\(\int \frac {x^3 (a+b \arctan (c x))}{d+e x} \, dx\) [134]

Optimal result

Integrand size = 19, antiderivative size = 297 \[ \int \frac {x^3 (a+b \arctan (c x))}{d+e x} \, dx=\frac {a d^2 x}{e^3}+\frac {b d x}{2 c e^2}-\frac {b x^2}{6 c e}-\frac {b d \arctan (c x)}{2 c^2 e^2}+\frac {b d^2 x \arctan (c x)}{e^3}-\frac {d x^2 (a+b \arctan (c x))}{2 e^2}+\frac {x^3 (a+b \arctan (c x))}{3 e}+\frac {d^3 (a+b \arctan (c x)) \log \left (\frac {2}{1-i c x}\right )}{e^4}-\frac {d^3 (a+b \arctan (c x)) \log \left (\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{e^4}-\frac {b d^2 \log \left (1+c^2 x^2\right )}{2 c e^3}+\frac {b \log \left (1+c^2 x^2\right )}{6 c^3 e}-\frac {i b d^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{2 e^4}+\frac {i b d^3 \operatorname {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 e^4} \]

[Out]

a*d^2*x/e^3+1/2*b*d*x/c/e^2-1/6*b*x^2/c/e-1/2*b*d*arctan(c*x)/c^2/e^2+b*d^2*x*arctan(c*x)/e^3-1/2*d*x^2*(a+b*a
rctan(c*x))/e^2+1/3*x^3*(a+b*arctan(c*x))/e+d^3*(a+b*arctan(c*x))*ln(2/(1-I*c*x))/e^4-d^3*(a+b*arctan(c*x))*ln
(2*c*(e*x+d)/(c*d+I*e)/(1-I*c*x))/e^4-1/2*b*d^2*ln(c^2*x^2+1)/c/e^3+1/6*b*ln(c^2*x^2+1)/c^3/e-1/2*I*b*d^3*poly
log(2,1-2/(1-I*c*x))/e^4+1/2*I*b*d^3*polylog(2,1-2*c*(e*x+d)/(c*d+I*e)/(1-I*c*x))/e^4

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.632, Rules used = {4996, 4930, 266, 4946, 327, 209, 272, 45, 4966, 2449, 2352, 2497} \[ \int \frac {x^3 (a+b \arctan (c x))}{d+e x} \, dx=\frac {d^3 \log \left (\frac {2}{1-i c x}\right ) (a+b \arctan (c x))}{e^4}-\frac {d^3 (a+b \arctan (c x)) \log \left (\frac {2 c (d+e x)}{(1-i c x) (c d+i e)}\right )}{e^4}-\frac {d x^2 (a+b \arctan (c x))}{2 e^2}+\frac {x^3 (a+b \arctan (c x))}{3 e}+\frac {a d^2 x}{e^3}-\frac {b d \arctan (c x)}{2 c^2 e^2}+\frac {b d^2 x \arctan (c x)}{e^3}-\frac {b d^2 \log \left (c^2 x^2+1\right )}{2 c e^3}+\frac {b \log \left (c^2 x^2+1\right )}{6 c^3 e}-\frac {i b d^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{2 e^4}+\frac {i b d^3 \operatorname {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 e^4}+\frac {b d x}{2 c e^2}-\frac {b x^2}{6 c e} \]

[In]

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

[Out]

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

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

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 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 327

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

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 2497

Int[Log[u_]*(Pq_)^(m_.), x_Symbol] :> With[{C = FullSimplify[Pq^m*((1 - u)/D[u, x])]}, Simp[C*PolyLog[2, 1 - u
], x] /; FreeQ[C, x]] /; IntegerQ[m] && PolyQ[Pq, x] && RationalFunctionQ[u, x] && LeQ[RationalFunctionExponen
ts[u, x][[2]], Expon[Pq, x]]

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 4966

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

Rule 4996

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Int[Ex
pandIntegrand[(a + b*ArcTan[c*x])^p, (f*x)^m*(d + e*x)^q, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[p,
0] && IntegerQ[q] && (GtQ[q, 0] || NeQ[a, 0] || IntegerQ[m])

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

Mathematica [A] (verified)

