Integrand size = 18, antiderivative size = 652 \[ \int (d+e x)^3 (a+b \arctan (c x))^3 \, dx=\frac {3 a b^2 d e^2 x}{c^2}-\frac {b^3 e^3 x}{4 c^3}+\frac {b^3 e^3 \arctan (c x)}{4 c^4}+\frac {3 b^3 d e^2 x \arctan (c x)}{c^2}+\frac {b^2 e^3 x^2 (a+b \arctan (c x))}{4 c^2}-\frac {3 b d e^2 (a+b \arctan (c x))^2}{2 c^3}+\frac {i b e^3 (a+b \arctan (c x))^2}{4 c^4}-\frac {3 i b e \left (6 c^2 d^2-e^2\right ) (a+b \arctan (c x))^2}{4 c^4}-\frac {3 b e \left (6 c^2 d^2-e^2\right ) x (a+b \arctan (c x))^2}{4 c^3}-\frac {3 b d e^2 x^2 (a+b \arctan (c x))^2}{2 c}-\frac {b e^3 x^3 (a+b \arctan (c x))^2}{4 c}+\frac {i d (c d-e) (c d+e) (a+b \arctan (c x))^3}{c^3}-\frac {\left (c^4 d^4-6 c^2 d^2 e^2+e^4\right ) (a+b \arctan (c x))^3}{4 c^4 e}+\frac {(d+e x)^4 (a+b \arctan (c x))^3}{4 e}+\frac {b^2 e^3 (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{2 c^4}-\frac {3 b^2 e \left (6 c^2 d^2-e^2\right ) (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{2 c^4}+\frac {3 b d (c d-e) (c d+e) (a+b \arctan (c x))^2 \log \left (\frac {2}{1+i c x}\right )}{c^3}-\frac {3 b^3 d e^2 \log \left (1+c^2 x^2\right )}{2 c^3}+\frac {i b^3 e^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{4 c^4}-\frac {3 i b^3 e \left (6 c^2 d^2-e^2\right ) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{4 c^4}+\frac {3 i b^2 d (c d-e) (c d+e) (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{c^3}+\frac {3 b^3 d (c d-e) (c d+e) \operatorname {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )}{2 c^3} \]
3*a*b^2*d*e^2*x/c^2-1/4*b^3*e^3*x/c^3+1/4*b^3*e^3*arctan(c*x)/c^4+3*b^3*d* e^2*x*arctan(c*x)/c^2+1/4*b^2*e^3*x^2*(a+b*arctan(c*x))/c^2-3/2*b*d*e^2*(a +b*arctan(c*x))^2/c^3-3/4*I*b^3*e*(6*c^2*d^2-e^2)*polylog(2,1-2/(1+I*c*x)) /c^4+3*I*b^2*d*(c*d-e)*(c*d+e)*(a+b*arctan(c*x))*polylog(2,1-2/(1+I*c*x))/ c^3-3/4*b*e*(6*c^2*d^2-e^2)*x*(a+b*arctan(c*x))^2/c^3-3/2*b*d*e^2*x^2*(a+b *arctan(c*x))^2/c-1/4*b*e^3*x^3*(a+b*arctan(c*x))^2/c+I*d*(c*d-e)*(c*d+e)* (a+b*arctan(c*x))^3/c^3-1/4*(c^4*d^4-6*c^2*d^2*e^2+e^4)*(a+b*arctan(c*x))^ 3/c^4/e+1/4*(e*x+d)^4*(a+b*arctan(c*x))^3/e+1/2*b^2*e^3*(a+b*arctan(c*x))* ln(2/(1+I*c*x))/c^4-3/2*b^2*e*(6*c^2*d^2-e^2)*(a+b*arctan(c*x))*ln(2/(1+I* c*x))/c^4+3*b*d*(c*d-e)*(c*d+e)*(a+b*arctan(c*x))^2*ln(2/(1+I*c*x))/c^3-3/ 2*b^3*d*e^2*ln(c^2*x^2+1)/c^3-3/4*I*b*e*(6*c^2*d^2-e^2)*(a+b*arctan(c*x))^ 2/c^4+1/4*I*b*e^3*(a+b*arctan(c*x))^2/c^4+1/4*I*b^3*e^3*polylog(2,1-2/(1+I *c*x))/c^4+3/2*b^3*d*(c*d-e)*(c*d+e)*polylog(3,1-2/(1+I*c*x))/c^3
