3.492 \(\int \frac{(a+b \log (c (d (e+f x)^p)^q))^2}{(g+h x)^{3/2}} \, dx\)

Optimal. Leaf size=330 \[ -\frac{8 b^2 \sqrt{f} p^2 q^2 \text{PolyLog}\left (2,1-\frac{2}{1-\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}}\right )}{h \sqrt{f g-e h}}-\frac{2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{h \sqrt{g+h x}}-\frac{8 b \sqrt{f} p q \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h \sqrt{f g-e h}}+\frac{8 b^2 \sqrt{f} p^2 q^2 \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right )^2}{h \sqrt{f g-e h}}-\frac{16 b^2 \sqrt{f} p^2 q^2 \log \left (\frac{2}{1-\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}}\right ) \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right )}{h \sqrt{f g-e h}} \]

[Out]

(8*b^2*Sqrt[f]*p^2*q^2*ArcTanh[(Sqrt[f]*Sqrt[g + h*x])/Sqrt[f*g - e*h]]^2)/(h*Sqrt[f*g - e*h]) - (8*b*Sqrt[f]*
p*q*ArcTanh[(Sqrt[f]*Sqrt[g + h*x])/Sqrt[f*g - e*h]]*(a + b*Log[c*(d*(e + f*x)^p)^q]))/(h*Sqrt[f*g - e*h]) - (
2*(a + b*Log[c*(d*(e + f*x)^p)^q])^2)/(h*Sqrt[g + h*x]) - (16*b^2*Sqrt[f]*p^2*q^2*ArcTanh[(Sqrt[f]*Sqrt[g + h*
x])/Sqrt[f*g - e*h]]*Log[2/(1 - (Sqrt[f]*Sqrt[g + h*x])/Sqrt[f*g - e*h])])/(h*Sqrt[f*g - e*h]) - (8*b^2*Sqrt[f
]*p^2*q^2*PolyLog[2, 1 - 2/(1 - (Sqrt[f]*Sqrt[g + h*x])/Sqrt[f*g - e*h])])/(h*Sqrt[f*g - e*h])

________________________________________________________________________________________

Rubi [A]  time = 1.62029, antiderivative size = 330, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 13, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.433, Rules used = {2398, 2411, 63, 208, 2348, 12, 1587, 6741, 5984, 5918, 2402, 2315, 2445} \[ -\frac{8 b^2 \sqrt{f} p^2 q^2 \text{PolyLog}\left (2,1-\frac{2}{1-\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}}\right )}{h \sqrt{f g-e h}}-\frac{2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{h \sqrt{g+h x}}-\frac{8 b \sqrt{f} p q \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h \sqrt{f g-e h}}+\frac{8 b^2 \sqrt{f} p^2 q^2 \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right )^2}{h \sqrt{f g-e h}}-\frac{16 b^2 \sqrt{f} p^2 q^2 \log \left (\frac{2}{1-\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}}\right ) \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right )}{h \sqrt{f g-e h}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[c*(d*(e + f*x)^p)^q])^2/(g + h*x)^(3/2),x]

[Out]

(8*b^2*Sqrt[f]*p^2*q^2*ArcTanh[(Sqrt[f]*Sqrt[g + h*x])/Sqrt[f*g - e*h]]^2)/(h*Sqrt[f*g - e*h]) - (8*b*Sqrt[f]*
p*q*ArcTanh[(Sqrt[f]*Sqrt[g + h*x])/Sqrt[f*g - e*h]]*(a + b*Log[c*(d*(e + f*x)^p)^q]))/(h*Sqrt[f*g - e*h]) - (
2*(a + b*Log[c*(d*(e + f*x)^p)^q])^2)/(h*Sqrt[g + h*x]) - (16*b^2*Sqrt[f]*p^2*q^2*ArcTanh[(Sqrt[f]*Sqrt[g + h*
x])/Sqrt[f*g - e*h]]*Log[2/(1 - (Sqrt[f]*Sqrt[g + h*x])/Sqrt[f*g - e*h])])/(h*Sqrt[f*g - e*h]) - (8*b^2*Sqrt[f
]*p^2*q^2*PolyLog[2, 1 - 2/(1 - (Sqrt[f]*Sqrt[g + h*x])/Sqrt[f*g - e*h])])/(h*Sqrt[f*g - e*h])

Rule 2398

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

Rule 2411

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 63

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 208

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

Rule 2348

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 12

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

Rule 1587

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]*D[Qq, x])/(q*Coeff[Q
q, x, q])]]] /; PolyQ[Pp, x] && PolyQ[Qq, x]

Rule 6741

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

Rule 5984

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 5918

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 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 2445

