\(\int (e+f x)^3 (a+b \arctan (c+d x)) \, dx\) [24]

Optimal result

Integrand size = 18, antiderivative size = 233 \[ \int (e+f x)^3 (a+b \arctan (c+d x)) \, dx=-\frac {b f \left (6 d^2 e^2-12 c d e f-\left (1-6 c^2\right ) f^2\right ) x}{4 d^3}-\frac {b f^2 (d e-c f) (c+d x)^2}{2 d^4}-\frac {b f^3 (c+d x)^3}{12 d^4}-\frac {b \left (d^4 e^4-4 c d^3 e^3 f-6 \left (1-c^2\right ) d^2 e^2 f^2+4 c \left (3-c^2\right ) d e f^3+\left (1-6 c^2+c^4\right ) f^4\right ) \arctan (c+d x)}{4 d^4 f}+\frac {(e+f x)^4 (a+b \arctan (c+d x))}{4 f}-\frac {b (d e-c f) (d e+f-c f) (d e-(1+c) f) \log \left (1+(c+d x)^2\right )}{2 d^4} \]

[Out]

-1/4*b*f*(6*d^2*e^2-12*c*d*e*f-(-6*c^2+1)*f^2)*x/d^3-1/2*b*f^2*(-c*f+d*e)*(d*x+c)^2/d^4-1/12*b*f^3*(d*x+c)^3/d
^4-1/4*b*(d^4*e^4-4*c*d^3*e^3*f-6*(-c^2+1)*d^2*e^2*f^2+4*c*(-c^2+3)*d*e*f^3+(c^4-6*c^2+1)*f^4)*arctan(d*x+c)/d
^4/f+1/4*(f*x+e)^4*(a+b*arctan(d*x+c))/f-1/2*b*(-c*f+d*e)*(-c*f+d*e+f)*(d*e-(1+c)*f)*ln(1+(d*x+c)^2)/d^4

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5155, 4972, 716, 649, 209, 266} \[ \int (e+f x)^3 (a+b \arctan (c+d x)) \, dx=\frac {(e+f x)^4 (a+b \arctan (c+d x))}{4 f}-\frac {b \arctan (c+d x) \left (-6 \left (1-c^2\right ) d^2 e^2 f^2+4 c \left (3-c^2\right ) d e f^3+\left (c^4-6 c^2+1\right ) f^4-4 c d^3 e^3 f+d^4 e^4\right )}{4 d^4 f}-\frac {b f x \left (-\left (1-6 c^2\right ) f^2-12 c d e f+6 d^2 e^2\right )}{4 d^3}-\frac {b f^2 (c+d x)^2 (d e-c f)}{2 d^4}-\frac {b (d e-c f) (-c f+d e+f) (d e-(c+1) f) \log \left ((c+d x)^2+1\right )}{2 d^4}-\frac {b f^3 (c+d x)^3}{12 d^4} \]

[In]

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

[Out]

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

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

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 649

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[d, Int[1/(a + c*x^2), x], x] + Dist[e, Int[x/
(a + c*x^2), x], x] /; FreeQ[{a, c, d, e}, x] &&  !NiceSqrtQ[(-a)*c]

Rule 716

Int[((d_) + (e_.)*(x_))^(m_)/((a_) + (c_.)*(x_)^2), x_Symbol] :> Int[PolynomialDivide[(d + e*x)^m, a + c*x^2,
x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[m, 1] && (NeQ[d, 0] || GtQ[m, 2])