Time = 2.01 (sec) , antiderivative size = 855, normalized size of antiderivative = 1.31 \[ \int (d+e x)^3 (a+b \arctan (c x))^3 \, dx=\frac {a^2 c \left (4 a c^3 d^3+3 b e \left (-6 c^2 d^2+e^2\right )\right ) x+6 a^2 c^3 d e (a c d-b e) x^2+a^2 c^3 e^2 (4 a c d-b e) x^3+a^3 c^4 e^3 x^4+3 a^2 b \left (6 c^2 d^2 e-e^3\right ) \arctan (c x)+3 a^2 b c^4 x \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right ) \arctan (c x)+a b^2 e^3 \left (1+c^2 x^2+\left (6 c x-2 c^3 x^3\right ) \arctan (c x)+3 \left (-1+c^4 x^4\right ) \arctan (c x)^2-4 \log \left (1+c^2 x^2\right )\right )-6 a^2 b c d \left (c^2 d^2-e^2\right ) \log \left (1+c^2 x^2\right )+18 a b^2 c^2 d^2 e \left (-2 c x \arctan (c x)+\left (1+c^2 x^2\right ) \arctan (c x)^2+\log \left (1+c^2 x^2\right )\right )+12 a b^2 c^3 d^3 \left (\arctan (c x) \left ((-i+c x) \arctan (c x)+2 \log \left (1+e^{2 i \arctan (c x)}\right )\right )-i \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )\right )+12 a b^2 c d e^2 \left (c x+\left (i+c^3 x^3\right ) \arctan (c x)^2-\arctan (c x) \left (1+c^2 x^2+2 \log \left (1+e^{2 i \arctan (c x)}\right )\right )+i \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )\right )+6 b^3 c^2 d^2 e \left (\arctan (c x) \left ((3 i-3 c x) \arctan (c x)+\left (1+c^2 x^2\right ) \arctan (c x)^2-6 \log \left (1+e^{2 i \arctan (c x)}\right )\right )+3 i \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )\right )+b^3 e^3 \left (-c x-\left (4 i-3 c x+c^3 x^3\right ) \arctan (c x)^2+\left (-1+c^4 x^4\right ) \arctan (c x)^3+\arctan (c x) \left (1+c^2 x^2+8 \log \left (1+e^{2 i \arctan (c x)}\right )\right )-4 i \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )\right )+2 b^3 c d e^2 \left (6 c x \arctan (c x)-3 \arctan (c x)^2-3 c^2 x^2 \arctan (c x)^2+2 i \arctan (c x)^3+2 c^3 x^3 \arctan (c x)^3-6 \arctan (c x)^2 \log \left (1+e^{2 i \arctan (c x)}\right )-3 \log \left (1+c^2 x^2\right )+6 i \arctan (c x) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )-3 \operatorname {PolyLog}\left (3,-e^{2 i \arctan (c x)}\right )\right )+2 b^3 c^3 d^3 \left (2 \arctan (c x)^2 \left ((-i+c x) \arctan (c x)+3 \log \left (1+e^{2 i \arctan (c x)}\right )\right )-6 i \arctan (c x) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )+3 \operatorname {PolyLog}\left (3,-e^{2 i \arctan (c x)}\right )\right )}{4 c^4} \]
