\(\int \frac {a+b \arctan (c x^3)}{(d+e x)^2} \, dx\) [31]

Optimal result

Integrand size = 18, antiderivative size = 906 \[ \int \frac {a+b \arctan \left (c x^3\right )}{(d+e x)^2} \, dx=-\frac {b c^{2/3} d e^3 \arctan \left (\sqrt [3]{c} x\right )}{c^2 d^6+e^6}+\frac {b c^2 d^5 \arctan \left (c x^3\right )}{e \left (c^2 d^6+e^6\right )}-\frac {a+b \arctan \left (c x^3\right )}{e (d+e x)}+\frac {b c^{2/3} d \left (\sqrt {3} c d^3+e^3\right ) \arctan \left (\sqrt {3}-2 \sqrt [3]{c} x\right )}{2 \left (c^2 d^6+e^6\right )}+\frac {b c^{2/3} d \left (\sqrt {3} c d^3-e^3\right ) \arctan \left (\sqrt {3}+2 \sqrt [3]{c} x\right )}{2 \left (c^2 d^6+e^6\right )}+\frac {\sqrt {3} b c^{5/3} e \left (\sqrt {-c^2} d^3+e^3\right ) \arctan \left (\frac {1+\frac {2 c^{2/3} x}{\sqrt [6]{-c^2}}}{\sqrt {3}}\right )}{2 \left (-c^2\right )^{2/3} \left (c^2 d^6+e^6\right )}-\frac {\sqrt {3} b c^{5/3} e \left (\sqrt {-c^2} d^3-e^3\right ) \arctan \left (\frac {c^{4/3}+2 \left (-c^2\right )^{5/6} x}{\sqrt {3} c^{4/3}}\right )}{2 \left (-c^2\right )^{2/3} \left (c^2 d^6+e^6\right )}+\frac {b c^{5/3} e \left (\sqrt {-c^2} d^3+e^3\right ) \log \left (\sqrt [6]{-c^2}-c^{2/3} x\right )}{2 \left (-c^2\right )^{2/3} \left (c^2 d^6+e^6\right )}-\frac {b c^{5/3} e \left (\sqrt {-c^2} d^3-e^3\right ) \log \left (\sqrt [6]{-c^2}+c^{2/3} x\right )}{2 \left (-c^2\right )^{2/3} \left (c^2 d^6+e^6\right )}+\frac {3 b c d^2 e^2 \log (d+e x)}{c^2 d^6+e^6}+\frac {b c^{5/3} d^4 \log \left (1+c^{2/3} x^2\right )}{2 \left (c^2 d^6+e^6\right )}-\frac {b c^{2/3} d \left (c d^3-\sqrt {3} e^3\right ) \log \left (1-\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2\right )}{4 \left (c^2 d^6+e^6\right )}-\frac {b c^{2/3} d \left (c d^3+\sqrt {3} e^3\right ) \log \left (1+\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2\right )}{4 \left (c^2 d^6+e^6\right )}+\frac {b c^{5/3} e \left (\sqrt {-c^2} d^3-e^3\right ) \log \left (\sqrt [3]{-c^2}-c^{2/3} \sqrt [6]{-c^2} x+c^{4/3} x^2\right )}{4 \left (-c^2\right )^{2/3} \left (c^2 d^6+e^6\right )}-\frac {b c^{5/3} e \left (\sqrt {-c^2} d^3+e^3\right ) \log \left (\sqrt [3]{-c^2}+c^{2/3} \sqrt [6]{-c^2} x+c^{4/3} x^2\right )}{4 \left (-c^2\right )^{2/3} \left (c^2 d^6+e^6\right )}-\frac {b c d^2 e^2 \log \left (1+c^2 x^6\right )}{2 \left (c^2 d^6+e^6\right )} \]

[Out]

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

Rubi [A] (verified)

Time = 0.91 (sec) , antiderivative size = 906, normalized size of antiderivative = 1.00, number of steps used = 34, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {4980, 6857, 1890, 1430, 649, 209, 266, 648, 631, 210, 642, 1525, 298, 31, 1483} \[ \int \frac {a+b \arctan \left (c x^3\right )}{(d+e x)^2} \, dx=\frac {b c^2 \arctan \left (c x^3\right ) d^5}{e \left (c^2 d^6+e^6\right )}+\frac {b c^{5/3} \log \left (c^{2/3} x^2+1\right ) d^4}{2 \left (c^2 d^6+e^6\right )}+\frac {3 b c e^2 \log (d+e x) d^2}{c^2 d^6+e^6}-\frac {b c e^2 \log \left (c^2 x^6+1\right ) d^2}{2 \left (c^2 d^6+e^6\right )}-\frac {b c^{2/3} e^3 \arctan \left (\sqrt [3]{c} x\right ) d}{c^2 d^6+e^6}+\frac {b c^{2/3} \left (\sqrt {3} c d^3+e^3\right ) \arctan \left (\sqrt {3}-2 \sqrt [3]{c} x\right ) d}{2 \left (c^2 d^6+e^6\right )}+\frac {b c^{2/3} \left (\sqrt {3} c d^3-e^3\right ) \arctan \left (2 \sqrt [3]{c} x+\sqrt {3}\right ) d}{2 \left (c^2 d^6+e^6\right )}-\frac {b c^{2/3} \left (c d^3-\sqrt {3} e^3\right ) \log \left (c^{2/3} x^2-\sqrt {3} \sqrt [3]{c} x+1\right ) d}{4 \left (c^2 d^6+e^6\right )}-\frac {b c^{2/3} \left (c d^3+\sqrt {3} e^3\right ) \log \left (c^{2/3} x^2+\sqrt {3} \sqrt [3]{c} x+1\right ) d}{4 \left (c^2 d^6+e^6\right )}-\frac {a+b \arctan \left (c x^3\right )}{e (d+e x)}+\frac {\sqrt {3} b c^{5/3} e \left (\sqrt {-c^2} d^3+e^3\right ) \arctan \left (\frac {\frac {2 c^{2/3} x}{\sqrt [6]{-c^2}}+1}{\sqrt {3}}\right )}{2 \left (-c^2\right )^{2/3} \left (c^2 d^6+e^6\right )}-\frac {\sqrt {3} b c^{5/3} e \left (\sqrt {-c^2} d^3-e^3\right ) \arctan \left (\frac {c^{4/3}+2 \left (-c^2\right )^{5/6} x}{\sqrt {3} c^{4/3}}\right )}{2 \left (-c^2\right )^{2/3} \left (c^2 d^6+e^6\right )}+\frac {b c^{5/3} e \left (\sqrt {-c^2} d^3+e^3\right ) \log \left (\sqrt [6]{-c^2}-c^{2/3} x\right )}{2 \left (-c^2\right )^{2/3} \left (c^2 d^6+e^6\right )}-\frac {b c^{5/3} e \left (\sqrt {-c^2} d^3-e^3\right ) \log \left (c^{2/3} x+\sqrt [6]{-c^2}\right )}{2 \left (-c^2\right )^{2/3} \left (c^2 d^6+e^6\right )}+\frac {b c^{5/3} e \left (\sqrt {-c^2} d^3-e^3\right ) \log \left (c^{4/3} x^2-c^{2/3} \sqrt [6]{-c^2} x+\sqrt [3]{-c^2}\right )}{4 \left (-c^2\right )^{2/3} \left (c^2 d^6+e^6\right )}-\frac {b c^{5/3} e \left (\sqrt {-c^2} d^3+e^3\right ) \log \left (c^{4/3} x^2+c^{2/3} \sqrt [6]{-c^2} x+\sqrt [3]{-c^2}\right )}{4 \left (-c^2\right )^{2/3} \left (c^2 d^6+e^6\right )} \]