Time = 2.49 (sec) , antiderivative size = 484, normalized size of antiderivative = 1.63 \[ \int \frac {x^3 (a+b \arctan (c x))}{d+e x} \, dx=-\frac {\frac {b e^3}{c^3}-6 a d^2 e x-\frac {3 b d e^2 x}{c}+3 a d e^2 x^2+\frac {b e^3 x^2}{c}-2 a e^3 x^3+\frac {3 b d e^2 \arctan (c x)}{c^2}+3 i b d^3 \pi \arctan (c x)-6 b d^2 e x \arctan (c x)+3 b d e^2 x^2 \arctan (c x)-2 b e^3 x^3 \arctan (c x)-6 i b d^3 \arctan \left (\frac {c d}{e}\right ) \arctan (c x)+3 i b d^3 \arctan (c x)^2+\frac {3 b d^2 e \arctan (c x)^2}{c}-\frac {3 b d^2 \sqrt {1+\frac {c^2 d^2}{e^2}} e e^{i \arctan \left (\frac {c d}{e}\right )} \arctan (c x)^2}{c}+3 b d^3 \pi \log \left (1+e^{-2 i \arctan (c x)}\right )-6 b d^3 \arctan (c x) \log \left (1+e^{2 i \arctan (c x)}\right )+6 b d^3 \arctan \left (\frac {c d}{e}\right ) \log \left (1-e^{2 i \left (\arctan \left (\frac {c d}{e}\right )+\arctan (c x)\right )}\right )+6 b d^3 \arctan (c x) \log \left (1-e^{2 i \left (\arctan \left (\frac {c d}{e}\right )+\arctan (c x)\right )}\right )+6 a d^3 \log (d+e x)+\frac {3 b d^2 e \log \left (1+c^2 x^2\right )}{c}-\frac {b e^3 \log \left (1+c^2 x^2\right )}{c^3}+\frac {3}{2} b d^3 \pi \log \left (1+c^2 x^2\right )-6 b d^3 \arctan \left (\frac {c d}{e}\right ) \log \left (\sin \left (\arctan \left (\frac {c d}{e}\right )+\arctan (c x)\right )\right )+3 i b d^3 \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )-3 i b d^3 \operatorname {PolyLog}\left (2,e^{2 i \left (\arctan \left (\frac {c d}{e}\right )+\arctan (c x)\right )}\right )}{6 e^4} \]

[In]

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

[Out]

-1/6*((b*e^3)/c^3 - 6*a*d^2*e*x - (3*b*d*e^2*x)/c + 3*a*d*e^2*x^2 + (b*e^3*x^2)/c - 2*a*e^3*x^3 + (3*b*d*e^2*A
rcTan[c*x])/c^2 + (3*I)*b*d^3*Pi*ArcTan[c*x] - 6*b*d^2*e*x*ArcTan[c*x] + 3*b*d*e^2*x^2*ArcTan[c*x] - 2*b*e^3*x
^3*ArcTan[c*x] - (6*I)*b*d^3*ArcTan[(c*d)/e]*ArcTan[c*x] + (3*I)*b*d^3*ArcTan[c*x]^2 + (3*b*d^2*e*ArcTan[c*x]^
2)/c - (3*b*d^2*Sqrt[1 + (c^2*d^2)/e^2]*e*E^(I*ArcTan[(c*d)/e])*ArcTan[c*x]^2)/c + 3*b*d^3*Pi*Log[1 + E^((-2*I
)*ArcTan[c*x])] - 6*b*d^3*ArcTan[c*x]*Log[1 + E^((2*I)*ArcTan[c*x])] + 6*b*d^3*ArcTan[(c*d)/e]*Log[1 - E^((2*I
)*(ArcTan[(c*d)/e] + ArcTan[c*x]))] + 6*b*d^3*ArcTan[c*x]*Log[1 - E^((2*I)*(ArcTan[(c*d)/e] + ArcTan[c*x]))] +
6*a*d^3*Log[d + e*x] + (3*b*d^2*e*Log[1 + c^2*x^2])/c - (b*e^3*Log[1 + c^2*x^2])/c^3 + (3*b*d^3*Pi*Log[1 + c^
2*x^2])/2 - 6*b*d^3*ArcTan[(c*d)/e]*Log[Sin[ArcTan[(c*d)/e] + ArcTan[c*x]]] + (3*I)*b*d^3*PolyLog[2, -E^((2*I)
*ArcTan[c*x])] - (3*I)*b*d^3*PolyLog[2, E^((2*I)*(ArcTan[(c*d)/e] + ArcTan[c*x]))])/e^4

Maple [A] (verified)

