\(\int \frac {(a+b \log (c (d (e+f x)^p)^q))^2}{\sqrt {g+h x}} \, dx\) [491]

Optimal result

Integrand size = 30, antiderivative size = 447 \[ \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{\sqrt {g+h x}} \, dx=\frac {16 b^2 p^2 q^2 \sqrt {g+h x}}{h}-\frac {16 b^2 \sqrt {f g-e h} p^2 q^2 \text {arctanh}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )}{\sqrt {f} h}-\frac {8 b^2 \sqrt {f g-e h} p^2 q^2 \text {arctanh}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )^2}{\sqrt {f} h}-\frac {8 b p q \sqrt {g+h x} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h}+\frac {8 b \sqrt {f g-e h} p q \text {arctanh}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{\sqrt {f} h}+\frac {2 \sqrt {g+h x} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{h}+\frac {16 b^2 \sqrt {f g-e h} p^2 q^2 \text {arctanh}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right ) \log \left (\frac {2}{1-\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}}\right )}{\sqrt {f} h}+\frac {8 b^2 \sqrt {f g-e h} p^2 q^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1-\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}}\right )}{\sqrt {f} h} \]

[Out]

-16*b^2*p^2*q^2*arctanh(f^(1/2)*(h*x+g)^(1/2)/(-e*h+f*g)^(1/2))*(-e*h+f*g)^(1/2)/h/f^(1/2)-8*b^2*p^2*q^2*arcta
nh(f^(1/2)*(h*x+g)^(1/2)/(-e*h+f*g)^(1/2))^2*(-e*h+f*g)^(1/2)/h/f^(1/2)+8*b*p*q*arctanh(f^(1/2)*(h*x+g)^(1/2)/
(-e*h+f*g)^(1/2))*(a+b*ln(c*(d*(f*x+e)^p)^q))*(-e*h+f*g)^(1/2)/h/f^(1/2)+16*b^2*p^2*q^2*arctanh(f^(1/2)*(h*x+g
)^(1/2)/(-e*h+f*g)^(1/2))*ln(2/(1-f^(1/2)*(h*x+g)^(1/2)/(-e*h+f*g)^(1/2)))*(-e*h+f*g)^(1/2)/h/f^(1/2)+8*b^2*p^
2*q^2*polylog(2,1-2/(1-f^(1/2)*(h*x+g)^(1/2)/(-e*h+f*g)^(1/2)))*(-e*h+f*g)^(1/2)/h/f^(1/2)+16*b^2*p^2*q^2*(h*x
+g)^(1/2)/h-8*b*p*q*(a+b*ln(c*(d*(f*x+e)^p)^q))*(h*x+g)^(1/2)/h+2*(a+b*ln(c*(d*(f*x+e)^p)^q))^2*(h*x+g)^(1/2)/
h

Rubi [A] (verified)

Time = 1.55 (sec) , antiderivative size = 447, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used = {2445, 2458, 2388, 65, 214, 2390, 12, 1601, 6873, 6131, 6055, 2449, 2352, 2356, 52, 2495} \[ \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{\sqrt {g+h x}} \, dx=\frac {8 b p q \sqrt {f g-e h} \text {arctanh}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{\sqrt {f} h}-\frac {8 b p q \sqrt {g+h x} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h}+\frac {2 \sqrt {g+h x} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{h}-\frac {8 b^2 p^2 q^2 \sqrt {f g-e h} \text {arctanh}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )^2}{\sqrt {f} h}-\frac {16 b^2 p^2 q^2 \sqrt {f g-e h} \text {arctanh}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )}{\sqrt {f} h}+\frac {16 b^2 p^2 q^2 \sqrt {f g-e h} \text {arctanh}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right ) \log \left (\frac {2}{1-\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}}\right )}{\sqrt {f} h}+\frac {8 b^2 p^2 q^2 \sqrt {f g-e h} \operatorname {PolyLog}\left (2,1-\frac {2}{1-\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}}\right )}{\sqrt {f} h}+\frac {16 b^2 p^2 q^2 \sqrt {g+h x}}{h} \]

[In]

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

[Out]

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

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 2445

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 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 2495

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]]

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1407\) vs. \(2(447)=894\).

