3.44 \(\int \frac{x^2 (a+b \tanh ^{-1}(c \sqrt{x}))}{d+e x} \, dx\)

Optimal. Leaf size=460 \[ \frac{b d^2 \text{PolyLog}\left (2,1-\frac{2}{c \sqrt{x}+1}\right )}{e^3}-\frac{b d^2 \text{PolyLog}\left (2,1-\frac{2 c \left (\sqrt{-d}-\sqrt{e} \sqrt{x}\right )}{\left (c \sqrt{x}+1\right ) \left (c \sqrt{-d}-\sqrt{e}\right )}\right )}{2 e^3}-\frac{b d^2 \text{PolyLog}\left (2,1-\frac{2 c \left (\sqrt{-d}+\sqrt{e} \sqrt{x}\right )}{\left (c \sqrt{x}+1\right ) \left (c \sqrt{-d}+\sqrt{e}\right )}\right )}{2 e^3}-\frac{2 d^2 \log \left (\frac{2}{c \sqrt{x}+1}\right ) \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )}{e^3}+\frac{d^2 \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right ) \log \left (\frac{2 c \left (\sqrt{-d}-\sqrt{e} \sqrt{x}\right )}{\left (c \sqrt{x}+1\right ) \left (c \sqrt{-d}-\sqrt{e}\right )}\right )}{e^3}+\frac{d^2 \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right ) \log \left (\frac{2 c \left (\sqrt{-d}+\sqrt{e} \sqrt{x}\right )}{\left (c \sqrt{x}+1\right ) \left (c \sqrt{-d}+\sqrt{e}\right )}\right )}{e^3}-\frac{d x \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )}{e^2}+\frac{x^2 \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )}{2 e}+\frac{b d \tanh ^{-1}\left (c \sqrt{x}\right )}{c^2 e^2}+\frac{b \sqrt{x}}{2 c^3 e}-\frac{b \tanh ^{-1}\left (c \sqrt{x}\right )}{2 c^4 e}-\frac{b d \sqrt{x}}{c e^2}+\frac{b x^{3/2}}{6 c e} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.783123, antiderivative size = 460, normalized size of antiderivative = 1., number of steps used = 20, number of rules used = 11, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.478, Rules used = {43, 5980, 5916, 302, 206, 321, 6044, 5920, 2402, 2315, 2447} \[ \frac{b d^2 \text{PolyLog}\left (2,1-\frac{2}{c \sqrt{x}+1}\right )}{e^3}-\frac{b d^2 \text{PolyLog}\left (2,1-\frac{2 c \left (\sqrt{-d}-\sqrt{e} \sqrt{x}\right )}{\left (c \sqrt{x}+1\right ) \left (c \sqrt{-d}-\sqrt{e}\right )}\right )}{2 e^3}-\frac{b d^2 \text{PolyLog}\left (2,1-\frac{2 c \left (\sqrt{-d}+\sqrt{e} \sqrt{x}\right )}{\left (c \sqrt{x}+1\right ) \left (c \sqrt{-d}+\sqrt{e}\right )}\right )}{2 e^3}-\frac{2 d^2 \log \left (\frac{2}{c \sqrt{x}+1}\right ) \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )}{e^3}+\frac{d^2 \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right ) \log \left (\frac{2 c \left (\sqrt{-d}-\sqrt{e} \sqrt{x}\right )}{\left (c \sqrt{x}+1\right ) \left (c \sqrt{-d}-\sqrt{e}\right )}\right )}{e^3}+\frac{d^2 \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right ) \log \left (\frac{2 c \left (\sqrt{-d}+\sqrt{e} \sqrt{x}\right )}{\left (c \sqrt{x}+1\right ) \left (c \sqrt{-d}+\sqrt{e}\right )}\right )}{e^3}-\frac{d x \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )}{e^2}+\frac{x^2 \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )}{2 e}+\frac{b d \tanh ^{-1}\left (c \sqrt{x}\right )}{c^2 e^2}+\frac{b \sqrt{x}}{2 c^3 e}-\frac{b \tanh ^{-1}\left (c \sqrt{x}\right )}{2 c^4 e}-\frac{b d \sqrt{x}}{c e^2}+\frac{b x^{3/2}}{6 c e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 43

