ArcTan[a*x])/(5 + m) + (a^6*c^3*x^(7 + m)*Arc Tan[a*x])/(7 + m) - (a*c^3*x^(2 + m)*Hypergeometric2F1[1, (2 + m)/2, (4 + m)/2, -(a^2*x^2)])/(2 + 3*m + m^2) - (3*a^3*c^3*x^(4 + m)*Hypergeometric2F 1[1, (4 + m)/2, (6 + m)/2, -(a^2*x^2)])/(12 + 7*m + m^2) - (3*a^5*c^3*x^(6 + m)*Hypergeometric2F1[1, (6 + m)/2, (8 + m)/2, -(a^2*x^2)])/((5 + m)*(6 + m)) - (a^7*c^3*x^(8 + m)*Hypergeometric2F1[1, (8 + m)/2, (10 + m)/2, -(a ^2*x^2)])/((7 + m)*(8 + m))
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_)*((d_) + (e_. )*(x_)^2)^(q_), x_Symbol] :> Int[ExpandIntegrand[(f*x)^m*(d + e*x^2)^q*(a + b*ArcTan[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2* d] && IGtQ[p, 0] && IGtQ[q, 1] && (EqQ[p, 1] || IntegerQ[m])
Result contains higher order function than in optimal. Order 9 vs. order 5.
Time = 120.87 (sec) , antiderivative size = 600, normalized size of antiderivative = 2.22
| method | result | size |
| meijerg | \(\frac {a^{-m -1} c^{3} \left (-\frac {4 x^{m} a^{m} \left (a^{6} m^{3} x^{6}+6 a^{6} m^{2} x^{6}+8 a^{6} m \,x^{6}-a^{4} m^{3} x^{4}-8 a^{4} m^{2} x^{4}-12 a^{4} m \,x^{4}+a^{2} m^{3} x^{2}+10 a^{2} m^{2} x^{2}+24 a^{2} m \,x^{2}-m^{3}-12 m^{2}-44 m -48\right )}{\left (7+m \right ) m \left (2+m \right ) \left (4+m \right ) \left (6+m \right )}+\frac {8 x^{8+m} a^{8+m} \arctan \left (\sqrt {a^{2} x^{2}}\right )}{\left (14+2 m \right ) \sqrt {a^{2} x^{2}}}+\frac {2 x^{m} a^{m} \left (-8-m \right ) \operatorname {LerchPhi}\left (-a^{2} x^{2}, 1, \frac {m}{2}\right )}{\left (8+m \right ) \left (7+m \right )}\right )}{4}+\frac {3 a^{-m -1} c^{3} \left (-\frac {4 x^{m} a^{m} \left (a^{4} m^{2} x^{4}+2 a^{4} m \,x^{4}-a^{2} m^{2} x^{2}-4 a^{2} m \,x^{2}+m^{2}+6 m +8\right )}{\left (5+m \right ) m \left (2+m \right ) \left (4+m \right )}+\frac {8 x^{6+m} a^{6+m} \arctan \left (\sqrt {a^{2} x^{2}}\right )}{\left (10+2 m \right ) \sqrt {a^{2} x^{2}}}+\frac {2 x^{m} a^{m} \operatorname {LerchPhi}\left (-a^{2} x^{2}, 1, \frac {m}{2}\right )}{5+m}\right )}{4}+\frac {3 a^{-m -1} c^{3} \left (-\frac {4 x^{m} a^{m} \left (a^{2} m \,x^{2}-m -2\right )}{\left (3+m \right ) m \left (2+m \right )}+\frac {8 x^{4+m} a^{4+m} \arctan \left (\sqrt {a^{2} x^{2}}\right )}{\left (6+2 m \right ) \sqrt {a^{2} x^{2}}}+\frac {2 x^{m} a^{m} \left (-m -4\right ) \operatorname {LerchPhi}\left (-a^{2} x^{2}, 1, \frac {m}{2}\right )}{\left (4+m \right ) \left (3+m \right )}\right )}{4}+\frac {a^{-m -1} c^{3} \left (\frac {4 x^{m} a^{m} \left (-m -2\right )}{\left (2+m \right ) \left (1+m \right ) m}+\frac {8 x^{2+m} a^{2+m} \arctan \left (\sqrt {a^{2} x^{2}}\right )}{\left (2+2 m \right ) \sqrt {a^{2} x^{2}}}+\frac {2 x^{m} a^{m} \operatorname {LerchPhi}\left (-a^{2} x^{2}, 1, \frac {m}{2}\right )}{1+m}\right )}{4}\) | \(600\) |
