3.1.44 \(\int \frac {x^2 (a+b \tanh ^{-1}(c \sqrt {x}))}{d+e x} \, dx\) [44]

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

[Out]

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

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Rubi [A]
time = 0.58, antiderivative size = 460, normalized size of antiderivative = 1.00, 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 = {45, 6127, 6037, 308, 212, 327, 6191, 6057, 2449, 2352, 2497} \begin {gather*} -\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 \tanh ^{-1}\left (c \sqrt {x}\right )}{2 c^4 e}+\frac {b \sqrt {x}}{2 c^3 e}+\frac {b d \tanh ^{-1}\left (c \sqrt {x}\right )}{c^2 e^2}+\frac {b d^2 \text {Li}_2\left (1-\frac {2}{\sqrt {x} c+1}\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 (\sqrt {x} c+1\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 (\sqrt {-d} c+\sqrt {e}\right ) \left (\sqrt {x} c+1\right )}\right )}{2 e^3}-\frac {b d \sqrt {x}}{c e^2}+\frac {b x^{3/2}}{6 c e} \end {gather*}

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 45

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

Rule 212

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

Rule 308

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 327

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

Rule 2352

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

Rule 2449

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> Dist[-e/g, Subst[Int[Log[2*d*x]/(1 - 2*
d*x), x], x, 1/(d + e*x)], x] /; FreeQ[{c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]

Rule 2497

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

Rule 6037

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

Rule 6057

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 6127

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 6191

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])

Rubi steps

\begin {align*} \int \frac {x^2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{d+e x} \, dx &=2 \text {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 \text {Subst}\left (\int x^3 \left (a+b \tanh ^{-1}(c x)\right ) \, dx,x,\sqrt {x}\right )}{e}-\frac {(2 d) \text {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) \text {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 ) \text {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) \text {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) \text {Subst}\left (\int \frac {x^2}{1-c^2 x^2} \, dx,x,\sqrt {x}\right )}{e^2}+\frac {\left (2 d^2\right ) \text {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) \text {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 \text {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 \text {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) \text {Subst}\left (\int \frac {1}{1-c^2 x^2} \, dx,x,\sqrt {x}\right )}{c e^2}-\frac {b \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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*}

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Mathematica [C] Result contains complex when optimal does not.
time = 2.66, size = 558, normalized size = 1.21 \begin {gather*} \frac {-6 a d e x+3 a e^2 x^2+6 a d^2 \log (d+e x)+\frac {b \left (2 c e \left (-3 c^2 d+2 e\right ) \sqrt {x}+c e^2 \sqrt {x} \left (-1+c^2 x\right )-6 \left (c^2 d-e\right ) e \left (-1+c^2 x\right ) \tanh ^{-1}\left (c \sqrt {x}\right )+3 e^2 \left (-1+c^2 x\right )^2 \tanh ^{-1}\left (c \sqrt {x}\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 (1+e^{-2 \tanh ^{-1}\left (c \sqrt {x}\right )}\right )\right )-\text {PolyLog}\left (2,-e^{-2 \tanh ^{-1}\left (c \sqrt {x}\right )}\right )\right )+3 c^4 d^2 \left (2 \tanh ^{-1}\left (c \sqrt {x}\right )^2-4 i \text {ArcSin}\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 \left (-i \text {ArcSin}\left (\sqrt {\frac {c^2 d}{c^2 d+e}}\right )+\tanh ^{-1}\left (c \sqrt {x}\right )\right ) \log \left (\frac {e^{-2 \tanh ^{-1}\left (c \sqrt {x}\right )} \left (-2 \sqrt {-c^2 d e}+e \left (-1+e^{2 \tanh ^{-1}\left (c \sqrt {x}\right )}\right )+c^2 d \left (1+e^{2 \tanh ^{-1}\left (c \sqrt {x}\right )}\right )\right )}{c^2 d+e}\right )+2 \left (i \text {ArcSin}\left (\sqrt {\frac {c^2 d}{c^2 d+e}}\right )+\tanh ^{-1}\left (c \sqrt {x}\right )\right ) \log \left (\frac {e^{-2 \tanh ^{-1}\left (c \sqrt {x}\right )} \left (2 \sqrt {-c^2 d e}+e \left (-1+e^{2 \tanh ^{-1}\left (c \sqrt {x}\right )}\right )+c^2 d \left (1+e^{2 \tanh ^{-1}\left (c \sqrt {x}\right )}\right )\right )}{c^2 d+e}\right )-\text {PolyLog}\left (2,\frac {\left (-c^2 d+e-2 \sqrt {-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 (-c^2 d+e+2 \sqrt {-c^2 d e}\right ) e^{-2 \tanh ^{-1}\left (c \sqrt {x}\right )}}{c^2 d+e}\right )\right )\right )}{c^4}}{6 e^3} \end {gather*}

Antiderivative was successfully verified.

