\(\int \frac {(d+e x^2)^3 (a+b \arctan (c x))}{x^8} \, dx\) [1148]

Optimal result

Integrand size = 21, antiderivative size = 224 \[ \int \frac {\left (d+e x^2\right )^3 (a+b \arctan (c x))}{x^8} \, dx=-\frac {b c d^3}{42 x^6}+\frac {b c d^2 \left (5 c^2 d-21 e\right )}{140 x^4}-\frac {b c d \left (5 c^4 d^2-21 c^2 d e+35 e^2\right )}{70 x^2}-\frac {d^3 (a+b \arctan (c x))}{7 x^7}-\frac {3 d^2 e (a+b \arctan (c x))}{5 x^5}-\frac {d e^2 (a+b \arctan (c x))}{x^3}-\frac {e^3 (a+b \arctan (c x))}{x}-\frac {1}{35} b c \left (5 c^6 d^3-21 c^4 d^2 e+35 c^2 d e^2-35 e^3\right ) \log (x)+\frac {1}{70} b c \left (5 c^6 d^3-21 c^4 d^2 e+35 c^2 d e^2-35 e^3\right ) \log \left (1+c^2 x^2\right ) \]

[Out]

-1/42*b*c*d^3/x^6+1/140*b*c*d^2*(5*c^2*d-21*e)/x^4-1/70*b*c*d*(5*c^4*d^2-21*c^2*d*e+35*e^2)/x^2-1/7*d^3*(a+b*a
rctan(c*x))/x^7-3/5*d^2*e*(a+b*arctan(c*x))/x^5-d*e^2*(a+b*arctan(c*x))/x^3-e^3*(a+b*arctan(c*x))/x-1/35*b*c*(
5*c^6*d^3-21*c^4*d^2*e+35*c^2*d*e^2-35*e^3)*ln(x)+1/70*b*c*(5*c^6*d^3-21*c^4*d^2*e+35*c^2*d*e^2-35*e^3)*ln(c^2
*x^2+1)

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {276, 5096, 12, 1813, 1634} \[ \int \frac {\left (d+e x^2\right )^3 (a+b \arctan (c x))}{x^8} \, dx=-\frac {d^3 (a+b \arctan (c x))}{7 x^7}-\frac {3 d^2 e (a+b \arctan (c x))}{5 x^5}-\frac {d e^2 (a+b \arctan (c x))}{x^3}-\frac {e^3 (a+b \arctan (c x))}{x}+\frac {b c d^2 \left (5 c^2 d-21 e\right )}{140 x^4}-\frac {b c d \left (5 c^4 d^2-21 c^2 d e+35 e^2\right )}{70 x^2}+\frac {1}{70} b c \left (5 c^6 d^3-21 c^4 d^2 e+35 c^2 d e^2-35 e^3\right ) \log \left (c^2 x^2+1\right )-\frac {1}{35} b c \log (x) \left (5 c^6 d^3-21 c^4 d^2 e+35 c^2 d e^2-35 e^3\right )-\frac {b c d^3}{42 x^6} \]

[In]

Int[((d + e*x^2)^3*(a + b*ArcTan[c*x]))/x^8,x]

[Out]

-1/42*(b*c*d^3)/x^6 + (b*c*d^2*(5*c^2*d - 21*e))/(140*x^4) - (b*c*d*(5*c^4*d^2 - 21*c^2*d*e + 35*e^2))/(70*x^2
) - (d^3*(a + b*ArcTan[c*x]))/(7*x^7) - (3*d^2*e*(a + b*ArcTan[c*x]))/(5*x^5) - (d*e^2*(a + b*ArcTan[c*x]))/x^
3 - (e^3*(a + b*ArcTan[c*x]))/x - (b*c*(5*c^6*d^3 - 21*c^4*d^2*e + 35*c^2*d*e^2 - 35*e^3)*Log[x])/35 + (b*c*(5
*c^6*d^3 - 21*c^4*d^2*e + 35*c^2*d*e^2 - 35*e^3)*Log[1 + c^2*x^2])/70

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 1634

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)
^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2]) &&
GtQ[Expon[Px, x], 2]

