3.49 \(\int \frac{a+b \tanh ^{-1}(c \sqrt{x})}{x^3 (d+e x)} \, dx\)

Optimal. Leaf size=506 \[ -\frac{b e^2 \text{PolyLog}\left (2,1-\frac{2}{c \sqrt{x}+1}\right )}{d^3}+\frac{b e^2 \text{PolyLog}\left (2,1-\frac{2 c \left (\sqrt{-d}-\sqrt{e} \sqrt{x}\right )}{\left (c \sqrt{x}+1\right ) \left (c \sqrt{-d}-\sqrt{e}\right )}\right )}{2 d^3}+\frac{b e^2 \text{PolyLog}\left (2,1-\frac{2 c \left (\sqrt{-d}+\sqrt{e} \sqrt{x}\right )}{\left (c \sqrt{x}+1\right ) \left (c \sqrt{-d}+\sqrt{e}\right )}\right )}{2 d^3}-\frac{b e^2 \text{PolyLog}\left (2,-c \sqrt{x}\right )}{d^3}+\frac{b e^2 \text{PolyLog}\left (2,c \sqrt{x}\right )}{d^3}+\frac{2 e^2 \log \left (\frac{2}{c \sqrt{x}+1}\right ) \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )}{d^3}-\frac{e^2 \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right ) \log \left (\frac{2 c \left (\sqrt{-d}-\sqrt{e} \sqrt{x}\right )}{\left (c \sqrt{x}+1\right ) \left (c \sqrt{-d}-\sqrt{e}\right )}\right )}{d^3}-\frac{e^2 \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right ) \log \left (\frac{2 c \left (\sqrt{-d}+\sqrt{e} \sqrt{x}\right )}{\left (c \sqrt{x}+1\right ) \left (c \sqrt{-d}+\sqrt{e}\right )}\right )}{d^3}+\frac{e \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )}{d^2 x}-\frac{a+b \tanh ^{-1}\left (c \sqrt{x}\right )}{2 d x^2}+\frac{a e^2 \log (x)}{d^3}-\frac{b c^2 e \tanh ^{-1}\left (c \sqrt{x}\right )}{d^2}-\frac{b c^3}{2 d \sqrt{x}}+\frac{b c^4 \tanh ^{-1}\left (c \sqrt{x}\right )}{2 d}+\frac{b c e}{d^2 \sqrt{x}}-\frac{b c}{6 d x^{3/2}} \]

[Out]

-(b*c)/(6*d*x^(3/2)) - (b*c^3)/(2*d*Sqrt[x]) + (b*c*e)/(d^2*Sqrt[x]) + (b*c^4*ArcTanh[c*Sqrt[x]])/(2*d) - (b*c
^2*e*ArcTanh[c*Sqrt[x]])/d^2 - (a + b*ArcTanh[c*Sqrt[x]])/(2*d*x^2) + (e*(a + b*ArcTanh[c*Sqrt[x]]))/(d^2*x) +
(2*e^2*(a + b*ArcTanh[c*Sqrt[x]])*Log[2/(1 + c*Sqrt[x])])/d^3 - (e^2*(a + b*ArcTanh[c*Sqrt[x]])*Log[(2*c*(Sqr
t[-d] - Sqrt[e]*Sqrt[x]))/((c*Sqrt[-d] - Sqrt[e])*(1 + c*Sqrt[x]))])/d^3 - (e^2*(a + b*ArcTanh[c*Sqrt[x]])*Log
[(2*c*(Sqrt[-d] + Sqrt[e]*Sqrt[x]))/((c*Sqrt[-d] + Sqrt[e])*(1 + c*Sqrt[x]))])/d^3 + (a*e^2*Log[x])/d^3 - (b*e
^2*PolyLog[2, 1 - 2/(1 + c*Sqrt[x])])/d^3 + (b*e^2*PolyLog[2, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*Sqrt[x]))/((c*Sqrt[
-d] - Sqrt[e])*(1 + c*Sqrt[x]))])/(2*d^3) + (b*e^2*PolyLog[2, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*Sqrt[x]))/((c*Sqrt[
-d] + Sqrt[e])*(1 + c*Sqrt[x]))])/(2*d^3) - (b*e^2*PolyLog[2, -(c*Sqrt[x])])/d^3 + (b*e^2*PolyLog[2, c*Sqrt[x]
])/d^3