(a^2*c*(4*a*c^3*d^3 + 3*b*e*(-6*c^2*d^2 + e^2))*x + 6*a^2*c^3*d*e*(a*c*d - b*e)*x^2 + a^2*c^3*e^2*(4*a*c*d - b*e)*x^3 + a^3*c^4*e^3*x^4 + 3*a^2*b*(6 *c^2*d^2*e - e^3)*ArcTan[c*x] + 3*a^2*b*c^4*x*(4*d^3 + 6*d^2*e*x + 4*d*e^2 *x^2 + e^3*x^3)*ArcTan[c*x] + a*b^2*e^3*(1 + c^2*x^2 + (6*c*x - 2*c^3*x^3) *ArcTan[c*x] + 3*(-1 + c^4*x^4)*ArcTan[c*x]^2 - 4*Log[1 + c^2*x^2]) - 6*a^ 2*b*c*d*(c^2*d^2 - e^2)*Log[1 + c^2*x^2] + 18*a*b^2*c^2*d^2*e*(-2*c*x*ArcT an[c*x] + (1 + c^2*x^2)*ArcTan[c*x]^2 + Log[1 + c^2*x^2]) + 12*a*b^2*c^3*d ^3*(ArcTan[c*x]*((-I + c*x)*ArcTan[c*x] + 2*Log[1 + E^((2*I)*ArcTan[c*x])] ) - I*PolyLog[2, -E^((2*I)*ArcTan[c*x])]) + 12*a*b^2*c*d*e^2*(c*x + (I + c ^3*x^3)*ArcTan[c*x]^2 - ArcTan[c*x]*(1 + c^2*x^2 + 2*Log[1 + E^((2*I)*ArcT an[c*x])]) + I*PolyLog[2, -E^((2*I)*ArcTan[c*x])]) + 6*b^3*c^2*d^2*e*(ArcT an[c*x]*((3*I - 3*c*x)*ArcTan[c*x] + (1 + c^2*x^2)*ArcTan[c*x]^2 - 6*Log[1 + E^((2*I)*ArcTan[c*x])]) + (3*I)*PolyLog[2, -E^((2*I)*ArcTan[c*x])]) + b ^3*e^3*(-(c*x) - (4*I - 3*c*x + c^3*x^3)*ArcTan[c*x]^2 + (-1 + c^4*x^4)*Ar cTan[c*x]^3 + ArcTan[c*x]*(1 + c^2*x^2 + 8*Log[1 + E^((2*I)*ArcTan[c*x])]) - (4*I)*PolyLog[2, -E^((2*I)*ArcTan[c*x])]) + 2*b^3*c*d*e^2*(6*c*x*ArcTan [c*x] - 3*ArcTan[c*x]^2 - 3*c^2*x^2*ArcTan[c*x]^2 + (2*I)*ArcTan[c*x]^3 + 2*c^3*x^3*ArcTan[c*x]^3 - 6*ArcTan[c*x]^2*Log[1 + E^((2*I)*ArcTan[c*x])] - 3*Log[1 + c^2*x^2] + (6*I)*ArcTan[c*x]*PolyLog[2, -E^((2*I)*ArcTan[c*x])] - 3*PolyLog[3, -E^((2*I)*ArcTan[c*x])]) + 2*b^3*c^3*d^3*(2*ArcTan[c*x]...
Time = 1.37 (sec) , antiderivative size = 645, normalized size of antiderivative = 0.99, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {5389, 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.
| \(\displaystyle \int (d+e x)^3 (a+b \arctan (c x))^3 \, dx\) |
| \(\Big \downarrow \) 5389 |
| \(\displaystyle \frac {(d+e x)^4 (a+b \arctan (c x))^3}{4 e}-\frac {3 b c \int \left (\frac {x^2 (a+b \arctan (c x))^2 e^4}{c^2}+\frac {4 d x (a+b \arctan (c x))^2 e^3}{c^2}+\frac {\left (6 c^2 d^2-e^2\right ) (a+b \arctan (c x))^2 e^2}{c^4}+\frac {\left (c^4 d^4-6 c^2 e^2 d^2+4 c^2 (c d-e) e (c d+e) x d+e^4\right ) (a+b \arctan (c x))^2}{c^4 \left (c^2 x^2+1\right )}\right )dx}{4 e}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {(d+e x)^4 (a+b \arctan (c x))^3}{4 e}-\frac {3 b c \left (-\frac {i e^4 (a+b \arctan (c x))^2}{3 c^5}-\frac {2 b e^4 \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{3 c^5}+\frac {2 d e^3 (a+b \arctan (c x))^2}{c^4}-\frac {4 i b d e (c d-e) (c d+e) \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right ) (a+b \arctan (c x))}{c^4}-\frac {4 i d e (c d-e) (c d+e) (a+b \arctan (c x))^3}{3 b c^4}-\frac {4 d e (c d-e) (c d+e) \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))^2}{c^4}-\frac {b e^4 x^2 (a+b \arctan (c x))}{3 c^3}+\frac {2 d e^3 x^2 (a+b \arctan (c x))^2}{c^2}+\frac {e^4 x^3 (a+b \arctan (c x))^2}{3 c^2}+\frac {i e^2 \left (6 c^2 d^2-e^2\right ) (a+b \arctan (c x))^2}{c^5}+\frac {2 b e^2 \left (6 c^2 d^2-e^2\right ) \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{c^5}+\frac {e^2 x \left (6 c^2 d^2-e^2\right ) (a+b \arctan (c x))^2}{c^4}+\frac {\left (c^4 d^4-6 c^2 d^2 e^2+e^4\right ) (a+b \arctan (c x))^3}{3 b c^5}-\frac {4 a b d e^3 x}{c^3}-\frac {b^2 e^4 \arctan (c x)}{3 c^5}-\frac {4 b^2 d e^3 x \arctan (c x)}{c^3}-\frac {i b^2 e^4 \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )}{3 c^5}-\frac {2 b^2 d e (c d-e) (c d+e) \operatorname {PolyLog}\left (3,1-\frac {2}{i c x+1}\right )}{c^4}+\frac {b^2 e^4 x}{3 c^4}+\frac {i b^2 e^2 \left (6 c^2 d^2-e^2\right ) \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )}{c^5}+\frac {2 b^2 d e^3 \log \left (c^2 x^2+1\right )}{c^4}\right )}{4 e}\) |
((d + e*x)^4*(a + b*ArcTan[c*x])^3)/(4*e) - (3*b*c*((-4*a*b*d*e^3*x)/c^3 + (b^2*e^4*x)/(3*c^4) - (b^2*e^4*ArcTan[c*x])/(3*c^5) - (4*b^2*d*e^3*x*ArcT an[c*x])/c^3 - (b*e^4*x^2*(a + b*ArcTan[c*x]))/(3*c^3) + (2*d*e^3*(a + b*A rcTan[c*x])^2)/c^4 - ((I/3)*e^4*(a + b*ArcTan[c*x])^2)/c^5 + (I*e^2*(6*c^2 *d^2 - e^2)*(a + b*ArcTan[c*x])^2)/c^5 + (e^2*(6*c^2*d^2 - e^2)*x*(a + b*A rcTan[c*x])^2)/c^4 + (2*d*e^3*x^2*(a + b*ArcTan[c*x])^2)/c^2 + (e^4*x^3*(a + b*ArcTan[c*x])^2)/(3*c^2) - (((4*I)/3)*d*(c*d - e)*e*(c*d + e)*(a + b*A rcTan[c*x])^3)/(b*c^4) + ((c^4*d^4 - 6*c^2*d^2*e^2 + e^4)*(a + b*ArcTan[c* x])^3)/(3*b*c^5) - (2*b*e^4*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(3*c^5 ) + (2*b*e^2*(6*c^2*d^2 - e^2)*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/c^5 - (4*d*(c*d - e)*e*(c*d + e)*(a + b*ArcTan[c*x])^2*Log[2/(1 + I*c*x)])/c^ 4 + (2*b^2*d*e^3*Log[1 + c^2*x^2])/c^4 - ((I/3)*b^2*e^4*PolyLog[2, 1 - 2/( 1 + I*c*x)])/c^5 + (I*b^2*e^2*(6*c^2*d^2 - e^2)*PolyLog[2, 1 - 2/(1 + I*c* x)])/c^5 - ((4*I)*b*d*(c*d - e)*e*(c*d + e)*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 + I*c*x)])/c^4 - (2*b^2*d*(c*d - e)*e*(c*d + e)*PolyLog[3, 1 - 2 /(1 + I*c*x)])/c^4))/(4*e)