Time = 12.63 (sec) , antiderivative size = 1407, normalized size of antiderivative = 3.15 \[ \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{\sqrt {g+h x}} \, dx=\frac {2 b p q \left (\frac {4 \sqrt {f} \sqrt {f g-e h} \text {arctanh}\left (\frac {\sqrt {f} \sqrt {\frac {f g-e h+h (e+f x)}{f}}}{\sqrt {f g-e h}}\right )}{h}+\frac {2 f \sqrt {\frac {f g-e h+h (e+f x)}{f}} (-2+\log (e+f x))}{h}\right ) \left (a+b q \left (-p \log (e+f x)+\log \left (d (e+f x)^p\right )\right )+b \left (-q \left (-p \log (e+f x)+\log \left (d (e+f x)^p\right )\right )-\log \left (d (e+f x)^p\right ) \left (q-\frac {q \left (-p \log (e+f x)+\log \left (d (e+f x)^p\right )\right )}{\log \left (d (e+f x)^p\right )}\right )+\log \left (c e^{q \left (-p \log (e+f x)+\log \left (d (e+f x)^p\right )\right )} \left (d (e+f x)^p\right )^{q-\frac {q \left (-p \log (e+f x)+\log \left (d (e+f x)^p\right )\right )}{\log \left (d (e+f x)^p\right )}}\right )\right )\right )}{f}+\frac {2 \sqrt {g+h x} \left (a+b q \left (-p \log (e+f x)+\log \left (d (e+f x)^p\right )\right )+b \left (-q \left (-p \log (e+f x)+\log \left (d (e+f x)^p\right )\right )-\log \left (d (e+f x)^p\right ) \left (q-\frac {q \left (-p \log (e+f x)+\log \left (d (e+f x)^p\right )\right )}{\log \left (d (e+f x)^p\right )}\right )+\log \left (c e^{q \left (-p \log (e+f x)+\log \left (d (e+f x)^p\right )\right )} \left (d (e+f x)^p\right )^{q-\frac {q \left (-p \log (e+f x)+\log \left (d (e+f x)^p\right )\right )}{\log \left (d (e+f x)^p\right )}}\right )\right )\right )^2}{h}+\frac {b^2 p^2 q^2 \left (-\frac {16 f^2 g \sqrt {e+f x} \sqrt {1+\frac {f g-e h}{h (e+f x)}} \sqrt {\frac {f g-e h+h (e+f x)}{f}} \arcsin \left (\frac {\sqrt {-f g+e h}}{\sqrt {h} \sqrt {e+f x}}\right )}{\sqrt {h} \sqrt {-f g+e h} (f g-e h+h (e+f x))}+\frac {16 e f \sqrt {h} \sqrt {e+f x} \sqrt {1+\frac {f g-e h}{h (e+f x)}} \sqrt {\frac {f g-e h+h (e+f x)}{f}} \arcsin \left (\frac {\sqrt {-f g+e h}}{\sqrt {h} \sqrt {e+f x}}\right )}{\sqrt {-f g+e h} (f g-e h+h (e+f x))}+\frac {2 f \sqrt {\frac {f g-e h+h (e+f x)}{f}} \left (8-4 \log (e+f x)+\log ^2(e+f x)\right )}{h}+\frac {2 e f \sqrt {\frac {f g-e h+h (e+f x)}{f}} \left (-\frac {4 \text {arctanh}\left (\frac {\sqrt {f g-e h+h (e+f x)}}{\sqrt {f g-e h}}\right ) \left (\log (e+f x)-\log \left (-\frac {h (e+f x)}{f g-e h}\right )\right )}{\sqrt {f g-e h}}+\frac {\sqrt {1+\frac {h (e+f x)}{f g-e h}} \left (\log ^2\left (-\frac {h (e+f x)}{f g-e h}\right )-4 \log \left (-\frac {h (e+f x)}{f g-e h}\right ) \log \left (\frac {1}{2} \left (1+\sqrt {1+\frac {h (e+f x)}{f g-e h}}\right )\right )+2 \log ^2\left (\frac {1}{2} \left (1+\sqrt {1+\frac {h (e+f x)}{f g-e h}}\right )\right )-4 \operatorname {PolyLog}\left (2,1+\frac {1}{2} \left (-1-\sqrt {1+\frac {h (e+f x)}{f g-e h}}\right )\right )\right )}{\sqrt {f g-e h+h (e+f x)}}\right )}{\sqrt {f g-e h+h (e+f x)}}-\frac {2 f^2 g \sqrt {\frac {f g-e h+h (e+f x)}{f}} \left (-\frac {4 \text {arctanh}\left (\frac {\sqrt {f g-e h+h (e+f x)}}{\sqrt {f g-e h}}\right ) \left (\log (e+f x)-\log \left (-\frac {h (e+f x)}{f g-e h}\right )\right )}{\sqrt {f g-e h}}+\frac {\sqrt {1+\frac {h (e+f x)}{f g-e h}} \left (\log ^2\left (-\frac {h (e+f x)}{f g-e h}\right )-4 \log \left (-\frac {h (e+f x)}{f g-e h}\right ) \log \left (\frac {1}{2} \left (1+\sqrt {1+\frac {h (e+f x)}{f g-e h}}\right )\right )+2 \log ^2\left (\frac {1}{2} \left (1+\sqrt {1+\frac {h (e+f x)}{f g-e h}}\right )\right )-4 \operatorname {PolyLog}\left (2,1+\frac {1}{2} \left (-1-\sqrt {1+\frac {h (e+f x)}{f g-e h}}\right )\right )\right )}{\sqrt {f g-e h+h (e+f x)}}\right )}{h \sqrt {f g-e h+h (e+f x)}}\right )}{f} \]