________________________________________________________________________________________

Rubi [A]  time = 0.872401, antiderivative size = 506, normalized size of antiderivative = 1., number of steps used = 24, number of rules used = 13, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.565, Rules used = {44, 1593, 5982, 5916, 325, 206, 5992, 5912, 6044, 5920, 2402, 2315, 2447} \[ -\frac{b e^2 \text{PolyLog}\left (2,1-\frac{2}{c \sqrt{x}+1}\right )}{d^3}+\frac{b e^2 \text{PolyLog}\left (2,1-\frac{2 c \left (\sqrt{-d}-\sqrt{e} \sqrt{x}\right )}{\left (c \sqrt{x}+1\right ) \left (c \sqrt{-d}-\sqrt{e}\right )}\right )}{2 d^3}+\frac{b e^2 \text{PolyLog}\left (2,1-\frac{2 c \left (\sqrt{-d}+\sqrt{e} \sqrt{x}\right )}{\left (c \sqrt{x}+1\right ) \left (c \sqrt{-d}+\sqrt{e}\right )}\right )}{2 d^3}-\frac{b e^2 \text{PolyLog}\left (2,-c \sqrt{x}\right )}{d^3}+\frac{b e^2 \text{PolyLog}\left (2,c \sqrt{x}\right )}{d^3}+\frac{2 e^2 \log \left (\frac{2}{c \sqrt{x}+1}\right ) \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )}{d^3}-\frac{e^2 \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right ) \log \left (\frac{2 c \left (\sqrt{-d}-\sqrt{e} \sqrt{x}\right )}{\left (c \sqrt{x}+1\right ) \left (c \sqrt{-d}-\sqrt{e}\right )}\right )}{d^3}-\frac{e^2 \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right ) \log \left (\frac{2 c \left (\sqrt{-d}+\sqrt{e} \sqrt{x}\right )}{\left (c \sqrt{x}+1\right ) \left (c \sqrt{-d}+\sqrt{e}\right )}\right )}{d^3}+\frac{e \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )}{d^2 x}-\frac{a+b \tanh ^{-1}\left (c \sqrt{x}\right )}{2 d x^2}+\frac{a e^2 \log (x)}{d^3}-\frac{b c^2 e \tanh ^{-1}\left (c \sqrt{x}\right )}{d^2}-\frac{b c^3}{2 d \sqrt{x}}+\frac{b c^4 \tanh ^{-1}\left (c \sqrt{x}\right )}{2 d}+\frac{b c e}{d^2 \sqrt{x}}-\frac{b c}{6 d x^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b*c)/(6*d*x^(3/2)) - (b*c^3)/(2*d*Sqrt[x]) + (b*c*e)/(d^2*Sqrt[x]) + (b*c^4*ArcTanh[c*Sqrt[x]])/(2*d) - (b*c
^2*e*ArcTanh[c*Sqrt[x]])/d^2 - (a + b*ArcTanh[c*Sqrt[x]])/(2*d*x^2) + (e*(a + b*ArcTanh[c*Sqrt[x]]))/(d^2*x) +
(2*e^2*(a + b*ArcTanh[c*Sqrt[x]])*Log[2/(1 + c*Sqrt[x])])/d^3 - (e^2*(a + b*ArcTanh[c*Sqrt[x]])*Log[(2*c*(Sqr
t[-d] - Sqrt[e]*Sqrt[x]))/((c*Sqrt[-d] - Sqrt[e])*(1 + c*Sqrt[x]))])/d^3 - (e^2*(a + b*ArcTanh[c*Sqrt[x]])*Log
[(2*c*(Sqrt[-d] + Sqrt[e]*Sqrt[x]))/((c*Sqrt[-d] + Sqrt[e])*(1 + c*Sqrt[x]))])/d^3 + (a*e^2*Log[x])/d^3 - (b*e
^2*PolyLog[2, 1 - 2/(1 + c*Sqrt[x])])/d^3 + (b*e^2*PolyLog[2, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*Sqrt[x]))/((c*Sqrt[
-d] - Sqrt[e])*(1 + c*Sqrt[x]))])/(2*d^3) + (b*e^2*PolyLog[2, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*Sqrt[x]))/((c*Sqrt[
-d] + Sqrt[e])*(1 + c*Sqrt[x]))])/(2*d^3) - (b*e^2*PolyLog[2, -(c*Sqrt[x])])/d^3 + (b*e^2*PolyLog[2, c*Sqrt[x]
])/d^3