[In]

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

[Out]

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

Rule 31

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

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 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[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 298

Int[(x_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> Dist[-(3*Rt[a, 3]*Rt[b, 3])^(-1), Int[1/(Rt[a, 3] + Rt[b, 3]*x),
x], x] + Dist[1/(3*Rt[a, 3]*Rt[b, 3]), Int[(Rt[a, 3] + Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3
]^2*x^2), x], x] /; FreeQ[{a, b}, x]

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

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

Rule 648

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

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 1430

Int[((d_) + (e_.)*(x_)^3)/((a_) + (c_.)*(x_)^6), x_Symbol] :> With[{q = Rt[c/a, 6]}, Dist[1/(3*a*q^2), Int[(q^
2*d - e*x)/(1 + q^2*x^2), x], x] + (Dist[1/(6*a*q^2), Int[(2*q^2*d - (Sqrt[3]*q^3*d - e)*x)/(1 - Sqrt[3]*q*x +
q^2*x^2), x], x] + Dist[1/(6*a*q^2), Int[(2*q^2*d + (Sqrt[3]*q^3*d + e)*x)/(1 + Sqrt[3]*q*x + q^2*x^2), x], x
])] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && PosQ[c/a]

Rule 1483

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[I
nt[(d + e*x)^q*(a + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, c, d, e, m, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[Sim
plify[m - n + 1], 0]

Rule 1525

Int[(((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(n_)))/((a_) + (c_.)*(x_)^(n2_)), x_Symbol] :> With[{q = Rt[(-a)*c,
2]}, Dist[-(e/2 + c*(d/(2*q))), Int[(f*x)^m/(q - c*x^n), x], x] + Dist[e/2 - c*(d/(2*q)), Int[(f*x)^m/(q + c*
x^n), x], x]] /; FreeQ[{a, c, d, e, f, m}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rule 1890

Int[(Pq_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = Sum[x^ii*((Coeff[Pq, x, ii] + Coeff[Pq, x, n/2 + ii
]*x^(n/2))/(a + b*x^n)), {ii, 0, n/2 - 1}]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ
[n/2, 0] && Expon[Pq, x] < n