[In]

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

[Out]

(2*b*p*q*((4*Sqrt[f]*Sqrt[f*g - e*h]*ArcTanh[(Sqrt[f]*Sqrt[(f*g - e*h + h*(e + f*x))/f])/Sqrt[f*g - e*h]])/h +
(2*f*Sqrt[(f*g - e*h + h*(e + f*x))/f]*(-2 + Log[e + f*x]))/h)*(a + b*q*(-(p*Log[e + f*x]) + Log[d*(e + f*x)^
p]) + b*(-(q*(-(p*Log[e + f*x]) + Log[d*(e + f*x)^p])) - Log[d*(e + f*x)^p]*(q - (q*(-(p*Log[e + f*x]) + Log[d
*(e + f*x)^p]))/Log[d*(e + f*x)^p]) + Log[c*E^(q*(-(p*Log[e + f*x]) + Log[d*(e + f*x)^p]))*(d*(e + f*x)^p)^(q
- (q*(-(p*Log[e + f*x]) + Log[d*(e + f*x)^p]))/Log[d*(e + f*x)^p])])))/f + (2*Sqrt[g + h*x]*(a + b*q*(-(p*Log[
e + f*x]) + Log[d*(e + f*x)^p]) + b*(-(q*(-(p*Log[e + f*x]) + Log[d*(e + f*x)^p])) - Log[d*(e + f*x)^p]*(q - (
q*(-(p*Log[e + f*x]) + Log[d*(e + f*x)^p]))/Log[d*(e + f*x)^p]) + Log[c*E^(q*(-(p*Log[e + f*x]) + Log[d*(e + f
*x)^p]))*(d*(e + f*x)^p)^(q - (q*(-(p*Log[e + f*x]) + Log[d*(e + f*x)^p]))/Log[d*(e + f*x)^p])]))^2)/h + (b^2*
p^2*q^2*((-16*f^2*g*Sqrt[e + f*x]*Sqrt[1 + (f*g - e*h)/(h*(e + f*x))]*Sqrt[(f*g - e*h + h*(e + f*x))/f]*ArcSin
[Sqrt[-(f*g) + e*h]/(Sqrt[h]*Sqrt[e + f*x])])/(Sqrt[h]*Sqrt[-(f*g) + e*h]*(f*g - e*h + h*(e + f*x))) + (16*e*f
*Sqrt[h]*Sqrt[e + f*x]*Sqrt[1 + (f*g - e*h)/(h*(e + f*x))]*Sqrt[(f*g - e*h + h*(e + f*x))/f]*ArcSin[Sqrt[-(f*g
) + e*h]/(Sqrt[h]*Sqrt[e + f*x])])/(Sqrt[-(f*g) + e*h]*(f*g - e*h + h*(e + f*x))) + (2*f*Sqrt[(f*g - e*h + h*(
e + f*x))/f]*(8 - 4*Log[e + f*x] + Log[e + f*x]^2))/h + (2*e*f*Sqrt[(f*g - e*h + h*(e + f*x))/f]*((-4*ArcTanh[
Sqrt[f*g - e*h + h*(e + f*x)]/Sqrt[f*g - e*h]]*(Log[e + f*x] - Log[-((h*(e + f*x))/(f*g - e*h))]))/Sqrt[f*g -
e*h] + (Sqrt[1 + (h*(e + f*x))/(f*g - e*h)]*(Log[-((h*(e + f*x))/(f*g - e*h))]^2 - 4*Log[-((h*(e + f*x))/(f*g
- e*h))]*Log[(1 + Sqrt[1 + (h*(e + f*x))/(f*g - e*h)])/2] + 2*Log[(1 + Sqrt[1 + (h*(e + f*x))/(f*g - e*h)])/2]
^2 - 4*PolyLog[2, 1 + (-1 - Sqrt[1 + (h*(e + f*x))/(f*g - e*h)])/2]))/Sqrt[f*g - e*h + h*(e + f*x)]))/Sqrt[f*g
- e*h + h*(e + f*x)] - (2*f^2*g*Sqrt[(f*g - e*h + h*(e + f*x))/f]*((-4*ArcTanh[Sqrt[f*g - e*h + h*(e + f*x)]/
Sqrt[f*g - e*h]]*(Log[e + f*x] - Log[-((h*(e + f*x))/(f*g - e*h))]))/Sqrt[f*g - e*h] + (Sqrt[1 + (h*(e + f*x))
/(f*g - e*h)]*(Log[-((h*(e + f*x))/(f*g - e*h))]^2 - 4*Log[-((h*(e + f*x))/(f*g - e*h))]*Log[(1 + Sqrt[1 + (h*
(e + f*x))/(f*g - e*h)])/2] + 2*Log[(1 + Sqrt[1 + (h*(e + f*x))/(f*g - e*h)])/2]^2 - 4*PolyLog[2, 1 + (-1 - Sq
rt[1 + (h*(e + f*x))/(f*g - e*h)])/2]))/Sqrt[f*g - e*h + h*(e + f*x)]))/(h*Sqrt[f*g - e*h + h*(e + f*x)])))/f