Input:
int(x^m*(a^2*c*x^2+c)^3*arctan(a*x),x,method=_RETURNVERBOSE)
Output:
1/4*a^(-m-1)*c^3*(-4*x^m*a^m*(a^6*m^3*x^6+6*a^6*m^2*x^6+8*a^6*m*x^6-a^4*m^ 3*x^4-8*a^4*m^2*x^4-12*a^4*m*x^4+a^2*m^3*x^2+10*a^2*m^2*x^2+24*a^2*m*x^2-m ^3-12*m^2-44*m-48)/(7+m)/m/(2+m)/(4+m)/(6+m)+8*x^(8+m)*a^(8+m)/(14+2*m)/(a ^2*x^2)^(1/2)*arctan((a^2*x^2)^(1/2))+2/(8+m)*x^m*a^m*(-8-m)/(7+m)*LerchPh i(-a^2*x^2,1,1/2*m))+3/4*a^(-m-1)*c^3*(-4*x^m*a^m*(a^4*m^2*x^4+2*a^4*m*x^4 -a^2*m^2*x^2-4*a^2*m*x^2+m^2+6*m+8)/(5+m)/m/(2+m)/(4+m)+8*x^(6+m)*a^(6+m)/ (10+2*m)/(a^2*x^2)^(1/2)*arctan((a^2*x^2)^(1/2))+2*x^m*a^m/(5+m)*LerchPhi( -a^2*x^2,1,1/2*m))+3/4*a^(-m-1)*c^3*(-4*x^m*a^m*(a^2*m*x^2-m-2)/(3+m)/m/(2 +m)+8*x^(4+m)*a^(4+m)/(6+2*m)/(a^2*x^2)^(1/2)*arctan((a^2*x^2)^(1/2))+2/(4 +m)*x^m*a^m*(-m-4)/(3+m)*LerchPhi(-a^2*x^2,1,1/2*m))+1/4*a^(-m-1)*c^3*(4/( 2+m)*x^m*a^m*(-m-2)/(1+m)/m+8*x^(2+m)*a^(2+m)/(2+2*m)/(a^2*x^2)^(1/2)*arct an((a^2*x^2)^(1/2))+2*x^m*a^m/(1+m)*LerchPhi(-a^2*x^2,1,1/2*m))
\[ \int x^m \left (c+a^2 c x^2\right )^3 \arctan (a x) \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{3} x^{m} \arctan \left (a x\right ) \,d x } \] Input:
integrate(x^m*(a^2*c*x^2+c)^3*arctan(a*x),x, algorithm="fricas")
Output:
integral((a^6*c^3*x^6 + 3*a^4*c^3*x^4 + 3*a^2*c^3*x^2 + c^3)*x^m*arctan(a* x), x)
\[ \int x^m \left (c+a^2 c x^2\right )^3 \arctan (a x) \, dx=c^{3} \left (\int x^{m} \operatorname {atan}{\left (a x \right )}\, dx + \int 3 a^{2} x^{2} x^{m} \operatorname {atan}{\left (a x \right )}\, dx + \int 3 a^{4} x^{4} x^{m} \operatorname {atan}{\left (a x \right )}\, dx + \int a^{6} x^{6} x^{m} \operatorname {atan}{\left (a x \right )}\, dx\right ) \] Input:
integrate(x**m*(a**2*c*x**2+c)**3*atan(a*x),x)
Output:
c**3*(Integral(x**m*atan(a*x), x) + Integral(3*a**2*x**2*x**m*atan(a*x), x ) + Integral(3*a**4*x**4*x**m*atan(a*x), x) + Integral(a**6*x**6*x**m*atan (a*x), x))
\[ \int x^m \left (c+a^2 c x^2\right )^3 \arctan (a x) \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{3} x^{m} \arctan \left (a x\right ) \,d x } \] Input:
integrate(x^m*(a^2*c*x^2+c)^3*arctan(a*x),x, algorithm="maxima")
Output:
(((a^6*c^3*m^3 + 9*a^6*c^3*m^2 + 23*a^6*c^3*m + 15*a^6*c^3)*x^7 + 3*(a^4*c ^3*m^3 + 11*a^4*c^3*m^2 + 31*a^4*c^3*m + 21*a^4*c^3)*x^5 + 3*(a^2*c^3*m^3 + 13*a^2*c^3*m^2 + 47*a^2*c^3*m + 35*a^2*c^3)*x^3 + (c^3*m^3 + 15*c^3*m^2 + 71*c^3*m + 105*c^3)*x)*x^m*arctan(a*x) - (m^4 + 16*m^3 + 86*m^2 + 176*m + 105)*integrate(((a^7*c^3*m^3 + 9*a^7*c^3*m^2 + 23*a^7*c^3*m + 15*a^7*c^3 )*x^7 + 3*(a^5*c^3*m^3 + 11*a^5*c^3*m^2 + 31*a^5*c^3*m + 21*a^5*c^3)*x^5 + 3*(a^3*c^3*m^3 + 13*a^3*c^3*m^2 + 47*a^3*c^3*m + 35*a^3*c^3)*x^3 + (a*c^3 *m^3 + 15*a*c^3*m^2 + 71*a*c^3*m + 105*a*c^3)*x)*x^m/(m^4 + 16*m^3 + (a^2* m^4 + 16*a^2*m^3 + 86*a^2*m^2 + 176*a^2*m + 105*a^2)*x^2 + 86*m^2 + 176*m + 105), x))/(m^4 + 16*m^3 + 86*m^2 + 176*m + 105)
\[ \int x^m \left (c+a^2 c x^2\right )^3 \arctan (a x) \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{3} x^{m} \arctan \left (a x\right ) \,d x } \] Input:
integrate(x^m*(a^2*c*x^2+c)^3*arctan(a*x),x, algorithm="giac")
Output:
integrate((a^2*c*x^2 + c)^3*x^m*arctan(a*x), x)
Timed out. \[ \int x^m \left (c+a^2 c x^2\right )^3 \arctan (a x) \, dx=\int x^m\,\mathrm {atan}\left (a\,x\right )\,{\left (c\,a^2\,x^2+c\right )}^3 \,d x \] Input:
int(x^m*atan(a*x)*(c + a^2*c*x^2)^3,x)
Output:
int(x^m*atan(a*x)*(c + a^2*c*x^2)^3, x)
\[ \int x^m \left (c+a^2 c x^2\right )^3 \arctan (a x) \, dx =\text {Too large to display} \] Input:
int(x^m*(a^2*c*x^2+c)^3*atan(a*x),x)
Output:
(c**3*(x**m*atan(a*x)*a**7*m**7*x**7 + 21*x**m*atan(a*x)*a**7*m**6*x**7 + 175*x**m*atan(a*x)*a**7*m**5*x**7 + 735*x**m*atan(a*x)*a**7*m**4*x**7 + 16 24*x**m*atan(a*x)*a**7*m**3*x**7 + 1764*x**m*atan(a*x)*a**7*m**2*x**7 + 72 0*x**m*atan(a*x)*a**7*m*x**7 + 3*x**m*atan(a*x)*a**5*m**7*x**5 + 69*x**m*a tan(a*x)*a**5*m**6*x**5 + 621*x**m*atan(a*x)*a**5*m**5*x**5 + 2775*x**m*at an(a*x)*a**5*m**4*x**5 + 6432*x**m*atan(a*x)*a**5*m**3*x**5 + 7236*x**m*at an(a*x)*a**5*m**2*x**5 + 3024*x**m*atan(a*x)*a**5*m*x**5 + 3*x**m*atan(a*x )*a**3*m**7*x**3 + 75*x**m*atan(a*x)*a**3*m**6*x**3 + 741*x**m*atan(a*x)*a **3*m**5*x**3 + 3657*x**m*atan(a*x)*a**3*m**4*x**3 + 9336*x**m*atan(a*x)*a **3*m**3*x**3 + 11388*x**m*atan(a*x)*a**3*m**2*x**3 + 5040*x**m*atan(a*x)* a**3*m*x**3 + x**m*atan(a*x)*a*m**7*x + 27*x**m*atan(a*x)*a*m**6*x + 295*x **m*atan(a*x)*a*m**5*x + 1665*x**m*atan(a*x)*a*m**4*x + 5104*x**m*atan(a*x )*a*m**3*x + 8028*x**m*atan(a*x)*a*m**2*x + 5040*x**m*atan(a*x)*a*m*x - x* *m*a**6*m**6*x**6 - 15*x**m*a**6*m**5*x**6 - 85*x**m*a**6*m**4*x**6 - 225* x**m*a**6*m**3*x**6 - 274*x**m*a**6*m**2*x**6 - 120*x**m*a**6*m*x**6 - 2*x **m*a**4*m**6*x**4 - 40*x**m*a**4*m**5*x**4 - 286*x**m*a**4*m**4*x**4 - 89 6*x**m*a**4*m**3*x**4 - 1224*x**m*a**4*m**2*x**4 - 576*x**m*a**4*m*x**4 - x**m*a**2*m**6*x**2 - 25*x**m*a**2*m**5*x**2 - 245*x**m*a**2*m**4*x**2 - 1 127*x**m*a**2*m**3*x**2 - 2274*x**m*a**2*m**2*x**2 - 1368*x**m*a**2*m*x**2 - 48*x**m*m**3 - 576*x**m*m**2 - 2112*x**m*m - 2304*x**m + 48*int(x**m...