Rule 4972

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

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]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \left (\frac {d e-c f}{d}+\frac {f x}{d}\right )^3 (a+b \arctan (x)) \, dx,x,c+d x\right )}{d} \\ & = \frac {(e+f x)^4 (a+b \arctan (c+d x))}{4 f}-\frac {b \text {Subst}\left (\int \frac {\left (\frac {d e-c f}{d}+\frac {f x}{d}\right )^4}{1+x^2} \, dx,x,c+d x\right )}{4 f} \\ & = \frac {(e+f x)^4 (a+b \arctan (c+d x))}{4 f}-\frac {b \text {Subst}\left (\int \left (\frac {f^2 \left (6 d^2 e^2-12 c d e f-\left (1-6 c^2\right ) f^2\right )}{d^4}+\frac {4 f^3 (d e-c f) x}{d^4}+\frac {f^4 x^2}{d^4}+\frac {d^4 e^4-4 c d^3 e^3 f-6 \left (1-c^2\right ) d^2 e^2 f^2+4 c \left (3-c^2\right ) d e f^3+\left (1-6 c^2+c^4\right ) f^4+4 f (d e-c f) (d e-f-c f) (d e+f-c f) x}{d^4 \left (1+x^2\right )}\right ) \, dx,x,c+d x\right )}{4 f} \\ & = -\frac {b f \left (6 d^2 e^2-12 c d e f-\left (1-6 c^2\right ) f^2\right ) x}{4 d^3}-\frac {b f^2 (d e-c f) (c+d x)^2}{2 d^4}-\frac {b f^3 (c+d x)^3}{12 d^4}+\frac {(e+f x)^4 (a+b \arctan (c+d x))}{4 f}-\frac {b \text {Subst}\left (\int \frac {d^4 e^4-4 c d^3 e^3 f-6 \left (1-c^2\right ) d^2 e^2 f^2+4 c \left (3-c^2\right ) d e f^3+\left (1-6 c^2+c^4\right ) f^4+4 f (d e-c f) (d e-f-c f) (d e+f-c f) x}{1+x^2} \, dx,x,c+d x\right )}{4 d^4 f} \\ & = -\frac {b f \left (6 d^2 e^2-12 c d e f-\left (1-6 c^2\right ) f^2\right ) x}{4 d^3}-\frac {b f^2 (d e-c f) (c+d x)^2}{2 d^4}-\frac {b f^3 (c+d x)^3}{12 d^4}+\frac {(e+f x)^4 (a+b \arctan (c+d x))}{4 f}-\frac {(b (d e-c f) (d e+f-c f) (d e-(1+c) f)) \text {Subst}\left (\int \frac {x}{1+x^2} \, dx,x,c+d x\right )}{d^4}-\frac {\left (b \left (d^4 e^4-4 c d^3 e^3 f-6 \left (1-c^2\right ) d^2 e^2 f^2+4 c \left (3-c^2\right ) d e f^3+\left (1-6 c^2+c^4\right ) f^4\right )\right ) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,c+d x\right )}{4 d^4 f} \\ & = -\frac {b f \left (6 d^2 e^2-12 c d e f-\left (1-6 c^2\right ) f^2\right ) x}{4 d^3}-\frac {b f^2 (d e-c f) (c+d x)^2}{2 d^4}-\frac {b f^3 (c+d x)^3}{12 d^4}-\frac {b \left (d^4 e^4-4 c d^3 e^3 f-6 \left (1-c^2\right ) d^2 e^2 f^2+4 c \left (3-c^2\right ) d e f^3+\left (1-6 c^2+c^4\right ) f^4\right ) \arctan (c+d x)}{4 d^4 f}+\frac {(e+f x)^4 (a+b \arctan (c+d x))}{4 f}-\frac {b (d e-c f) (d e+f-c f) (d e-(1+c) f) \log \left (1+(c+d x)^2\right )}{2 d^4} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.23 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.67 \[ \int (e+f x)^3 (a+b \arctan (c+d x)) \, dx=\frac {(e+f x)^4 (a+b \arctan (c+d x))-\frac {b \left (6 d f^2 \left (6 d^2 e^2-12 c d e f+\left (-1+6 c^2\right ) f^2\right ) x+12 f^3 (d e-c f) (c+d x)^2+2 f^4 (c+d x)^3-3 i (d e-(-i+c) f)^4 \log (i-c-d x)+3 i (d e-(i+c) f)^4 \log (i+c+d x)\right )}{6 d^4}}{4 f} \]

[In]

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

[Out]

((e + f*x)^4*(a + b*ArcTan[c + d*x]) - (b*(6*d*f^2*(6*d^2*e^2 - 12*c*d*e*f + (-1 + 6*c^2)*f^2)*x + 12*f^3*(d*e
- c*f)*(c + d*x)^2 + 2*f^4*(c + d*x)^3 - (3*I)*(d*e - (-I + c)*f)^4*Log[I - c - d*x] + (3*I)*(d*e - (I + c)*f
)^4*Log[I + c + d*x]))/(6*d^4))/(4*f)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(463\) vs. \(2(221)=442\).

Time = 0.25 (sec) , antiderivative size = 464, normalized size of antiderivative = 1.99

