\(\int \frac {(d+e x)^{3/2} \log (a+b x)}{a+b x} \, dx\) [204]

Optimal result

Integrand size = 23, antiderivative size = 381 \[ \int \frac {(d+e x)^{3/2} \log (a+b x)}{a+b x} \, dx=-\frac {16 (b d-a e) \sqrt {d+e x}}{3 b^2}-\frac {4 (d+e x)^{3/2}}{9 b}+\frac {16 (b d-a e)^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{3 b^{5/2}}+\frac {2 (b d-a e)^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )^2}{b^{5/2}}+\frac {2 (b d-a e) \sqrt {d+e x} \log (a+b x)}{b^2}+\frac {2 (d+e x)^{3/2} \log (a+b x)}{3 b}-\frac {2 (b d-a e)^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right ) \log (a+b x)}{b^{5/2}}-\frac {4 (b d-a e)^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right ) \log \left (\frac {2}{1-\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}}\right )}{b^{5/2}}-\frac {2 (b d-a e)^{3/2} \operatorname {PolyLog}\left (2,1-\frac {2}{1-\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}}\right )}{b^{5/2}} \]

[Out]

-4/9*(e*x+d)^(3/2)/b+16/3*(-a*e+b*d)^(3/2)*arctanh(b^(1/2)*(e*x+d)^(1/2)/(-a*e+b*d)^(1/2))/b^(5/2)+2*(-a*e+b*d
)^(3/2)*arctanh(b^(1/2)*(e*x+d)^(1/2)/(-a*e+b*d)^(1/2))^2/b^(5/2)+2/3*(e*x+d)^(3/2)*ln(b*x+a)/b-2*(-a*e+b*d)^(
3/2)*arctanh(b^(1/2)*(e*x+d)^(1/2)/(-a*e+b*d)^(1/2))*ln(b*x+a)/b^(5/2)-4*(-a*e+b*d)^(3/2)*arctanh(b^(1/2)*(e*x
+d)^(1/2)/(-a*e+b*d)^(1/2))*ln(2/(1-b^(1/2)*(e*x+d)^(1/2)/(-a*e+b*d)^(1/2)))/b^(5/2)-2*(-a*e+b*d)^(3/2)*polylo
g(2,1-2/(1-b^(1/2)*(e*x+d)^(1/2)/(-a*e+b*d)^(1/2)))/b^(5/2)-16/3*(-a*e+b*d)*(e*x+d)^(1/2)/b^2+2*(-a*e+b*d)*ln(
b*x+a)*(e*x+d)^(1/2)/b^2

Rubi [A] (verified)

Time = 1.03 (sec) , antiderivative size = 381, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.609, Rules used = {2458, 2388, 65, 214, 2390, 12, 1601, 6873, 6131, 6055, 2449, 2352, 2356, 52} \[ \int \frac {(d+e x)^{3/2} \log (a+b x)}{a+b x} \, dx=\frac {2 (b d-a e)^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )^2}{b^{5/2}}+\frac {16 (b d-a e)^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{3 b^{5/2}}-\frac {2 (b d-a e)^{3/2} \log (a+b x) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{5/2}}-\frac {4 (b d-a e)^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right ) \log \left (\frac {2}{1-\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}}\right )}{b^{5/2}}-\frac {2 (b d-a e)^{3/2} \operatorname {PolyLog}\left (2,1-\frac {2}{1-\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}}\right )}{b^{5/2}}-\frac {16 \sqrt {d+e x} (b d-a e)}{3 b^2}+\frac {2 \sqrt {d+e x} (b d-a e) \log (a+b x)}{b^2}+\frac {2 (d+e x)^{3/2} \log (a+b x)}{3 b}-\frac {4 (d+e x)^{3/2}}{9 b} \]

[In]

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

[Out]

(-16*(b*d - a*e)*Sqrt[d + e*x])/(3*b^2) - (4*(d + e*x)^(3/2))/(9*b) + (16*(b*d - a*e)^(3/2)*ArcTanh[(Sqrt[b]*S
qrt[d + e*x])/Sqrt[b*d - a*e]])/(3*b^(5/2)) + (2*(b*d - a*e)^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d -
a*e]]^2)/b^(5/2) + (2*(b*d - a*e)*Sqrt[d + e*x]*Log[a + b*x])/b^2 + (2*(d + e*x)^(3/2)*Log[a + b*x])/(3*b) - (
2*(b*d - a*e)^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]]*Log[a + b*x])/b^(5/2) - (4*(b*d - a*e)^(3
/2)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]]*Log[2/(1 - (Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e])])/b^(
5/2) - (2*(b*d - a*e)^(3/2)*PolyLog[2, 1 - 2/(1 - (Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e])])/b^(5/2)

