3.31 \(\int \frac {a+b \tan ^{-1}(c x^3)}{(d+e x)^2} \, dx\)

Optimal. Leaf size=906 \[ \frac {b c^2 \tan ^{-1}\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 \tan ^{-1}\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 ) \tan ^{-1}\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 ) \tan ^{-1}\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 \tan ^{-1}\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 ) \tan ^{-1}\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 ) \tan ^{-1}\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 )} \]

[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]  time = 1.48, antiderivative size = 906, normalized size of antiderivative = 1.00, number of steps used = 35, number of rules used = 16, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.889, Rules used = {5205, 12, 6725, 1876, 1416, 635, 203, 260, 634, 617, 204, 628, 1511, 292, 31, 1469} \[ \frac {b c^2 \tan ^{-1}\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 \tan ^{-1}\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 ) \tan ^{-1}\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 ) \tan ^{-1}\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 \tan ^{-1}\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 ) \tan ^{-1}\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 ) \tan ^{-1}\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 )} \]

Antiderivative was successfully verified.

[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 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 31

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

Rule 203

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

Rule 204

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

Rule 260

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 292

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 617

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 628

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 634

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 635

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 1416

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 1469

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 1511

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 1876

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 5205

Int[((a_.) + ArcTan[u_]*(b_.))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^(m + 1)*(a + b*ArcTan[
u]))/(d*(m + 1)), x] - Dist[b/(d*(m + 1)), Int[SimplifyIntegrand[((c + d*x)^(m + 1)*D[u, x])/(1 + u^2), x], x]
, x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1] && InverseFunctionFreeQ[u, x] &&  !FunctionOfQ[(c + d*x)^(m +
1), u, x] && FalseQ[PowerVariableExpn[u, m + 1, x]]

Rule 6725

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*} \int \frac {a+b \tan ^{-1}\left (c x^3\right )}{(d+e x)^2} \, dx &=-\frac {a+b \tan ^{-1}\left (c x^3\right )}{e (d+e x)}+\frac {b \int \frac {3 c x^2}{(d+e x) \left (1+c^2 x^6\right )} \, dx}{e}\\ &=-\frac {a+b \tan ^{-1}\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 \tan ^{-1}\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 \tan ^{-1}\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 \tan ^{-1}\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 \tan ^{-1}\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 \tan ^{-1}\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) \operatorname {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 \tan ^{-1}\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 ) \operatorname {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 ) \operatorname {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 \tan ^{-1}\left (\sqrt [3]{c} x\right )}{c^2 d^6+e^6}+\frac {b c^2 d^5 \tan ^{-1}\left (c x^3\right )}{e \left (c^2 d^6+e^6\right )}-\frac {a+b \tan ^{-1}\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 ) \operatorname {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 ) \operatorname {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 \tan ^{-1}\left (\sqrt [3]{c} x\right )}{c^2 d^6+e^6}+\frac {b c^2 d^5 \tan ^{-1}\left (c x^3\right )}{e \left (c^2 d^6+e^6\right )}-\frac {a+b \tan ^{-1}\left (c x^3\right )}{e (d+e x)}+\frac {b c^{2/3} d \left (\sqrt {3} c d^3+e^3\right ) \tan ^{-1}\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 ) \tan ^{-1}\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 ) \operatorname {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 ) \operatorname {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 \tan ^{-1}\left (\sqrt [3]{c} x\right )}{c^2 d^6+e^6}+\frac {b c^2 d^5 \tan ^{-1}\left (c x^3\right )}{e \left (c^2 d^6+e^6\right )}-\frac {a+b \tan ^{-1}\left (c x^3\right )}{e (d+e x)}+\frac {b c^{2/3} d \left (\sqrt {3} c d^3+e^3\right ) \tan ^{-1}\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 ) \tan ^{-1}\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 ) \tan ^{-1}\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 ) \tan ^{-1}\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*}