Int[((a_.) + Log[(c_.)*((d_.)*((e_.) + (f_.)*(x_))^(m_.))^(n_)]*(b_.))^(p_.)*(u_.), x_Symbol] :> Subst[Int[u*(
a + b*Log[c*d^n*(e + f*x)^(m*n)])^p, x], c*d^n*(e + f*x)^(m*n), c*(d*(e + f*x)^m)^n] /; FreeQ[{a, b, c, d, e,
f, m, n, p}, x] &&  !IntegerQ[n] &&  !(EqQ[d, 1] && EqQ[m, 1]) && IntegralFreeQ[IntHide[u*(a + b*Log[c*d^n*(e
+ f*x)^(m*n)])^p, x]]

Rubi steps

\begin{align*} \int \frac{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{(g+h x)^{3/2}} \, dx &=\operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^2}{(g+h x)^{3/2}} \, dx,c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac{2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{h \sqrt{g+h x}}+\operatorname{Subst}\left (\frac{(4 b f p q) \int \frac{a+b \log \left (c d^q (e+f x)^{p q}\right )}{(e+f x) \sqrt{g+h x}} \, dx}{h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac{2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{h \sqrt{g+h x}}+\operatorname{Subst}\left (\frac{(4 b p q) \operatorname{Subst}\left (\int \frac{a+b \log \left (c d^q x^{p q}\right )}{x \sqrt{\frac{f g-e h}{f}+\frac{h x}{f}}} \, dx,x,e+f x\right )}{h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac{8 b \sqrt{f} p q \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h \sqrt{f g-e h}}-\frac{2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{h \sqrt{g+h x}}-\operatorname{Subst}\left (\frac{\left (4 b^2 p^2 q^2\right ) \operatorname{Subst}\left (\int -\frac{2 \sqrt{f} \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g-\frac{e h}{f}+\frac{h x}{f}}}{\sqrt{f g-e h}}\right )}{\sqrt{f g-e h} x} \, dx,x,e+f x\right )}{h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac{8 b \sqrt{f} p q \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h \sqrt{f g-e h}}-\frac{2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{h \sqrt{g+h x}}+\operatorname{Subst}\left (\frac{\left (8 b^2 \sqrt{f} p^2 q^2\right ) \operatorname{Subst}\left (\int \frac{\tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g-\frac{e h}{f}+\frac{h x}{f}}}{\sqrt{f g-e h}}\right )}{x} \, dx,x,e+f x\right )}{h \sqrt{f g-e h}},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac{8 b \sqrt{f} p q \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h \sqrt{f g-e h}}-\frac{2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{h \sqrt{g+h x}}+\operatorname{Subst}\left (\frac{\left (16 b^2 f^{3/2} p^2 q^2\right ) \operatorname{Subst}\left (\int \frac{x \tanh ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{f g-e h}}\right )}{e h+f \left (-g+x^2\right )} \, dx,x,\sqrt{g+h x}\right )}{h \sqrt{f g-e h}},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac{8 b \sqrt{f} p q \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h \sqrt{f g-e h}}-\frac{2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{h \sqrt{g+h x}}+\operatorname{Subst}\left (\frac{\left (16 b^2 f^{3/2} p^2 q^2\right ) \operatorname{Subst}\left (\int \frac{x \tanh ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{f g-e h}}\right )}{-f g+e h+f x^2} \, dx,x,\sqrt{g+h x}\right )}{h \sqrt{f g-e h}},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{8 b^2 \sqrt{f} p^2 q^2 \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right )^2}{h \sqrt{f g-e h}}-\frac{8 b \sqrt{f} p q \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h \sqrt{f g-e h}}-\frac{2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{h \sqrt{g+h x}}-\operatorname{Subst}\left (\frac{\left (16 b^2 f p^2 q^2\right ) \operatorname{Subst}\left (\int \frac{\tanh ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{f g-e h}}\right )}{1-\frac{\sqrt{f} x}{\sqrt{f g-e h}}} \, dx,x,\sqrt{g+h x}\right )}{h (f g-e h)},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{8 b^2 \sqrt{f} p^2 q^2 \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right )^2}{h \sqrt{f g-e h}}-\frac{8 b \sqrt{f} p q \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h \sqrt{f g-e h}}-\frac{2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{h \sqrt{g+h x}}-\frac{16 b^2 \sqrt{f} p^2 q^2 \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right ) \log \left (\frac{2}{1-\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}}\right )}{h \sqrt{f g-e h}}+\operatorname{Subst}\left (\frac{\left (16 b^2 f p^2 q^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (\frac{2}{1-\frac{\sqrt{f} x}{\sqrt{f g-e h}}}\right )}{1-\frac{f x^2}{f g-e h}} \, dx,x,\sqrt{g+h x}\right )}{h (f g-e h)},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{8 b^2 \sqrt{f} p^2 q^2 \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right )^2}{h \sqrt{f g-e h}}-\frac{8 b \sqrt{f} p q \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h \sqrt{f g-e h}}-\frac{2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{h \sqrt{g+h x}}-\frac{16 b^2 \sqrt{f} p^2 q^2 \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right ) \log \left (\frac{2}{1-\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}}\right )}{h \sqrt{f g-e h}}-\operatorname{Subst}\left (\frac{\left (16 b^2 \sqrt{f} p^2 q^2\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1-\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}}\right )}{h \sqrt{f g-e h}},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{8 b^2 \sqrt{f} p^2 q^2 \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right )^2}{h \sqrt{f g-e h}}-\frac{8 b \sqrt{f} p q \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h \sqrt{f g-e h}}-\frac{2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{h \sqrt{g+h x}}-\frac{16 b^2 \sqrt{f} p^2 q^2 \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right ) \log \left (\frac{2}{1-\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}}\right )}{h \sqrt{f g-e h}}-\frac{8 b^2 \sqrt{f} p^2 q^2 \text{Li}_2\left (1-\frac{2}{1-\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}}\right )}{h \sqrt{f g-e h}}\\ \end{align*}