Time = 0.37 (sec) , antiderivative size = 377, normalized size of antiderivative = 1.27

method	result	size
parts	\(\frac {a \,x^{3}}{3 e}-\frac {a d \,x^{2}}{2 e^{2}}+\frac {a \,d^{2} x}{e^{3}}-\frac {a \,d^{3} \ln \left (e x +d \right )}{e^{4}}+\frac {b \left (\frac {c^{4} \arctan \left (c x \right ) x^{3}}{3 e}-\frac {c^{4} \arctan \left (c x \right ) x^{2} d}{2 e^{2}}+\frac {c^{4} \arctan \left (c x \right ) x \,d^{2}}{e^{3}}-\frac {c^{4} \arctan \left (c x \right ) d^{3} \ln \left (e c x +c d \right )}{e^{4}}-\frac {c \left (\frac {\ln \left (c^{2} d^{2}-2 c d \left (e c x +c d \right )+e^{2}+\left (e c x +c d \right )^{2}\right ) c^{2} d^{2}}{2 e^{2}}+\frac {\arctan \left (c x \right ) c d}{2 e}-\frac {\ln \left (c^{2} d^{2}-2 c d \left (e c x +c d \right )+e^{2}+\left (e c x +c d \right )^{2}\right )}{6}-\frac {5 c d \left (e c x +c d \right )}{6 e^{2}}+\frac {\left (e c x +c d \right )^{2}}{6 e^{2}}-\frac {c^{3} d^{3} \left (-\frac {i \ln \left (e c x +c d \right ) \left (\ln \left (\frac {-e c x +i e}{c d +i e}\right )-\ln \left (\frac {e c x +i e}{-c d +i e}\right )\right )}{2 e}-\frac {i \left (\operatorname {dilog}\left (\frac {-e c x +i e}{c d +i e}\right )-\operatorname {dilog}\left (\frac {e c x +i e}{-c d +i e}\right )\right )}{2 e}\right )}{e^{2}}\right )}{e}\right )}{c^{4}}\)	\(377\)
derivativedivides	\(\frac {\frac {a \,c^{4} d^{2} x}{e^{3}}-\frac {a \,c^{4} d \,x^{2}}{2 e^{2}}+\frac {a \,c^{4} x^{3}}{3 e}-\frac {a \,c^{4} d^{3} \ln \left (e c x +c d \right )}{e^{4}}+b c \left (\frac {\arctan \left (c x \right ) c^{3} d^{2} x}{e^{3}}-\frac {\arctan \left (c x \right ) c^{3} d \,x^{2}}{2 e^{2}}+\frac {\arctan \left (c x \right ) c^{3} x^{3}}{3 e}-\frac {\arctan \left (c x \right ) c^{3} d^{3} \ln \left (e c x +c d \right )}{e^{4}}-\frac {\frac {\ln \left (c^{2} d^{2}-2 c d \left (e c x +c d \right )+e^{2}+\left (e c x +c d \right )^{2}\right ) c^{2} d^{2}}{2 e^{2}}+\frac {\arctan \left (c x \right ) c d}{2 e}-\frac {\ln \left (c^{2} d^{2}-2 c d \left (e c x +c d \right )+e^{2}+\left (e c x +c d \right )^{2}\right )}{6}-\frac {5 c d \left (e c x +c d \right )}{6 e^{2}}+\frac {\left (e c x +c d \right )^{2}}{6 e^{2}}-\frac {c^{3} d^{3} \left (\frac {i \ln \left (e c x +c d \right ) \left (-\ln \left (\frac {-e c x +i e}{c d +i e}\right )+\ln \left (\frac {e c x +i e}{-c d +i e}\right )\right )}{2 e}-\frac {i \left (\operatorname {dilog}\left (\frac {-e c x +i e}{c d +i e}\right )-\operatorname {dilog}\left (\frac {e c x +i e}{-c d +i e}\right )\right )}{2 e}\right )}{e^{2}}}{e}\right )}{c^{4}}\)	\(393\)
default	\(\frac {\frac {a \,c^{4} d^{2} x}{e^{3}}-\frac {a \,c^{4} d \,x^{2}}{2 e^{2}}+\frac {a \,c^{4} x^{3}}{3 e}-\frac {a \,c^{4} d^{3} \ln \left (e c x +c d \right )}{e^{4}}+b c \left (\frac {\arctan \left (c x \right ) c^{3} d^{2} x}{e^{3}}-\frac {\arctan \left (c x \right ) c^{3} d \,x^{2}}{2 e^{2}}+\frac {\arctan \left (c x \right ) c^{3} x^{3}}{3 e}-\frac {\arctan \left (c x \right ) c^{3} d^{3} \ln \left (e c x +c d \right )}{e^{4}}-\frac {\frac {\ln \left (c^{2} d^{2}-2 c d \left (e c x +c d \right )+e^{2}+\left (e c x +c d \right )^{2}\right ) c^{2} d^{2}}{2 e^{2}}+\frac {\arctan \left (c x \right ) c d}{2 e}-\frac {\ln \left (c^{2} d^{2}-2 c d \left (e c x +c d \right )+e^{2}+\left (e c x +c d \right )^{2}\right )}{6}-\frac {5 c d \left (e c x +c d \right )}{6 e^{2}}+\frac {\left (e c x +c d \right )^{2}}{6 e^{2}}-\frac {c^{3} d^{3} \left (\frac {i \ln \left (e c x +c d \right ) \left (-\ln \left (\frac {-e c x +i e}{c d +i e}\right )+\ln \left (\frac {e c x +i e}{-c d +i e}\right )\right )}{2 e}-\frac {i \left (\operatorname {dilog}\left (\frac {-e c x +i e}{c d +i e}\right )-\operatorname {dilog}\left (\frac {e c x +i e}{-c d +i e}\right )\right )}{2 e}\right )}{e^{2}}}{e}\right )}{c^{4}}\)	\(393\)
risch	\(-\frac {b \,x^{2}}{6 c e}+\frac {b \ln \left (c^{2} x^{2}+1\right )}{6 c^{3} e}+\frac {b d x}{2 c \,e^{2}}-\frac {b d \arctan \left (c x \right )}{2 c^{2} e^{2}}-\frac {b \,d^{2} \ln \left (c^{2} x^{2}+1\right )}{2 c \,e^{3}}-\frac {a d}{2 c^{2} e^{2}}-\frac {a \,d^{3} \ln \left (i c d -\left (-i c x +1\right ) e +e \right )}{e^{4}}-\frac {a d \,x^{2}}{2 e^{2}}+\frac {i b \ln \left (-i c x +1\right ) x^{3}}{6 e}-\frac {i b \,d^{3} \operatorname {dilog}\left (\frac {-i c d +\left (-i c x +1\right ) e -e}{-i c d -e}\right )}{2 e^{4}}-\frac {i b \ln \left (i c x +1\right ) x^{3}}{6 e}+\frac {i b \,d^{3} \operatorname {dilog}\left (\frac {i c d +\left (i c x +1\right ) e -e}{i c d -e}\right )}{2 e^{4}}+\frac {b \,d^{2}}{c \,e^{3}}+\frac {a \,d^{2} x}{e^{3}}-\frac {11 b}{18 c^{3} e}+\frac {a \,x^{3}}{3 e}-\frac {i a}{3 c^{3} e}+\frac {i a \,d^{2}}{c \,e^{3}}+\frac {i b d \ln \left (i c x +1\right ) x^{2}}{4 e^{2}}-\frac {i b \ln \left (i c x +1\right ) x \,d^{2}}{2 e^{3}}+\frac {i b \,d^{3} \ln \left (i c x +1\right ) \ln \left (\frac {i c d +\left (i c x +1\right ) e -e}{i c d -e}\right )}{2 e^{4}}-\frac {i b d \ln \left (-i c x +1\right ) x^{2}}{4 e^{2}}+\frac {i b \ln \left (-i c x +1\right ) x \,d^{2}}{2 e^{3}}-\frac {i b \,d^{3} \ln \left (-i c x +1\right ) \ln \left (\frac {-i c d +\left (-i c x +1\right ) e -e}{-i c d -e}\right )}{2 e^{4}}\)	\(480\)