Rule 12

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

Rule 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + n + 1))), x] + Dist[n*((b*c - a*d)/(b*(m + n + 1))), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 214

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

Rule 1601

Int[(Pp_)/(Qq_), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x]}, Simp[Coeff[Pp, x, p]*(Log[RemoveConte
nt[Qq, x]]/(q*Coeff[Qq, x, q])), x] /; EqQ[p, q - 1] && EqQ[Pp, Simplify[(Coeff[Pp, x, p]/(q*Coeff[Qq, x, q]))
*D[Qq, x]]]] /; PolyQ[Pp, x] && PolyQ[Qq, 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 2356

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[(d + e*x)^(q + 1)
*((a + b*Log[c*x^n])^p/(e*(q + 1))), x] - Dist[b*n*(p/(e*(q + 1))), Int[((d + e*x)^(q + 1)*(a + b*Log[c*x^n])^
(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && GtQ[p, 0] && NeQ[q, -1] && (EqQ[p, 1] || (Integers
Q[2*p, 2*q] &&  !IGtQ[q, 0]) || (EqQ[p, 2] && NeQ[q, 1]))

Rule 2388

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_.))/(x_), x_Symbol] :> Dist[d, Int[(d
+ e*x)^(q - 1)*((a + b*Log[c*x^n])^p/x), x], x] + Dist[e, Int[(d + e*x)^(q - 1)*(a + b*Log[c*x^n])^p, x], x] /
; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && GtQ[q, 0] && IntegerQ[2*q]

Rule 2390

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^(r_.))^(q_.))/(x_), x_Symbol] :> With[{u = IntHi
de[(d + e*x^r)^q/x, x]}, Simp[u*(a + b*Log[c*x^n]), x] - Dist[b*n, Int[Dist[1/x, u, x], x], x]] /; FreeQ[{a, b
, c, d, e, n, r}, x] && IntegerQ[q - 1/2]

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 2458

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_.)*(x_))^(q_.)*((h_.) + (i_.)*(x_))
^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[(g*(x/e))^q*((e*h - d*i)/e + i*(x/e))^r*(a + b*Log[c*x^n])^p, x], x,
d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - d*g, 0] && (IGtQ[p, 0] || IGtQ[
r, 0]) && IntegerQ[2*r]