[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 = 0.65, size = 706, normalized size = 1.53

method	result	size
derivativedivides	\(\frac {-\frac {a \,c^{6} d x}{e^{2}}+\frac {a \,c^{6} x^{2}}{2 e}+\frac {a \,c^{6} d^{2} \ln \left (c^{2} e x +c^{2} d \right )}{e^{3}}-\frac {b \,c^{6} \arctanh \left (c \sqrt {x}\right ) d x}{e^{2}}+\frac {b \,c^{6} \arctanh \left (c \sqrt {x}\right ) x^{2}}{2 e}+\frac {b \,c^{6} \arctanh \left (c \sqrt {x}\right ) d^{2} \ln \left (c^{2} e x +c^{2} d \right )}{e^{3}}-\frac {b \,c^{5} d \sqrt {x}}{e^{2}}+\frac {b \,c^{5} x^{\frac {3}{2}}}{6 e}+\frac {b \,c^{3} \sqrt {x}}{2 e}-\frac {b \,c^{4} \ln \left (c \sqrt {x}-1\right ) d}{2 e^{2}}+\frac {b \,c^{2} \ln \left (c \sqrt {x}-1\right )}{4 e}+\frac {b \,c^{4} \ln \left (1+c \sqrt {x}\right ) d}{2 e^{2}}-\frac {b \,c^{2} \ln \left (1+c \sqrt {x}\right )}{4 e}-\frac {b \,c^{6} d^{2} \ln \left (1+c \sqrt {x}\right ) \ln \left (c^{2} e x +c^{2} d \right )}{2 e^{3}}+\frac {b \,c^{6} d^{2} \ln \left (1+c \sqrt {x}\right ) \ln \left (\frac {c \sqrt {-d e}-e \left (1+c \sqrt {x}\right )+e}{c \sqrt {-d e}+e}\right )}{2 e^{3}}+\frac {b \,c^{6} d^{2} \ln \left (1+c \sqrt {x}\right ) \ln \left (\frac {c \sqrt {-d e}+e \left (1+c \sqrt {x}\right )-e}{c \sqrt {-d e}-e}\right )}{2 e^{3}}+\frac {b \,c^{6} d^{2} \dilog \left (\frac {c \sqrt {-d e}-e \left (1+c \sqrt {x}\right )+e}{c \sqrt {-d e}+e}\right )}{2 e^{3}}+\frac {b \,c^{6} d^{2} \dilog \left (\frac {c \sqrt {-d e}+e \left (1+c \sqrt {x}\right )-e}{c \sqrt {-d e}-e}\right )}{2 e^{3}}+\frac {b \,c^{6} d^{2} \ln \left (c \sqrt {x}-1\right ) \ln \left (c^{2} e x +c^{2} d \right )}{2 e^{3}}-\frac {b \,c^{6} d^{2} \ln \left (c \sqrt {x}-1\right ) \ln \left (\frac {c \sqrt {-d e}-e \left (c \sqrt {x}-1\right )-e}{c \sqrt {-d e}-e}\right )}{2 e^{3}}-\frac {b \,c^{6} d^{2} \ln \left (c \sqrt {x}-1\right ) \ln \left (\frac {c \sqrt {-d e}+e \left (c \sqrt {x}-1\right )+e}{c \sqrt {-d e}+e}\right )}{2 e^{3}}-\frac {b \,c^{6} d^{2} \dilog \left (\frac {c \sqrt {-d e}-e \left (c \sqrt {x}-1\right )-e}{c \sqrt {-d e}-e}\right )}{2 e^{3}}-\frac {b \,c^{6} d^{2} \dilog \left (\frac {c \sqrt {-d e}+e \left (c \sqrt {x}-1\right )+e}{c \sqrt {-d e}+e}\right )}{2 e^{3}}}{c^{6}}\)	\(706\)
default	\(\frac {-\frac {a \,c^{6} d x}{e^{2}}+\frac {a \,c^{6} x^{2}}{2 e}+\frac {a \,c^{6} d^{2} \ln \left (c^{2} e x +c^{2} d \right )}{e^{3}}-\frac {b \,c^{6} \arctanh \left (c \sqrt {x}\right ) d x}{e^{2}}+\frac {b \,c^{6} \arctanh \left (c \sqrt {x}\right ) x^{2}}{2 e}+\frac {b \,c^{6} \arctanh \left (c \sqrt {x}\right ) d^{2} \ln \left (c^{2} e x +c^{2} d \right )}{e^{3}}-\frac {b \,c^{5} d \sqrt {x}}{e^{2}}+\frac {b \,c^{5} x^{\frac {3}{2}}}{6 e}+\frac {b \,c^{3} \sqrt {x}}{2 e}-\frac {b \,c^{4} \ln \left (c \sqrt {x}-1\right ) d}{2 e^{2}}+\frac {b \,c^{2} \ln \left (c \sqrt {x}-1\right )}{4 e}+\frac {b \,c^{4} \ln \left (1+c \sqrt {x}\right ) d}{2 e^{2}}-\frac {b \,c^{2} \ln \left (1+c \sqrt {x}\right )}{4 e}-\frac {b \,c^{6} d^{2} \ln \left (1+c \sqrt {x}\right ) \ln \left (c^{2} e x +c^{2} d \right )}{2 e^{3}}+\frac {b \,c^{6} d^{2} \ln \left (1+c \sqrt {x}\right ) \ln \left (\frac {c \sqrt {-d e}-e \left (1+c \sqrt {x}\right )+e}{c \sqrt {-d e}+e}\right )}{2 e^{3}}+\frac {b \,c^{6} d^{2} \ln \left (1+c \sqrt {x}\right ) \ln \left (\frac {c \sqrt {-d e}+e \left (1+c \sqrt {x}\right )-e}{c \sqrt {-d e}-e}\right )}{2 e^{3}}+\frac {b \,c^{6} d^{2} \dilog \left (\frac {c \sqrt {-d e}-e \left (1+c \sqrt {x}\right )+e}{c \sqrt {-d e}+e}\right )}{2 e^{3}}+\frac {b \,c^{6} d^{2} \dilog \left (\frac {c \sqrt {-d e}+e \left (1+c \sqrt {x}\right )-e}{c \sqrt {-d e}-e}\right )}{2 e^{3}}+\frac {b \,c^{6} d^{2} \ln \left (c \sqrt {x}-1\right ) \ln \left (c^{2} e x +c^{2} d \right )}{2 e^{3}}-\frac {b \,c^{6} d^{2} \ln \left (c \sqrt {x}-1\right ) \ln \left (\frac {c \sqrt {-d e}-e \left (c \sqrt {x}-1\right )-e}{c \sqrt {-d e}-e}\right )}{2 e^{3}}-\frac {b \,c^{6} d^{2} \ln \left (c \sqrt {x}-1\right ) \ln \left (\frac {c \sqrt {-d e}+e \left (c \sqrt {x}-1\right )+e}{c \sqrt {-d e}+e}\right )}{2 e^{3}}-\frac {b \,c^{6} d^{2} \dilog \left (\frac {c \sqrt {-d e}-e \left (c \sqrt {x}-1\right )-e}{c \sqrt {-d e}-e}\right )}{2 e^{3}}-\frac {b \,c^{6} d^{2} \dilog \left (\frac {c \sqrt {-d e}+e \left (c \sqrt {x}-1\right )+e}{c \sqrt {-d e}+e}\right )}{2 e^{3}}}{c^{6}}\)	\(706\)