[In]

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

[Out]

1/3*a/e*x^3-1/2*a/e^2*d*x^2+a*d^2*x/e^3-a/e^4*d^3*ln(e*x+d)+b/c^4*(1/3*c^4*arctan(c*x)/e*x^3-1/2*c^4*arctan(c*
x)/e^2*x^2*d+c^4*arctan(c*x)/e^3*x*d^2-c^4*arctan(c*x)*d^3/e^4*ln(c*e*x+c*d)-c/e*(1/2/e^2*ln(c^2*d^2-2*c*d*(c*
e*x+c*d)+e^2+(c*e*x+c*d)^2)*c^2*d^2+1/2/e*arctan(c*x)*c*d-1/6*ln(c^2*d^2-2*c*d*(c*e*x+c*d)+e^2+(c*e*x+c*d)^2)-
5/6/e^2*c*d*(c*e*x+c*d)+1/6/e^2*(c*e*x+c*d)^2-1/e^2*c^3*d^3*(-1/2*I*ln(c*e*x+c*d)*(ln((I*e-e*c*x)/(c*d+I*e))-l
n((I*e+e*c*x)/(I*e-c*d)))/e-1/2*I*(dilog((I*e-e*c*x)/(c*d+I*e))-dilog((I*e+e*c*x)/(I*e-c*d)))/e)))

Fricas [F]

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

[In]

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

[Out]

integral((b*x^3*arctan(c*x) + a*x^3)/(e*x + d), x)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

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

[Out]

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