Rule 1813

Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[x^((m - 1)/2)*SubstFor[x^2,
Pq, x]*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x^2] && IntegerQ[(m - 1)/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 (a+b \arctan (c x))}{7 x^7}-\frac {3 d^2 e (a+b \arctan (c x))}{5 x^5}-\frac {d e^2 (a+b \arctan (c x))}{x^3}-\frac {e^3 (a+b \arctan (c x))}{x}-(b c) \int \frac {-5 d^3-21 d^2 e x^2-35 d e^2 x^4-35 e^3 x^6}{35 x^7 \left (1+c^2 x^2\right )} \, dx \\ & = -\frac {d^3 (a+b \arctan (c x))}{7 x^7}-\frac {3 d^2 e (a+b \arctan (c x))}{5 x^5}-\frac {d e^2 (a+b \arctan (c x))}{x^3}-\frac {e^3 (a+b \arctan (c x))}{x}-\frac {1}{35} (b c) \int \frac {-5 d^3-21 d^2 e x^2-35 d e^2 x^4-35 e^3 x^6}{x^7 \left (1+c^2 x^2\right )} \, dx \\ & = -\frac {d^3 (a+b \arctan (c x))}{7 x^7}-\frac {3 d^2 e (a+b \arctan (c x))}{5 x^5}-\frac {d e^2 (a+b \arctan (c x))}{x^3}-\frac {e^3 (a+b \arctan (c x))}{x}-\frac {1}{70} (b c) \text {Subst}\left (\int \frac {-5 d^3-21 d^2 e x-35 d e^2 x^2-35 e^3 x^3}{x^4 \left (1+c^2 x\right )} \, dx,x,x^2\right ) \\ & = -\frac {d^3 (a+b \arctan (c x))}{7 x^7}-\frac {3 d^2 e (a+b \arctan (c x))}{5 x^5}-\frac {d e^2 (a+b \arctan (c x))}{x^3}-\frac {e^3 (a+b \arctan (c x))}{x}-\frac {1}{70} (b c) \text {Subst}\left (\int \left (-\frac {5 d^3}{x^4}+\frac {d^2 \left (5 c^2 d-21 e\right )}{x^3}-\frac {d \left (5 c^4 d^2-21 c^2 d e+35 e^2\right )}{x^2}+\frac {5 c^6 d^3-21 c^4 d^2 e+35 c^2 d e^2-35 e^3}{x}+\frac {-5 c^8 d^3+21 c^6 d^2 e-35 c^4 d e^2+35 c^2 e^3}{1+c^2 x}\right ) \, dx,x,x^2\right ) \\ & = -\frac {b c d^3}{42 x^6}+\frac {b c d^2 \left (5 c^2 d-21 e\right )}{140 x^4}-\frac {b c d \left (5 c^4 d^2-21 c^2 d e+35 e^2\right )}{70 x^2}-\frac {d^3 (a+b \arctan (c x))}{7 x^7}-\frac {3 d^2 e (a+b \arctan (c x))}{5 x^5}-\frac {d e^2 (a+b \arctan (c x))}{x^3}-\frac {e^3 (a+b \arctan (c x))}{x}-\frac {1}{35} b c \left (5 c^6 d^3-21 c^4 d^2 e+35 c^2 d e^2-35 e^3\right ) \log (x)+\frac {1}{70} b c \left (5 c^6 d^3-21 c^4 d^2 e+35 c^2 d e^2-35 e^3\right ) \log \left (1+c^2 x^2\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.03 \[ \int \frac {\left (d+e x^2\right )^3 (a+b \arctan (c x))}{x^8} \, dx=-\frac {d^3 (a+b \arctan (c x))}{7 x^7}-\frac {3 d^2 e (a+b \arctan (c x))}{5 x^5}-\frac {d e^2 (a+b \arctan (c x))}{x^3}-\frac {e^3 (a+b \arctan (c x))}{x}+\frac {1}{2} b c e^3 \left (2 \log (x)-\log \left (1+c^2 x^2\right )\right )-\frac {1}{2} b c d e^2 \left (\frac {1}{x^2}+2 c^2 \log (x)-c^2 \log \left (1+c^2 x^2\right )\right )-\frac {3}{20} b c d^2 e \left (\frac {1}{x^4}-\frac {2 c^2}{x^2}-4 c^4 \log (x)+2 c^4 \log \left (1+c^2 x^2\right )\right )-\frac {1}{84} b c d^3 \left (\frac {2}{x^6}-\frac {3 c^2}{x^4}+\frac {6 c^4}{x^2}+12 c^6 \log (x)-6 c^6 \log \left (1+c^2 x^2\right )\right ) \]

