\(\int (e+f x)^2 (a+b \arctan (c+d x))^3 \, dx\) [36]

Optimal result

Integrand size = 20, antiderivative size = 564 \[ \int (e+f x)^2 (a+b \arctan (c+d x))^3 \, dx=\frac {a b^2 f^2 x}{d^2}+\frac {b^3 f^2 (c+d x) \arctan (c+d x)}{d^3}-\frac {b f^2 (a+b \arctan (c+d x))^2}{2 d^3}-\frac {3 i b f (d e-c f) (a+b \arctan (c+d x))^2}{d^3}-\frac {3 b f (d e-c f) (c+d x) (a+b \arctan (c+d x))^2}{d^3}-\frac {b f^2 (c+d x)^2 (a+b \arctan (c+d x))^2}{2 d^3}+\frac {i \left (3 d^2 e^2-6 c d e f-\left (1-3 c^2\right ) f^2\right ) (a+b \arctan (c+d x))^3}{3 d^3}-\frac {(d e-c f) \left (d^2 e^2-2 c d e f-\left (3-c^2\right ) f^2\right ) (a+b \arctan (c+d x))^3}{3 d^3 f}+\frac {(e+f x)^3 (a+b \arctan (c+d x))^3}{3 f}-\frac {6 b^2 f (d e-c f) (a+b \arctan (c+d x)) \log \left (\frac {2}{1+i (c+d x)}\right )}{d^3}+\frac {b \left (3 d^2 e^2-6 c d e f-\left (1-3 c^2\right ) f^2\right ) (a+b \arctan (c+d x))^2 \log \left (\frac {2}{1+i (c+d x)}\right )}{d^3}-\frac {b^3 f^2 \log \left (1+(c+d x)^2\right )}{2 d^3}-\frac {3 i b^3 f (d e-c f) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i (c+d x)}\right )}{d^3}+\frac {i b^2 \left (3 d^2 e^2-6 c d e f-\left (1-3 c^2\right ) f^2\right ) (a+b \arctan (c+d x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i (c+d x)}\right )}{d^3}+\frac {b^3 \left (3 d^2 e^2-6 c d e f-\left (1-3 c^2\right ) f^2\right ) \operatorname {PolyLog}\left (3,1-\frac {2}{1+i (c+d x)}\right )}{2 d^3} \]

[Out]

a*b^2*f^2*x/d^2+b^3*f^2*(d*x+c)*arctan(d*x+c)/d^3-1/2*b*f^2*(a+b*arctan(d*x+c))^2/d^3-3*I*b*f*(-c*f+d*e)*(a+b*
arctan(d*x+c))^2/d^3-3*b*f*(-c*f+d*e)*(d*x+c)*(a+b*arctan(d*x+c))^2/d^3-1/2*b*f^2*(d*x+c)^2*(a+b*arctan(d*x+c)
)^2/d^3+1/3*I*(3*d^2*e^2-6*c*d*e*f-(-3*c^2+1)*f^2)*(a+b*arctan(d*x+c))^3/d^3-1/3*(-c*f+d*e)*(d^2*e^2-2*c*d*e*f
-(-c^2+3)*f^2)*(a+b*arctan(d*x+c))^3/d^3/f+1/3*(f*x+e)^3*(a+b*arctan(d*x+c))^3/f-6*b^2*f*(-c*f+d*e)*(a+b*arcta
n(d*x+c))*ln(2/(1+I*(d*x+c)))/d^3+b*(3*d^2*e^2-6*c*d*e*f-(-3*c^2+1)*f^2)*(a+b*arctan(d*x+c))^2*ln(2/(1+I*(d*x+
c)))/d^3-1/2*b^3*f^2*ln(1+(d*x+c)^2)/d^3-3*I*b^3*f*(-c*f+d*e)*polylog(2,1-2/(1+I*(d*x+c)))/d^3+I*b^2*(3*d^2*e^
2-6*c*d*e*f-(-3*c^2+1)*f^2)*(a+b*arctan(d*x+c))*polylog(2,1-2/(1+I*(d*x+c)))/d^3+1/2*b^3*(3*d^2*e^2-6*c*d*e*f-
(-3*c^2+1)*f^2)*polylog(3,1-2/(1+I*(d*x+c)))/d^3

Rubi [A] (verified)

Time = 0.66 (sec) , antiderivative size = 564, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {5155, 4974, 4930, 5040, 4964, 2449, 2352, 4946, 5036, 266, 5004, 5104, 5114, 6745} \[ \int (e+f x)^2 (a+b \arctan (c+d x))^3 \, dx=\frac {i b^2 \left (-\left (1-3 c^2\right ) f^2-6 c d e f+3 d^2 e^2\right ) \operatorname {PolyLog}\left (2,1-\frac {2}{i (c+d x)+1}\right ) (a+b \arctan (c+d x))}{d^3}-\frac {6 b^2 f (d e-c f) \log \left (\frac {2}{1+i (c+d x)}\right ) (a+b \arctan (c+d x))}{d^3}+\frac {i \left (-\left (1-3 c^2\right ) f^2-6 c d e f+3 d^2 e^2\right ) (a+b \arctan (c+d x))^3}{3 d^3}-\frac {(d e-c f) \left (-\left (3-c^2\right ) f^2-2 c d e f+d^2 e^2\right ) (a+b \arctan (c+d x))^3}{3 d^3 f}+\frac {b \left (-\left (1-3 c^2\right ) f^2-6 c d e f+3 d^2 e^2\right ) \log \left (\frac {2}{1+i (c+d x)}\right ) (a+b \arctan (c+d x))^2}{d^3}-\frac {3 i b f (d e-c f) (a+b \arctan (c+d x))^2}{d^3}-\frac {3 b f (c+d x) (d e-c f) (a+b \arctan (c+d x))^2}{d^3}-\frac {b f^2 (a+b \arctan (c+d x))^2}{2 d^3}-\frac {b f^2 (c+d x)^2 (a+b \arctan (c+d x))^2}{2 d^3}+\frac {(e+f x)^3 (a+b \arctan (c+d x))^3}{3 f}+\frac {a b^2 f^2 x}{d^2}+\frac {b^3 f^2 (c+d x) \arctan (c+d x)}{d^3}+\frac {b^3 \left (-\left (1-3 c^2\right ) f^2-6 c d e f+3 d^2 e^2\right ) \operatorname {PolyLog}\left (3,1-\frac {2}{i (c+d x)+1}\right )}{2 d^3}-\frac {3 i b^3 f (d e-c f) \operatorname {PolyLog}\left (2,1-\frac {2}{i (c+d x)+1}\right )}{d^3}-\frac {b^3 f^2 \log \left ((c+d x)^2+1\right )}{2 d^3} \]