Rule 6055

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

Rule 6131

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

Rule 6873

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =!= u]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\left (\frac {b d-a e}{b}+\frac {e x}{b}\right )^{3/2} \log (x)}{x} \, dx,x,a+b x\right )}{b} \\ & = \frac {e \text {Subst}\left (\int \sqrt {\frac {b d-a e}{b}+\frac {e x}{b}} \log (x) \, dx,x,a+b x\right )}{b^2}+\frac {(b d-a e) \text {Subst}\left (\int \frac {\sqrt {\frac {b d-a e}{b}+\frac {e x}{b}} \log (x)}{x} \, dx,x,a+b x\right )}{b^2} \\ & = \frac {2 (d+e x)^{3/2} \log (a+b x)}{3 b}-\frac {2 \text {Subst}\left (\int \frac {\left (\frac {b d-a e}{b}+\frac {e x}{b}\right )^{3/2}}{x} \, dx,x,a+b x\right )}{3 b}+\frac {(e (b d-a e)) \text {Subst}\left (\int \frac {\log (x)}{\sqrt {\frac {b d-a e}{b}+\frac {e x}{b}}} \, dx,x,a+b x\right )}{b^3}+\frac {(b d-a e)^2 \text {Subst}\left (\int \frac {\log (x)}{x \sqrt {\frac {b d-a e}{b}+\frac {e x}{b}}} \, dx,x,a+b x\right )}{b^3} \\ & = -\frac {4 (d+e x)^{3/2}}{9 b}+\frac {2 (b d-a e) \sqrt {d+e x} \log (a+b x)}{b^2}+\frac {2 (d+e x)^{3/2} \log (a+b x)}{3 b}-\frac {2 (b d-a e)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right ) \log (a+b x)}{b^{5/2}}-\frac {(2 (b d-a e)) \text {Subst}\left (\int \frac {\sqrt {\frac {b d-a e}{b}+\frac {e x}{b}}}{x} \, dx,x,a+b x\right )}{3 b^2}-\frac {(2 (b d-a e)) \text {Subst}\left (\int \frac {\sqrt {\frac {b d-a e}{b}+\frac {e x}{b}}}{x} \, dx,x,a+b x\right )}{b^2}-\frac {(b d-a e)^2 \text {Subst}\left (\int -\frac {2 \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d-\frac {a e}{b}+\frac {e x}{b}}}{\sqrt {b d-a e}}\right )}{\sqrt {b d-a e} x} \, dx,x,a+b x\right )}{b^3} \\ & = -\frac {16 (b d-a e) \sqrt {d+e x}}{3 b^2}-\frac {4 (d+e x)^{3/2}}{9 b}+\frac {2 (b d-a e) \sqrt {d+e x} \log (a+b x)}{b^2}+\frac {2 (d+e x)^{3/2} \log (a+b x)}{3 b}-\frac {2 (b d-a e)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right ) \log (a+b x)}{b^{5/2}}+\frac {\left (2 (b d-a e)^{3/2}\right ) \text {Subst}\left (\int \frac {\tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d-\frac {a e}{b}+\frac {e x}{b}}}{\sqrt {b d-a e}}\right )}{x} \, dx,x,a+b x\right )}{b^{5/2}}-\frac {\left (2 (b d-a e)^2\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {\frac {b d-a e}{b}+\frac {e x}{b}}} \, dx,x,a+b x\right )}{3 b^3}-\frac {\left (2 (b d-a e)^2\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {\frac {b d-a e}{b}+\frac {e x}{b}}} \, dx,x,a+b x\right )}{b^3} \\ & = -\frac {16 (b d-a e) \sqrt {d+e x}}{3 b^2}-\frac {4 (d+e x)^{3/2}}{9 b}+\frac {2 (b d-a e) \sqrt {d+e x} \log (a+b x)}{b^2}+\frac {2 (d+e x)^{3/2} \log (a+b x)}{3 b}-\frac {2 (b d-a e)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right ) \log (a+b x)}{b^{5/2}}+\frac {\left (4 (b d-a e)^{3/2}\right ) \text {Subst}\left (\int \frac {x \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {b d-a e}}\right )}{a e+b \left (-d+x^2\right )} \, dx,x,\sqrt {d+e x}\right )}{b^{3/2}}-\frac {\left (4 (b d-a e)^2\right ) \text {Subst}\left (\int \frac {1}{-\frac {b d-a e}{e}+\frac {b x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{3 b^2 e}-\frac {\left (4 (b d-a e)^2\right ) \text {Subst}\left (\int \frac {1}{-\frac {b d-a e}{e}+\frac {b x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{b^2 e} \\ & = -\frac {16 (b d-a e) \sqrt {d+e x}}{3 b^2}-\frac {4 (d+e x)^{3/2}}{9 b}+\frac {16 (b d-a e)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{3 b^{5/2}}+\frac {2 (b d-a e) \sqrt {d+e x} \log (a+b x)}{b^2}+\frac {2 (d+e x)^{3/2} \log (a+b x)}{3 b}-\frac {2 (b d-a e)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right ) \log (a+b x)}{b^{5/2}}+\frac {\left (4 (b d-a e)^{3/2}\right ) \text {Subst}\left (\int \frac {x \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {b d-a e}}\right )}{-b d+a e+b x^2} \, dx,x,\sqrt {d+e x}\right )}{b^{3/2}} \\ & = -\frac {16 (b d-a e) \sqrt {d+e x}}{3 b^2}-\frac {4 (d+e x)^{3/2}}{9 b}+\frac {16 (b d-a e)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{3 b^{5/2}}+\frac {2 (b d-a