Rule 44

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 5982

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

Rule 5916

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

Rule 325

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

Rule 206

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

Rule 5992

Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))*(x_)^(m_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[a
+ b*ArcTanh[c*x], x^m/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IntegerQ[m] &&  !(EqQ[m, 1] && NeQ[
a, 0])

Rule 5912

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))/(x_), x_Symbol] :> Simp[a*Log[x], x] + (-Simp[(b*PolyLog[2, -(c*x)])/2
, x] + Simp[(b*PolyLog[2, c*x])/2, x]) /; FreeQ[{a, b, c}, x]

Rule 6044

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.), x_Symbol] :> Wit
h[{u = ExpandIntegrand[(a + b*ArcTanh[c*x])^p, (f*x)^m*(d + e*x^2)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a,
b, c, d, e, f, m}, x] && IntegerQ[q] && IGtQ[p, 0] && (GtQ[q, 0] || IntegerQ[m])

Rule 5920

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

Rule 2402

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 2315

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

Rule 2447

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]]

Rubi steps

\begin{align*} \int \frac{a+b \tanh ^{-1}\left (c \sqrt{x}\right )}{x^3 (d+e x)} \, dx &=2 \operatorname{Subst}\left (\int \frac{a+b \tanh ^{-1}(c x)}{d x^5+e x^7} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \frac{a+b \tanh ^{-1}(c x)}{x^5 \left (d+e x^2\right )} \, dx,x,\sqrt{x}\right )\\ &=\frac{2 \operatorname{Subst}\left (\int \frac{a+b \tanh ^{-1}(c x)}{x^5} \, dx,x,\sqrt{x}\right )}{d}-\frac{(2 e) \operatorname{Subst}\left (\int \frac{a+b \tanh ^{-1}(c x)}{x^3 \left (d+e x^2\right )} \, dx,x,\sqrt{x}\right )}{d}\\ &=-\frac{a+b \tanh ^{-1}\left (c \sqrt{x}\right )}{2 d x^2}+\frac{(b c) \operatorname{Subst}\left (\int \frac{1}{x^4 \left (1-c^2 x^2\right )} \, dx,x,\sqrt{x}\right )}{2 d}-\frac{(2 e) \operatorname{Subst}\left (\int \frac{a+b \tanh ^{-1}(c x)}{x^3} \, dx,x,\sqrt{x}\right )}{d^2}+\frac{\left (2 e^2\right ) \operatorname{Subst}\left (\int \frac{a+b \tanh ^{-1}(c x)}{x \left (d+e x^2\right )} \, dx,x,\sqrt{x}\right )}{d^2}\\ &=-\frac{b c}{6 d x^{3/2}}-\frac{a+b \tanh ^{-1}\left (c \sqrt{x}\right )}{2 d x^2}+\frac{e \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )}{d^2 x}+\frac{\left (b c^3\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \left (1-c^2 x^2\right )} \, dx,x,\sqrt{x}\right )}{2 d}-\frac{(b c e) \operatorname{Subst}\left (\int \frac{1}{x^2 \left (1-c^2 x^2\right )} \, dx,x,\sqrt{x}\right )}{d^2}+\frac{\left (2 e^2\right ) \operatorname{Subst}\left (\int \left (\frac{a+b \tanh ^{-1}(c x)}{d x}-\frac{e x \left (a+b \tanh ^{-1}(c x)\right )}{d \left (d+e x^2\right )}\right ) \, dx,x,\sqrt{x}\right )}{d^2}\\ &=-\frac{b c}{6 d x^{3/2}}-\frac{b c^3}{2 d \sqrt{x}}+\frac{b c e}{d^2 \sqrt{x}}-\frac{a+b \tanh ^{-1}\left (c \sqrt{x}\right )}{2 d x^2}+\frac{e \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )}{d^2 x}+\frac{\left (b c^5\right ) \operatorname{Subst}\left (\int \frac{1}{1-c^2 x^2} \, dx,x,\sqrt{x}\right )}{2 d}-\frac{\left (b c^3 e\right ) \operatorname{Subst}\left (\int \frac{1}{1-c^2 x^2} \, dx,x,\sqrt{x}\right )}{d^2}+\frac{\left (2 e^2\right ) \operatorname{Subst}\left (\int \frac{a+b \tanh ^{-1}(c x)}{x} \, dx,x,\sqrt{x}\right )}{d^3}-\frac{\left (2 e^3\right ) \operatorname{Subst}\left (\int \frac{x \left (a+b \tanh ^{-1}(c x)\right )}{d+e x^2} \, dx,x,\sqrt{x}\right )}{d^3}\\ &=-\frac{b c}{6 d x^{3/2}}-\frac{b c^3}{2 d \sqrt{x}}+\frac{b c e}{d^2 \sqrt{x}}+\frac{b c^4 \tanh ^{-1}\left (c \sqrt{x}\right )}{2 d}-\frac{b c^2 e \tanh ^{-1}\left (c \sqrt{x}\right )}{d^2}-\frac{a+b \tanh ^{-1}\left (c \sqrt{x}\right )}{2 d x^2}+\frac{e \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )}{d^2 x}+\frac{a e^2 \log (x)}{d^3}-\frac{b e^2 \text{Li}_2\left (-c \sqrt{x}\right )}{d^3}+\frac{b e^2 \text{Li}_2\left (c \sqrt{x}\right )}{d^3}-\frac{\left (2 e^3\right ) \operatorname{Subst}\left (\int \left (-\frac{a+b \tanh ^{-1}(c x)}{2 \sqrt{e} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{a+b \tanh ^{-1}(c x)}{2 \sqrt{e} \left (\sqrt{-d}+\sqrt{e} x\right )}\right ) \, dx,x,\sqrt{x}\right )}{d^3}\\ &=-\frac{b c}{6 d x^{3/2}}-\frac{b c^3}{2 d \sqrt{x}}+\frac{b c e}{d^2 \sqrt{x}}+\frac{b c^4 \tanh ^{-1}\left (c \sqrt{x}\right )}{2 d}-\frac{b c^2 e \tanh ^{-1}\left (c \sqrt{x}\right )}{d^2}-\frac{a+b \tanh ^{-1}\left (c \sqrt{x}\right )}{2 d x^2}+\frac{e \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )}{d^2 x}+\frac{a e^2 \log (x)}{d^3}-\frac{b e^2 \text{Li}_2\left (-c \sqrt{x}\right )}{d^3}+\frac{b e^2 \text{Li}_2\left (c \sqrt{x}\right )}{d^3}+\frac{e^{5/2} \operatorname{Subst}\left (\int \frac{a+b \tanh ^{-1}(c x)}{\sqrt{-d}-\sqrt{e} x} \, dx,x,\sqrt{x}\right )}{d^3}-\frac{e^{5/2} \operatorname{Subst}\left (\int \frac{a+b \tanh ^{-1}(c x)}{\sqrt{-d}+\sqrt{e} x} \, dx,x,\sqrt{x}\right )}{d^3}\\ &=-\frac{b c}{6 d x^{3/2}}-\frac{b c^3}{2 d \sqrt{x}}+\frac{b c e}{d^2 \sqrt{x}}+\frac{b c^4 \tanh ^{-1}\left (c \sqrt{x}\right )}{2 d}-\frac{b c^2 e \tanh ^{-1}\left (c \sqrt{x}\right )}{d^2}-\frac{a+b \tanh ^{-1}\left (c \sqrt{x}\right )}{2 d x^2}+\frac{e \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )}{d^2 x}+\frac{2 e^2 \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right ) \log \left (\frac{2}{1+c \sqrt{x}}\right )}{d^3}-\frac{e^2 \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right ) \log \left (\frac{2 c \left (\sqrt{-d}-\sqrt{e} \sqrt{x}\right )}{\left (c \sqrt{-d}-\sqrt{e}\right ) \left (1+c \sqrt{x}\right )}\right )}{d^3}-\frac{e^2 \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right ) \log \left (\frac{2 c \left (\sqrt{-d}+\sqrt{e} \sqrt{x}\right )}{\left (c \sqrt{-d}+\sqrt{e}\right ) \left (1+c \sqrt{x}\right )}\right )}{d^3}+\frac{a e^2 \log (x)}{d^3}-\frac{b e^2 \text{Li}_2\left (-c \sqrt{x}\right )}{d^3}+\frac{b e^2 \text{Li}_2\left (c \sqrt{x}\right )}{d^3}-2 \frac{\left (b c e^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (\frac{2}{1+c x}\right )}{1-c^2 x^2} \, dx,x,\sqrt{x}\right )}{d^3}+\frac{\left (b c e^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (\frac{2 c \left (\sqrt{-d}-\sqrt{e} x\right )}{\left (c \sqrt{-d}-\sqrt{e}\right ) (1+c x)}\right )}{1-c^2 x^2} \, dx,x,\sqrt{x}\right )}{d^3}+\frac{\left (b c e^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (\frac{2 c \left (\sqrt{-d}+\sqrt{e} x\right )}{\left (c \sqrt{-d}+\sqrt{e}\right ) (1+c x)}\right )}{1-c^2 x^2} \, dx,x,\sqrt{x}\right )}{d^3}\\ &=-\frac{b c}{6 d x^{3/2}}-\frac{b c^3}{2 d \sqrt{x}}+\frac{b c e}{d^2 \sqrt{x}}+\frac{b c^4 \tanh ^{-1}\left (c \sqrt{x}\right )}{2 d}-\frac{b c^2 e \tanh ^{-1}\left (c \sqrt{x}\right )}{d^2}-\frac{a+b \tanh ^{-1}\left (c \sqrt{x}\right )}{2 d x^2}+\frac{e \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )}{d^2 x}+\frac{2 e^2 \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right ) \log \left (\frac{2}{1+c \sqrt{x}}\right )}{d^3}-\frac{e^2 \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right ) \log \left (\frac{2 c \left (\sqrt{-d}-\sqrt{e} \sqrt{x}\right )}{\left (c \sqrt{-d}-\sqrt{e}\right ) \left (1+c \sqrt{x}\right )}\right )}{d^3}-\frac{e^2 \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right ) \log \left (\frac{2 c \left (\sqrt{-d}+\sqrt{e} \sqrt{x}\right )}{\left (c \sqrt{-d}+\sqrt{e}\right ) \left (1+c \sqrt{x}\right )}\right )}{d^3}+\frac{a e^2 \log (x)}{d^3}+\frac{b e^2 \text{Li}_2\left (1-\frac{2 c \left (\sqrt{-d}-\sqrt{e} \sqrt{x}\right )}{\left (c \sqrt{-d}-\sqrt{e}\right ) \left (1+c \sqrt{x}\right )}\right )}{2 d^3}+\frac{b e^2 \text{Li}_2\left (1-\frac{2 c \left (\sqrt{-d}+\sqrt{e} \sqrt{x}\right )}{\left (c \sqrt{-d}+\sqrt{e}\right ) \left (1+c \sqrt{x}\right )}\right )}{2 d^3}-\frac{b e^2 \text{Li}_2\left (-c \sqrt{x}\right )}{d^3}+\frac{b e^2 \text{Li}_2\left (c \sqrt{x}\right )}{d^3}-2 \frac{\left (b e^2\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+c \sqrt{x}}\right )}{d^3}\\ &=-\frac{b c}{6 d x^{3/2}}-\frac{b c^3}{2 d \sqrt{x}}+\frac{b c e}{d^2 \sqrt{x}}+\frac{b c^4 \tanh ^{-1}\left (c \sqrt{x}\right )}{2 d}-\frac{b c^2 e \tanh ^{-1}\left (c \sqrt{x}\right )}{d^2}-\frac{a+b \tanh ^{-1}\left (c \sqrt{x}\right )}{2 d x^2}+\frac{e \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )}{d^2 x}+\frac{2 e^2 \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right ) \log \left (\frac{2}{1+c \sqrt{x}}\right )}{d^3}-\frac{e^2 \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right ) \log \left (\frac{2 c \left (\sqrt{-d}-\sqrt{e} \sqrt{x}\right )}{\left (c \sqrt{-d}-\sqrt{e}\right ) \left (1+c \sqrt{x}\right )}\right )}{d^3}-\frac{e^2 \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right ) \log \left (\frac{2 c \left (\sqrt{-d}+\sqrt{e} \sqrt{x}\right )}{\left (c \sqrt{-d}+\sqrt{e}\right ) \left (1+c \sqrt{x}\right )}\right )}{d^3}+\frac{a e^2 \log (x)}{d^3}-\frac{b e^2 \text{Li}_2\left (1-\frac{2}{1+c \sqrt{x}}\right )}{d^3}+\frac{b e^2 \text{Li}_2\left (1-\frac{2 c \left (\sqrt{-d}-\sqrt{e} \sqrt{x}\right )}{\left (c \sqrt{-d}-\sqrt{e}\right ) \left (1+c \sqrt{x}\right )}\right )}{2 d^3}+\frac{b e^2 \text{Li}_2\left (1-\frac{2 c \left (\sqrt{-d}+\sqrt{e} \sqrt{x}\right )}{\left (c \sqrt{-d}+\sqrt{e}\right ) \left (1+c \sqrt{x}\right )}\right )}{2 d^3}-\frac{b e^2 \text{Li}_2\left (-c \sqrt{x}\right )}{d^3}+\frac{b e^2 \text{Li}_2\left (c \sqrt{x}\right )}{d^3}\\ \end{align*}