Rule 4980

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

Rule 6857

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {a+b \arctan \left (c x^3\right )}{e (d+e x)}+\frac {(3 b c) \int \frac {x^2}{(d+e x) \left (1+c^2 x^6\right )} \, dx}{e} \\ & = -\frac {a+b \arctan \left (c x^3\right )}{e (d+e x)}+\frac {(3 b c) \int \left (\frac {d^2 e^4}{\left (c^2 d^6+e^6\right ) (d+e x)}+\frac {(d-e x) \left (-e^4+c^2 d^4 x^2+c^2 d^2 e^2 x^4\right )}{\left (c^2 d^6+e^6\right ) \left (1+c^2 x^6\right )}\right ) \, dx}{e} \\ & = -\frac {a+b \arctan \left (c x^3\right )}{e (d+e x)}+\frac {3 b c d^2 e^2 \log (d+e x)}{c^2 d^6+e^6}+\frac {(3 b c) \int \frac {(d-e x) \left (-e^4+c^2 d^4 x^2+c^2 d^2 e^2 x^4\right )}{1+c^2 x^6} \, dx}{e \left (c^2 d^6+e^6\right )} \\ & = -\frac {a+b \arctan \left (c x^3\right )}{e (d+e x)}+\frac {3 b c d^2 e^2 \log (d+e x)}{c^2 d^6+e^6}+\frac {(3 b c) \int \left (\frac {-d e^4-c^2 d^4 e x^3}{1+c^2 x^6}+\frac {x \left (e^5+c^2 d^3 e^2 x^3\right )}{1+c^2 x^6}+\frac {x^2 \left (c^2 d^5-c^2 d^2 e^3 x^3\right )}{1+c^2 x^6}\right ) \, dx}{e \left (c^2 d^6+e^6\right )} \\ & = -\frac {a+b \arctan \left (c x^3\right )}{e (d+e x)}+\frac {3 b c d^2 e^2 \log (d+e x)}{c^2 d^6+e^6}+\frac {(3 b c) \int \frac {-d e^4-c^2 d^4 e x^3}{1+c^2 x^6} \, dx}{e \left (c^2 d^6+e^6\right )}+\frac {(3 b c) \int \frac {x \left (e^5+c^2 d^3 e^2 x^3\right )}{1+c^2 x^6} \, dx}{e \left (c^2 d^6+e^6\right )}+\frac {(3 b c) \int \frac {x^2 \left (c^2 d^5-c^2 d^2 e^3 x^3\right )}{1+c^2 x^6} \, dx}{e \left (c^2 d^6+e^6\right )} \\ & = -\frac {a+b \arctan \left (c x^3\right )}{e (d+e x)}+\frac {3 b c d^2 e^2 \log (d+e x)}{c^2 d^6+e^6}+\frac {\left (b \sqrt [3]{c}\right ) \int \frac {-2 c^{2/3} d e^4-\left (c^2 d^4 e-\sqrt {3} c d e^4\right ) x}{1-\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2} \, dx}{2 e \left (c^2 d^6+e^6\right )}+\frac {\left (b \sqrt [3]{c}\right ) \int \frac {-2 c^{2/3} d e^4+\left (-c^2 d^4 e-\sqrt {3} c d e^4\right ) x}{1+\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2} \, dx}{2 e \left (c^2 d^6+e^6\right )}+\frac {\left (b \sqrt [3]{c}\right ) \int \frac {-c^{2/3} d e^4+c^2 d^4 e x}{1+c^{2/3} x^2} \, dx}{e \left (c^2 d^6+e^6\right )}+\frac {(b c) \text {Subst}\left (\int \frac {c^2 d^5-c^2 d^2 e^3 x}{1+c^2 x^2} \, dx,x,x^3\right )}{e \left (c^2 d^6+e^6\right )}-\frac {\left (3 b c^3 e \left (d^3+\frac {e^3}{\sqrt {-c^2}}\right )\right ) \int \frac {x}{\sqrt {-c^2}-c^2 x^3} \, dx}{2 \left (c^2 d^6+e^6\right )}+\frac {\left (3 b c e \left (c^2 d^3+\sqrt {-c^2} e^3\right )\right ) \int \frac {x}{\sqrt {-c^2}+c^2 x^3} \, dx}{2 \left (c^2 d^6+e^6\right )} \\ & = -\frac {a+b \arctan \left (c x^3\right )}{e (d+e x)}+\frac {3 b c d^2 e^2 \log (d+e x)}{c^2 d^6+e^6}+\frac {\left (b c^{7/3} d^4\right ) \int \frac {x}{1+c^{2/3} x^2} \, dx}{c^2 d^6+e^6}+\frac {\left (b c^3 d^5\right ) \text {Subst}\left (\int \frac {1}{1+c^2 x^2} \, dx,x,x^3\right )}{e \left (c^2 d^6+e^6\right )}-\frac {\left (b c^3 d^2 e^2\right ) \text {Subst}\left (\int \frac {x}{1+c^2 x^2} \, dx,x,x^3\right )}{c^2 d^6+e^6}-\frac {\left (b c d e^3\right ) \int \frac {1}{1+c^{2/3} x^2} \, dx}{c^2 d^6+e^6}+\frac {\left (b c d \left (\sqrt {3} c d^3-e^3\right )\right ) \int \frac {1}{1+\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2} \, dx}{4 \left (c^2 d^6+e^6\right )}+\frac {\left (b \sqrt [3]{c} \sqrt [3]{-c^2} e \left (\sqrt {-c^2} d^3-e^3\right )\right ) \int \frac {1}{\sqrt [6]{-c^2}+c^{2/3} x} \, dx}{2 \left (c^2 d^6+e^6\right )}-\frac {\left (b \sqrt [3]{c} \sqrt [3]{-c^2} e \left (\sqrt {-c^2} d^3-e^3\right )\right ) \int \frac {\sqrt [6]{-c^2}+c^{2/3} x}{\sqrt [3]{-c^2}-c^{2/3} \sqrt [6]{-c^2} x+c^{4/3} x^2} \, dx}{2 \left (c^2 d^6+e^6\right )}-\frac {\left (b c d \left (\sqrt {3} c d^3+e^3\right )\right ) \int \frac {1}{1-\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2} \, dx}{4 \left (c^2 d^6+e^6\right )}-\frac {\left (b c^{7/3} e \left (\sqrt {-c^2} d^3+e^3\right )\right ) \int \frac {1}{\sqrt [6]{-c^2}-c^{2/3} x} \, dx}{2 \left (-c^2\right )^{2/3} \left (c^2 d^6+e^6\right )}+\frac {\left (b c^{7/3} e \left (\sqrt {-c^2} d^3+e^3\right )\right ) \int \frac {\sqrt [6]{-c^2}-c^{2/3} x}{\sqrt [3]{-c^2}+c^{2/3} \sqrt [6]{-c^2} x+c^{4/3} x^2} \, dx}{2 \left (-c^2\right )^{2/3} \left (c^2 d^6+e^6\right )}-\frac {\left (b c^{2/3} d \left (c d^3-\sqrt {3} e^3\right )\right ) \int \frac {-\sqrt {3} \sqrt [3]{c}+2 c^{2/3} x}{1-\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2} \, dx}{4 \left (c^2 d^6+e^6\right )}-\frac {\left (b c^{2/3} d \left (c d^3+\sqrt {3} e^3\right )\right ) \int \frac {\sqrt {3} \sqrt [3]{c}+2 c^{2/3} x}{1+\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2} \, dx}{4 \left (c^2 d^6+e^6\right )} \\ & = -\frac {b c^{2/3} d e^3 \arctan \left (\sqrt [3]{c} x\right )}{c^2 d^6+e^6}+\frac {b c^2 d^5 \arctan \left (c x^3\right )}{e \left (c^2 d^6+e^6\right )}-\frac {a+b \arctan \left (c x^3\right )}{e (d+e x)}+\frac {b c^{5/3} e \left (\sqrt {-c^2} d^3+e^3\right ) \log \left (\sqrt [6]{-c^2}-c^{2/3} x\right )}{2 \left (-c^2\right )^{2/3} \left (c^2 d^6+e^6\right )}+\frac {b \sqrt [3]{-c^2} e \left (\sqrt {-c^2} d^3-e^3\right ) \log \left (\sqrt [6]{-c^2}+c^{2/3} x\right )}{2 \sqrt [3]{c} \left (c^2 d^6+e^6\right )}+\frac {3 b c d^2 e^2 \log (d+e x)}{c^2 d^6+e^6}+\frac {b c^{5/3} d^4 \log \left (1+c^{2/3} x^2\right )}{2 \left (c^2 d^6+e^6\right )}-\frac {b c^{2/3} d \left (c d^3-\sqrt {3} e^3\right ) \log \left (1-\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2\right )}{4 \left (c^2 d^6+e^6\right )}-\frac {b c^{2/3} d \left (c d^3+\sqrt {3} e^3\right ) \log \left (1+\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2\right )}{4 \left (c^2 d^6+e^6\right )}-\frac {b c d^2 e^2 \log \left (1+c^2 x^6\right )}{2 \left (c^2 d^6+e^6\right )}-\frac {\left (b \sqrt [3]{-c^2} e \left (\sqrt {-c^2} d^3-e^3\right )\right ) \int \frac {-c^{2/3} \sqrt [6]{-c^2}+2 c^{4/3} x}{\sqrt [3]{-c^2}-c^{2/3} \sqrt [6]{-c^2} x+c^{4/3} x^2} \, dx}{4 \sqrt [3]{c} \left (c^2 d^6+e^6\right )}-\frac {\left (b c^{5/3} e \left (\sqrt {-c^2} d^3+e^3\right )\right ) \int \frac {c^{2/3} \sqrt [6]{-c^2}+2 c^{4/3} x}{\sqrt [3]{-c^2}+c^{2/3} \sqrt [6]{-c^2} x+c^{4/3} x^2} \, dx}{4 \left (-c^2\right )^{2/3} \left (c^2 d^6+e^6\right )}+\frac {\left (3 b c^{7/3} e \left (\sqrt {-c^2} d^3+e^3\right )\right ) \int \frac {1}{\sqrt [3]{-c^2}+c^{2/3} \sqrt [6]{-c^2} x+c^{4/3} x^2} \, dx}{4 \sqrt {-c^2} \left (c^2 d^6+e^6\right )}-\frac {\left (b c^{2/3} d \left (3 c d^3-\sqrt {3} e^3\right )\right ) \text {Subst}\left (\int \frac {1}{-\frac {1}{3}-x^2} \, dx,x,1+\frac {2 \sqrt [3]{c} x}{\sqrt {3}}\right )}{6 \left (c^2 d^6+e^6\right )}-\frac {\left (b c^{2/3} d \left (3 c d^3+\sqrt {3} e^3\right )\right ) \text {Subst}\left (\int \frac {1}{-\frac {1}{3}-x^2} \, dx,x,1-\frac {2 \sqrt [3]{c} x}{\sqrt {3}}\right )}{6 \left (c^2 d^6+e^6\right )}+\frac {\left (3 b \sqrt [3]{c} e \left (c^2 d^3+\sqrt {-c^2} e^3\right )\right ) \int \frac {1}{\sqrt [3]{-c^2}-c^{2/3} \sqrt [6]{-c^2} x+c^{4/3} x^2} \, dx}{4 \left (c^2 d^6+e^6\right )} \\ & = -\frac {b c^{2/3} d e^3 \arctan \left (\sqrt [3]{c} x\right )}{c^2 d^6+e^6}+\frac {b c^2 d^5 \arctan \left (c x^3\right )}{e \left (c^2 d^6+e^6\right )}-\frac {a+b \arctan \left (c x^3\right )}{e (d+e x)}+\frac {b c^{2/3} d \left (\sqrt {3} c d^3+e^3\right ) \arctan \left (\sqrt {3}-2 \sqrt [3]{c} x\right )}{2 \left (c^2 d^6+e^6\right )}+\frac {b c^{2/3} d \left (\sqrt {3} c d^3-e^3\right ) \arctan \left (\sqrt {3}+2 \sqrt [3]{c} x\right )}{2 \left (c^2 d^6+e^6\right )}+\frac {b c^{5/3} e \left (\sqrt {-c^2} d^3+e^3\right ) \log \left (\sqrt [6]{-c^2}-c^{2/3} x\right )}{2 \left (-c^2\right )^{2/3} \left (c^2 d^6+e^6\right )}+\frac {b \sqrt [3]{-c^2} e \left (\sqrt {-c^2} d^3-e^3\right ) \log \left (\sqrt [6]{-c^2}+c^{2/3} x\right )}{2 \sqrt [3]{c} \left (c^2 d^6+e^6\right )}+\frac {3 b c d^2 e^2 \log (d+e x)}{c^2 d^6+e^6}+\frac {b c^{5/3} d^4 \log \left (1+c^{2/3} x^2\right )}{2 \left (c^2 d^6+e^6\right )}-\frac {b c^{2/3} d \left (c d^3-\sqrt {3} e^3\right ) \log \left (1-\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2\right )}{4 \left (c^2 d^6+e^6\right )}-\frac {b c^{2/3} d \left (c d^3+\sqrt {3} e^3\right ) \log \left (1+\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2\right )}{4 \left (c^2 d^6+e^6\right )}-\frac {b \sqrt [3]{-c^2} e \left (\sqrt {-c^2} d^3-e^3\right ) \log \left (\sqrt [3]{-c^2}-c^{2/3} \sqrt [6]{-c^2} x+c^{4/3} x^2\right )}{4 \sqrt [3]{c} \left (c^2 d^6+e^6\right )}-\frac {b c^{5/3} e \left (\sqrt {-c^2} d^3+e^3\right ) \log \left (\sqrt [3]{-c^2}+c^{2/3} \sqrt [6]{-c^2} x+c^{4/3} x^2\right )}{4 \left (-c^2\right )^{2/3} \left (c^2 d^6+e^6\right )}-\frac {b c d^2 e^2 \log \left (1+c^2 x^6\right )}{2 \left (c^2 d^6+e^6\right )}-\frac {\left (3 b \sqrt [3]{-c^2} e \left (\sqrt {-c^2} d^3-e^3\right )\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 c^{2/3} x}{\sqrt [6]{-c^2}}\right )}{2 \sqrt [3]{c} \left (c^2 d^6+e^6\right )}-\frac {\left (3 b c^{5/3} e \left (\sqrt {-c^2} d^3+e^3\right )\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 c^{2/3} x}{\sqrt [6]{-c^2}}\right )}{2 \left (-c^2\right )^{2/3} \left (c^2 d^6+e^6\right )} \\ & = -\frac {b c^{2/3} d e^3 \arctan \left (\sqrt [3]{c} x\right )}{c^2 d^6+e^6}+\frac {b c^2 d^5 \arctan \left (c x^3\right )}{e \left (c^2 d^6+e^6\right )}-\frac {a+b \arctan \left (c x^3\right )}{e (d+e x)}+\frac {b c^{2/3} d \left (\sqrt {3} c d^3+e^3\right ) \arctan \left (\sqrt {3}-2 \sqrt [3]{c} x\right )}{2 \left (c^2 d^6+e^6\right )}+\frac {b c^{2/3} d \left (\sqrt {3} c d^3-e^3\right ) \arctan \left (\sqrt {3}+2 \sqrt [3]{c} x\right )}{2 \left (c^2 d^6+e^6\right )}+\frac {\sqrt {3} b c^{5/3} e \left (\sqrt {-c^2} d^3+e^3\right ) \arctan \left (\frac {1+\frac {2 c^{2/3} x}{\sqrt [6]{-c^2}}}{\sqrt {3}}\right )}{2 \left (-c^2\right )^{2/3} \left (c^2 d^6+e^6\right )}+\frac {\sqrt {3} b \sqrt [3]{-c^2} e \left (\sqrt {-c^2} d^3-e^3\right ) \arctan \left (\frac {c^{4/3}+2 \left (-c^2\right )^{5/6} x}{\sqrt {3} c^{4/3}}\right )}{2 \sqrt [3]{c} \left (c^2 d^6+e^6\right )}+\frac {b c^{5/3} e \left (\sqrt {-c^2} d^3+e^3\right ) \log \left (\sqrt [6]{-c^2}-c^{2/3} x\right )}{2 \left (-c^2\right )^{2/3} \left (c^2 d^6+e^6\right )}+\frac {b \sqrt [3]{-c^2} e \left (\sqrt {-c^2} d^3-e^3\right ) \log \left (\sqrt [6]{-c^2}+c^{2/3} x\right )}{2 \sqrt [3]{c} \left (c^2 d^6+e^6\right )}+\frac {3 b c d^2 e^2 \log (d+e x)}{c^2 d^6+e^6}+\frac {b c^{5/3} d^4 \log \left (1+c^{2/3} x^2\right )}{2 \left (c^2 d^6+e^6\right )}-\frac {b c^{2/3} d \left (c d^3-\sqrt {3} e^3\right ) \log \left (1-\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2\right )}{4 \left (c^2 d^6+e^6\right )}-\frac {b c^{2/3} d \left (c d^3+\sqrt {3} e^3\right ) \log \left (1+\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2\right )}{4 \left (c^2 d^6+e^6\right )}-\frac {b \sqrt [3]{-c^2} e \left (\sqrt {-c^2} d^3-e^3\right ) \log \left (\sqrt [3]{-c^2}-c^{2/3} \sqrt [6]{-c^2} x+c^{4/3} x^2\right )}{4 \sqrt [3]{c} \left (c^2 d^6+e^6\right )}-\frac {b c^{5/3} e \left (\sqrt {-c^2} d^3+e^3\right ) \log \left (\sqrt [3]{-c^2}+c^{2/3} \sqrt [6]{-c^2} x+c^{4/3} x^2\right )}{4 \left (-c^2\right )^{2/3} \left (c^2 d^6+e^6\right )}-\frac {b c d^2 e^2 \log \left (1+c^2 x^6\right )}{2 \left (c^2 d^6+e^6\right )} \\ \end{align*}

