\(\int x^m (d+e x^2)^3 (a+b \arctan (c x)) \, dx\) [1228]

Optimal result

Integrand size = 21, antiderivative size = 378 \[ \int x^m \left (d+e x^2\right )^3 (a+b \arctan (c x)) \, dx=-\frac {b e \left (e^2 \left (15+8 m+m^2\right )-3 c^2 d e \left (21+10 m+m^2\right )+3 c^4 d^2 \left (35+12 m+m^2\right )\right ) x^{2+m}}{c^5 (2+m) (3+m) (5+m) (7+m)}+\frac {b e^2 \left (e (5+m)-3 c^2 d (7+m)\right ) x^{4+m}}{c^3 (4+m) (5+m) (7+m)}-\frac {b e^3 x^{6+m}}{c (6+m) (7+m)}+\frac {d^3 x^{1+m} (a+b \arctan (c x))}{1+m}+\frac {3 d^2 e x^{3+m} (a+b \arctan (c x))}{3+m}+\frac {3 d e^2 x^{5+m} (a+b \arctan (c x))}{5+m}+\frac {e^3 x^{7+m} (a+b \arctan (c x))}{7+m}+\frac {b \left (e^3 \left (15+23 m+9 m^2+m^3\right )-3 c^2 d e^2 \left (21+31 m+11 m^2+m^3\right )+3 c^4 d^2 e \left (35+47 m+13 m^2+m^3\right )-c^6 d^3 \left (105+71 m+15 m^2+m^3\right )\right ) x^{2+m} \operatorname {Hypergeometric2F1}\left (1,\frac {2+m}{2},\frac {4+m}{2},-c^2 x^2\right )}{c^5 (1+m) (2+m) (3+m) (5+m) (7+m)} \]

[Out]

-b*e*(e^2*(m^2+8*m+15)-3*c^2*d*e*(m^2+10*m+21)+3*c^4*d^2*(m^2+12*m+35))*x^(2+m)/c^5/(2+m)/(7+m)/(m^2+8*m+15)+b
*e^2*(e*(5+m)-3*c^2*d*(7+m))*x^(4+m)/c^3/(4+m)/(5+m)/(7+m)-b*e^3*x^(6+m)/c/(6+m)/(7+m)+d^3*x^(1+m)*(a+b*arctan
(c*x))/(1+m)+3*d^2*e*x^(3+m)*(a+b*arctan(c*x))/(3+m)+3*d*e^2*x^(5+m)*(a+b*arctan(c*x))/(5+m)+e^3*x^(7+m)*(a+b*
arctan(c*x))/(7+m)+b*(e^3*(m^3+9*m^2+23*m+15)-3*c^2*d*e^2*(m^3+11*m^2+31*m+21)+3*c^4*d^2*e*(m^3+13*m^2+47*m+35
)-c^6*d^3*(m^3+15*m^2+71*m+105))*x^(2+m)*hypergeom([1, 1+1/2*m],[2+1/2*m],-c^2*x^2)/c^5/(m^2+12*m+35)/(m^3+6*m
^2+11*m+6)

Rubi [A] (verified)

Time = 1.33 (sec) , antiderivative size = 374, normalized size of antiderivative = 0.99, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {276, 5096, 1816, 371} \[ \int x^m \left (d+e x^2\right )^3 (a+b \arctan (c x)) \, dx=\frac {d^3 x^{m+1} (a+b \arctan (c x))}{m+1}+\frac {3 d^2 e x^{m+3} (a+b \arctan (c x))}{m+3}+\frac {3 d e^2 x^{m+5} (a+b \arctan (c x))}{m+5}+\frac {e^3 x^{m+7} (a+b \arctan (c x))}{m+7}-\frac {b e^2 x^{m+4} \left (\frac {3 c^2 d}{m+5}-\frac {e}{m+7}\right )}{c^3 (m+4)}-\frac {b e x^{m+2} \left (3 c^4 d^2 \left (m^2+12 m+35\right )-3 c^2 d e \left (m^2+10 m+21\right )+e^2 \left (m^2+8 m+15\right )\right )}{c^5 (m+2) (m+3) (m+5) (m+7)}+\frac {b x^{m+2} \left (c^6 \left (-d^3\right ) \left (m^3+15 m^2+71 m+105\right )+3 c^4 d^2 e \left (m^3+13 m^2+47 m+35\right )-3 c^2 d e^2 \left (m^3+11 m^2+31 m+21\right )+e^3 \left (m^3+9 m^2+23 m+15\right )\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {m+2}{2},\frac {m+4}{2},-c^2 x^2\right )}{c^5 (m+1) (m+2) (m+3) (m+5) (m+7)}-\frac {b e^3 x^{m+6}}{c (m+6) (m+7)} \]