method	result	size
parts	\(\frac {5 b \,f^{2} c^{2} e}{2 d^{3}}-\frac {3 b f c \,e^{2}}{2 d^{2}}+\frac {b \,f^{2} \ln \left (1+\left (d x +c \right )^{2}\right ) e}{2 d^{3}}+\frac {b \,f^{3} \ln \left (1+\left (d x +c \right )^{2}\right ) c^{3}}{2 d^{4}}-\frac {b \,f^{3} \ln \left (1+\left (d x +c \right )^{2}\right ) c}{2 d^{4}}+b \,f^{2} \arctan \left (d x +c \right ) e \,x^{3}+\frac {3 b f \arctan \left (d x +c \right ) e^{2} x^{2}}{2}+b \arctan \left (d x +c \right ) x \,e^{3}-\frac {b \ln \left (1+\left (d x +c \right )^{2}\right ) e^{3}}{2 d}+\frac {b \,f^{3} c}{4 d^{4}}-\frac {13 b \,f^{3} c^{3}}{12 d^{4}}+\frac {b \,f^{3} \arctan \left (d x +c \right ) x^{4}}{4}+\frac {3 b f \ln \left (1+\left (d x +c \right )^{2}\right ) c \,e^{2}}{2 d^{2}}-\frac {3 b \,f^{2} \ln \left (1+\left (d x +c \right )^{2}\right ) c^{2} e}{2 d^{3}}-\frac {f^{3} b \,x^{3}}{12 d}+\frac {f^{3} b x}{4 d^{3}}-\frac {f^{3} b \arctan \left (d x +c \right )}{4 d^{4}}+\frac {f^{2} b \,c^{3} e \arctan \left (d x +c \right )}{d^{3}}-\frac {3 f^{2} b c e \arctan \left (d x +c \right )}{d^{3}}-\frac {3 f b \,c^{2} e^{2} \arctan \left (d x +c \right )}{2 d^{2}}+\frac {2 f^{2} b c e x}{d^{2}}+\frac {a \left (f x +e \right )^{4}}{4 f}+\frac {f^{3} b c \,x^{2}}{4 d^{2}}-\frac {f^{2} b e \,x^{2}}{2 d}-\frac {3 f^{3} b \,c^{2} x}{4 d^{3}}-\frac {3 f b \,e^{2} x}{2 d}+\frac {b c \,e^{3} \arctan \left (d x +c \right )}{d}-\frac {f^{3} b \,c^{4} \arctan \left (d x +c \right )}{4 d^{4}}+\frac {3 f^{3} b \,c^{2} \arctan \left (d x +c \right )}{2 d^{4}}+\frac {3 f b \,e^{2} \arctan \left (d x +c \right )}{2 d^{2}}\)	\(464\)
derivativedivides	\(\frac {\frac {a \left (c f -d e -f \left (d x +c \right )\right )^{4}}{4 d^{3} f}-\frac {b \left (-\frac {f^{3} \arctan \left (d x +c \right ) c^{4}}{4}+f^{2} \arctan \left (d x +c \right ) c^{3} d e +f^{3} \arctan \left (d x +c \right ) c^{3} \left (d x +c \right )-\frac {3 f \arctan \left (d x +c \right ) c^{2} d^{2} e^{2}}{2}-3 f^{2} \arctan \left (d x +c \right ) c^{2} d e \left (d x +c \right )-\frac {3 f^{3} \arctan \left (d x +c \right ) c^{2} \left (d x +c \right )^{2}}{2}+\arctan \left (d x +c \right ) c \,d^{3} e^{3}+3 f \arctan \left (d x +c \right ) c \,d^{2} e^{2} \left (d x +c \right )+3 f^{2} \arctan \left (d x +c \right ) c d e \left (d x +c \right )^{2}+f^{3} \arctan \left (d x +c \right ) c \left (d x +c \right )^{3}-\frac {\arctan \left (d x +c \right ) d^{4} e^{4}}{4 f}-\arctan \left (d x +c \right ) d^{3} e^{3} \left (d x +c \right )-\frac {3 f \arctan \left (d x +c \right ) d^{2} e^{2} \left (d x +c \right )^{2}}{2}-f^{2} \arctan \left (d x +c \right ) d e \left (d x +c \right )^{3}-\frac {f^{3} \arctan \left (d x +c \right ) \left (d x +c \right )^{4}}{4}+\frac {6 c^{2} f^{4} \left (d x +c \right )-12 c d e \,f^{3} \left (d x +c \right )-2 c \,f^{4} \left (d x +c \right )^{2}+6 d^{2} e^{2} f^{2} \left (d x +c \right )+2 d e \,f^{3} \left (d x +c \right )^{2}+\frac {f^{4} \left (d x +c \right )^{3}}{3}-f^{4} \left (d x +c \right )+\frac {\left (-4 c^{3} f^{4}+12 c^{2} d e \,f^{3}-12 c \,d^{2} e^{2} f^{2}+4 f \,e^{3} d^{3}+4 c \,f^{4}-4 e \,f^{3} d \right ) \ln \left (1+\left (d x +c \right )^{2}\right )}{2}+\left (c^{4} f^{4}-4 c^{3} d e \,f^{3}+6 c^{2} d^{2} e^{2} f^{2}-4 c \,d^{3} e^{3} f +d^{4} e^{4}-6 c^{2} f^{4}+12 c d e \,f^{3}-6 e^{2} f^{2} d^{2}+f^{4}\right ) \arctan \left (d x +c \right )}{4 f}\right )}{d^{3}}}{d}\)	\(565\)
default	\(\frac {\frac {a \left (c f -d e -f \left (d x +c \right )\right )^{4}}{4 d^{3} f}-\frac {b \left (-\frac {f^{3} \arctan \left (d x +c \right ) c^{4}}{4}+f^{2} \arctan \left (d x +c \right ) c^{3} d e +f^{3} \arctan \left (d x +c \right ) c^{3} \left (d x +c \right )-\frac {3 f \arctan \left (d x +c \right ) c^{2} d^{2} e^{2}}{2}-3 f^{2} \arctan \left (d x +c \right ) c^{2} d e \left (d x +c \right )-\frac {3 f^{3} \arctan \left (d x +c \right ) c^{2} \left (d x +c \right )^{2}}{2}+\arctan \left (d x +c \right ) c \,d^{3} e^{3}+3 f \arctan \left (d x +c \right ) c \,d^{2} e^{2} \left (d x +c \right )+3 f^{2} \arctan \left (d x +c \right ) c d e \left (d x +c \right )^{2}+f^{3} \arctan \left (d x +c \right ) c \left (d x +c \right )^{3}-\frac {\arctan \left (d x +c \right ) d^{4} e^{4}}{4 f}-\arctan \left (d x +c \right ) d^{3} e^{3} \left (d x +c \right )-\frac {3 f \arctan \left (d x +c \right ) d^{2} e^{2} \left (d x +c \right )^{2}}{2}-f^{2} \arctan \left (d x +c \right ) d e \left (d x +c \right )^{3}-\frac {f^{3} \arctan \left (d x +c \right ) \left (d x +c \right )^{4}}{4}+\frac {6 c^{2} f^{4} \left (d x +c \right )-12 c d e \,f^{3} \left (d x +c \right )-2 c \,f^{4} \left (d x +c \right )^{2}+6 d^{2} e^{2} f^{2} \left (d x +c \right )+2 d e \,f^{3} \left (d x +c \right )^{2}+\frac {f^{4} \left (d x +c \right )^{3}}{3}-f^{4} \left (d x +c \right )+\frac {\left (-4 c^{3} f^{4}+12 c^{2} d e \,f^{3}-12 c \,d^{2} e^{2} f^{2}+4 f \,e^{3} d^{3}+4 c \,f^{4}-4 e \,f^{3} d \right ) \ln \left (1+\left (d x +c \right )^{2}\right )}{2}+\left (c^{4} f^{4}-4 c^{3} d e \,f^{3}+6 c^{2} d^{2} e^{2} f^{2}-4 c \,d^{3} e^{3} f +d^{4} e^{4}-6 c^{2} f^{4}+12 c d e \,f^{3}-6 e^{2} f^{2} d^{2}+f^{4}\right ) \arctan \left (d x +c \right )}{4 f}\right )}{d^{3}}}{d}\)	\(565\)
parallelrisch	\(\frac {-42 b \,c^{2} d e \,f^{2}+36 b c \,d^{2} e^{2} f +3 x b d \,f^{3}-x^{3} b \,d^{3} f^{3}+6 \ln \left (d^{2} x^{2}+2 c d x +c^{2}+1\right ) b \,c^{3} f^{3}-6 \ln \left (d^{2} x^{2}+2 c d x +c^{2}+1\right ) b \,d^{3} e^{3}-6 \ln \left (d^{2} x^{2}+2 c d x +c^{2}+1\right ) b c \,f^{3}-3 \arctan \left (d x +c \right ) b \,c^{4} f^{3}+18 \arctan \left (d x +c \right ) b \,c^{2} f^{3}+3 x^{4} a \,d^{4} f^{3}+12 x a \,d^{4} e^{3}-24 a c \,d^{3} e^{3}+18 x^{2} a \,d^{4} e^{2} f +6 \ln \left (d^{2} x^{2}+2 c d x +c^{2}+1\right ) b d e \,f^{2}+12 \arctan \left (d x +c \right ) b c \,d^{3} e^{3}+18 \arctan \left (d x +c \right ) b \,d^{2} e^{2} f -9 x b \,c^{2} d \,f^{3}-18 x b \,d^{3} e^{2} f +12 x^{3} a \,d^{4} e \,f^{2}+3 x^{4} \arctan \left (d x +c \right ) b \,d^{4} f^{3}+12 x \arctan \left (d x +c \right ) b \,d^{4} e^{3}+3 x^{2} b c \,d^{2} f^{3}-6 x^{2} b \,d^{3} e \,f^{2}+15 b \,c^{3} f^{3}-18 f \,e^{2} a \,d^{2}+6 f^{2} e b d -9 b c \,f^{3}-3 \arctan \left (d x +c \right ) b \,f^{3}+12 \arctan \left (d x +c \right ) b \,c^{3} d e \,f^{2}-36 \arctan \left (d x +c \right ) b c d e \,f^{2}+24 x b c \,d^{2} e \,f^{2}+12 x^{3} \arctan \left (d x +c \right ) b \,d^{4} e \,f^{2}+18 x^{2} \arctan \left (d x +c \right ) b \,d^{4} e^{2} f -18 \arctan \left (d x +c \right ) b \,c^{2} d^{2} e^{2} f -18 \ln \left (d^{2} x^{2}+2 c d x +c^{2}+1\right ) b \,c^{2} d e \,f^{2}+18 \ln \left (d^{2} x^{2}+2 c d x +c^{2}+1\right ) b c \,d^{2} e^{2} f -18 a \,c^{2} d^{2} e^{2} f}{12 d^{4}}\)	\(574\)
risch	\(\frac {x^{4} f^{3} a}{4}+x a \,e^{3}-\frac {i \left (f x +e \right )^{4} b \ln \left (1+i \left (d x +c \right )\right )}{8 f}+\frac {i f^{3} b \,x^{4} \ln \left (1-i \left (d x +c \right )\right )}{8}+\frac {i b \,e^{3} x \ln \left (1-i \left (d x +c \right )\right )}{2}+\frac {i b \,e^{4} \ln \left (d^{2} x^{2}+2 c d x +c^{2}+1\right )}{16 f}-\frac {f^{3} b \,x^{3}}{12 d}+\frac {f^{3} b x}{4 d^{3}}-\frac {f^{3} b \arctan \left (d x +c \right )}{4 d^{4}}-\frac {b \,e^{4} \arctan \left (d x +c \right )}{8 f}-\frac {b \,e^{3} \ln \left (d^{2} x^{2}+2 c d x +c^{2}+1\right )}{2 d}+x^{3} f^{2} e a +\frac {3 x^{2} f \,e^{2} a}{2}-\frac {3 f^{2} b \,c^{2} e \ln \left (d^{2} x^{2}+2 c d x +c^{2}+1\right )}{2 d^{3}}+\frac {3 f b c \,e^{2} \ln \left (d^{2} x^{2}+2 c d x +c^{2}+1\right )}{2 d^{2}}+\frac {f^{2} b \,c^{3} e \arctan \left (d x +c \right )}{d^{3}}-\frac {3 f^{2} b c e \arctan \left (d x +c \right )}{d^{3}}-\frac {3 f b \,c^{2} e^{2} \arctan \left (d x +c \right )}{2 d^{2}}+\frac {2 f^{2} b c e x}{d^{2}}+\frac {f^{3} b c \,x^{2}}{4 d^{2}}-\frac {f^{2} b e \,x^{2}}{2 d}-\frac {3 f^{3} b \,c^{2} x}{4 d^{3}}-\frac {3 f b \,e^{2} x}{2 d}+\frac {b c \,e^{3} \arctan \left (d x +c \right )}{d}-\frac {f^{3} b \,c^{4} \arctan \left (d x +c \right )}{4 d^{4}}+\frac {3 f^{3} b \,c^{2} \arctan \left (d x +c \right )}{2 d^{4}}+\frac {f^{3} b \,c^{3} \ln \left (d^{2} x^{2}+2 c d x +c^{2}+1\right )}{2 d^{4}}-\frac {f^{3} b c \ln \left (d^{2} x^{2}+2 c d x +c^{2}+1\right )}{2 d^{4}}+\frac {3 f b \,e^{2} \arctan \left (d x +c \right )}{2 d^{2}}+\frac {f^{2} b e \ln \left (d^{2} x^{2}+2 c d x +c^{2}+1\right )}{2 d^{3}}+\frac {i f^{2} b e \,x^{3} \ln \left (1-i \left (d x +c \right )\right )}{2}+\frac {3 i f b \,e^{2} x^{2} \ln \left (1-i \left (d x +c \right )\right )}{4}\)	\(581\)