Mathematica [A] (verified)

Time = 10.48 (sec) , antiderivative size = 536, normalized size of antiderivative = 0.59 \[ \int \frac {a+b \arctan \left (c x^3\right )}{(d+e x)^2} \, dx=\frac {-4 a \sqrt [3]{c} \left (c^2 d^6+e^6\right )-4 b c d \left (c^{4/3} d^4-c^{2/3} d^2 e^2+e^4\right ) (d+e x) \arctan \left (\sqrt [3]{c} x\right )-4 b \sqrt [3]{c} \left (c^2 d^6+e^6\right ) \arctan \left (c x^3\right )-2 b c^{2/3} \left (2 c^{5/3} d^5-\sqrt {3} c^{4/3} d^4 e+c d^3 e^2-\sqrt [3]{c} d e^4+\sqrt {3} e^5\right ) (d+e x) \arctan \left (\sqrt {3}-2 \sqrt [3]{c} x\right )+2 b c^{2/3} \left (2 c^{5/3} d^5+\sqrt {3} c^{4/3} d^4 e+c d^3 e^2-\sqrt [3]{c} d e^4-\sqrt {3} e^5\right ) (d+e x) \arctan \left (\sqrt {3}+2 \sqrt [3]{c} x\right )+12 b c^{4/3} d^2 e^3 (d+e x) \log (d+e x)+2 b e \left (c^2 d^4+c^{2/3} e^4\right ) (d+e x) \log \left (1+c^{2/3} x^2\right )-b c^{2/3} e \left (c^{4/3} d^4-\sqrt {3} c d^3 e-\sqrt {3} \sqrt [3]{c} d e^3+e^4\right ) (d+e x) \log \left (1-\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2\right )-b c^{2/3} e \left (c^{4/3} d^4+\sqrt {3} c d^3 e+\sqrt {3} \sqrt [3]{c} d e^3+e^4\right ) (d+e x) \log \left (1+\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2\right )-2 b c^{4/3} d^2 e^3 (d+e x) \log \left (1+c^2 x^6\right )}{4 \sqrt [3]{c} e \left (c^2 d^6+e^6\right ) (d+e x)} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 1.73 (sec) , antiderivative size = 862, normalized size of antiderivative = 0.95

method	result	size
default	\(\text {Expression too large to display}\)	\(862\)
parts	\(\text {Expression too large to display}\)	\(862\)