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,method=_RETURNVERBOSE)

[Out]

2/c^6*(-1/2*a*c^6/e^2*d*x+1/4*a*c^6/e*x^2+1/2*a*c^6*d^2/e^3*ln(c^2*e*x+c^2*d)-1/2*b*c^6*arctanh(c*x^(1/2))/e^2
*d*x+1/4*b*c^6*arctanh(c*x^(1/2))/e*x^2+1/2*b*c^6*arctanh(c*x^(1/2))*d^2/e^3*ln(c^2*e*x+c^2*d)-1/2*b*c^5/e^2*d
*x^(1/2)+1/12*b*c^5/e*x^(3/2)+1/4*b*c^3/e*x^(1/2)-1/4*b*c^4/e^2*ln(c*x^(1/2)-1)*d+1/8*b*c^2*ln(c*x^(1/2)-1)/e+
1/4*b*c^4/e^2*ln(1+c*x^(1/2))*d-1/8*b*c^2*ln(1+c*x^(1/2))/e-1/4*b*c^6*d^2/e^3*ln(1+c*x^(1/2))*ln(c^2*e*x+c^2*d
)+1/4*b*c^6*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))+1/4*b*c^6*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))+1/4*b*c^6*d^2/e^3*dilog((c*(-d*e)^
(1/2)-e*(1+c*x^(1/2))+e)/(c*(-d*e)^(1/2)+e))+1/4*b*c^6*d^2/e^3*dilog((c*(-d*e)^(1/2)+e*(1+c*x^(1/2))-e)/(c*(-d
*e)^(1/2)-e))+1/4*b*c^6*d^2/e^3*ln(c*x^(1/2)-1)*ln(c^2*e*x+c^2*d)-1/4*b*c^6*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))-1/4*b*c^6*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))-1/4*b*c^6*d^2/e^3*dilog((c*(-d*e)^(1/2)-e*(c*x^(1/2)-1)-e)/(c*(-d*e)^(1/2)-e))-1
/4*b*c^6*d^2/e^3*dilog((c*(-d*e)^(1/2)+e*(c*x^(1/2)-1)+e)/(c*(-d*e)^(1/2)+e)))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)/(x*e + d), x)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^2\,\left (a+b\,\mathrm {atanh}\left (c\,\sqrt {x}\right )\right )}{d+e\,x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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