[In]

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

[Out]

5/2*b/d^3*f^2*c^2*e-3/2*b/d^2*f*c*e^2+1/2*b/d^3*f^2*ln(1+(d*x+c)^2)*e+1/2*b/d^4*f^3*ln(1+(d*x+c)^2)*c^3-1/2*b/
d^4*f^3*ln(1+(d*x+c)^2)*c+b*f^2*arctan(d*x+c)*e*x^3+3/2*b*f*arctan(d*x+c)*e^2*x^2+b*arctan(d*x+c)*x*e^3-1/2*b/
d*ln(1+(d*x+c)^2)*e^3+1/4*b/d^4*f^3*c-13/12*b/d^4*f^3*c^3+1/4*b*f^3*arctan(d*x+c)*x^4+3/2*b/d^2*f*ln(1+(d*x+c)
^2)*c*e^2-3/2*b/d^3*f^2*ln(1+(d*x+c)^2)*c^2*e-1/12/d*f^3*b*x^3+1/4/d^3*f^3*b*x-1/4/d^4*f^3*b*arctan(d*x+c)+1/d
^3*f^2*b*c^3*e*arctan(d*x+c)-3/d^3*f^2*b*c*e*arctan(d*x+c)-3/2/d^2*f*b*c^2*e^2*arctan(d*x+c)+2/d^2*f^2*b*c*e*x
+1/4*a*(f*x+e)^4/f+1/4/d^2*f^3*b*c*x^2-1/2/d*f^2*b*e*x^2-3/4/d^3*f^3*b*c^2*x-3/2/d*f*b*e^2*x+1/d*b*c*e^3*arcta
n(d*x+c)-1/4/d^4*f^3*b*c^4*arctan(d*x+c)+3/2/d^4*f^3*b*c^2*arctan(d*x+c)+3/2/d^2*f*b*e^2*arctan(d*x+c)

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 317, normalized size of antiderivative = 1.36 \[ \int (e+f x)^3 (a+b \arctan (c+d x)) \, dx=\frac {3 \, a d^{4} f^{3} x^{4} + {\left (12 \, a d^{4} e f^{2} - b d^{3} f^{3}\right )} x^{3} + 3 \, {\left (6 \, a d^{4} e^{2} f - 2 \, b d^{3} e f^{2} + b c d^{2} f^{3}\right )} x^{2} + 3 \, {\left (4 \, a d^{4} e^{3} - 6 \, b d^{3} e^{2} f + 8 \, b c d^{2} e f^{2} - {\left (3 \, b c^{2} - b\right )} d f^{3}\right )} x + 3 \, {\left (b d^{4} f^{3} x^{4} + 4 \, b d^{4} e f^{2} x^{3} + 6 \, b d^{4} e^{2} f x^{2} + 4 \, b d^{4} e^{3} x + 4 \, b c d^{3} e^{3} - 6 \, {\left (b c^{2} - b\right )} d^{2} e^{2} f + 4 \, {\left (b c^{3} - 3 \, b c\right )} d e f^{2} - {\left (b c^{4} - 6 \, b c^{2} + b\right )} f^{3}\right )} \arctan \left (d x + c\right ) - 6 \, {\left (b d^{3} e^{3} - 3 \, b c d^{2} e^{2} f + {\left (3 \, b c^{2} - b\right )} d e f^{2} - {\left (b c^{3} - b c\right )} f^{3}\right )} \log \left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right )}{12 \, d^{4}} \]

[In]

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

[Out]

1/12*(3*a*d^4*f^3*x^4 + (12*a*d^4*e*f^2 - b*d^3*f^3)*x^3 + 3*(6*a*d^4*e^2*f - 2*b*d^3*e*f^2 + b*c*d^2*f^3)*x^2
+ 3*(4*a*d^4*e^3 - 6*b*d^3*e^2*f + 8*b*c*d^2*e*f^2 - (3*b*c^2 - b)*d*f^3)*x + 3*(b*d^4*f^3*x^4 + 4*b*d^4*e*f^
2*x^3 + 6*b*d^4*e^2*f*x^2 + 4*b*d^4*e^3*x + 4*b*c*d^3*e^3 - 6*(b*c^2 - b)*d^2*e^2*f + 4*(b*c^3 - 3*b*c)*d*e*f^
2 - (b*c^4 - 6*b*c^2 + b)*f^3)*arctan(d*x + c) - 6*(b*d^3*e^3 - 3*b*c*d^2*e^2*f + (3*b*c^2 - b)*d*e*f^2 - (b*c
^3 - b*c)*f^3)*log(d^2*x^2 + 2*c*d*x + c^2 + 1))/d^4