[In]

Int[x^m*(d + e*x^2)^3*(a + b*ArcTan[c*x]),x]

[Out]

-((b*e*(e^2*(15 + 8*m + m^2) - 3*c^2*d*e*(21 + 10*m + m^2) + 3*c^4*d^2*(35 + 12*m + m^2))*x^(2 + m))/(c^5*(2 +
m)*(3 + m)*(5 + m)*(7 + m))) - (b*e^2*((3*c^2*d)/(5 + m) - e/(7 + m))*x^(4 + m))/(c^3*(4 + m)) - (b*e^3*x^(6
+ m))/(c*(6 + m)*(7 + m)) + (d^3*x^(1 + m)*(a + b*ArcTan[c*x]))/(1 + m) + (3*d^2*e*x^(3 + m)*(a + b*ArcTan[c*x
]))/(3 + m) + (3*d*e^2*x^(5 + m)*(a + b*ArcTan[c*x]))/(5 + m) + (e^3*x^(7 + m)*(a + b*ArcTan[c*x]))/(7 + m) +
(b*(e^3*(15 + 23*m + 9*m^2 + m^3) - 3*c^2*d*e^2*(21 + 31*m + 11*m^2 + m^3) + 3*c^4*d^2*e*(35 + 47*m + 13*m^2 +
m^3) - c^6*d^3*(105 + 71*m + 15*m^2 + m^3))*x^(2 + m)*Hypergeometric2F1[1, (2 + m)/2, (4 + m)/2, -(c^2*x^2)])
/(c^5*(1 + m)*(2 + m)*(3 + m)*(5 + m)*(7 + m))

Rule 276

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rule 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 1816

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*Pq*(a + b*x
^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rule 5096

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]}, Dist[a + b*ArcTan[c*x], u, x] - Dist[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] &&
!ILtQ[(m - 1)/2, 0]))