[In]

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

[Out]

(a*b^2*f^2*x)/d^2 + (b^3*f^2*(c + d*x)*ArcTan[c + d*x])/d^3 - (b*f^2*(a + b*ArcTan[c + d*x])^2)/(2*d^3) - ((3*
I)*b*f*(d*e - c*f)*(a + b*ArcTan[c + d*x])^2)/d^3 - (3*b*f*(d*e - c*f)*(c + d*x)*(a + b*ArcTan[c + d*x])^2)/d^
3 - (b*f^2*(c + d*x)^2*(a + b*ArcTan[c + d*x])^2)/(2*d^3) + ((I/3)*(3*d^2*e^2 - 6*c*d*e*f - (1 - 3*c^2)*f^2)*(
a + b*ArcTan[c + d*x])^3)/d^3 - ((d*e - c*f)*(d^2*e^2 - 2*c*d*e*f - (3 - c^2)*f^2)*(a + b*ArcTan[c + d*x])^3)/
(3*d^3*f) + ((e + f*x)^3*(a + b*ArcTan[c + d*x])^3)/(3*f) - (6*b^2*f*(d*e - c*f)*(a + b*ArcTan[c + d*x])*Log[2
/(1 + I*(c + d*x))])/d^3 + (b*(3*d^2*e^2 - 6*c*d*e*f - (1 - 3*c^2)*f^2)*(a + b*ArcTan[c + d*x])^2*Log[2/(1 + I
*(c + d*x))])/d^3 - (b^3*f^2*Log[1 + (c + d*x)^2])/(2*d^3) - ((3*I)*b^3*f*(d*e - c*f)*PolyLog[2, 1 - 2/(1 + I*
(c + d*x))])/d^3 + (I*b^2*(3*d^2*e^2 - 6*c*d*e*f - (1 - 3*c^2)*f^2)*(a + b*ArcTan[c + d*x])*PolyLog[2, 1 - 2/(
1 + I*(c + d*x))])/d^3 + (b^3*(3*d^2*e^2 - 6*c*d*e*f - (1 - 3*c^2)*f^2)*PolyLog[3, 1 - 2/(1 + I*(c + d*x))])/(
2*d^3)

Rule 266

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 2352

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLog[2, 1 - c*x], x] /; FreeQ[{c, d, e
}, x] && EqQ[e + c*d, 0]

Rule 2449

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> Dist[-e/g, Subst[Int[Log[2*d*x]/(1 - 2*
d*x), x], x, 1/(d + e*x)], x] /; FreeQ[{c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]

Rule 4930

Int[((a_.) + ArcTan[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*ArcTan[c*x^n])^p, x] - Dist[b*c
*n*p, Int[x^n*((a + b*ArcTan[c*x^n])^(p - 1)/(1 + c^2*x^(2*n))), x], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[p, 0
] && (EqQ[n, 1] || EqQ[p, 1])

Rule 4946

Int[((a_.) + ArcTan[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*((a + b*ArcTan[c*x^
n])^p/(m + 1)), x] - Dist[b*c*n*(p/(m + 1)), Int[x^(m + n)*((a + b*ArcTan[c*x^n])^(p - 1)/(1 + c^2*x^(2*n))),
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0] && (EqQ[p, 1] || (EqQ[n, 1] && IntegerQ[m])) && NeQ[m, -1]

Rule 4964

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcTan[c*x])^p)*(
Log[2/(1 + e*(x/d))]/e), x] + Dist[b*c*(p/e), Int[(a + b*ArcTan[c*x])^(p - 1)*(Log[2/(1 + e*(x/d))]/(1 + c^2*x
^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 + e^2, 0]

Rule 4974

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[(d + e*x)^(q + 1)*((a
+ b*ArcTan[c*x])^p/(e*(q + 1))), x] - Dist[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] && N
eQ[q, -1]

Rule 5004

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(a + b*ArcTan[c*x])^(p +
1)/(b*c*d*(p + 1)), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && NeQ[p, -1]

Rule 5036

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[f^2/
e, Int[(f*x)^(m - 2)*(a + b*ArcTan[c*x])^p, x], x] - Dist[d*(f^2/e), Int[(f*x)^(m - 2)*((a + b*ArcTan[c*x])^p/
(d + e*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && GtQ[m, 1]

Rule 5040

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(-I)*((a + b*ArcT
an[c*x])^(p + 1)/(b*e*(p + 1))), x] - Dist[1/(c*d), Int[(a + b*ArcTan[c*x])^p/(I - c*x), x], x] /; FreeQ[{a, b
, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0]

Rule 5104

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_) + (g_.)*(x_))^(m_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> I
nt[ExpandIntegrand[(a + b*ArcTan[c*x])^p/(d + e*x^2), (f + g*x)^m, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] &
& IGtQ[p, 0] && EqQ[e, c^2*d] && IGtQ[m, 0]

Rule 5114

Int[(Log[u_]*((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(-I)*(a + b*Ar
cTan[c*x])^p*(PolyLog[2, 1 - u]/(2*c*d)), x] + Dist[b*p*(I/2), Int[(a + b*ArcTan[c*x])^(p - 1)*(PolyLog[2, 1 -
u]/(d + e*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[e, c^2*d] && EqQ[(1 - u)^2 - (1 - 2
*(I/(I - c*x)))^2, 0]

Rule 5155

Int[((a_.) + ArcTan[(c_) + (d_.)*(x_)]*(b_.))^(p_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Dist[1/d, Subst[I
nt[((d*e - c*f)/d + f*(x/d))^m*(a + b*ArcTan[x])^p, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x]
&& IGtQ[p, 0]