[In]

Integrate[((d + e*x^2)^3*(a + b*ArcTan[c*x]))/x^8,x]

[Out]

-1/7*(d^3*(a + b*ArcTan[c*x]))/x^7 - (3*d^2*e*(a + b*ArcTan[c*x]))/(5*x^5) - (d*e^2*(a + b*ArcTan[c*x]))/x^3 -
(e^3*(a + b*ArcTan[c*x]))/x + (b*c*e^3*(2*Log[x] - Log[1 + c^2*x^2]))/2 - (b*c*d*e^2*(x^(-2) + 2*c^2*Log[x] -
c^2*Log[1 + c^2*x^2]))/2 - (3*b*c*d^2*e*(x^(-4) - (2*c^2)/x^2 - 4*c^4*Log[x] + 2*c^4*Log[1 + c^2*x^2]))/20 -
(b*c*d^3*(2/x^6 - (3*c^2)/x^4 + (6*c^4)/x^2 + 12*c^6*Log[x] - 6*c^6*Log[1 + c^2*x^2]))/84

Maple [A] (verified)

Time = 0.27 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.10

method	result	size
parts	\(a \left (-\frac {e^{3}}{x}-\frac {d^{3}}{7 x^{7}}-\frac {d \,e^{2}}{x^{3}}-\frac {3 d^{2} e}{5 x^{5}}\right )+b \,c^{7} \left (-\frac {\arctan \left (c x \right ) e^{3}}{c^{7} x}-\frac {\arctan \left (c x \right ) d^{3}}{7 c^{7} x^{7}}-\frac {\arctan \left (c x \right ) e^{2} d}{c^{7} x^{3}}-\frac {3 \arctan \left (c x \right ) d^{2} e}{5 c^{7} x^{5}}-\frac {\left (5 c^{6} d^{3}-21 c^{4} d^{2} e +35 e^{2} d \,c^{2}-35 e^{3}\right ) \ln \left (c x \right )-\frac {d^{2} \left (5 c^{2} d -21 e \right )}{4 x^{4}}+\frac {5 d^{3}}{6 x^{6}}+\frac {d \left (5 c^{4} d^{2}-21 c^{2} d e +35 e^{2}\right )}{2 x^{2}}+\frac {\left (-5 c^{6} d^{3}+21 c^{4} d^{2} e -35 e^{2} d \,c^{2}+35 e^{3}\right ) \ln \left (c^{2} x^{2}+1\right )}{2}}{35 c^{6}}\right )\)	\(247\)
derivativedivides	\(c^{7} \left (\frac {a \left (-\frac {3 d^{2} e}{5 c \,x^{5}}-\frac {e^{3}}{c x}-\frac {d^{3}}{7 c \,x^{7}}-\frac {d \,e^{2}}{c \,x^{3}}\right )}{c^{6}}+\frac {b \left (-\frac {3 \arctan \left (c x \right ) d^{2} e}{5 c \,x^{5}}-\frac {\arctan \left (c x \right ) e^{3}}{c x}-\frac {\arctan \left (c x \right ) d^{3}}{7 c \,x^{7}}-\frac {\arctan \left (c x \right ) d \,e^{2}}{c \,x^{3}}-\frac {\left (-5 c^{6} d^{3}+21 c^{4} d^{2} e -35 e^{2} d \,c^{2}+35 e^{3}\right ) \ln \left (c^{2} x^{2}+1\right )}{70}-\frac {\left (5 c^{6} d^{3}-21 c^{4} d^{2} e +35 e^{2} d \,c^{2}-35 e^{3}\right ) \ln \left (c x \right )}{35}+\frac {d^{2} \left (5 c^{2} d -21 e \right )}{140 x^{4}}-\frac {d^{3}}{42 x^{6}}-\frac {d \left (5 c^{4} d^{2}-21 c^{2} d e +35 e^{2}\right )}{70 x^{2}}\right )}{c^{6}}\right )\)	\(261\)