e)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )^2}{b^{5/2}}+\frac {2 (b d-a e) \sqrt {d+e x} \log (a+b x)}{b^2}+\frac {2 (d+e x)^{3/2} \log (a+b x)}{3 b}-\frac {2 (b d-a e)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right ) \log (a+b x)}{b^{5/2}}-\frac {(4 (b d-a e)) \text {Subst}\left (\int \frac {\tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {b d-a e}}\right )}{1-\frac {\sqrt {b} x}{\sqrt {b d-a e}}} \, dx,x,\sqrt {d+e x}\right )}{b^2} \\ & = -\frac {16 (b d-a e) \sqrt {d+e x}}{3 b^2}-\frac {4 (d+e x)^{3/2}}{9 b}+\frac {16 (b d-a e)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{3 b^{5/2}}+\frac {2 (b d-a e)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )^2}{b^{5/2}}+\frac {2 (b d-a e) \sqrt {d+e x} \log (a+b x)}{b^2}+\frac {2 (d+e x)^{3/2} \log (a+b x)}{3 b}-\frac {2 (b d-a e)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right ) \log (a+b x)}{b^{5/2}}-\frac {4 (b d-a e)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right ) \log \left (\frac {2}{1-\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}}\right )}{b^{5/2}}+\frac {(4 (b d-a e)) \text {Subst}\left (\int \frac {\log \left (\frac {2}{1-\frac {\sqrt {b} x}{\sqrt {b d-a e}}}\right )}{1-\frac {b x^2}{b d-a e}} \, dx,x,\sqrt {d+e x}\right )}{b^2} \\ & = -\frac {16 (b d-a e) \sqrt {d+e x}}{3 b^2}-\frac {4 (d+e x)^{3/2}}{9 b}+\frac {16 (b d-a e)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{3 b^{5/2}}+\frac {2 (b d-a e)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )^2}{b^{5/2}}+\frac {2 (b d-a e) \sqrt {d+e x} \log (a+b x)}{b^2}+\frac {2 (d+e x)^{3/2} \log (a+b x)}{3 b}-\frac {2 (b d-a e)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right ) \log (a+b x)}{b^{5/2}}-\frac {4 (b d-a e)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right ) \log \left (\frac {2}{1-\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}}\right )}{b^{5/2}}-\frac {\left (4 (b d-a e)^{3/2}\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}}\right )}{b^{5/2}} \\ & = -\frac {16 (b d-a e) \sqrt {d+e x}}{3 b^2}-\frac {4 (d+e x)^{3/2}}{9 b}+\frac {16 (b d-a e)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{3 b^{5/2}}+\frac {2 (b d-a e)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )^2}{b^{5/2}}+\frac {2 (b d-a e) \sqrt {d+e x} \log (a+b x)}{b^2}+\frac {2 (d+e x)^{3/2} \log (a+b x)}{3 b}-\frac {2 (b d-a e)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right ) \log (a+b x)}{b^{5/2}}-\frac {4 (b d-a e)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right ) \log \left (\frac {2}{1-\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}}\right )}{b^{5/2}}-\frac {2 (b d-a e)^{3/2} \text {Li}_2\left (1-\frac {2}{1-\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}}\right )}{b^{5/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.59 (sec) , antiderivative size = 556, normalized size of antiderivative = 1.46 \[ \int \frac {(d+e x)^{3/2} \log (a+b x)}{a+b x} \, dx=\frac {-72 \sqrt {b} (b d-a e) \sqrt {d+e x}-8 \sqrt {b} \sqrt {d+e x} (4 b d-3 a e+b e x)+96 (b d-a e)^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )+36 \sqrt {b} (b d-a e) \sqrt {d+e x} \log (a+b x)+12 b^{3/2} (d+e x)^{3/2} \log (a+b x)+18 (b d-a e)^{3/2} \log (a+b x) \log \left (\sqrt {b d-a e}-\sqrt {b} \sqrt {d+e x}\right )-18 (b d-a e)^{3/2} \log (a+b x) \log \left (\sqrt {b d-a e}+\sqrt {b} \sqrt {d+e x}\right )-9 (b d-a e)^{3/2} \left (\log \left (\sqrt {b d-a e}-\sqrt {b} \sqrt {d+e x}\right ) \left (\log \left (\sqrt {b d-a e}-\sqrt {b} \sqrt {d+e x}\right )+2 \log \left (\frac {1}{2} \left (1+\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )\right )\right )+2 \operatorname {PolyLog}\left (2,\frac {1}{2}-\frac {\sqrt {b} \sqrt {d+e x}}{2 \sqrt {b d-a e}}\right )\right )+9 (b d-a e)^{3/2} \left (\log \left (\sqrt {b d-a e}+\sqrt {b} \sqrt {d+e x}\right ) \left (\log \left (\sqrt {b d-a e}+\sqrt {b} \sqrt {d+e x}\right )+2 \log \left (\frac {1}{2}-\frac {\sqrt {b} \sqrt {d+e x}}{2 \sqrt {b d-a e}}\right )\right )+2 \operatorname {PolyLog}\left (2,\frac {1}{2} \left (1+\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )\right )\right )}{18 b^{5/2}} \]