Rule 6745

Int[(u_)*PolyLog[n_, v_], x_Symbol] :> With[{w = DerivativeDivides[v, u*v, x]}, Simp[w*PolyLog[n + 1, v], x] /
;  !FalseQ[w]] /; FreeQ[n, x]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \left (\frac {d e-c f}{d}+\frac {f x}{d}\right )^2 (a+b \arctan (x))^3 \, dx,x,c+d x\right )}{d} \\ & = \frac {(e+f x)^3 (a+b \arctan (c+d x))^3}{3 f}-\frac {b \text {Subst}\left (\int \left (\frac {3 f^2 (d e-c f) (a+b \arctan (x))^2}{d^3}+\frac {f^3 x (a+b \arctan (x))^2}{d^3}+\frac {\left ((d e-c f) \left (d^2 e^2-2 c d e f-3 f^2+c^2 f^2\right )+f \left (3 d^2 e^2-6 c d e f-\left (1-3 c^2\right ) f^2\right ) x\right ) (a+b \arctan (x))^2}{d^3 \left (1+x^2\right )}\right ) \, dx,x,c+d x\right )}{f} \\ & = \frac {(e+f x)^3 (a+b \arctan (c+d x))^3}{3 f}-\frac {b \text {Subst}\left (\int \frac {\left ((d e-c f) \left (d^2 e^2-2 c d e f-3 f^2+c^2 f^2\right )+f \left (3 d^2 e^2-6 c d e f-\left (1-3 c^2\right ) f^2\right ) x\right ) (a+b \arctan (x))^2}{1+x^2} \, dx,x,c+d x\right )}{d^3 f}-\frac {\left (b f^2\right ) \text {Subst}\left (\int x (a+b \arctan (x))^2 \, dx,x,c+d x\right )}{d^3}-\frac {(3 b f (d e-c f)) \text {Subst}\left (\int (a+b \arctan (x))^2 \, dx,x,c+d x\right )}{d^3} \\ & = -\frac {3 b f (d e-c f) (c+d x) (a+b \arctan (c+d x))^2}{d^3}-\frac {b f^2 (c+d x)^2 (a+b \arctan (c+d x))^2}{2 d^3}+\frac {(e+f x)^3 (a+b \arctan (c+d x))^3}{3 f}-\frac {b \text {Subst}\left (\int \left (\frac {(d e-c f) \left (d^2 e^2-2 c d e f-\left (3-c^2\right ) f^2\right ) (a+b \arctan (x))^2}{1+x^2}+\frac {f \left (3 d^2 e^2-6 c d e f-\left (1-3 c^2\right ) f^2\right ) x (a+b \arctan (x))^2}{1+x^2}\right ) \, dx,x,c+d x\right )}{d^3 f}+\frac {\left (b^2 f^2\right ) \text {Subst}\left (\int \frac {x^2 (a+b \arctan (x))}{1+x^2} \, dx,x,c+d x\right )}{d^3}+\frac {\left (6 b^2 f (d e-c f)\right ) \text {Subst}\left (\int \frac {x (a+b \arctan (x))}{1+x^2} \, dx,x,c+d x\right )}{d^3} \\ & = -\frac {3 i b f (d e-c f) (a+b \arctan (c+d x))^2}{d^3}-\frac {3 b f (d e-c f) (c+d x) (a+b \arctan (c+d x))^2}{d^3}-\frac {b f^2 (c+d x)^2 (a+b \arctan (c+d x))^2}{2 d^3}+\frac {(e+f x)^3 (a+b \arctan (c+d x))^3}{3 f}+\frac {\left (b^2 f^2\right ) \text {Subst}(\int (a+b \arctan (x)) \, dx,x,c+d x)}{d^3}-\frac {\left (b^2 f^2\right ) \text {Subst}\left (\int \frac {a+b \arctan (x)}{1+x^2} \, dx,x,c+d x\right )}{d^3}-\frac {\left (6 b^2 f (d e-c f)\right ) \text {Subst}\left (\int \frac {a+b \arctan (x)}{i-x} \, dx,x,c+d x\right )}{d^3}-\frac {\left (b \left (3 d^2 e^2-6 c d e f-\left (1-3 c^2\right ) f^2\right )\right ) \text {Subst}\left (\int \frac {x (a+b \arctan (x))^2}{1+x^2} \, dx,x,c+d x\right )}{d^3}-\frac {\left (b (d e-c f) \left (d^2 e^2-2 c d e f-\left (3-c^2\right ) f^2\right )\right ) \text {Subst}\left (\int \frac {(a+b \arctan (x))^2}{1+x^2} \, dx,x,c+d x\right )}{d^3 f} \\ & = \frac {a b^2 f^2 x}{d^2}-\frac {b f^2 (a+b \arctan (c+d x))^2}{2 d^3}-\frac {3 i b f (d e-c f) (a+b \arctan (c+d x))^2}{d^3}-\frac {3 b f (d e-c f) (c+d x) (a+b \arctan (c+d x))^2}{d^3}-\frac {b f^2 (c+d x)^2 (a+b \arctan (c+d x))^2}{2 d^3}+\frac {i \left (3 d^2 e^2-6 c d e f-\left (1-3 c^2\right ) f^2\right ) (a+b \arctan (c+d x))^3}{3 d^3}-\frac {(d e-c f) \left (d^2 e^2-2 c d e f-\left (3-c^2\right ) f^2\right ) (a+b \arctan (c+d x))^3}{3 d^3 f}+\frac {(e+f x)^3 (a+b \arctan (c+d x))^3}{3 f}-\frac {6 b^2 f (d e-c f) (a+b \arctan (c+d x)) \log \left (\frac {2}{1+i (c+d x)}\right )}{d^3}+\frac {\left (b^3 f^2\right ) \text {Subst}(\int \arctan (x) \, dx,x,c+d x)}{d^3}+\frac {\left (6 b^3 f (d e-c f)\right ) \text {Subst}\left (\int \frac {\log \left (\frac {2}{1+i x}\right )}{1+x^2} \, dx,x,c+d x\right )}{d^3}+\frac {\left (b \left (3 d^2 e^2-6 c d e f-\left (1-3 c^2\right ) f^2\right )\right ) \text {Subst}\left (\int \frac {(a+b \arctan (x))^2}{i-x} \, dx,x,c+d x\right )}{d^3} \\ & = \frac {a b^2 f^2 x}{d^2}+\frac {b^3 f^2 (c+d x) \arctan (c+d x)}{d^3}-\frac {b f^2 (a+b \arctan (c+d x))^2}{2 d^3}-\frac {3 i b f (d e-c f) (a+b \arctan (c+d x))^2}{d^3}-\frac {3 b f (d e-c f) (c+d x) (a+b \arctan (c+d x))^2}{d^3}-\frac {b f^2 (c+d x)^2 (a+b \arctan (c+d x))^2}{2 d^3}+\frac {i \left (3 d^2 e^2-6 c d e f-\left (1-3 c^2\right ) f^2\right ) (a+b \arctan (c+d x))^3}{3 d^3}-\frac {(d e-c f) \left (d^2 e^2-2 c d e f-\left (3-c^2\right ) f^2\right ) (a+b \arctan (c+d x))^3}{3 d^3 f}+\frac {(e+f x)^3 (a+b \arctan (c+d x))^3}{3 f}-\frac {6 b^2 f (d e-c f) (a+b \arctan (c+d x)) \log \left (\frac {2}{1+i (c+d x)}\right )}{d^3}+\frac {b \left (3 d^2 e^2-6 c d e f-\left (1-3 c^2\right ) f^2\right ) (a+b \arctan (c+d x))^2 \log \left (\frac {2}{1+i (c+d x)}\right )}{d^3}-\frac {\left (b^3 f^2\right ) \text {Subst}\left (\int \frac {x}{1+x^2} \, dx,x,c+d x\right )}{d^3}-\frac {\left (6 i b^3 f (d e-c f)\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i (c+d x)}\right )}{d^3}-\frac {\left (2 b^2 \left (3 d^2 e^2-6 c d e f-\left (1-3 c^2\right ) f^2\right )\right ) \text {Subst}\left (\int \frac {(a+b \arctan (x)) \log \left (\frac {2}{1+i x}\right )}{1+x^2} \, dx,x,c+d x\right )}{d^3} \\ & = \frac {a b^2 f^2 x}{d^2}+\frac {b^3 f^2 (c+d x) \arctan (c+d x)}{d^3}-\frac {b f^2 (a+b \arctan (c+d x))^2}{2 d^3}-\frac {3 i b f (d e-c f) (a+b \arctan (c+d x))^2}{d^3}-\frac {3 b f (d e-c f) (c+d x) (a+b \arctan (c+d x))^2}{d^3}-\frac {b f^2 (c+d x)^2 (a+b \arctan (c+d x))^2}{2 d^3}+\frac {i \left (3 d^2 e^2-6 c d e f-\left (1-3 c^2\right ) f^2\right ) (a+b \arctan (c+d x))^3}{3 d^3}-\frac {(d e-c f) \left (d^2 e^2-2 c d e f-\left (3-c^2\right ) f^2\right ) (a+b \arctan (c+d x))^3}{3 d^3 f}+\frac {(e+f x)^3 (a+b \arctan (c+d x))^3}{3 f}-\frac {6 b^2 f (d e-c f) (a+b \arctan (c+d x)) \log \left (\frac {2}{1+i (c+d x)}\right )}{d^3}+\frac {b \left (3 d^2 e^2-6 c d e f-\left (1-3 c^2\right ) f^2\right ) (a+b \arctan (c+d x))^2 \log \left (\frac {2}{1+i (c+d x)}\right )}{d^3}-\frac {b^3 f^2 \log \left (1+(c+d x)^2\right )}{2 d^3}-\frac {3 i b^3 f (d e-c f) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i (c+d x)}\right )}{d^3}+\frac {i b^2 \left (3 d^2 e^2-6 c d e f-\left (1-3 c^2\right ) f^2\right ) (a+b \arctan (c+d x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i (c+d x)}\right )}{d^3}-\frac {\left (i b^3 \left (3 d^2 e^2-6 c d e f-\left (1-3 c^2\right ) f^2\right )\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1+i x}\right )}{1+x^2} \, dx,x,c+d x\right )}{d^3} \\ & = \frac {a b^2 f^2 x}{d^2}+\frac {b^3 f^2 (c+d x) \arctan (c+d x)}{d^3}-\frac {b f^2 (a+b \arctan (c+d x))^2}{2 d^3}-\frac {3 i b f (d e-c f) (a+b \arctan (c+d x))^2}{d^3}-\frac {3 b f (d e-c f) (c+d x) (a+b \arctan (c+d x))^2}{d^3}-\frac {b f^2 (c+d x)^2 (a+b \arctan (c+d x))^2}{2 d^3}+\frac {i \left (3 d^2 e^2-6 c d e f-\left (1-3 c^2\right ) f^2\right ) (a+b \arctan (c+d x))^3}{3 d^3}-\frac {(d e-c f) \left (d^2 e^2-2 c d e f-\left (3-c^2\right ) f^2\right ) (a+b \arctan (c+d x))^3}{3 d^3 f}+\frac {(e+f x)^3 (a+b \arctan (c+d x))^3}{3 f}-\frac {6 b^2 f (d e-c f) (a+b \arctan (c+d x)) \log \left (\frac {2}{1+i (c+d x)}\right )}{d^3}+\frac {b \left (3 d^2 e^2-6 c d e f-\left (1-3 c^2\right ) f^2\right ) (a+b \arctan (c+d x))^2 \log \left (\frac {2}{1+i (c+d x)}\right )}{d^3}-\frac {b^3 f^2 \log \left (1+(c+d x)^2\right )}{2 d^3}-\frac {3 i b^3 f (d e-c f) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i (c+d x)}\right )}{d^3}+\frac {i b^2 \left (3 d^2 e^2-6 c d e f-\left (1-3 c^2\right ) f^2\right ) (a+b \arctan (c+d x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i (c+d x)}\right )}{d^3}+\frac {b^3 \left (3 d^2 e^2-6 c d e f-\left (1-3 c^2\right ) f^2\right ) \operatorname {PolyLog}\left (3,1-\frac {2}{1+i (c+d x)}\right )}{2 d^3} \\ \end{align*}