default	\(c^{7} \left (\frac {a \left (-\frac {3 d^{2} e}{5 c \,x^{5}}-\frac {e^{3}}{c x}-\frac {d^{3}}{7 c \,x^{7}}-\frac {d \,e^{2}}{c \,x^{3}}\right )}{c^{6}}+\frac {b \left (-\frac {3 \arctan \left (c x \right ) d^{2} e}{5 c \,x^{5}}-\frac {\arctan \left (c x \right ) e^{3}}{c x}-\frac {\arctan \left (c x \right ) d^{3}}{7 c \,x^{7}}-\frac {\arctan \left (c x \right ) d \,e^{2}}{c \,x^{3}}-\frac {\left (-5 c^{6} d^{3}+21 c^{4} d^{2} e -35 e^{2} d \,c^{2}+35 e^{3}\right ) \ln \left (c^{2} x^{2}+1\right )}{70}-\frac {\left (5 c^{6} d^{3}-21 c^{4} d^{2} e +35 e^{2} d \,c^{2}-35 e^{3}\right ) \ln \left (c x \right )}{35}+\frac {d^{2} \left (5 c^{2} d -21 e \right )}{140 x^{4}}-\frac {d^{3}}{42 x^{6}}-\frac {d \left (5 c^{4} d^{2}-21 c^{2} d e +35 e^{2}\right )}{70 x^{2}}\right )}{c^{6}}\right )\)	\(261\)
parallelrisch	\(-\frac {420 a d \,e^{2} x^{4}+252 a \,d^{2} e \,x^{2}+420 x^{4} \arctan \left (c x \right ) b d \,e^{2}+252 x^{2} \arctan \left (c x \right ) b \,d^{2} e -15 x^{3} d^{3} c^{3} b +30 x^{5} c^{5} d^{3} b +10 b c \,d^{3} x +60 d^{3} a +420 a \,e^{3} x^{6}+60 b \,d^{3} \arctan \left (c x \right )-126 x^{5} c^{3} d^{2} e b -30 b \,c^{7} d^{3} x^{7}+60 \ln \left (x \right ) b \,c^{7} d^{3} x^{7}-30 \ln \left (c^{2} x^{2}+1\right ) b \,c^{7} d^{3} x^{7}-420 \ln \left (x \right ) b c \,e^{3} x^{7}+210 \ln \left (c^{2} x^{2}+1\right ) b c \,e^{3} x^{7}+126 b \,c^{5} d^{2} e \,x^{7}-210 b \,c^{3} d \,e^{2} x^{7}-252 \ln \left (x \right ) b \,c^{5} d^{2} e \,x^{7}+126 \ln \left (c^{2} x^{2}+1\right ) b \,c^{5} d^{2} e \,x^{7}+420 \ln \left (x \right ) b \,c^{3} d \,e^{2} x^{7}-210 \ln \left (c^{2} x^{2}+1\right ) b \,c^{3} d \,e^{2} x^{7}+420 x^{6} \arctan \left (c x \right ) b \,e^{3}+63 b c e \,d^{2} x^{3}+210 b c \,e^{2} d \,x^{5}}{420 x^{7}}\)	\(342\)
risch	\(\frac {i b \left (35 e^{3} x^{6}+35 x^{4} e^{2} d +21 e \,d^{2} x^{2}+5 d^{3}\right ) \ln \left (i c x +1\right )}{70 x^{7}}-\frac {60 \ln \left (x \right ) b \,c^{7} d^{3} x^{7}-30 \ln \left (c^{2} x^{2}+1\right ) b \,c^{7} d^{3} x^{7}-252 \ln \left (x \right ) b \,c^{5} d^{2} e \,x^{7}+126 \ln \left (c^{2} x^{2}+1\right ) b \,c^{5} d^{2} e \,x^{7}+420 \ln \left (x \right ) b \,c^{3} d \,e^{2} x^{7}-210 \ln \left (c^{2} x^{2}+1\right ) b \,c^{3} d \,e^{2} x^{7}+30 x^{5} c^{5} d^{3} b -420 \ln \left (x \right ) b c \,e^{3} x^{7}+210 \ln \left (c^{2} x^{2}+1\right ) b c \,e^{3} x^{7}+30 i b \,d^{3} \ln \left (-i c x +1\right )-126 x^{5} c^{3} d^{2} e b +210 i b \,e^{2} d \ln \left (-i c x +1\right ) x^{4}+420 a \,e^{3} x^{6}-15 x^{3} d^{3} c^{3} b +210 b c \,e^{2} d \,x^{5}+210 i b \,e^{3} \ln \left (-i c x +1\right ) x^{6}+420 a d \,e^{2} x^{4}+63 b c e \,d^{2} x^{3}+126 i b \,d^{2} e \ln \left (-i c x +1\right ) x^{2}+252 a \,d^{2} e \,x^{2}+10 b c \,d^{3} x +60 d^{3} a}{420 x^{7}}\)	\(372\)