[In]

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

[Out]

(-72*Sqrt[b]*(b*d - a*e)*Sqrt[d + e*x] - 8*Sqrt[b]*Sqrt[d + e*x]*(4*b*d - 3*a*e + b*e*x) + 96*(b*d - a*e)^(3/2
)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]] + 36*Sqrt[b]*(b*d - a*e)*Sqrt[d + e*x]*Log[a + b*x] + 12*b^
(3/2)*(d + e*x)^(3/2)*Log[a + b*x] + 18*(b*d - a*e)^(3/2)*Log[a + b*x]*Log[Sqrt[b*d - a*e] - Sqrt[b]*Sqrt[d +
e*x]] - 18*(b*d - a*e)^(3/2)*Log[a + b*x]*Log[Sqrt[b*d - a*e] + Sqrt[b]*Sqrt[d + e*x]] - 9*(b*d - a*e)^(3/2)*(
Log[Sqrt[b*d - a*e] - Sqrt[b]*Sqrt[d + e*x]]*(Log[Sqrt[b*d - a*e] - Sqrt[b]*Sqrt[d + e*x]] + 2*Log[(1 + (Sqrt[
b]*Sqrt[d + e*x])/Sqrt[b*d - a*e])/2]) + 2*PolyLog[2, 1/2 - (Sqrt[b]*Sqrt[d + e*x])/(2*Sqrt[b*d - a*e])]) + 9*
(b*d - a*e)^(3/2)*(Log[Sqrt[b*d - a*e] + Sqrt[b]*Sqrt[d + e*x]]*(Log[Sqrt[b*d - a*e] + Sqrt[b]*Sqrt[d + e*x]]
+ 2*Log[1/2 - (Sqrt[b]*Sqrt[d + e*x])/(2*Sqrt[b*d - a*e])]) + 2*PolyLog[2, (1 + (Sqrt[b]*Sqrt[d + e*x])/Sqrt[b
*d - a*e])/2]))/(18*b^(5/2))

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 1.05 (sec) , antiderivative size = 416, normalized size of antiderivative = 1.09