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 346, normalized size of antiderivative = 1.48 \[ \int (e+f x)^3 (a+b \arctan (c+d x)) \, dx=\frac {1}{4} \, a f^{3} x^{4} + a e f^{2} x^{3} + \frac {3}{2} \, a e^{2} f x^{2} + \frac {3}{2} \, {\left (x^{2} \arctan \left (d x + c\right ) - d {\left (\frac {x}{d^{2}} + \frac {{\left (c^{2} - 1\right )} \arctan \left (\frac {d^{2} x + c d}{d}\right )}{d^{3}} - \frac {c \log \left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right )}{d^{3}}\right )}\right )} b e^{2} f + \frac {1}{2} \, {\left (2 \, x^{3} \arctan \left (d x + c\right ) - d {\left (\frac {d x^{2} - 4 \, c x}{d^{3}} - \frac {2 \, {\left (c^{3} - 3 \, c\right )} \arctan \left (\frac {d^{2} x + c d}{d}\right )}{d^{4}} + \frac {{\left (3 \, c^{2} - 1\right )} \log \left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right )}{d^{4}}\right )}\right )} b e f^{2} + \frac {1}{12} \, {\left (3 \, x^{4} \arctan \left (d x + c\right ) - d {\left (\frac {d^{2} x^{3} - 3 \, c d x^{2} + 3 \, {\left (3 \, c^{2} - 1\right )} x}{d^{4}} + \frac {3 \, {\left (c^{4} - 6 \, c^{2} + 1\right )} \arctan \left (\frac {d^{2} x + c d}{d}\right )}{d^{5}} - \frac {6 \, {\left (c^{3} - c\right )} \log \left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right )}{d^{5}}\right )}\right )} b f^{3} + a e^{3} x + \frac {{\left (2 \, {\left (d x + c\right )} \arctan \left (d x + c\right ) - \log \left ({\left (d x + c\right )}^{2} + 1\right )\right )} b e^{3}}{2 \, d} \]

[In]

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

[Out]

1/4*a*f^3*x^4 + a*e*f^2*x^3 + 3/2*a*e^2*f*x^2 + 3/2*(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))*b*e^2*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))*b*e*
f^2 + 1/12*(3*x^4*arctan(d*x + c) - d*((d^2*x^3 - 3*c*d*x^2 + 3*(3*c^2 - 1)*x)/d^4 + 3*(c^4 - 6*c^2 + 1)*arcta
n((d^2*x + c*d)/d)/d^5 - 6*(c^3 - c)*log(d^2*x^2 + 2*c*d*x + c^2 + 1)/d^5))*b*f^3 + a*e^3*x + 1/2*(2*(d*x + c)
*arctan(d*x + c) - log((d*x + c)^2 + 1))*b*e^3/d

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [B] (verification not implemented)

Time = 1.11 (sec) , antiderivative size = 787, normalized size of antiderivative = 3.38 \[ \int (e+f x)^3 (a+b \arctan (c+d x)) \, dx=\mathrm {atan}\left (c+d\,x\right )\,\left (b\,e^3\,x+\frac {3\,b\,e^2\,f\,x^2}{2}+b\,e\,f^2\,x^3+\frac {b\,f^3\,x^4}{4}\right )+x\,\left (\frac {e\,\left (6\,a\,c^2\,f^2+12\,a\,c\,d\,e\,f+2\,a\,d^2\,e^2-3\,b\,d\,e\,f+6\,a\,f^2\right )}{2\,d^2}-\frac {\left (4\,c^2+4\right )\,\left (\frac {f^2\,\left (8\,a\,c\,f-b\,f+12\,a\,d\,e\right )}{4\,d}-\frac {2\,a\,c\,f^3}{d}\right )}{4\,d^2}+\frac {2\,c\,\left (\frac {2\,c\,\left (\frac {f^2\,\left (8\,a\,c\,f-b\,f+12\,a\,d\,e\right )}{4\,d}-\frac {2\,a\,c\,f^3}{d}\right )}{d}-\frac {4\,a\,c^2\,f^3+24\,a\,c\,d\,e\,f^2+12\,a\,d^2\,e^2\,f-4\,b\,d\,e\,f^2+4\,a\,f^3}{4\,d^2}+\frac {a\,f^3\,\left (4\,c^2+4\right )}{4\,d^2}\right )}{d}\right )-x^2\,\left (\frac {c\,\left (\frac {f^2\,\left (8\,a\,c\,f-b\,f+12\,a\,d\,e\right )}{4\,d}-\frac {2\,a\,c\,f^3}{d}\right )}{d}-\frac {4\,a\,c^2\,f^3+24\,a\,c\,d\,e\,f^2+12\,a\,d^2\,e^2\,f-4\,b\,d\,e\,f^2+4\,a\,f^3}{8\,d^2}+\frac {a\,f^3\,\left (4\,c^2+4\right )}{8\,d^2}\right )+x^3\,\left (\frac {f^2\,\left (8\,a\,c\,f-b\,f+12\,a\,d\,e\right )}{12\,d}-\frac {2\,a\,c\,f^3}{3\,d}\right )+\frac {a\,f^3\,x^4}{4}-\frac {\ln \left (c^2+2\,c\,d\,x+d^2\,x^2+1\right )\,\left (-64\,b\,c^3\,d^4\,f^3+192\,b\,c^2\,d^5\,e\,f^2-192\,b\,c\,d^6\,e^2\,f+64\,b\,c\,d^4\,f^3+64\,b\,d^7\,e^3-64\,b\,d^5\,e\,f^2\right )}{128\,d^8}-\frac {b\,\mathrm {atan}\left (\frac {4\,d^3\,\left (\frac {c\,\left (c^4\,f^3-4\,c^3\,d\,e\,f^2+6\,c^2\,d^2\,e^2\,f-6\,c^2\,f^3-4\,c\,d^3\,e^3+12\,c\,d\,e\,f^2-6\,d^2\,e^2\,f+f^3\right )}{4\,d^3}+\frac {x\,\left (c^4\,f^3-4\,c^3\,d\,e\,f^2+6\,c^2\,d^2\,e^2\,f-6\,c^2\,f^3-4\,c\,d^3\,e^3+12\,c\,d\,e\,f^2-6\,d^2\,e^2\,f+f^3\right )}{4\,d^2}\right )}{c^4\,f^3-4\,c^3\,d\,e\,f^2+6\,c^2\,d^2\,e^2\,f-6\,c^2\,f^3-4\,c\,d^3\,e^3+12\,c\,d\,e\,f^2-6\,d^2\,e^2\,f+f^3}\right )\,\left (c^4\,f^3-4\,c^3\,d\,e\,f^2+6\,c^2\,d^2\,e^2\,f-6\,c^2\,f^3-4\,c\,d^3\,e^3+12\,c\,d\,e\,f^2-6\,d^2\,e^2\,f+f^3\right )}{4\,d^4} \]