[In]

int((e*x^2+d)^3*(a+b*arctan(c*x))/x^8,x,method=_RETURNVERBOSE)

[Out]

a*(-e^3/x-1/7*d^3/x^7-d*e^2/x^3-3/5*d^2*e/x^5)+b*c^7*(-arctan(c*x)/c^7*e^3/x-1/7*arctan(c*x)*d^3/c^7/x^7-arcta
n(c*x)/c^7*e^2*d/x^3-3/5*arctan(c*x)/c^7*d^2*e/x^5-1/35/c^6*((5*c^6*d^3-21*c^4*d^2*e+35*c^2*d*e^2-35*e^3)*ln(c
*x)-1/4*d^2*(5*c^2*d-21*e)/x^4+5/6*d^3/x^6+1/2*d*(5*c^4*d^2-21*c^2*d*e+35*e^2)/x^2+1/2*(-5*c^6*d^3+21*c^4*d^2*
e-35*c^2*d*e^2+35*e^3)*ln(c^2*x^2+1)))

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.08 \[ \int \frac {\left (d+e x^2\right )^3 (a+b \arctan (c x))}{x^8} \, dx=-\frac {420 \, a e^{3} x^{6} - 6 \, {\left (5 \, b c^{7} d^{3} - 21 \, b c^{5} d^{2} e + 35 \, b c^{3} d e^{2} - 35 \, b c e^{3}\right )} x^{7} \log \left (c^{2} x^{2} + 1\right ) + 12 \, {\left (5 \, b c^{7} d^{3} - 21 \, b c^{5} d^{2} e + 35 \, b c^{3} d e^{2} - 35 \, b c e^{3}\right )} x^{7} \log \left (x\right ) + 420 \, a d e^{2} x^{4} + 10 \, b c d^{3} x + 252 \, a d^{2} e x^{2} + 6 \, {\left (5 \, b c^{5} d^{3} - 21 \, b c^{3} d^{2} e + 35 \, b c d e^{2}\right )} x^{5} + 60 \, a d^{3} - 3 \, {\left (5 \, b c^{3} d^{3} - 21 \, b c d^{2} e\right )} x^{3} + 12 \, {\left (35 \, b e^{3} x^{6} + 35 \, b d e^{2} x^{4} + 21 \, b d^{2} e x^{2} + 5 \, b d^{3}\right )} \arctan \left (c x\right )}{420 \, x^{7}} \]

[In]

integrate((e*x^2+d)^3*(a+b*arctan(c*x))/x^8,x, algorithm="fricas")

[Out]

-1/420*(420*a*e^3*x^6 - 6*(5*b*c^7*d^3 - 21*b*c^5*d^2*e + 35*b*c^3*d*e^2 - 35*b*c*e^3)*x^7*log(c^2*x^2 + 1) +
12*(5*b*c^7*d^3 - 21*b*c^5*d^2*e + 35*b*c^3*d*e^2 - 35*b*c*e^3)*x^7*log(x) + 420*a*d*e^2*x^4 + 10*b*c*d^3*x +
252*a*d^2*e*x^2 + 6*(5*b*c^5*d^3 - 21*b*c^3*d^2*e + 35*b*c*d*e^2)*x^5 + 60*a*d^3 - 3*(5*b*c^3*d^3 - 21*b*c*d^2
*e)*x^3 + 12*(35*b*e^3*x^6 + 35*b*d*e^2*x^4 + 21*b*d^2*e*x^2 + 5*b*d^3)*arctan(c*x))/x^7

Sympy [A] (verification not implemented)