Maple [F]

\[\int \frac {{\left (a +b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )\right )}^{2}}{\sqrt {h x +g}}d x\]

[In]

int((a+b*ln(c*(d*(f*x+e)^p)^q))^2/(h*x+g)^(1/2),x)

[Out]

int((a+b*ln(c*(d*(f*x+e)^p)^q))^2/(h*x+g)^(1/2),x)

Fricas [F]

\[ \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{\sqrt {g+h x}} \, dx=\int { \frac {{\left (b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}^{2}}{\sqrt {h x + g}} \,d x } \]

[In]

integrate((a+b*log(c*(d*(f*x+e)^p)^q))^2/(h*x+g)^(1/2),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*x + g), x)

Sympy [F]

\[ \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{\sqrt {g+h x}} \, dx=\int \frac {\left (a + b \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}\right )^{2}}{\sqrt {g + h x}}\, dx \]

[In]

integrate((a+b*ln(c*(d*(f*x+e)**p)**q))**2/(h*x+g)**(1/2),x)

[Out]

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{\sqrt {g+h x}} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((a+b*log(c*(d*(f*x+e)^p)^q))^2/(h*x+g)^(1/2),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(e*h-f*g>0)', see `assume?` for
more detail

Giac [F]

\[ \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{\sqrt {g+h x}} \, dx=\int { \frac {{\left (b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}^{2}}{\sqrt {h x + g}} \,d x } \]

[In]

integrate((a+b*log(c*(d*(f*x+e)^p)^q))^2/(h*x+g)^(1/2),x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{\sqrt {g+h x}} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )\right )}^2}{\sqrt {g+h\,x}} \,d x \]

[In]

int((a + b*log(c*(d*(e + f*x)^p)^q))^2/(g + h*x)^(1/2),x)

[Out]

int((a + b*log(c*(d*(e + f*x)^p)^q))^2/(g + h*x)^(1/2), x)