[In]

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

[Out]

-a/(e*x+d)/e+b*(-1/(e*x+d)/e*arctan(c*x^3)+3*c/e*(d^2*e^3/(c^2*d^6+e^6)*ln(e*x+d)+(-1/12*ln(x^2+3^(1/2)*(1/c^2
)^(1/6)*x+(1/c^2)^(1/3))*(1/c^2)^(2/3)*c^2*d^4*e-1/6*(1/c^2)^(7/6)*arctan(2*x/(1/c^2)^(1/6)+3^(1/2))*c^2*d*e^4
-1/12*ln(x^2-3^(1/2)*(1/c^2)^(1/6)*x+(1/c^2)^(1/3))*(1/c^2)^(2/3)*c^2*d^4*e-1/6*(1/c^2)^(7/6)*arctan(2*x/(1/c^
2)^(1/6)-3^(1/2))*c^2*d*e^4+1/3*(1/c^2)^(4/3)*arctan(2*x/(1/c^2)^(1/6)-3^(1/2))*3^(1/2)*c^2*e^5+1/6*ln(x^2+(1/
c^2)^(1/3))*(1/c^2)^(2/3)*c^2*d^4*e-1/3*(1/c^2)^(7/6)*arctan(x/(1/c^2)^(1/6))*c^2*d*e^4+1/3/(1/c^2)^(1/6)*arct
an(x/(1/c^2)^(1/6))*d^3*e^2+1/6/(1/c^2)^(1/6)*arctan(2*x/(1/c^2)^(1/6)+3^(1/2))*d^3*e^2-1/6*(1/c^2)^(1/3)*arct
an(2*x/(1/c^2)^(1/6)+3^(1/2))*3^(1/2)*e^5+1/3*(1/c^2)^(1/2)*arctan(2*x/(1/c^2)^(1/6)+3^(1/2))*c^2*d^5+1/6/(1/c
^2)^(1/6)*arctan(2*x/(1/c^2)^(1/6)-3^(1/2))*d^3*e^2-1/6*(1/c^2)^(1/3)*arctan(2*x/(1/c^2)^(1/6)-3^(1/2))*3^(1/2
)*e^5+1/3*(1/c^2)^(1/2)*arctan(2*x/(1/c^2)^(1/6)-3^(1/2))*c^2*d^5-1/3*(1/c^2)^(1/2)*arctan(x/(1/c^2)^(1/6))*c^
2*d^5-1/12*ln(x^2+3^(1/2)*(1/c^2)^(1/6)*x+(1/c^2)^(1/3))*(1/c^2)^(1/3)*e^5-1/6*ln(x^2+3^(1/2)*(1/c^2)^(1/6)*x+
(1/c^2)^(1/3))*d^2*e^3-1/12*ln(x^2-3^(1/2)*(1/c^2)^(1/6)*x+(1/c^2)^(1/3))*(1/c^2)^(1/3)*e^5-1/6*ln(x^2-3^(1/2)
*(1/c^2)^(1/6)*x+(1/c^2)^(1/3))*d^2*e^3+1/6*ln(x^2+(1/c^2)^(1/3))*(1/c^2)^(1/3)*e^5-1/6*ln(x^2+(1/c^2)^(1/3))*
d^2*e^3+1/6*(1/c^2)^(2/3)*arctan(2*x/(1/c^2)^(1/6)+3^(1/2))*3^(1/2)*c^2*d^4*e+1/12*ln(x^2-3^(1/2)*(1/c^2)^(1/6
)*x+(1/c^2)^(1/3))*3^(1/2)*(1/c^2)^(7/6)*c^2*d*e^4+1/12*ln(x^2-3^(1/2)*(1/c^2)^(1/6)*x+(1/c^2)^(1/3))*3^(1/2)*
(1/c^2)^(5/6)*c^2*d^3*e^2-1/6*(1/c^2)^(2/3)*arctan(2*x/(1/c^2)^(1/6)-3^(1/2))*3^(1/2)*c^2*d^4*e-1/12*ln(x^2+3^
(1/2)*(1/c^2)^(1/6)*x+(1/c^2)^(1/3))*3^(1/2)*(1/c^2)^(7/6)*c^2*d*e^4-1/12*ln(x^2+3^(1/2)*(1/c^2)^(1/6)*x+(1/c^
2)^(1/3))*3^(1/2)*(1/c^2)^(5/6)*c^2*d^3*e^2)/(c^2*d^6+e^6)))