method	result	size
derivativedivides	\(\frac {\frac {2 \left (e x +d \right )^{\frac {3}{2}} \ln \left (\frac {\left (e x +d \right ) b +a e -b d}{e}\right )}{3}-\frac {4 b \left (-\frac {-\frac {b \left (e x +d \right )^{\frac {3}{2}}}{3}+\sqrt {e x +d}\, a e -\sqrt {e x +d}\, b d}{b^{2}}+\frac {\left (a^{2} e^{2}-2 a d e b +b^{2} d^{2}\right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{b^{2} \sqrt {\left (a e -b d \right ) b}}\right )}{3}}{b}-\frac {2 \left (\sqrt {e x +d}\, \ln \left (\frac {\left (e x +d \right ) b +a e -b d}{e}\right )-2 b \left (\frac {\sqrt {e x +d}}{b}+\frac {\left (-a e +b d \right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{b \sqrt {\left (a e -b d \right ) b}}\right )\right ) \left (a e -b d \right )}{b^{2}}+2 \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (b \,\textit {\_Z}^{2}+a e -b d \right )}{\sum }\frac {\left (\ln \left (\sqrt {e x +d}-\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {\left (e x +d \right ) b +a e -b d}{e}\right )-2 b \left (\frac {\ln \left (\sqrt {e x +d}-\underline {\hspace {1.25 ex}}\alpha \right )^{2}}{4 \underline {\hspace {1.25 ex}}\alpha b}+\frac {\underline {\hspace {1.25 ex}}\alpha \ln \left (\sqrt {e x +d}-\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {\sqrt {e x +d}+\underline {\hspace {1.25 ex}}\alpha }{2 \underline {\hspace {1.25 ex}}\alpha }\right )}{2 a e -2 b d}+\frac {\underline {\hspace {1.25 ex}}\alpha \operatorname {dilog}\left (\frac {\sqrt {e x +d}+\underline {\hspace {1.25 ex}}\alpha }{2 \underline {\hspace {1.25 ex}}\alpha }\right )}{2 a e -2 b d}\right )\right ) \left (a^{2} e^{2}-2 a d e b +b^{2} d^{2}\right )}{2 b^{3} \underline {\hspace {1.25 ex}}\alpha }\right )\)	\(416\)
default	\(\frac {\frac {2 \left (e x +d \right )^{\frac {3}{2}} \ln \left (\frac {\left (e x +d \right ) b +a e -b d}{e}\right )}{3}-\frac {4 b \left (-\frac {-\frac {b \left (e x +d \right )^{\frac {3}{2}}}{3}+\sqrt {e x +d}\, a e -\sqrt {e x +d}\, b d}{b^{2}}+\frac {\left (a^{2} e^{2}-2 a d e b +b^{2} d^{2}\right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{b^{2} \sqrt {\left (a e -b d \right ) b}}\right )}{3}}{b}-\frac {2 \left (\sqrt {e x +d}\, \ln \left (\frac {\left (e x +d \right ) b +a e -b d}{e}\right )-2 b \left (\frac {\sqrt {e x +d}}{b}+\frac {\left (-a e +b d \right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{b \sqrt {\left (a e -b d \right ) b}}\right )\right ) \left (a e -b d \right )}{b^{2}}+2 \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (b \,\textit {\_Z}^{2}+a e -b d \right )}{\sum }\frac {\left (\ln \left (\sqrt {e x +d}-\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {\left (e x +d \right ) b +a e -b d}{e}\right )-2 b \left (\frac {\ln \left (\sqrt {e x +d}-\underline {\hspace {1.25 ex}}\alpha \right )^{2}}{4 \underline {\hspace {1.25 ex}}\alpha b}+\frac {\underline {\hspace {1.25 ex}}\alpha \ln \left (\sqrt {e x +d}-\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {\sqrt {e x +d}+\underline {\hspace {1.25 ex}}\alpha }{2 \underline {\hspace {1.25 ex}}\alpha }\right )}{2 a e -2 b d}+\frac {\underline {\hspace {1.25 ex}}\alpha \operatorname {dilog}\left (\frac {\sqrt {e x +d}+\underline {\hspace {1.25 ex}}\alpha }{2 \underline {\hspace {1.25 ex}}\alpha }\right )}{2 a e -2 b d}\right )\right ) \left (a^{2} e^{2}-2 a d e b +b^{2} d^{2}\right )}{2 b^{3} \underline {\hspace {1.25 ex}}\alpha }\right )\)	\(416\)

[In]

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

[Out]

2*(1/3*(e*x+d)^(3/2)*ln(((e*x+d)*b+a*e-b*d)/e)-2/3*b*(-1/b^2*(-1/3*b*(e*x+d)^(3/2)+(e*x+d)^(1/2)*a*e-(e*x+d)^(
1/2)*b*d)+(a^2*e^2-2*a*b*d*e+b^2*d^2)/b^2/((a*e-b*d)*b)^(1/2)*arctan(b*(e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2))))/b-
2*((e*x+d)^(1/2)*ln(((e*x+d)*b+a*e-b*d)/e)-2*b*((e*x+d)^(1/2)/b+(-a*e+b*d)/b/((a*e-b*d)*b)^(1/2)*arctan(b*(e*x
+d)^(1/2)/((a*e-b*d)*b)^(1/2))))*(a*e-b*d)/b^2+2*Sum(1/2*(ln((e*x+d)^(1/2)-_alpha)*ln(((e*x+d)*b+a*e-b*d)/e)-2
*b*(1/4/_alpha/b*ln((e*x+d)^(1/2)-_alpha)^2+1/2*_alpha/(a*e-b*d)*ln((e*x+d)^(1/2)-_alpha)*ln(1/2*((e*x+d)^(1/2
)+_alpha)/_alpha)+1/2*_alpha/(a*e-b*d)*dilog(1/2*((e*x+d)^(1/2)+_alpha)/_alpha)))*(a^2*e^2-2*a*b*d*e+b^2*d^2)/
b^3/_alpha,_alpha=RootOf(_Z^2*b+a*e-b*d))

Fricas [F]

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

[In]

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

[Out]

integral((e*x + d)^(3/2)*log(b*x + a)/(b*x + a), x)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {(d+e x)^{3/2} \log (a+b x)}{a+b x} \, dx=\text {Exception raised: ValueError} \]

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(a*e-b*d>0)', see `assume?` for
more detail

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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