Mathematica [A]  time = 2.67282, size = 394, normalized size = 0.78 \[ -\frac{b \left (3 e^2 x^2 \left (\text{PolyLog}\left (2,-\frac{\left (c^2 d+e\right ) e^{2 \tanh ^{-1}\left (c \sqrt{x}\right )}}{c^2 d-2 c \sqrt{-d} \sqrt{e}-e}\right )+\text{PolyLog}\left (2,-\frac{\left (c^2 d+e\right ) e^{2 \tanh ^{-1}\left (c \sqrt{x}\right )}}{c^2 d+2 c \sqrt{-d} \sqrt{e}-e}\right )+2 \tanh ^{-1}\left (c \sqrt{x}\right ) \left (\log \left (\frac{\left (c^2 d+e\right ) e^{2 \tanh ^{-1}\left (c \sqrt{x}\right )}}{c^2 d-2 c \sqrt{-d} \sqrt{e}-e}+1\right )+\log \left (\frac{\left (c^2 d+e\right ) e^{2 \tanh ^{-1}\left (c \sqrt{x}\right )}}{c^2 d+2 c \sqrt{-d} \sqrt{e}-e}+1\right )-\tanh ^{-1}\left (c \sqrt{x}\right )\right )\right )+6 e^2 x^2 \text{PolyLog}\left (2,e^{-2 \tanh ^{-1}\left (c \sqrt{x}\right )}\right )-3 \tanh ^{-1}\left (c \sqrt{x}\right ) \left (d \left (c^2 x-1\right ) \left (c^2 d x+d-2 e x\right )+2 e^2 x^2 \tanh ^{-1}\left (c \sqrt{x}\right )+4 e^2 x^2 \log \left (1-e^{-2 \tanh ^{-1}\left (c \sqrt{x}\right )}\right )\right )+c d \sqrt{x} \left (3 c^2 d x+d-6 e x\right )\right )+3 a d^2+6 a e^2 x^2 \log (d+e x)-6 a d e x-6 a e^2 x^2 \log (x)}{6 d^3 x^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [A]  time = 0.072, size = 741, normalized size = 1.5 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arctanh(c*x^(1/2)))/x^3/(e*x+d),x)