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 464, normalized size of antiderivative = 0.51 \[ \int \frac {a+b \arctan \left (c x^3\right )}{(d+e x)^2} \, dx=\frac {1}{4} \, {\left ({\left (\frac {12 \, d^{2} e^{2} \log \left (e x + d\right )}{c^{2} d^{6} + e^{6}} - \frac {\frac {4 \, {\left (c^{\frac {8}{3}} d^{5} - c^{2} d^{3} e^{2} + c^{\frac {4}{3}} d e^{4}\right )} \arctan \left (c^{\frac {1}{3}} x\right )}{c^{\frac {5}{3}}} - \frac {2 \, {\left (\sqrt {3} c^{\frac {8}{3}} d^{4} e + 2 \, c^{3} d^{5} + c^{\frac {7}{3}} d^{3} e^{2} - \sqrt {3} c^{\frac {4}{3}} e^{5} - c^{\frac {5}{3}} d e^{4}\right )} \arctan \left (\frac {2 \, c^{\frac {2}{3}} x + \sqrt {3} c^{\frac {1}{3}}}{c^{\frac {1}{3}}}\right )}{c^{2}} + \frac {2 \, {\left (\sqrt {3} c^{\frac {8}{3}} d^{4} e - 2 \, c^{3} d^{5} - c^{\frac {7}{3}} d^{3} e^{2} - \sqrt {3} c^{\frac {4}{3}} e^{5} + c^{\frac {5}{3}} d e^{4}\right )} \arctan \left (\frac {2 \, c^{\frac {2}{3}} x - \sqrt {3} c^{\frac {1}{3}}}{c^{\frac {1}{3}}}\right )}{c^{2}} + \frac {{\left (\sqrt {3} c^{\frac {7}{3}} d^{3} e^{2} + c^{\frac {8}{3}} d^{4} e + \sqrt {3} c^{\frac {5}{3}} d e^{4} + 2 \, c^{2} d^{2} e^{3} + c^{\frac {4}{3}} e^{5}\right )} \log \left (c^{\frac {2}{3}} x^{2} + \sqrt {3} c^{\frac {1}{3}} x + 1\right )}{c^{2}} - \frac {{\left (\sqrt {3} c^{\frac {7}{3}} d^{3} e^{2} - c^{\frac {8}{3}} d^{4} e + \sqrt {3} c^{\frac {5}{3}} d e^{4} - 2 \, c^{2} d^{2} e^{3} - c^{\frac {4}{3}} e^{5}\right )} \log \left (c^{\frac {2}{3}} x^{2} - \sqrt {3} c^{\frac {1}{3}} x + 1\right )}{c^{2}} - \frac {2 \, {\left (c^{\frac {8}{3}} d^{4} e - c^{2} d^{2} e^{3} + c^{\frac {4}{3}} e^{5}\right )} \log \left (c^{\frac {2}{3}} x^{2} + 1\right )}{c^{2}}}{c^{2} d^{6} e + e^{7}}\right )} c - \frac {4 \, \arctan \left (c x^{3}\right )}{e^{2} x + d e}\right )} b - \frac {a}{e^{2} x + d e} \]

[In]

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

[Out]

1/4*((12*d^2*e^2*log(e*x + d)/(c^2*d^6 + e^6) - (4*(c^(8/3)*d^5 - c^2*d^3*e^2 + c^(4/3)*d*e^4)*arctan(c^(1/3)*
x)/c^(5/3) - 2*(sqrt(3)*c^(8/3)*d^4*e + 2*c^3*d^5 + c^(7/3)*d^3*e^2 - sqrt(3)*c^(4/3)*e^5 - c^(5/3)*d*e^4)*arc
tan((2*c^(2/3)*x + sqrt(3)*c^(1/3))/c^(1/3))/c^2 + 2*(sqrt(3)*c^(8/3)*d^4*e - 2*c^3*d^5 - c^(7/3)*d^3*e^2 - sq
rt(3)*c^(4/3)*e^5 + c^(5/3)*d*e^4)*arctan((2*c^(2/3)*x - sqrt(3)*c^(1/3))/c^(1/3))/c^2 + (sqrt(3)*c^(7/3)*d^3*
e^2 + c^(8/3)*d^4*e + sqrt(3)*c^(5/3)*d*e^4 + 2*c^2*d^2*e^3 + c^(4/3)*e^5)*log(c^(2/3)*x^2 + sqrt(3)*c^(1/3)*x
+ 1)/c^2 - (sqrt(3)*c^(7/3)*d^3*e^2 - c^(8/3)*d^4*e + sqrt(3)*c^(5/3)*d*e^4 - 2*c^2*d^2*e^3 - c^(4/3)*e^5)*lo
g(c^(2/3)*x^2 - sqrt(3)*c^(1/3)*x + 1)/c^2 - 2*(c^(8/3)*d^4*e - c^2*d^2*e^3 + c^(4/3)*e^5)*log(c^(2/3)*x^2 + 1
)/c^2)/(c^2*d^6*e + e^7))*c - 4*arctan(c*x^3)/(e^2*x + d*e))*b - a/(e^2*x + d*e)

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [B] (verification not implemented)

Time = 0.84 (sec) , antiderivative size = 2105, normalized size of antiderivative = 2.32 \[ \int \frac {a+b \arctan \left (c x^3\right )}{(d+e x)^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