Mathematica [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1844\) vs. \(2(564)=1128\).

Time = 13.86 (sec) , antiderivative size = 1844, normalized size of antiderivative = 3.27 \[ \int (e+f x)^2 (a+b \arctan (c+d x))^3 \, dx=\frac {a^2 \left (a d^2 e^2-3 b d e f+2 b c f^2\right ) x}{d^2}-\frac {a^2 f (-2 a d e+b f) x^2}{2 d}+\frac {1}{3} a^3 f^2 x^3+\frac {\left (3 a^2 b c d^2 e^2+3 a^2 b d e f-3 a^2 b c^2 d e f-3 a^2 b c f^2+a^2 b c^3 f^2\right ) \arctan (c+d x)}{d^3}+a^2 b x \left (3 e^2+3 e f x+f^2 x^2\right ) \arctan (c+d x)+\frac {\left (-3 a^2 b d^2 e^2+6 a^2 b c d e f+a^2 b f^2-3 a^2 b c^2 f^2\right ) \log \left (1+c^2+2 c d x+d^2 x^2\right )}{2 d^3}+\frac {3 a b^2 e^2 \left (-i \arctan (c+d x)^2+(c+d x) \arctan (c+d x)^2+2 \arctan (c+d x) \log \left (1+e^{2 i \arctan (c+d x)}\right )-i \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c+d x)}\right )\right )}{d}+6 a b^2 e f \left (-\frac {(c+d x) \arctan (c+d x)}{d^2}+\frac {i c \arctan (c+d x)^2}{d^2}-\frac {c (c+d x) \arctan (c+d x)^2}{d^2}+\frac {\left (1+(c+d x)^2\right ) \arctan (c+d x)^2}{2 d^2}-\frac {2 c \arctan (c+d x) \log \left (1+e^{2 i \arctan (c+d x)}\right )}{d^2}-\frac {\log \left (\frac {1}{\sqrt {1+(c+d x)^2}}\right )}{d^2}+\frac {i c \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c+d x)}\right )}{d^2}\right )+\frac {b^3 e^2 \left (-i \arctan (c+d x)^3+(c+d x) \arctan (c+d x)^3+3 \arctan (c+d x)^2 \log \left (1+e^{2 i \arctan (c+d x)}\right )-3 i \arctan (c+d x) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c+d x)}\right )+\frac {3}{2} \operatorname {PolyLog}\left (3,-e^{2 i \arctan (c+d x)}\right )\right )}{d}+\frac {b^3 e f \left (\arctan (c+d x) \left (3 i \arctan (c+d x)+2 i c \arctan (c+d x)^2+\left (1+(c+d x)^2\right ) \arctan (c+d x)^2-(c+d x) \arctan (c+d x) (3+2 c \arctan (c+d x))-6 \log \left (1+e^{2 i \arctan (c+d x)}\right )-6 c \arctan (c+d x) \log \left (1+e^{2 i \arctan (c+d x)}\right )\right )+3 i (1+2 c \arctan (c+d x)) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c+d x)}\right )-3 c \operatorname {PolyLog}\left (3,-e^{2 i \arctan (c+d x)}\right )\right )}{d^2}+\frac {a b^2 f^2 \left (1+(c+d x)^2\right )^{3/2} \left (\frac {c+d x}{\sqrt {1+(c+d x)^2}}+\frac {6 c (c+d x) \arctan (c+d x)}{\sqrt {1+(c+d x)^2}}+\frac {3 (c+d x) \arctan (c+d x)^2}{\sqrt {1+(c+d x)^2}}+\frac {3 c^2 (c+d x) \arctan (c+d x)^2}{\sqrt {1+(c+d x)^2}}+i \arctan (c+d x)^2 \cos (3 \arctan (c+d x))-3 i c^2 \arctan (c+d x)^2 \cos (3 \arctan (c+d x))-2 \arctan (c+d x) \cos (3 \arctan (c+d x)) \log \left (1+e^{2 i \arctan (c+d x)}\right )+6 c^2 \arctan (c+d x) \cos (3 \arctan (c+d x)) \log \left (1+e^{2 i \arctan (c+d x)}\right )+6 c \cos (3 \arctan (c+d x)) \log \left (\frac {1}{\sqrt {1+(c+d x)^2}}\right )+\frac {\arctan (c+d x) \left (-4+\left (3 i-12 c-9 i c^2\right ) \arctan (c+d x)\right )+6 \left (-1+3 c^2\right ) \arctan (c+d x) \log \left (1+e^{2 i \arctan (c+d x)}\right )+18 c \log \left (\frac {1}{\sqrt {1+(c+d x)^2}}\right )}{\sqrt {1+(c+d x)^2}}-\frac {4 i \left (-1+3 c^2\right ) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c+d x)}\right )}{\left (1+(c+d x)^2\right )^{3/2}}+\sin (3 \arctan (c+d x))+6 c \arctan (c+d x) \sin (3 \arctan (c+d x))-\arctan (c+d x)^2 \sin (3 \arctan (c+d x))+3 c^2 \arctan (c+d x)^2 \sin (3 \arctan (c+d x))\right )}{4 d^3}+\frac {b^3 f^2 \left (-i \left (3 c-\arctan (c+d x)+3 c^2 \arctan (c+d x)\right ) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c+d x)}\right )+\frac {1}{12} \left (1+(c+d x)^2\right )^{3/2} \left (\frac {3 (c+d x) \arctan (c+d x)}{\sqrt {1+(c+d x)^2}}+\frac {9 c (c+d x) \arctan (c+d x)^2}{\sqrt {1+(c+d x)^2}}+\frac {3 (c+d x) \arctan (c+d x)^3}{\sqrt {1+(c+d x)^2}}+\frac {3 c^2 (c+d x) \arctan (c+d x)^3}{\sqrt {1+(c+d x)^2}}-9 i c \arctan (c+d x)^2 \cos (3 \arctan (c+d x))+i \arctan (c+d x)^3 \cos (3 \arctan (c+d x))-3 i c^2 \arctan (c+d x)^3 \cos (3 \arctan (c+d x))+18 c \arctan (c+d x) \cos (3 \arctan (c+d x)) \log \left (1+e^{2 i \arctan (c+d x)}\right )-3 \arctan (c+d x)^2 \cos (3 \arctan (c+d x)) \log \left (1+e^{2 i \arctan (c+d x)}\right )+9 c^2 \arctan (c+d x)^2 \cos (3 \arctan (c+d x)) \log \left (1+e^{2 i \arctan (c+d x)}\right )+3 \cos (3 \arctan (c+d x)) \log \left (\frac {1}{\sqrt {1+(c+d x)^2}}\right )+\frac {3 \left (\arctan (c+d x)^2 \left (-2-9 i c+i \arctan (c+d x)-4 c \arctan (c+d x)-3 i c^2 \arctan (c+d x)\right )+3 \arctan (c+d x) \left (6 c-\arctan (c+d x)+3 c^2 \arctan (c+d x)\right ) \log \left (1+e^{2 i \arctan (c+d x)}\right )+3 \log \left (\frac {1}{\sqrt {1+(c+d x)^2}}\right )\right )}{\sqrt {1+(c+d x)^2}}+\frac {6 \left (-1+3 c^2\right ) \operatorname {PolyLog}\left (3,-e^{2 i \arctan (c+d x)}\right )}{\left (1+(c+d x)^2\right )^{3/2}}+3 \arctan (c+d x) \sin (3 \arctan (c+d x))+9 c \arctan (c+d x)^2 \sin (3 \arctan (c+d x))-\arctan (c+d x)^3 \sin (3 \arctan (c+d x))+3 c^2 \arctan (c+d x)^3 \sin (3 \arctan (c+d x))\right )\right )}{d^3} \]