Rubi steps \begin{align*} \text {integral}& = \frac {d^3 x^{1+m} (a+b \arctan (c x))}{1+m}+\frac {3 d^2 e x^{3+m} (a+b \arctan (c x))}{3+m}+\frac {3 d e^2 x^{5+m} (a+b \arctan (c x))}{5+m}+\frac {e^3 x^{7+m} (a+b \arctan (c x))}{7+m}-(b c) \int \frac {x^{1+m} \left (\frac {d^3}{1+m}+\frac {3 d^2 e x^2}{3+m}+\frac {3 d e^2 x^4}{5+m}+\frac {e^3 x^6}{7+m}\right )}{1+c^2 x^2} \, dx \\ & = \frac {d^3 x^{1+m} (a+b \arctan (c x))}{1+m}+\frac {3 d^2 e x^{3+m} (a+b \arctan (c x))}{3+m}+\frac {3 d e^2 x^{5+m} (a+b \arctan (c x))}{5+m}+\frac {e^3 x^{7+m} (a+b \arctan (c x))}{7+m}-(b c) \int \left (\frac {e \left (e^2 \left (15+8 m+m^2\right )-3 c^2 d e \left (21+10 m+m^2\right )+3 c^4 d^2 \left (35+12 m+m^2\right )\right ) x^{1+m}}{c^6 (3+m) (5+m) (7+m)}+\frac {e^2 \left (\frac {3 c^2 d}{5+m}-\frac {e}{7+m}\right ) x^{3+m}}{c^4}+\frac {e^3 x^{5+m}}{c^2 (7+m)}+\frac {\left (105 c^6 d^3-105 c^4 d^2 e+63 c^2 d e^2-15 e^3+71 c^6 d^3 m-141 c^4 d^2 e m+93 c^2 d e^2 m-23 e^3 m+15 c^6 d^3 m^2-39 c^4 d^2 e m^2+33 c^2 d e^2 m^2-9 e^3 m^2+c^6 d^3 m^3-3 c^4 d^2 e m^3+3 c^2 d e^2 m^3-e^3 m^3\right ) x^{1+m}}{c^6 (1+m) (3+m) (5+m) (7+m) \left (1+c^2 x^2\right )}\right ) \, dx \\ & = -\frac {b e \left (e^2 \left (15+8 m+m^2\right )-3 c^2 d e \left (21+10 m+m^2\right )+3 c^4 d^2 \left (35+12 m+m^2\right )\right ) x^{2+m}}{c^5 (2+m) (3+m) (5+m) (7+m)}-\frac {b e^2 \left (\frac {3 c^2 d}{5+m}-\frac {e}{7+m}\right ) x^{4+m}}{c^3 (4+m)}-\frac {b e^3 x^{6+m}}{c (6+m) (7+m)}+\frac {d^3 x^{1+m} (a+b \arctan (c x))}{1+m}+\frac {3 d^2 e x^{3+m} (a+b \arctan (c x))}{3+m}+\frac {3 d e^2 x^{5+m} (a+b \arctan (c x))}{5+m}+\frac {e^3 x^{7+m} (a+b \arctan (c x))}{7+m}+\frac {\left (b \left (e^3 \left (15+23 m+9 m^2+m^3\right )-3 c^2 d e^2 \left (21+31 m+11 m^2+m^3\right )+3 c^4 d^2 e \left (35+47 m+13 m^2+m^3\right )-c^6 d^3 \left (105+71 m+15 m^2+m^3\right )\right )\right ) \int \frac {x^{1+m}}{1+c^2 x^2} \, dx}{c^5 (1+m) (3+m) (5+m) (7+m)} \\ & = -\frac {b e \left (e^2 \left (15+8 m+m^2\right )-3 c^2 d e \left (21+10 m+m^2\right )+3 c^4 d^2 \left (35+12 m+m^2\right )\right ) x^{2+m}}{c^5 (2+m) (3+m) (5+m) (7+m)}-\frac {b e^2 \left (\frac {3 c^2 d}{5+m}-\frac {e}{7+m}\right ) x^{4+m}}{c^3 (4+m)}-\frac {b e^3 x^{6+m}}{c (6+m) (7+m)}+\frac {d^3 x^{1+m} (a+b \arctan (c x))}{1+m}+\frac {3 d^2 e x^{3+m} (a+b \arctan (c x))}{3+m}+\frac {3 d e^2 x^{5+m} (a+b \arctan (c x))}{5+m}+\frac {e^3 x^{7+m} (a+b \arctan (c x))}{7+m}+\frac {b \left (e^3 \left (15+23 m+9 m^2+m^3\right )-3 c^2 d e^2 \left (21+31 m+11 m^2+m^3\right )+3 c^4 d^2 e \left (35+47 m+13 m^2+m^3\right )-c^6 d^3 \left (105+71 m+15 m^2+m^3\right )\right ) x^{2+m} \operatorname {Hypergeometric2F1}\left (1,\frac {2+m}{2},\frac {4+m}{2},-c^2 x^2\right )}{c^5 (1+m) (2+m) (3+m) (5+m) (7+m)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.40 (sec) , antiderivative size = 264, normalized size of antiderivative = 0.70 \[ \int x^m \left (d+e x^2\right )^3 (a+b \arctan (c x)) \, dx=x^{1+m} \left (\frac {d^3 (a+b \arctan (c x))}{1+m}+\frac {3 d^2 e x^2 (a+b \arctan (c x))}{3+m}+\frac {3 d e^2 x^4 (a+b \arctan (c x))}{5+m}+\frac {e^3 x^6 (a+b \arctan (c x))}{7+m}-\frac {b c e^3 x^7 \operatorname {Hypergeometric2F1}\left (1,4+\frac {m}{2},5+\frac {m}{2},-c^2 x^2\right )}{(7+m) (8+m)}-\frac {b c d^3 x \operatorname {Hypergeometric2F1}\left (1,\frac {2+m}{2},\frac {4+m}{2},-c^2 x^2\right )}{2+3 m+m^2}-\frac {3 b c d^2 e x^3 \operatorname {Hypergeometric2F1}\left (1,\frac {4+m}{2},\frac {6+m}{2},-c^2 x^2\right )}{12+7 m+m^2}-\frac {3 b c d e^2 x^5 \operatorname {Hypergeometric2F1}\left (1,\frac {6+m}{2},\frac {8+m}{2},-c^2 x^2\right )}{(5+m) (6+m)}\right ) \]

[In]

Integrate[x^m*(d + e*x^2)^3*(a + b*ArcTan[c*x]),x]

[Out]

x^(1 + m)*((d^3*(a + b*ArcTan[c*x]))/(1 + m) + (3*d^2*e*x^2*(a + b*ArcTan[c*x]))/(3 + m) + (3*d*e^2*x^4*(a + b
*ArcTan[c*x]))/(5 + m) + (e^3*x^6*(a + b*ArcTan[c*x]))/(7 + m) - (b*c*e^3*x^7*Hypergeometric2F1[1, 4 + m/2, 5
+ m/2, -(c^2*x^2)])/((7 + m)*(8 + m)) - (b*c*d^3*x*Hypergeometric2F1[1, (2 + m)/2, (4 + m)/2, -(c^2*x^2)])/(2
+ 3*m + m^2) - (3*b*c*d^2*e*x^3*Hypergeometric2F1[1, (4 + m)/2, (6 + m)/2, -(c^2*x^2)])/(12 + 7*m + m^2) - (3*
b*c*d*e^2*x^5*Hypergeometric2F1[1, (6 + m)/2, (8 + m)/2, -(c^2*x^2)])/((5 + m)*(6 + m)))