3.1.15.3.1 Defintions of rubi rules used
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_)*((d_) + (e_.)*(x_))^(q_.), x_Sy mbol] :> Simp[(d + e*x)^(q + 1)*((a + b*ArcTan[c*x])^p/(e*(q + 1))), x] - S imp[b*c*(p/(e*(q + 1))) Int[ExpandIntegrand[(a + b*ArcTan[c*x])^(p - 1), (d + e*x)^(q + 1)/(1 + c^2*x^2), x], x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 1] && IntegerQ[q] && NeQ[q, -1]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 93.44 (sec) , antiderivative size = 3122, normalized size of antiderivative = 4.79
| method | result | size |
| parts | \(\text {Expression too large to display}\) | \(3122\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(3153\) |
| default | \(\text {Expression too large to display}\) | \(3153\) |
1/4*a^3*(e*x+d)^4/e+b^3/c*(1/4*c*e^3*arctan(c*x)^3*x^4+c*e^2*arctan(c*x)^3 *x^3*d+3/2*c*e*arctan(c*x)^3*x^2*d^2+arctan(c*x)^3*c*x*d^3+1/4*c/e*arctan( c*x)^3*d^4-3/4/c^3/e*(2*I*e^3*c*d*Pi*csgn(I*(1+I*c*x)/(c^2*x^2+1)^(1/2))*c sgn(I*(1+I*c*x)^2/(c^2*x^2+1))^2*arctan(c*x)^2-I*e*c^3*d^3*Pi*csgn(I*(1+(1 +I*c*x)^2/(c^2*x^2+1)))^2*csgn(I*(1+(1+I*c*x)^2/(c^2*x^2+1))^2)*arctan(c*x )^2-2*I*e*c^3*d^3*Pi*csgn(I*(1+I*c*x)/(c^2*x^2+1)^(1/2))*csgn(I*(1+I*c*x)^ 2/(c^2*x^2+1))^2*arctan(c*x)^2+I*e*c^3*d^3*Pi*csgn(I*(1+I*c*x)/(c^2*x^2+1) ^(1/2))^2*csgn(I*(1+I*c*x)^2/(c^2*x^2+1))*arctan(c*x)^2-I*e*c^3*d^3*Pi*csg n(I/(1+(1+I*c*x)^2/(c^2*x^2+1))^2)*csgn(I*(1+I*c*x)^2/(c^2*x^2+1)/(1+(1+I* c*x)^2/(c^2*x^2+1))^2)^2*arctan(c*x)^2+2*I*e*c^3*d^3*Pi*csgn(I*(1+(1+I*c*x )^2/(c^2*x^2+1)))*csgn(I*(1+(1+I*c*x)^2/(c^2*x^2+1))^2)^2*arctan(c*x)^2-I* e*c^3*d^3*Pi*csgn(I*(1+I*c*x)^2/(c^2*x^2+1))*csgn(I*(1+I*c*x)^2/(c^2*x^2+1 )/(1+(1+I*c*x)^2/(c^2*x^2+1))^2)^2*arctan(c*x)^2+I*e^3*c*d*Pi*csgn(I*(1+(1 +I*c*x)^2/(c^2*x^2+1)))^2*csgn(I*(1+(1+I*c*x)^2/(c^2*x^2+1))^2)*arctan(c*x )^2-2*I*e^3*c*d*Pi*csgn(I*(1+(1+I*c*x)^2/(c^2*x^2+1)))*csgn(I*(1+(1+I*c*x) ^2/(c^2*x^2+1))^2)^2*arctan(c*x)^2+I*e^3*c*d*Pi*csgn(I/(1+(1+I*c*x)^2/(c^2 *x^2+1))^2)*csgn(I*(1+I*c*x)^2/(c^2*x^2+1)/(1+(1+I*c*x)^2/(c^2*x^2+1))^2)^ 2*arctan(c*x)^2+I*e^3*c*d*Pi*csgn(I*(1+I*c*x)^2/(c^2*x^2+1))*csgn(I*(1+I*c *x)^2/(c^2*x^2+1)/(1+(1+I*c*x)^2/(c^2*x^2+1))^2)^2*arctan(c*x)^2-I*e^3*c*d *Pi*csgn(I*(1+I*c*x)/(c^2*x^2+1)^(1/2))^2*csgn(I*(1+I*c*x)^2/(c^2*x^2+1...