[In]

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

[Out]

(a^2*(a*d^2*e^2 - 3*b*d*e*f + 2*b*c*f^2)*x)/d^2 - (a^2*f*(-2*a*d*e + b*f)*x^2)/(2*d) + (a^3*f^2*x^3)/3 + ((3*a
^2*b*c*d^2*e^2 + 3*a^2*b*d*e*f - 3*a^2*b*c^2*d*e*f - 3*a^2*b*c*f^2 + a^2*b*c^3*f^2)*ArcTan[c + d*x])/d^3 + a^2
*b*x*(3*e^2 + 3*e*f*x + f^2*x^2)*ArcTan[c + d*x] + ((-3*a^2*b*d^2*e^2 + 6*a^2*b*c*d*e*f + a^2*b*f^2 - 3*a^2*b*
c^2*f^2)*Log[1 + c^2 + 2*c*d*x + d^2*x^2])/(2*d^3) + (3*a*b^2*e^2*((-I)*ArcTan[c + d*x]^2 + (c + d*x)*ArcTan[c
+ d*x]^2 + 2*ArcTan[c + d*x]*Log[1 + E^((2*I)*ArcTan[c + d*x])] - I*PolyLog[2, -E^((2*I)*ArcTan[c + d*x])]))/
d + 6*a*b^2*e*f*(-(((c + d*x)*ArcTan[c + d*x])/d^2) + (I*c*ArcTan[c + d*x]^2)/d^2 - (c*(c + d*x)*ArcTan[c + d*
x]^2)/d^2 + ((1 + (c + d*x)^2)*ArcTan[c + d*x]^2)/(2*d^2) - (2*c*ArcTan[c + d*x]*Log[1 + E^((2*I)*ArcTan[c + d
*x])])/d^2 - Log[1/Sqrt[1 + (c + d*x)^2]]/d^2 + (I*c*PolyLog[2, -E^((2*I)*ArcTan[c + d*x])])/d^2) + (b^3*e^2*(
(-I)*ArcTan[c + d*x]^3 + (c + d*x)*ArcTan[c + d*x]^3 + 3*ArcTan[c + d*x]^2*Log[1 + E^((2*I)*ArcTan[c + d*x])]
- (3*I)*ArcTan[c + d*x]*PolyLog[2, -E^((2*I)*ArcTan[c + d*x])] + (3*PolyLog[3, -E^((2*I)*ArcTan[c + d*x])])/2)
)/d + (b^3*e*f*(ArcTan[c + d*x]*((3*I)*ArcTan[c + d*x] + (2*I)*c*ArcTan[c + d*x]^2 + (1 + (c + d*x)^2)*ArcTan[
c + d*x]^2 - (c + d*x)*ArcTan[c + d*x]*(3 + 2*c*ArcTan[c + d*x]) - 6*Log[1 + E^((2*I)*ArcTan[c + d*x])] - 6*c*
ArcTan[c + d*x]*Log[1 + E^((2*I)*ArcTan[c + d*x])]) + (3*I)*(1 + 2*c*ArcTan[c + d*x])*PolyLog[2, -E^((2*I)*Arc
Tan[c + d*x])] - 3*c*PolyLog[3, -E^((2*I)*ArcTan[c + d*x])]))/d^2 + (a*b^2*f^2*(1 + (c + d*x)^2)^(3/2)*((c + d
*x)/Sqrt[1 + (c + d*x)^2] + (6*c*(c + d*x)*ArcTan[c + d*x])/Sqrt[1 + (c + d*x)^2] + (3*(c + d*x)*ArcTan[c + d*
x]^2)/Sqrt[1 + (c + d*x)^2] + (3*c^2*(c + d*x)*ArcTan[c + d*x]^2)/Sqrt[1 + (c + d*x)^2] + I*ArcTan[c + d*x]^2*
Cos[3*ArcTan[c + d*x]] - (3*I)*c^2*ArcTan[c + d*x]^2*Cos[3*ArcTan[c + d*x]] - 2*ArcTan[c + d*x]*Cos[3*ArcTan[c
+ d*x]]*Log[1 + E^((2*I)*ArcTan[c + d*x])] + 6*c^2*ArcTan[c + d*x]*Cos[3*ArcTan[c + d*x]]*Log[1 + E^((2*I)*Ar
cTan[c + d*x])] + 6*c*Cos[3*ArcTan[c + d*x]]*Log[1/Sqrt[1 + (c + d*x)^2]] + (ArcTan[c + d*x]*(-4 + (3*I - 12*c
- (9*I)*c^2)*ArcTan[c + d*x]) + 6*(-1 + 3*c^2)*ArcTan[c + d*x]*Log[1 + E^((2*I)*ArcTan[c + d*x])] + 18*c*Log[
1/Sqrt[1 + (c + d*x)^2]])/Sqrt[1 + (c + d*x)^2] - ((4*I)*(-1 + 3*c^2)*PolyLog[2, -E^((2*I)*ArcTan[c + d*x])])/
(1 + (c + d*x)^2)^(3/2) + Sin[3*ArcTan[c + d*x]] + 6*c*ArcTan[c + d*x]*Sin[3*ArcTan[c + d*x]] - ArcTan[c + d*x
]^2*Sin[3*ArcTan[c + d*x]] + 3*c^2*ArcTan[c + d*x]^2*Sin[3*ArcTan[c + d*x]]))/(4*d^3) + (b^3*f^2*((-I)*(3*c -
ArcTan[c + d*x] + 3*c^2*ArcTan[c + d*x])*PolyLog[2, -E^((2*I)*ArcTan[c + d*x])] + ((1 + (c + d*x)^2)^(3/2)*((3
*(c + d*x)*ArcTan[c + d*x])/Sqrt[1 + (c + d*x)^2] + (9*c*(c + d*x)*ArcTan[c + d*x]^2)/Sqrt[1 + (c + d*x)^2] +
(3*(c + d*x)*ArcTan[c + d*x]^3)/Sqrt[1 + (c + d*x)^2] + (3*c^2*(c + d*x)*ArcTan[c + d*x]^3)/Sqrt[1 + (c + d*x)
^2] - (9*I)*c*ArcTan[c + d*x]^2*Cos[3*ArcTan[c + d*x]] + I*ArcTan[c + d*x]^3*Cos[3*ArcTan[c + d*x]] - (3*I)*c^
2*ArcTan[c + d*x]^3*Cos[3*ArcTan[c + d*x]] + 18*c*ArcTan[c + d*x]*Cos[3*ArcTan[c + d*x]]*Log[1 + E^((2*I)*ArcT
an[c + d*x])] - 3*ArcTan[c + d*x]^2*Cos[3*ArcTan[c + d*x]]*Log[1 + E^((2*I)*ArcTan[c + d*x])] + 9*c^2*ArcTan[c
+ d*x]^2*Cos[3*ArcTan[c + d*x]]*Log[1 + E^((2*I)*ArcTan[c + d*x])] + 3*Cos[3*ArcTan[c + d*x]]*Log[1/Sqrt[1 +
(c + d*x)^2]] + (3*(ArcTan[c + d*x]^2*(-2 - (9*I)*c + I*ArcTan[c + d*x] - 4*c*ArcTan[c + d*x] - (3*I)*c^2*ArcT
an[c + d*x]) + 3*ArcTan[c + d*x]*(6*c - ArcTan[c + d*x] + 3*c^2*ArcTan[c + d*x])*Log[1 + E^((2*I)*ArcTan[c + d
*x])] + 3*Log[1/Sqrt[1 + (c + d*x)^2]]))/Sqrt[1 + (c + d*x)^2] + (6*(-1 + 3*c^2)*PolyLog[3, -E^((2*I)*ArcTan[c
+ d*x])])/(1 + (c + d*x)^2)^(3/2) + 3*ArcTan[c + d*x]*Sin[3*ArcTan[c + d*x]] + 9*c*ArcTan[c + d*x]^2*Sin[3*Ar
cTan[c + d*x]] - ArcTan[c + d*x]^3*Sin[3*ArcTan[c + d*x]] + 3*c^2*ArcTan[c + d*x]^3*Sin[3*ArcTan[c + d*x]]))/1
2))/d^3