Time = 0.75 (sec) , antiderivative size = 362, normalized size of antiderivative = 1.62 \[ \int \frac {\left (d+e x^2\right )^3 (a+b \arctan (c x))}{x^8} \, dx=\begin {cases} - \frac {a d^{3}}{7 x^{7}} - \frac {3 a d^{2} e}{5 x^{5}} - \frac {a d e^{2}}{x^{3}} - \frac {a e^{3}}{x} - \frac {b c^{7} d^{3} \log {\left (x \right )}}{7} + \frac {b c^{7} d^{3} \log {\left (x^{2} + \frac {1}{c^{2}} \right )}}{14} - \frac {b c^{5} d^{3}}{14 x^{2}} + \frac {3 b c^{5} d^{2} e \log {\left (x \right )}}{5} - \frac {3 b c^{5} d^{2} e \log {\left (x^{2} + \frac {1}{c^{2}} \right )}}{10} + \frac {b c^{3} d^{3}}{28 x^{4}} + \frac {3 b c^{3} d^{2} e}{10 x^{2}} - b c^{3} d e^{2} \log {\left (x \right )} + \frac {b c^{3} d e^{2} \log {\left (x^{2} + \frac {1}{c^{2}} \right )}}{2} - \frac {b c d^{3}}{42 x^{6}} - \frac {3 b c d^{2} e}{20 x^{4}} - \frac {b c d e^{2}}{2 x^{2}} + b c e^{3} \log {\left (x \right )} - \frac {b c e^{3} \log {\left (x^{2} + \frac {1}{c^{2}} \right )}}{2} - \frac {b d^{3} \operatorname {atan}{\left (c x \right )}}{7 x^{7}} - \frac {3 b d^{2} e \operatorname {atan}{\left (c x \right )}}{5 x^{5}} - \frac {b d e^{2} \operatorname {atan}{\left (c x \right )}}{x^{3}} - \frac {b e^{3} \operatorname {atan}{\left (c x \right )}}{x} & \text {for}\: c \neq 0 \\a \left (- \frac {d^{3}}{7 x^{7}} - \frac {3 d^{2} e}{5 x^{5}} - \frac {d e^{2}}{x^{3}} - \frac {e^{3}}{x}\right ) & \text {otherwise} \end {cases} \]

[In]

integrate((e*x**2+d)**3*(a+b*atan(c*x))/x**8,x)

[Out]

Piecewise((-a*d**3/(7*x**7) - 3*a*d**2*e/(5*x**5) - a*d*e**2/x**3 - a*e**3/x - b*c**7*d**3*log(x)/7 + b*c**7*d
**3*log(x**2 + c**(-2))/14 - b*c**5*d**3/(14*x**2) + 3*b*c**5*d**2*e*log(x)/5 - 3*b*c**5*d**2*e*log(x**2 + c**
(-2))/10 + b*c**3*d**3/(28*x**4) + 3*b*c**3*d**2*e/(10*x**2) - b*c**3*d*e**2*log(x) + b*c**3*d*e**2*log(x**2 +
c**(-2))/2 - b*c*d**3/(42*x**6) - 3*b*c*d**2*e/(20*x**4) - b*c*d*e**2/(2*x**2) + b*c*e**3*log(x) - b*c*e**3*l
og(x**2 + c**(-2))/2 - b*d**3*atan(c*x)/(7*x**7) - 3*b*d**2*e*atan(c*x)/(5*x**5) - b*d*e**2*atan(c*x)/x**3 - b
*e**3*atan(c*x)/x, Ne(c, 0)), (a*(-d**3/(7*x**7) - 3*d**2*e/(5*x**5) - d*e**2/x**3 - e**3/x), True))

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.10 \[ \int \frac {\left (d+e x^2\right )^3 (a+b \arctan (c x))}{x^8} \, dx=\frac {1}{84} \, {\left ({\left (6 \, c^{6} \log \left (c^{2} x^{2} + 1\right ) - 6 \, c^{6} \log \left (x^{2}\right ) - \frac {6 \, c^{4} x^{4} - 3 \, c^{2} x^{2} + 2}{x^{6}}\right )} c - \frac {12 \, \arctan \left (c x\right )}{x^{7}}\right )} b d^{3} - \frac {3}{20} \, {\left ({\left (2 \, c^{4} \log \left (c^{2} x^{2} + 1\right ) - 2 \, c^{4} \log \left (x^{2}\right ) - \frac {2 \, c^{2} x^{2} - 1}{x^{4}}\right )} c + \frac {4 \, \arctan \left (c x\right )}{x^{5}}\right )} b d^{2} e + \frac {1}{2} \, {\left ({\left (c^{2} \log \left (c^{2} x^{2} + 1\right ) - c^{2} \log \left (x^{2}\right ) - \frac {1}{x^{2}}\right )} c - \frac {2 \, \arctan \left (c x\right )}{x^{3}}\right )} b d e^{2} - \frac {1}{2} \, {\left (c {\left (\log \left (c^{2} x^{2} + 1\right ) - \log \left (x^{2}\right )\right )} + \frac {2 \, \arctan \left (c x\right )}{x}\right )} b e^{3} - \frac {a e^{3}}{x} - \frac {a d e^{2}}{x^{3}} - \frac {3 \, a d^{2} e}{5 \, x^{5}} - \frac {a d^{3}}{7 \, x^{7}} \]