\[ \int (d+e x)^3 (a+b \arctan (c x))^3 \, dx=\int { {\left (e x + d\right )}^{3} {\left (b \arctan \left (c x\right ) + a\right )}^{3} \,d x } \]
integral(a^3*e^3*x^3 + 3*a^3*d*e^2*x^2 + 3*a^3*d^2*e*x + a^3*d^3 + (b^3*e^ 3*x^3 + 3*b^3*d*e^2*x^2 + 3*b^3*d^2*e*x + b^3*d^3)*arctan(c*x)^3 + 3*(a*b^ 2*e^3*x^3 + 3*a*b^2*d*e^2*x^2 + 3*a*b^2*d^2*e*x + a*b^2*d^3)*arctan(c*x)^2 + 3*(a^2*b*e^3*x^3 + 3*a^2*b*d*e^2*x^2 + 3*a^2*b*d^2*e*x + a^2*b*d^3)*arc tan(c*x), x)
\[ \int (d+e x)^3 (a+b \arctan (c x))^3 \, dx=\int \left (a + b \operatorname {atan}{\left (c x \right )}\right )^{3} \left (d + e x\right )^{3}\, dx \]
\[ \int (d+e x)^3 (a+b \arctan (c x))^3 \, dx=\int { {\left (e x + d\right )}^{3} {\left (b \arctan \left (c x\right ) + a\right )}^{3} \,d x } \]
1/4*a^3*e^3*x^4 + a^3*d*e^2*x^3 + 7/32*b^3*d^3*arctan(c*x)^4/c + 112*b^3*c ^2*e^3*integrate(1/128*x^5*arctan(c*x)^3/(c^2*x^2 + 1), x) + 12*b^3*c^2*e^ 3*integrate(1/128*x^5*arctan(c*x)*log(c^2*x^2 + 1)^2/(c^2*x^2 + 1), x) + 3 84*a*b^2*c^2*e^3*integrate(1/128*x^5*arctan(c*x)^2/(c^2*x^2 + 1), x) + 336 *b^3*c^2*d*e^2*integrate(1/128*x^4*arctan(c*x)^3/(c^2*x^2 + 1), x) + 12*b^ 3*c^2*e^3*integrate(1/128*x^5*arctan(c*x)*log(c^2*x^2 + 1)/(c^2*x^2 + 1), x) + 36*b^3*c^2*d*e^2*integrate(1/128*x^4*arctan(c*x)*log(c^2*x^2 + 1)^2/( c^2*x^2 + 1), x) + 1152*a*b^2*c^2*d*e^2*integrate(1/128*x^4*arctan(c*x)^2/ (c^2*x^2 + 1), x) + 336*b^3*c^2*d^2*e*integrate(1/128*x^3*arctan(c*x)^3/(c ^2*x^2 + 1), x) + 48*b^3*c^2*d*e^2*integrate(1/128*x^4*arctan(c*x)*log(c^2 *x^2 + 1)/(c^2*x^2 + 1), x) + 36*b^3*c^2*d^2*e*integrate(1/128*x^3*arctan( c*x)*log(c^2*x^2 + 1)^2/(c^2*x^2 + 1), x) + 1152*a*b^2*c^2*d^2*e*integrate (1/128*x^3*arctan(c*x)^2/(c^2*x^2 + 1), x) + 112*b^3*c^2*d^3*integrate(1/1 28*x^2*arctan(c*x)^3/(c^2*x^2 + 1), x) + 72*b^3*c^2*d^2*e*integrate(1/128* x^3*arctan(c*x)*log(c^2*x^2 + 1)/(c^2*x^2 + 1), x) + 12*b^3*c^2*d^3*integr ate(1/128*x^2*arctan(c*x)*log(c^2*x^2 + 1)^2/(c^2*x^2 + 1), x) + 384*a*b^2 *c^2*d^3*integrate(1/128*x^2*arctan(c*x)^2/(c^2*x^2 + 1), x) + 48*b^3*c^2* d^3*integrate(1/128*x^2*arctan(c*x)*log(c^2*x^2 + 1)/(c^2*x^2 + 1), x) + 3 /2*a^3*d^2*e*x^2 + a*b^2*d^3*arctan(c*x)^3/c - 12*b^3*c*e^3*integrate(1/12 8*x^4*arctan(c*x)^2/(c^2*x^2 + 1), x) + 3*b^3*c*e^3*integrate(1/128*x^4...
\[ \int (d+e x)^3 (a+b \arctan (c x))^3 \, dx=\int { {\left (e x + d\right )}^{3} {\left (b \arctan \left (c x\right ) + a\right )}^{3} \,d x } \]
Timed out. \[ \int (d+e x)^3 (a+b \arctan (c x))^3 \, dx=\int {\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^3\,{\left (d+e\,x\right )}^3 \,d x \]