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 28.49 (sec) , antiderivative size = 5843, normalized size of antiderivative = 10.36

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(5843\)
default	\(\text {Expression too large to display}\)	\(5843\)
parts	\(\text {Expression too large to display}\)	\(6026\)

[In]

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

[Out]

result too large to display

Fricas [F]

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

[In]

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

[Out]

integral(a^3*f^2*x^2 + 2*a^3*e*f*x + a^3*e^2 + (b^3*f^2*x^2 + 2*b^3*e*f*x + b^3*e^2)*arctan(d*x + c)^3 + 3*(a*
b^2*f^2*x^2 + 2*a*b^2*e*f*x + a*b^2*e^2)*arctan(d*x + c)^2 + 3*(a^2*b*f^2*x^2 + 2*a^2*b*e*f*x + a^2*b*e^2)*arc
tan(d*x + c), x)

Sympy [F(-1)]

Timed out. \[ \int (e+f x)^2 (a+b \arctan (c+d x))^3 \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

7/8*b^3*c^2*e^2*arctan(d*x + c)^3*arctan((d^2*x + c*d)/d)/d + 3*a*b^2*c^2*e^2*arctan(d*x + c)^2*arctan((d^2*x
+ c*d)/d)/d - (3*arctan(d*x + c)*arctan((d^2*x + c*d)/d)^2/d - arctan((d^2*x + c*d)/d)^3/d)*a*b^2*c^2*e^2 - 7/
32*(6*arctan(d*x + c)^2*arctan((d^2*x + c*d)/d)^2/d - 4*arctan(d*x + c)*arctan((d^2*x + c*d)/d)^3/d + arctan((
d^2*x + c*d)/d)^4/d)*b^3*c^2*e^2 + 1/3*a^3*f^2*x^3 + 7/8*b^3*e^2*arctan(d*x + c)^3*arctan((d^2*x + c*d)/d)/d +
28*b^3*d^2*f^2*integrate(1/32*x^4*arctan(d*x + c)^3/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 3*b^3*d^2*f^2*integra
te(1/32*x^4*arctan(d*x + c)*log(d^2*x^2 + 2*c*d*x + c^2 + 1)^2/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 96*a*b^2*d^
2*f^2*integrate(1/32*x^4*arctan(d*x + c)^2/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 56*b^3*d^2*e*f*integrate(1/32*x
^3*arctan(d*x + c)^3/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 56*b^3*c*d*f^2*integrate(1/32*x^3*arctan(d*x + c)^3/(
d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 4*b^3*d^2*f^2*integrate(1/32*x^4*arctan(d*x + c)*log(d^2*x^2 + 2*c*d*x + c^
2 + 1)/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 6*b^3*d^2*e*f*integrate(1/32*x^3*arctan(d*x + c)*log(d^2*x^2 + 2*c*
d*x + c^2 + 1)^2/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 6*b^3*c*d*f^2*integrate(1/32*x^3*arctan(d*x + c)*log(d^2*
x^2 + 2*c*d*x + c^2 + 1)^2/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 192*a*b^2*d^2*e*f*integrate(1/32*x^3*arctan(d*x
+ c)^2/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 192*a*b^2*c*d*f^2*integrate(1/32*x^3*arctan(d*x + c)^2/(d^2*x^2 +
2*c*d*x + c^2 + 1), x) + 28*b^3*d^2*e^2*integrate(1/32*x^2*arctan(d*x + c)^3/(d^2*x^2 + 2*c*d*x + c^2 + 1), x)
+ 112*b^3*c*d*e*f*integrate(1/32*x^2*arctan(d*x + c)^3/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 28*b^3*c^2*f^2*int
egrate(1/32*x^2*arctan(d*x + c)^3/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 12*b^3*d^2*e*f*integrate(1/32*x^3*arctan
(d*x + c)*log(d^2*x^2 + 2*c*d*x + c^2 + 1)/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 4*b^3*c*d*f^2*integrate(1/32*x^
3*arctan(d*x + c)*log(d^2*x^2 + 2*c*d*x + c^2 + 1)/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 3*b^3*d^2*e^2*integrate
(1/32*x^2*arctan(d*x + c)*log(d^2*x^2 + 2*c*d*x + c^2 + 1)^2/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 12*b^3*c*d*e*
f*integrate(1/32*x^2*arctan(d*x + c)*log(d^2*x^2 + 2*c*d*x + c^2 + 1)^2/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 3*
b^3*c^2*f^2*integrate(1/32*x^2*arctan(d*x + c)*log(d^2*x^2 + 2*c*d*x + c^2 + 1)^2/(d^2*x^2 + 2*c*d*x + c^2 + 1
), x) + 96*a*b^2*d^2*e^2*integrate(1/32*x^2*arctan(d*x + c)^2/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 384*a*b^2*c*