Maple [F]

\[\int x^{m} \left (e \,x^{2}+d \right )^{3} \left (a +b \arctan \left (c x \right )\right )d x\]

[In]

int(x^m*(e*x^2+d)^3*(a+b*arctan(c*x)),x)

[Out]

int(x^m*(e*x^2+d)^3*(a+b*arctan(c*x)),x)

Fricas [F]

\[ \int x^m \left (d+e x^2\right )^3 (a+b \arctan (c x)) \, dx=\int { {\left (e x^{2} + d\right )}^{3} {\left (b \arctan \left (c x\right ) + a\right )} x^{m} \,d x } \]

[In]

integrate(x^m*(e*x^2+d)^3*(a+b*arctan(c*x)),x, algorithm="fricas")

[Out]

integral((a*e^3*x^6 + 3*a*d*e^2*x^4 + 3*a*d^2*e*x^2 + a*d^3 + (b*e^3*x^6 + 3*b*d*e^2*x^4 + 3*b*d^2*e*x^2 + b*d
^3)*arctan(c*x))*x^m, x)

Sympy [F]

\[ \int x^m \left (d+e x^2\right )^3 (a+b \arctan (c x)) \, dx=\int x^{m} \left (a + b \operatorname {atan}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{3}\, dx \]

[In]

integrate(x**m*(e*x**2+d)**3*(a+b*atan(c*x)),x)

[Out]

Integral(x**m*(a + b*atan(c*x))*(d + e*x**2)**3, x)

Maxima [F]

\[ \int x^m \left (d+e x^2\right )^3 (a+b \arctan (c x)) \, dx=\int { {\left (e x^{2} + d\right )}^{3} {\left (b \arctan \left (c x\right ) + a\right )} x^{m} \,d x } \]

[In]

integrate(x^m*(e*x^2+d)^3*(a+b*arctan(c*x)),x, algorithm="maxima")

[Out]

a*e^3*x^(m + 7)/(m + 7) + 3*a*d*e^2*x^(m + 5)/(m + 5) + 3*a*d^2*e*x^(m + 3)/(m + 3) + a*d^3*x^(m + 1)/(m + 1)
+ (((b*e^3*m^3 + 9*b*e^3*m^2 + 23*b*e^3*m + 15*b*e^3)*x^7 + 3*(b*d*e^2*m^3 + 11*b*d*e^2*m^2 + 31*b*d*e^2*m + 2
1*b*d*e^2)*x^5 + 3*(b*d^2*e*m^3 + 13*b*d^2*e*m^2 + 47*b*d^2*e*m + 35*b*d^2*e)*x^3 + (b*d^3*m^3 + 15*b*d^3*m^2
+ 71*b*d^3*m + 105*b*d^3)*x)*x^m*arctan(c*x) - (m^4 + 16*m^3 + 86*m^2 + 176*m + 105)*integrate(((b*c*e^3*m^3 +
9*b*c*e^3*m^2 + 23*b*c*e^3*m + 15*b*c*e^3)*x^7 + 3*(b*c*d*e^2*m^3 + 11*b*c*d*e^2*m^2 + 31*b*c*d*e^2*m + 21*b*
c*d*e^2)*x^5 + 3*(b*c*d^2*e*m^3 + 13*b*c*d^2*e*m^2 + 47*b*c*d^2*e*m + 35*b*c*d^2*e)*x^3 + (b*c*d^3*m^3 + 15*b*
c*d^3*m^2 + 71*b*c*d^3*m + 105*b*c*d^3)*x)*x^m/(m^4 + 16*m^3 + (c^2*m^4 + 16*c^2*m^3 + 86*c^2*m^2 + 176*c^2*m
+ 105*c^2)*x^2 + 86*m^2 + 176*m + 105), x))/(m^4 + 16*m^3 + 86*m^2 + 176*m + 105)

Giac [F]

\[ \int x^m \left (d+e x^2\right )^3 (a+b \arctan (c x)) \, dx=\int { {\left (e x^{2} + d\right )}^{3} {\left (b \arctan \left (c x\right ) + a\right )} x^{m} \,d x } \]

[In]

integrate(x^m*(e*x^2+d)^3*(a+b*arctan(c*x)),x, algorithm="giac")

[Out]

sage0*x

Mupad [F(-1)]

Timed out. \[ \int x^m \left (d+e x^2\right )^3 (a+b \arctan (c x)) \, dx=\int x^m\,\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )\,{\left (e\,x^2+d\right )}^3 \,d x \]

[In]

int(x^m*(a + b*atan(c*x))*(d + e*x^2)^3,x)

[Out]

int(x^m*(a + b*atan(c*x))*(d + e*x^2)^3, x)