[Out]

-1/2*a/d/x^2+1/2*b/d^3*e^2*ln(c*x^(1/2)-1)*ln((c*(-d*e)^(1/2)-e*(c*x^(1/2)-1)-e)/(c*(-d*e)^(1/2)-e))+1/2*b/d^3
*e^2*ln(1+c*x^(1/2))*ln(c^2*e*x+c^2*d)-1/2*b/d^3*e^2*ln(1+c*x^(1/2))*ln((c*(-d*e)^(1/2)-e*(1+c*x^(1/2))+e)/(c*
(-d*e)^(1/2)+e))-1/2*b/d^3*e^2*ln(1+c*x^(1/2))*ln((c*(-d*e)^(1/2)+e*(1+c*x^(1/2))-e)/(c*(-d*e)^(1/2)-e))-b*arc
tanh(c*x^(1/2))*e^2/d^3*ln(c^2*e*x+c^2*d)+2*b*arctanh(c*x^(1/2))/d^3*e^2*ln(c*x^(1/2))-1/2*c^2*b/d^2*ln(1+c*x^
(1/2))*e+1/2*c^2*b/d^2*ln(c*x^(1/2)-1)*e+1/2*b/d^3*e^2*ln(c*x^(1/2)-1)*ln((c*(-d*e)^(1/2)+e*(c*x^(1/2)-1)+e)/(
c*(-d*e)^(1/2)+e))-b/d^3*e^2*ln(c*x^(1/2))*ln(1+c*x^(1/2))-1/2*b/d^3*e^2*ln(c*x^(1/2)-1)*ln(c^2*e*x+c^2*d)+a/d
^2*e/x-1/2*b*arctanh(c*x^(1/2))/d/x^2+2*a/d^3*e^2*ln(c*x^(1/2))-b/d^3*e^2*dilog(c*x^(1/2))-b/d^3*e^2*dilog(1+c
*x^(1/2))+1/2*b/d^3*e^2*dilog((c*(-d*e)^(1/2)-e*(c*x^(1/2)-1)-e)/(c*(-d*e)^(1/2)-e))+1/2*b/d^3*e^2*dilog((c*(-
d*e)^(1/2)+e*(c*x^(1/2)-1)+e)/(c*(-d*e)^(1/2)+e))-a*e^2/d^3*ln(c^2*e*x+c^2*d)-1/2*b/d^3*e^2*dilog((c*(-d*e)^(1
/2)-e*(1+c*x^(1/2))+e)/(c*(-d*e)^(1/2)+e))-1/2*b/d^3*e^2*dilog((c*(-d*e)^(1/2)+e*(1+c*x^(1/2))-e)/(c*(-d*e)^(1
/2)-e))-1/4*c^4*b/d*ln(c*x^(1/2)-1)+1/4*c^4*b/d*ln(1+c*x^(1/2))+b*arctanh(c*x^(1/2))/d^2*e/x-1/6*b*c/d/x^(3/2)
+b*c*e/d^2/x^(1/2)-1/2*b*c^3/d/x^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{2} \, a{\left (\frac{2 \, e^{2} \log \left (e x + d\right )}{d^{3}} - \frac{2 \, e^{2} \log \left (x\right )}{d^{3}} - \frac{2 \, e x - d}{d^{2} x^{2}}\right )} + b \int \frac{\log \left (c \sqrt{x} + 1\right )}{2 \,{\left (e x^{\frac{7}{2}} + d x^{\frac{5}{2}}\right )} \sqrt{x}}\,{d x} - b \int \frac{\log \left (-c \sqrt{x} + 1\right )}{2 \,{\left (e x^{\frac{7}{2}} + d x^{\frac{5}{2}}\right )} \sqrt{x}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctanh(c*x^(1/2)))/x^3/(e*x+d),x, algorithm="maxima")

[Out]

-1/2*a*(2*e^2*log(e*x + d)/d^3 - 2*e^2*log(x)/d^3 - (2*e*x - d)/(d^2*x^2)) + b*integrate(1/2*log(c*sqrt(x) + 1
)/((e*x^(7/2) + d*x^(5/2))*sqrt(x)), x) - b*integrate(1/2*log(-c*sqrt(x) + 1)/((e*x^(7/2) + d*x^(5/2))*sqrt(x)
), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b \operatorname{artanh}\left (c \sqrt{x}\right ) + a}{e x^{4} + d x^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctanh(c*x^(1/2)))/x^3/(e*x+d),x, algorithm="fricas")

[Out]

integral((b*arctanh(c*sqrt(x)) + a)/(e*x^4 + d*x^3), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*atanh(c*x**(1/2)))/x**3/(e*x+d),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \operatorname{artanh}\left (c \sqrt{x}\right ) + a}{{\left (e x + d\right )} x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctanh(c*x^(1/2)))/x^3/(e*x+d),x, algorithm="giac")

[Out]

integrate((b*arctanh(c*sqrt(x)) + a)/((e*x + d)*x^3), x)