[In]

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

[Out]

atan(c + d*x)*((b*f^3*x^4)/4 + b*e^3*x + (3*b*e^2*f*x^2)/2 + b*e*f^2*x^3) + x*((e*(6*a*f^2 + 6*a*c^2*f^2 + 2*a
*d^2*e^2 - 3*b*d*e*f + 12*a*c*d*e*f))/(2*d^2) - ((4*c^2 + 4)*((f^2*(8*a*c*f - b*f + 12*a*d*e))/(4*d) - (2*a*c*
f^3)/d))/(4*d^2) + (2*c*((2*c*((f^2*(8*a*c*f - b*f + 12*a*d*e))/(4*d) - (2*a*c*f^3)/d))/d - (4*a*f^3 + 4*a*c^2
*f^3 - 4*b*d*e*f^2 + 12*a*d^2*e^2*f + 24*a*c*d*e*f^2)/(4*d^2) + (a*f^3*(4*c^2 + 4))/(4*d^2)))/d) - x^2*((c*((f
^2*(8*a*c*f - b*f + 12*a*d*e))/(4*d) - (2*a*c*f^3)/d))/d - (4*a*f^3 + 4*a*c^2*f^3 - 4*b*d*e*f^2 + 12*a*d^2*e^2
*f + 24*a*c*d*e*f^2)/(8*d^2) + (a*f^3*(4*c^2 + 4))/(8*d^2)) + x^3*((f^2*(8*a*c*f - b*f + 12*a*d*e))/(12*d) - (
2*a*c*f^3)/(3*d)) + (a*f^3*x^4)/4 - (log(c^2 + d^2*x^2 + 2*c*d*x + 1)*(64*b*d^7*e^3 - 64*b*c^3*d^4*f^3 + 64*b*
c*d^4*f^3 - 64*b*d^5*e*f^2 - 192*b*c*d^6*e^2*f + 192*b*c^2*d^5*e*f^2))/(128*d^8) - (b*atan((4*d^3*((c*(f^3 - 6
*c^2*f^3 + c^4*f^3 - 4*c*d^3*e^3 - 6*d^2*e^2*f + 6*c^2*d^2*e^2*f + 12*c*d*e*f^2 - 4*c^3*d*e*f^2))/(4*d^3) + (x
*(f^3 - 6*c^2*f^3 + c^4*f^3 - 4*c*d^3*e^3 - 6*d^2*e^2*f + 6*c^2*d^2*e^2*f + 12*c*d*e*f^2 - 4*c^3*d*e*f^2))/(4*
d^2)))/(f^3 - 6*c^2*f^3 + c^4*f^3 - 4*c*d^3*e^3 - 6*d^2*e^2*f + 6*c^2*d^2*e^2*f + 12*c*d*e*f^2 - 4*c^3*d*e*f^2
))*(f^3 - 6*c^2*f^3 + c^4*f^3 - 4*c*d^3*e^3 - 6*d^2*e^2*f + 6*c^2*d^2*e^2*f + 12*c*d*e*f^2 - 4*c^3*d*e*f^2))/(
4*d^4)