symsum(log((729*b^6*c^14*d*e^2 + 54432*root(64*c^2*d^6*e^6*z^6 + 64*e^12*z^6 + 192*b*c*d^2*e^8*z^5 + 48*b^2*c^
2*d^4*e^4*z^4 - 16*b^3*c*e^6*z^3 + 12*b^4*c^2*d^2*e^2*z^2 + b^6*c^2, z, k)^6*c^12*e^15*x + 729*b^6*c^14*e^3*x
- 31104*root(64*c^2*d^6*e^6*z^6 + 64*e^12*z^6 + 192*b*c*d^2*e^8*z^5 + 48*b^2*c^2*d^4*e^4*z^4 - 16*b^3*c*e^6*z^
3 + 12*b^4*c^2*d^2*e^2*z^2 + b^6*c^2, z, k)^6*c^14*d^7*e^8 - 243*root(64*c^2*d^6*e^6*z^6 + 64*e^12*z^6 + 192*b
*c*d^2*e^8*z^5 + 48*b^2*c^2*d^4*e^4*z^4 - 16*b^3*c*e^6*z^3 + 12*b^4*c^2*d^2*e^2*z^2 + b^6*c^2, z, k)*b^5*c^15*
d^5 + 62208*root(64*c^2*d^6*e^6*z^6 + 64*e^12*z^6 + 192*b*c*d^2*e^8*z^5 + 48*b^2*c^2*d^4*e^4*z^4 - 16*b^3*c*e^
6*z^3 + 12*b^4*c^2*d^2*e^2*z^2 + b^6*c^2, z, k)^6*c^12*d*e^14 + 5832*root(64*c^2*d^6*e^6*z^6 + 64*e^12*z^6 + 1
92*b*c*d^2*e^8*z^5 + 48*b^2*c^2*d^4*e^4*z^4 - 16*b^3*c*e^6*z^3 + 12*b^4*c^2*d^2*e^2*z^2 + b^6*c^2, z, k)^2*b^4
*c^14*d^3*e^4 - 1944*root(64*c^2*d^6*e^6*z^6 + 64*e^12*z^6 + 192*b*c*d^2*e^8*z^5 + 48*b^2*c^2*d^4*e^4*z^4 - 16
*b^3*c*e^6*z^3 + 12*b^4*c^2*d^2*e^2*z^2 + b^6*c^2, z, k)^3*b^3*c^15*d^7*e^2 + 15552*root(64*c^2*d^6*e^6*z^6 +
64*e^12*z^6 + 192*b*c*d^2*e^8*z^5 + 48*b^2*c^2*d^4*e^4*z^4 - 16*b^3*c*e^6*z^3 + 12*b^4*c^2*d^2*e^2*z^2 + b^6*c
^2, z, k)^4*b^2*c^14*d^5*e^6 - 10692*root(64*c^2*d^6*e^6*z^6 + 64*e^12*z^6 + 192*b*c*d^2*e^8*z^5 + 48*b^2*c^2*
d^4*e^4*z^4 - 16*b^3*c*e^6*z^3 + 12*b^4*c^2*d^2*e^2*z^2 + b^6*c^2, z, k)^3*b^3*c^13*d*e^8 + 101088*root(64*c^2
*d^6*e^6*z^6 + 64*e^12*z^6 + 192*b*c*d^2*e^8*z^5 + 48*b^2*c^2*d^4*e^4*z^4 - 16*b^3*c*e^6*z^3 + 12*b^4*c^2*d^2*
e^2*z^2 + b^6*c^2, z, k)^5*b*c^13*d^3*e^10 - 3888*root(64*c^2*d^6*e^6*z^6 + 64*e^12*z^6 + 192*b*c*d^2*e^8*z^5
+ 48*b^2*c^2*d^4*e^4*z^4 - 16*b^3*c*e^6*z^3 + 12*b^4*c^2*d^2*e^2*z^2 + b^6*c^2, z, k)^5*b*c^15*d^9*e^4 - 12636
*root(64*c^2*d^6*e^6*z^6 + 64*e^12*z^6 + 192*b*c*d^2*e^8*z^5 + 48*b^2*c^2*d^4*e^4*z^4 - 16*b^3*c*e^6*z^3 + 12*
b^4*c^2*d^2*e^2*z^2 + b^6*c^2, z, k)^3*b^3*c^13*e^9*x - 38880*root(64*c^2*d^6*e^6*z^6 + 64*e^12*z^6 + 192*b*c*
d^2*e^8*z^5 + 48*b^2*c^2*d^4*e^4*z^4 - 16*b^3*c*e^6*z^3 + 12*b^4*c^2*d^2*e^2*z^2 + b^6*c^2, z, k)^6*c^14*d^6*e
^9*x + 116640*root(64*c^2*d^6*e^6*z^6 + 64*e^12*z^6 + 192*b*c*d^2*e^8*z^5 + 48*b^2*c^2*d^4*e^4*z^4 - 16*b^3*c*
e^6*z^3 + 12*b^4*c^2*d^2*e^2*z^2 + b^6*c^2, z, k)^5*b*c^13*d^2*e^11*x - 11664*root(64*c^2*d^6*e^6*z^6 + 64*e^1
2*z^6 + 192*b*c*d^2*e^8*z^5 + 48*b^2*c^2*d^4*e^4*z^4 - 16*b^3*c*e^6*z^3 + 12*b^4*c^2*d^2*e^2*z^2 + b^6*c^2, z,
k)^5*b*c^15*d^8*e^5*x + 11664*root(64*c^2*d^6*e^6*z^6 + 64*e^12*z^6 + 192*b*c*d^2*e^8*z^5 + 48*b^2*c^2*d^4*e^
4*z^4 - 16*b^3*c*e^6*z^3 + 12*b^4*c^2*d^2*e^2*z^2 + b^6*c^2, z, k)^2*b^4*c^14*d^2*e^5*x - 3888*root(64*c^2*d^6
*e^6*z^6 + 64*e^12*z^6 + 192*b*c*d^2*e^8*z^5 + 48*b^2*c^2*d^4*e^4*z^4 - 16*b^3*c*e^6*z^3 + 12*b^4*c^2*d^2*e^2*
z^2 + b^6*c^2, z, k)^3*b^3*c^15*d^6*e^3*x + 38880*root(64*c^2*d^6*e^6*z^6 + 64*e^12*z^6 + 192*b*c*d^2*e^8*z^5
+ 48*b^2*c^2*d^4*e^4*z^4 - 16*b^3*c*e^6*z^3 + 12*b^4*c^2*d^2*e^2*z^2 + b^6*c^2, z, k)^4*b^2*c^14*d^4*e^7*x - 2
43*root(64*c^2*d^6*e^6*z^6 + 64*e^12*z^6 + 192*b*c*d^2*e^8*z^5 + 48*b^2*c^2*d^4*e^4*z^4 - 16*b^3*c*e^6*z^3 + 1
2*b^4*c^2*d^2*e^2*z^2 + b^6*c^2, z, k)*b^5*c^15*d^4*e*x)/e^4)*root(64*c^2*d^6*e^6*z^6 + 64*e^12*z^6 + 192*b*c*
d^2*e^8*z^5 + 48*b^2*c^2*d^4*e^4*z^4 - 16*b^3*c*e^6*z^3 + 12*b^4*c^2*d^2*e^2*z^2 + b^6*c^2, z, k), k, 1, 6) -
a/(d*e + e^2*x) - (b*atan(c*x^3))/(d*e + e^2*x) + (3*b*c*d^2*e^2*log(d + e*x))/(e^6 + c^2*d^6)