\(\int (g+h x)^{3/2} (a+b \log (c (d (e+f x)^p)^q))^2 \, dx\) [489]

Optimal result

Integrand size = 30, antiderivative size = 635 \[ \int (g+h x)^{3/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \, dx=\frac {368 b^2 (f g-e h)^2 p^2 q^2 \sqrt {g+h x}}{75 f^2 h}+\frac {128 b^2 (f g-e h) p^2 q^2 (g+h x)^{3/2}}{225 f h}+\frac {16 b^2 p^2 q^2 (g+h x)^{5/2}}{125 h}-\frac {368 b^2 (f g-e h)^{5/2} p^2 q^2 \text {arctanh}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )}{75 f^{5/2} h}-\frac {8 b^2 (f g-e h)^{5/2} p^2 q^2 \text {arctanh}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )^2}{5 f^{5/2} h}-\frac {8 b (f g-e h)^2 p q \sqrt {g+h x} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{5 f^2 h}-\frac {8 b (f g-e h) p q (g+h x)^{3/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{15 f h}-\frac {8 b p q (g+h x)^{5/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{25 h}+\frac {8 b (f g-e h)^{5/2} p q \text {arctanh}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{5 f^{5/2} h}+\frac {2 (g+h x)^{5/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{5 h}+\frac {16 b^2 (f g-e h)^{5/2} p^2 q^2 \text {arctanh}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right ) \log \left (\frac {2}{1-\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}}\right )}{5 f^{5/2} h}+\frac {8 b^2 (f g-e h)^{5/2} p^2 q^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1-\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}}\right )}{5 f^{5/2} h} \]

[Out]

128/225*b^2*(-e*h+f*g)*p^2*q^2*(h*x+g)^(3/2)/f/h+16/125*b^2*p^2*q^2*(h*x+g)^(5/2)/h-368/75*b^2*(-e*h+f*g)^(5/2
)*p^2*q^2*arctanh(f^(1/2)*(h*x+g)^(1/2)/(-e*h+f*g)^(1/2))/f^(5/2)/h-8/5*b^2*(-e*h+f*g)^(5/2)*p^2*q^2*arctanh(f
^(1/2)*(h*x+g)^(1/2)/(-e*h+f*g)^(1/2))^2/f^(5/2)/h-8/15*b*(-e*h+f*g)*p*q*(h*x+g)^(3/2)*(a+b*ln(c*(d*(f*x+e)^p)
^q))/f/h-8/25*b*p*q*(h*x+g)^(5/2)*(a+b*ln(c*(d*(f*x+e)^p)^q))/h+8/5*b*(-e*h+f*g)^(5/2)*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))/f^(5/2)/h+2/5*(h*x+g)^(5/2)*(a+b*ln(c*(d*(f*x+e)^p)^q
))^2/h+16/5*b^2*(-e*h+f*g)^(5/2)*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)))/f^(5/2)/h+8/5*b^2*(-e*h+f*g)^(5/2)*p^2*q^2*polylog(2,1-2/(1-f^(1/2)*(h*x+g)^(1/2)/
(-e*h+f*g)^(1/2)))/f^(5/2)/h+368/75*b^2*(-e*h+f*g)^2*p^2*q^2*(h*x+g)^(1/2)/f^2/h-8/5*b*(-e*h+f*g)^2*p*q*(a+b*l
n(c*(d*(f*x+e)^p)^q))*(h*x+g)^(1/2)/f^2/h

Rubi [A] (verified)

