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

Optimal. Leaf size=381 \[ -\frac {16 (b d-a e) \sqrt {d+e x}}{3 b^2}-\frac {4 (d+e x)^{3/2}}{9 b}+\frac {16 (b d-a e)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{3 b^{5/2}}+\frac {2 (b d-a e)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )^2}{b^{5/2}}+\frac {2 (b d-a e) \sqrt {d+e x} \log (a+b x)}{b^2}+\frac {2 (d+e x)^{3/2} \log (a+b x)}{3 b}-\frac {2 (b d-a e)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right ) \log (a+b x)}{b^{5/2}}-\frac {4 (b d-a e)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right ) \log \left (\frac {2}{1-\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}}\right )}{b^{5/2}}-\frac {2 (b d-a e)^{3/2} \text {Li}_2\left (1-\frac {2}{1-\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}}\right )}{b^{5/2}} \]

[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]
time = 1.08, antiderivative size = 381, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 14, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.609, Rules used = {2458, 2388, 65, 214, 2390, 12, 1601, 6873, 6131, 6055, 2449, 2352, 2356, 52} \begin {gather*} -\frac {2 (b d-a e)^{3/2} \text {PolyLog}\left (2,1-\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} \tanh ^{-1}\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} \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} \log (a+b x) \tanh ^{-1}\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} \log \left (\frac {2}{1-\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}}\right ) \tanh ^{-1}\left (\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} \end {gather*}

Antiderivative was successfully verified.

[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*} \int \frac {(d+e x)^{3/2} \log (a+b x)}{a+b x} \, dx &=\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 [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 4.12, size = 407, normalized size = 1.07 \begin {gather*} \frac {\sqrt {e} \sqrt {a+b x} \sqrt {\frac {b (d+e x)}{e (a+b x)}} \left (-\frac {12 b \sqrt {e} \sqrt {a+b x} (d+e x) \, _3F_2\left (-\frac {1}{2},-\frac {1}{2},-\frac {1}{2};\frac {1}{2},\frac {1}{2};\frac {-b d+a e}{e (a+b x)}\right )}{\sqrt {\frac {b (d+e x)}{b d-a e}}}-3 e^{3/2} (a+b x)^{3/2} \sqrt {\frac {b (d+e x)}{e (a+b x)}} \, _3F_2\left (-\frac {1}{2},1,1;2,2;\frac {e (a+b x)}{-b d+a e}\right )+2 \left (\sqrt {e} \sqrt {a+b x} \sqrt {\frac {b (d+e x)}{e (a+b x)}} \left (b e x \sqrt {\frac {b (d+e x)}{b d-a e}}+a e \left (1-3 \sqrt {\frac {b (d+e x)}{b d-a e}}\right )+b d \left (-1+4 \sqrt {\frac {b (d+e x)}{b d-a e}}\right )\right )-3 (b d-a e)^{3/2} \sqrt {\frac {b (d+e x)}{b d-a e}} \sinh ^{-1}\left (\frac {\sqrt {b d-a e}}{\sqrt {e} \sqrt {a+b x}}\right )\right ) \log (a+b x)\right )}{3 b^3 \sqrt {d+e x} \sqrt {\frac {b (d+e x)}{b d-a e}}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Sqrt[e]*Sqrt[a + b*x]*Sqrt[(b*(d + e*x))/(e*(a + b*x))]*((-12*b*Sqrt[e]*Sqrt[a + b*x]*(d + e*x)*Hypergeometri
cPFQ[{-1/2, -1/2, -1/2}, {1/2, 1/2}, (-(b*d) + a*e)/(e*(a + b*x))])/Sqrt[(b*(d + e*x))/(b*d - a*e)] - 3*e^(3/2
)*(a + b*x)^(3/2)*Sqrt[(b*(d + e*x))/(e*(a + b*x))]*HypergeometricPFQ[{-1/2, 1, 1}, {2, 2}, (e*(a + b*x))/(-(b
*d) + a*e)] + 2*(Sqrt[e]*Sqrt[a + b*x]*Sqrt[(b*(d + e*x))/(e*(a + b*x))]*(b*e*x*Sqrt[(b*(d + e*x))/(b*d - a*e)
] + a*e*(1 - 3*Sqrt[(b*(d + e*x))/(b*d - a*e)]) + b*d*(-1 + 4*Sqrt[(b*(d + e*x))/(b*d - a*e)])) - 3*(b*d - a*e
)^(3/2)*Sqrt[(b*(d + e*x))/(b*d - a*e)]*ArcSinh[Sqrt[b*d - a*e]/(Sqrt[e]*Sqrt[a + b*x])])*Log[a + b*x]))/(3*b^
3*Sqrt[d + e*x]*Sqrt[(b*(d + e*x))/(b*d - a*e)])

________________________________________________________________________________________

Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\left (e x +d \right )^{\frac {3}{2}} \ln \left (b x +a \right )}{b x +a}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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(b*d-%e*a>0)', see `assume?` fo
r more detai

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\ln \left (a+b\,x\right )\,{\left (d+e\,x\right )}^{3/2}}{a+b\,x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________