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 5980

Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[f^2
/e, Int[(f*x)^(m - 2)*(a + b*ArcTanh[c*x])^p, x], x] - Dist[(d*f^2)/e, 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] && GtQ[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 302

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

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 321

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

Mathematica [C]  time = 2.96978, size = 558, normalized size = 1.21 \[ \frac{\frac{b \left (3 c^4 d^2 \left (-\text{PolyLog}\left (2,\frac{\left (-2 \sqrt{-c^2 d e}+c^2 (-d)+e\right ) e^{-2 \tanh ^{-1}\left (c \sqrt{x}\right )}}{c^2 d+e}\right )-\text{PolyLog}\left (2,\frac{\left (2 \sqrt{-c^2 d e}+c^2 (-d)+e\right ) e^{-2 \tanh ^{-1}\left (c \sqrt{x}\right )}}{c^2 d+e}\right )-4 i \sin ^{-1}\left (\sqrt{\frac{c^2 d}{c^2 d+e}}\right ) \tanh ^{-1}\left (\frac{c e \sqrt{x}}{\sqrt{-c^2 d e}}\right )+2 \log \left (\frac{e^{-2 \tanh ^{-1}\left (c \sqrt{x}\right )} \left (-2 \sqrt{-c^2 d e}+c^2 d \left (e^{2 \tanh ^{-1}\left (c \sqrt{x}\right )}+1\right )+e \left (e^{2 \tanh ^{-1}\left (c \sqrt{x}\right )}-1\right )\right )}{c^2 d+e}\right ) \left (\tanh ^{-1}\left (c \sqrt{x}\right )-i \sin ^{-1}\left (\sqrt{\frac{c^2 d}{c^2 d+e}}\right )\right )+2 \log \left (\frac{e^{-2 \tanh ^{-1}\left (c \sqrt{x}\right )} \left (2 \sqrt{-c^2 d e}+c^2 d \left (e^{2 \tanh ^{-1}\left (c \sqrt{x}\right )}+1\right )+e \left (e^{2 \tanh ^{-1}\left (c \sqrt{x}\right )}-1\right )\right )}{c^2 d+e}\right ) \left (\tanh ^{-1}\left (c \sqrt{x}\right )+i \sin ^{-1}\left (\sqrt{\frac{c^2 d}{c^2 d+e}}\right )\right )+2 \tanh ^{-1}\left (c \sqrt{x}\right )^2\right )-6 c^4 d^2 \left (\tanh ^{-1}\left (c \sqrt{x}\right ) \left (\tanh ^{-1}\left (c \sqrt{x}\right )+2 \log \left (e^{-2 \tanh ^{-1}\left (c \sqrt{x}\right )}+1\right )\right )-\text{PolyLog}\left (2,-e^{-2 \tanh ^{-1}\left (c \sqrt{x}\right )}\right )\right )+2 c e \sqrt{x} \left (2 e-3 c^2 d\right )-6 e \left (c^2 x-1\right ) \left (c^2 d-e\right ) \tanh ^{-1}\left (c \sqrt{x}\right )+c e^2 \sqrt{x} \left (c^2 x-1\right )+3 e^2 \left (c^2 x-1\right )^2 \tanh ^{-1}\left (c \sqrt{x}\right )\right )}{c^4}+6 a d^2 \log (d+e x)-6 a d e x+3 a e^2 x^2}{6 e^3} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [A]  time = 0.06, size = 651, normalized size = 1.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b x^{2} \operatorname{artanh}\left (c \sqrt{x}\right ) + a x^{2}}{e x + d}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*x^2*arctanh(c*sqrt(x)) + a*x^2)/(e*x + d), 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(x**2*(a+b*atanh(c*x**(1/2)))/(e*x+d),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \operatorname{artanh}\left (c \sqrt{x}\right ) + a\right )} x^{2}}{e x + d}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*arctanh(c*sqrt(x)) + a)*x^2/(e*x + d), x)