[In]

integrate((e*x^2+d)^3*(a+b*arctan(c*x))/x^8,x, algorithm="maxima")

[Out]

1/84*((6*c^6*log(c^2*x^2 + 1) - 6*c^6*log(x^2) - (6*c^4*x^4 - 3*c^2*x^2 + 2)/x^6)*c - 12*arctan(c*x)/x^7)*b*d^
3 - 3/20*((2*c^4*log(c^2*x^2 + 1) - 2*c^4*log(x^2) - (2*c^2*x^2 - 1)/x^4)*c + 4*arctan(c*x)/x^5)*b*d^2*e + 1/2
*((c^2*log(c^2*x^2 + 1) - c^2*log(x^2) - 1/x^2)*c - 2*arctan(c*x)/x^3)*b*d*e^2 - 1/2*(c*(log(c^2*x^2 + 1) - lo
g(x^2)) + 2*arctan(c*x)/x)*b*e^3 - a*e^3/x - a*d*e^2/x^3 - 3/5*a*d^2*e/x^5 - 1/7*a*d^3/x^7

Giac [F]

\[ \int \frac {\left (d+e x^2\right )^3 (a+b \arctan (c x))}{x^8} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{3} {\left (b \arctan \left (c x\right ) + a\right )}}{x^{8}} \,d x } \]

[In]

integrate((e*x^2+d)^3*(a+b*arctan(c*x))/x^8,x, algorithm="giac")

[Out]

sage0*x

Mupad [B] (verification not implemented)

Time = 0.83 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.05 \[ \int \frac {\left (d+e x^2\right )^3 (a+b \arctan (c x))}{x^8} \, dx=\ln \left (c^2\,x^2+1\right )\,\left (\frac {b\,c^7\,d^3}{14}-\frac {3\,b\,c^5\,d^2\,e}{10}+\frac {b\,c^3\,d\,e^2}{2}-\frac {b\,c\,e^3}{2}\right )-\ln \left (x\right )\,\left (\frac {b\,c^7\,d^3}{7}-\frac {3\,b\,c^5\,d^2\,e}{5}+b\,c^3\,d\,e^2-b\,c\,e^3\right )-\frac {5\,a\,d^3-x^3\,\left (\frac {5\,b\,c^3\,d^3}{4}-\frac {21\,b\,c\,d^2\,e}{4}\right )+x^5\,\left (\frac {5\,b\,c^5\,d^3}{2}-\frac {21\,b\,c^3\,d^2\,e}{2}+\frac {35\,b\,c\,d\,e^2}{2}\right )+35\,a\,e^3\,x^6+\frac {5\,b\,c\,d^3\,x}{6}+21\,a\,d^2\,e\,x^2+35\,a\,d\,e^2\,x^4}{35\,x^7}-\frac {\mathrm {atan}\left (c\,x\right )\,\left (\frac {b\,d^3}{7}+\frac {3\,b\,d^2\,e\,x^2}{5}+b\,d\,e^2\,x^4+b\,e^3\,x^6\right )}{x^7} \]

[In]

int(((a + b*atan(c*x))*(d + e*x^2)^3)/x^8,x)

[Out]

log(c^2*x^2 + 1)*((b*c^7*d^3)/14 - (b*c*e^3)/2 + (b*c^3*d*e^2)/2 - (3*b*c^5*d^2*e)/10) - log(x)*((b*c^7*d^3)/7
- b*c*e^3 + b*c^3*d*e^2 - (3*b*c^5*d^2*e)/5) - (5*a*d^3 - x^3*((5*b*c^3*d^3)/4 - (21*b*c*d^2*e)/4) + x^5*((5*
b*c^5*d^3)/2 + (35*b*c*d*e^2)/2 - (21*b*c^3*d^2*e)/2) + 35*a*e^3*x^6 + (5*b*c*d^3*x)/6 + 21*a*d^2*e*x^2 + 35*a
*d*e^2*x^4)/(35*x^7) - (atan(c*x)*((b*d^3)/7 + b*e^3*x^6 + (3*b*d^2*e*x^2)/5 + b*d*e^2*x^4))/x^7