Mathematica [C]  time = 3.69164, size = 356, normalized size = 1.08 \[ \frac{2 \left (\frac{b^2 p^2 q^2 \left (h (e+f x) \sqrt{\frac{f (g+h x)}{f g-e h}} \, _4F_3\left (1,1,1,\frac{3}{2};2,2,2;\frac{h (e+f x)}{e h-f g}\right )+(f g-e h) \log (e+f x) \left (\log (e+f x) \left (\sqrt{\frac{f (g+h x)}{f g-e h}}-1\right )-4 \sqrt{\frac{f (g+h x)}{f g-e h}} \log \left (\frac{1}{2} \left (\sqrt{\frac{f (g+h x)}{f g-e h}}+1\right )\right )\right )\right )}{\sqrt{g+h x} (f g-e h)}-\frac{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )-b p q \log (e+f x)\right )^2}{\sqrt{g+h x}}+\frac{2 b p q \left (\sqrt{g+h x} \sqrt{f g-e h} \log (e+f x)+2 \sqrt{f} (g+h x) \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right )\right ) \left (-a-b \log \left (c \left (d (e+f x)^p\right )^q\right )+b p q \log (e+f x)\right )}{(g+h x) \sqrt{f g-e h}}\right )}{h} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Log[c*(d*(e + f*x)^p)^q])^2/(g + h*x)^(3/2),x]

[Out]

(2*((2*b*p*q*(2*Sqrt[f]*(g + h*x)*ArcTanh[(Sqrt[f]*Sqrt[g + h*x])/Sqrt[f*g - e*h]] + Sqrt[f*g - e*h]*Sqrt[g +
h*x]*Log[e + f*x])*(-a + b*p*q*Log[e + f*x] - b*Log[c*(d*(e + f*x)^p)^q]))/(Sqrt[f*g - e*h]*(g + h*x)) - (a -
b*p*q*Log[e + f*x] + b*Log[c*(d*(e + f*x)^p)^q])^2/Sqrt[g + h*x] + (b^2*p^2*q^2*(h*(e + f*x)*Sqrt[(f*(g + h*x)
)/(f*g - e*h)]*HypergeometricPFQ[{1, 1, 1, 3/2}, {2, 2, 2}, (h*(e + f*x))/(-(f*g) + e*h)] + (f*g - e*h)*Log[e
+ f*x]*((-1 + Sqrt[(f*(g + h*x))/(f*g - e*h)])*Log[e + f*x] - 4*Sqrt[(f*(g + h*x))/(f*g - e*h)]*Log[(1 + Sqrt[
(f*(g + h*x))/(f*g - e*h)])/2])))/((f*g - e*h)*Sqrt[g + h*x])))/h

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Maple [F]  time = 0.669, size = 0, normalized size = 0. \begin{align*} \int{ \left ( a+b\ln \left ( c \left ( d \left ( fx+e \right ) ^{p} \right ) ^{q} \right ) \right ) ^{2} \left ( hx+g \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*ln(c*(d*(f*x+e)^p)^q))^2/(h*x+g)^(3/2),x)

[Out]

int((a+b*ln(c*(d*(f*x+e)^p)^q))^2/(h*x+g)^(3/2),x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(d*(f*x+e)^p)^q))^2/(h*x+g)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(d*(f*x+e)^p)^q))^2/(h*x+g)^(3/2),x, algorithm="fricas")

[Out]

integral((sqrt(h*x + g)*b^2*log(((f*x + e)^p*d)^q*c)^2 + 2*sqrt(h*x + g)*a*b*log(((f*x + e)^p*d)^q*c) + sqrt(h
*x + g)*a^2)/(h^2*x^2 + 2*g*h*x + g^2), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b \log{\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}\right )^{2}}{\left (g + h x\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*ln(c*(d*(f*x+e)**p)**q))**2/(h*x+g)**(3/2),x)

[Out]

Integral((a + b*log(c*(d*(e + f*x)**p)**q))**2/(g + h*x)**(3/2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}^{2}}{{\left (h x + g\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(d*(f*x+e)^p)^q))^2/(h*x+g)^(3/2),x, algorithm="giac")

[Out]

integrate((b*log(((f*x + e)^p*d)^q*c) + a)^2/(h*x + g)^(3/2), x)