d*e*f*integrate(1/32*x^2*arctan(d*x + c)^2/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 96*a*b^2*c^2*f^2*integrate(1/32
*x^2*arctan(d*x + c)^2/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 56*b^3*c*d*e^2*integrate(1/32*x*arctan(d*x + c)^3/(
d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 56*b^3*c^2*e*f*integrate(1/32*x*arctan(d*x + c)^3/(d^2*x^2 + 2*c*d*x + c^2
+ 1), x) + 12*b^3*d^2*e^2*integrate(1/32*x^2*arctan(d*x + c)*log(d^2*x^2 + 2*c*d*x + c^2 + 1)/(d^2*x^2 + 2*c*d
*x + c^2 + 1), x) + 12*b^3*c*d*e*f*integrate(1/32*x^2*arctan(d*x + c)*log(d^2*x^2 + 2*c*d*x + c^2 + 1)/(d^2*x^
2 + 2*c*d*x + c^2 + 1), x) + 6*b^3*c*d*e^2*integrate(1/32*x*arctan(d*x + c)*log(d^2*x^2 + 2*c*d*x + c^2 + 1)^2
/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 6*b^3*c^2*e*f*integrate(1/32*x*arctan(d*x + c)*log(d^2*x^2 + 2*c*d*x + c^
2 + 1)^2/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 192*a*b^2*c*d*e^2*integrate(1/32*x*arctan(d*x + c)^2/(d^2*x^2 + 2
*c*d*x + c^2 + 1), x) + 192*a*b^2*c^2*e*f*integrate(1/32*x*arctan(d*x + c)^2/(d^2*x^2 + 2*c*d*x + c^2 + 1), x)
+ 12*b^3*c*d*e^2*integrate(1/32*x*arctan(d*x + c)*log(d^2*x^2 + 2*c*d*x + c^2 + 1)/(d^2*x^2 + 2*c*d*x + c^2 +
1), x) + 3*b^3*c^2*e^2*integrate(1/32*arctan(d*x + c)*log(d^2*x^2 + 2*c*d*x + c^2 + 1)^2/(d^2*x^2 + 2*c*d*x +
c^2 + 1), x) + a^3*e*f*x^2 + 3*a*b^2*e^2*arctan(d*x + c)^2*arctan((d^2*x + c*d)/d)/d - 4*b^3*d*f^2*integrate(
1/32*x^3*arctan(d*x + c)^2/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + b^3*d*f^2*integrate(1/32*x^3*log(d^2*x^2 + 2*c*
d*x + c^2 + 1)^2/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) - 12*b^3*d*e*f*integrate(1/32*x^2*arctan(d*x + c)^2/(d^2*x^
2 + 2*c*d*x + c^2 + 1), x) + 3*b^3*d*e*f*integrate(1/32*x^2*log(d^2*x^2 + 2*c*d*x + c^2 + 1)^2/(d^2*x^2 + 2*c*
d*x + c^2 + 1), x) - 12*b^3*d*e^2*integrate(1/32*x*arctan(d*x + c)^2/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 3*b^3
*d*e^2*integrate(1/32*x*log(d^2*x^2 + 2*c*d*x + c^2 + 1)^2/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) - (3*arctan(d*x +
c)*arctan((d^2*x + c*d)/d)^2/d - arctan((d^2*x + c*d)/d)^3/d)*a*b^2*e^2 - 7/32*(6*arctan(d*x + c)^2*arctan((d
^2*x + c*d)/d)^2/d - 4*arctan(d*x + c)*arctan((d^2*x + c*d)/d)^3/d + arctan((d^2*x + c*d)/d)^4/d)*b^3*e^2 + 3*
(x^2*arctan(d*x + c) - d*(x/d^2 + (c^2 - 1)*arctan((d^2*x + c*d)/d)/d^3 - c*log(d^2*x^2 + 2*c*d*x + c^2 + 1)/d
^3))*a^2*b*e*f + 1/2*(2*x^3*arctan(d*x + c) - d*((d*x^2 - 4*c*x)/d^3 - 2*(c^3 - 3*c)*arctan((d^2*x + c*d)/d)/d
^4 + (3*c^2 - 1)*log(d^2*x^2 + 2*c*d*x + c^2 + 1)/d^4))*a^2*b*f^2 + a^3*e^2*x + 28*b^3*f^2*integrate(1/32*x^2*
arctan(d*x + c)^3/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 3*b^3*f^2*integrate(1/32*x^2*arctan(d*x + c)*log(d^2*x^2
+ 2*c*d*x + c^2 + 1)^2/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 96*a*b^2*f^2*integrate(1/32*x^2*arctan(d*x + c)^2/
(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 56*b^3*e*f*integrate(1/32*x*arctan(d*x + c)^3/(d^2*x^2 + 2*c*d*x + c^2 + 1
), x) + 6*b^3*e*f*integrate(1/32*x*arctan(d*x + c)*log(d^2*x^2 + 2*c*d*x + c^2 + 1)^2/(d^2*x^2 + 2*c*d*x + c^2
+ 1), x) + 192*a*b^2*e*f*integrate(1/32*x*arctan(d*x + c)^2/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 3*b^3*e^2*int
egrate(1/32*arctan(d*x + c)*log(d^2*x^2 + 2*c*d*x + c^2 + 1)^2/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 3/2*(2*(d*x
+ c)*arctan(d*x + c) - log((d*x + c)^2 + 1))*a^2*b*e^2/d + 1/24*(b^3*f^2*x^3 + 3*b^3*e*f*x^2 + 3*b^3*e^2*x)*a
rctan(d*x + c)^3 - 1/32*(b^3*f^2*x^3 + 3*b^3*e*f*x^2 + 3*b^3*e^2*x)*arctan(d*x + c)*log(d^2*x^2 + 2*c*d*x + c^
2 + 1)^2

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

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

[Out]

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