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Mathematica [A]  time = 16.81, size = 536, normalized size = 0.59 \[ \frac {-4 a \sqrt [3]{c} \left (c^2 d^6+e^6\right )+12 b c^{4/3} d^2 e^3 (d+e x) \log (d+e x)-b c^{2/3} e \left (c^{4/3} d^4-\sqrt {3} c d^3 e-\sqrt {3} \sqrt [3]{c} d e^3+e^4\right ) \log \left (c^{2/3} x^2-\sqrt {3} \sqrt [3]{c} x+1\right ) (d+e x)-b c^{2/3} e \left (c^{4/3} d^4+\sqrt {3} c d^3 e+\sqrt {3} \sqrt [3]{c} d e^3+e^4\right ) \log \left (c^{2/3} x^2+\sqrt {3} \sqrt [3]{c} x+1\right ) (d+e x)-4 b c d \left (c^{4/3} d^4-c^{2/3} d^2 e^2+e^4\right ) \tan ^{-1}\left (\sqrt [3]{c} x\right ) (d+e x)-2 b c^{2/3} \left (2 c^{5/3} d^5-\sqrt {3} c^{4/3} d^4 e+c d^3 e^2-\sqrt [3]{c} d e^4+\sqrt {3} e^5\right ) \tan ^{-1}\left (\sqrt {3}-2 \sqrt [3]{c} x\right ) (d+e x)+2 b c^{2/3} \left (2 c^{5/3} d^5+\sqrt {3} c^{4/3} d^4 e+c d^3 e^2-\sqrt [3]{c} d e^4-\sqrt {3} e^5\right ) \tan ^{-1}\left (2 \sqrt [3]{c} x+\sqrt {3}\right ) (d+e x)-4 b \sqrt [3]{c} \left (c^2 d^6+e^6\right ) \tan ^{-1}\left (c x^3\right )+2 b e \left (c^{2/3} e^4+c^2 d^4\right ) \log \left (c^{2/3} x^2+1\right ) (d+e x)-2 b c^{4/3} d^2 e^3 \log \left (c^2 x^6+1\right ) (d+e x)}{4 \sqrt [3]{c} e \left (c^2 d^6+e^6\right ) (d+e x)} \]

Antiderivative was successfully verified.

[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))

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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maple [A]  time = 0.23, size = 1220, normalized size = 1.35 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.41, size = 464, normalized size = 0.51 \[ \frac {1}{4} \, {\left ({\left (\frac {12 \, d^{2} e^{2} \log \left (e x + d\right )}{c^{2} d^{6} + e^{6}} - \frac {\frac {4 \, {\left (c^{\frac {8}{3}} d^{5} - c^{2} d^{3} e^{2} + c^{\frac {4}{3}} d e^{4}\right )} \arctan \left (c^{\frac {1}{3}} x\right )}{c^{\frac {5}{3}}} - \frac {2 \, {\left (\sqrt {3} c^{\frac {8}{3}} d^{4} e + 2 \, c^{3} d^{5} + c^{\frac {7}{3}} d^{3} e^{2} - \sqrt {3} c^{\frac {4}{3}} e^{5} - c^{\frac {5}{3}} d e^{4}\right )} \arctan \left (\frac {2 \, c^{\frac {2}{3}} x + \sqrt {3} c^{\frac {1}{3}}}{c^{\frac {1}{3}}}\right )}{c^{2}} + \frac {2 \, {\left (\sqrt {3} c^{\frac {8}{3}} d^{4} e - 2 \, c^{3} d^{5} - c^{\frac {7}{3}} d^{3} e^{2} - \sqrt {3} c^{\frac {4}{3}} e^{5} + c^{\frac {5}{3}} d e^{4}\right )} \arctan \left (\frac {2 \, c^{\frac {2}{3}} x - \sqrt {3} c^{\frac {1}{3}}}{c^{\frac {1}{3}}}\right )}{c^{2}} + \frac {{\left (\sqrt {3} c^{\frac {7}{3}} d^{3} e^{2} + c^{\frac {8}{3}} d^{4} e + \sqrt {3} c^{\frac {5}{3}} d e^{4} + 2 \, c^{2} d^{2} e^{3} + c^{\frac {4}{3}} e^{5}\right )} \log \left (c^{\frac {2}{3}} x^{2} + \sqrt {3} c^{\frac {1}{3}} x + 1\right )}{c^{2}} - \frac {{\left (\sqrt {3} c^{\frac {7}{3}} d^{3} e^{2} - c^{\frac {8}{3}} d^{4} e + \sqrt {3} c^{\frac {5}{3}} d e^{4} - 2 \, c^{2} d^{2} e^{3} - c^{\frac {4}{3}} e^{5}\right )} \log \left (c^{\frac {2}{3}} x^{2} - \sqrt {3} c^{\frac {1}{3}} x + 1\right )}{c^{2}} - \frac {2 \, {\left (c^{\frac {8}{3}} d^{4} e - c^{2} d^{2} e^{3} + c^{\frac {4}{3}} e^{5}\right )} \log \left (c^{\frac {2}{3}} x^{2} + 1\right )}{c^{2}}}{c^{2} d^{6} e + e^{7}}\right )} c - \frac {4 \, \arctan \left (c x^{3}\right )}{e^{2} x + d e}\right )} b - \frac {a}{e^{2} x + d e} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [B]  time = 0.78, size = 2105, normalized size = 2.32 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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