Time = 3.08 (sec) , antiderivative size = 635, normalized size of antiderivative = 1.00, number of steps used = 29, 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} \[ \int (g+h x)^{3/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \, dx=\frac {8 b p q (f g-e h)^{5/2} \text {arctanh}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{5 f^{5/2} h}-\frac {8 b p q \sqrt {g+h x} (f g-e h)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{5 f^2 h}-\frac {8 b p q (g+h x)^{3/2} (f g-e h) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{15 f h}+\frac {2 (g+h x)^{5/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{5 h}-\frac {8 b p q (g+h x)^{5/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{25 h}-\frac {8 b^2 p^2 q^2 (f g-e h)^{5/2} \text {arctanh}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )^2}{5 f^{5/2} h}-\frac {368 b^2 p^2 q^2 (f g-e h)^{5/2} \text {arctanh}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )}{75 f^{5/2} h}+\frac {16 b^2 p^2 q^2 (f g-e h)^{5/2} \text {arctanh}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right ) \log \left (\frac {2}{1-\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}}\right )}{5 f^{5/2} h}+\frac {8 b^2 p^2 q^2 (f g-e h)^{5/2} \operatorname {PolyLog}\left (2,1-\frac {2}{1-\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}}\right )}{5 f^{5/2} h}+\frac {368 b^2 p^2 q^2 \sqrt {g+h x} (f g-e h)^2}{75 f^2 h}+\frac {128 b^2 p^2 q^2 (g+h x)^{3/2} (f g-e h)}{225 f h}+\frac {16 b^2 p^2 q^2 (g+h x)^{5/2}}{125 h} \]

[In]

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

[Out]

(368*b^2*(f*g - e*h)^2*p^2*q^2*Sqrt[g + h*x])/(75*f^2*h) + (128*b^2*(f*g - e*h)*p^2*q^2*(g + h*x)^(3/2))/(225*
f*h) + (16*b^2*p^2*q^2*(g + h*x)^(5/2))/(125*h) - (368*b^2*(f*g - e*h)^(5/2)*p^2*q^2*ArcTanh[(Sqrt[f]*Sqrt[g +
h*x])/Sqrt[f*g - e*h]])/(75*f^(5/2)*h) - (8*b^2*(f*g - e*h)^(5/2)*p^2*q^2*ArcTanh[(Sqrt[f]*Sqrt[g + h*x])/Sqr
t[f*g - e*h]]^2)/(5*f^(5/2)*h) - (8*b*(f*g - e*h)^2*p*q*Sqrt[g + h*x]*(a + b*Log[c*(d*(e + f*x)^p)^q]))/(5*f^2
*h) - (8*b*(f*g - e*h)*p*q*(g + h*x)^(3/2)*(a + b*Log[c*(d*(e + f*x)^p)^q]))/(15*f*h) - (8*b*p*q*(g + h*x)^(5/
2)*(a + b*Log[c*(d*(e + f*x)^p)^q]))/(25*h) + (8*b*(f*g - e*h)^(5/2)*p*q*ArcTanh[(Sqrt[f]*Sqrt[g + h*x])/Sqrt[
f*g - e*h]]*(a + b*Log[c*(d*(e + f*x)^p)^q]))/(5*f^(5/2)*h) + (2*(g + h*x)^(5/2)*(a + b*Log[c*(d*(e + f*x)^p)^
q])^2)/(5*h) + (16*b^2*(f*g - e*h)^(5/2)*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])])/(5*f^(5/2)*h) + (8*b^2*(f*g - e*h)^(5/2)*p^2*q^2*PolyLog[2, 1 - 2/(1
- (Sqrt[f]*Sqrt[g + h*x])/Sqrt[f*g - e*h])])/(5*f^(5/2)*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 (g+h x)^{3/2} \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^2 \, dx,c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right ) \\ & = \frac {2 (g+h x)^{5/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{5 h}-\text {Subst}\left (\frac {(4 b f p q) \int \frac {(g+h x)^{5/2} \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{e+f x} \, dx}{5 h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right ) \\ & = \frac {2 (g+h x)^{5/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{5 h}-\text {Subst}\left (\frac {(4 b p q) \text {Subst}\left (\int \frac {\left (\frac {f g-e h}{f}+\frac {h x}{f}\right )^{5/2} \left (a+b \log \left (c d^q x^{p q}\right )\right )}{x} \, dx,x,e+f x\right )}{5 h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right ) \\ & = \frac {2 (g+h x)^{5/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{5 h}-\text {Subst}\left (\frac {(4 b p q) \text {Subst}\left (\int \left (\frac {f g-e h}{f}+\frac {h x}{f}\right )^{3/2} \left (a+b \log \left (c d^q x^{p q}\right )\right ) \, dx,x,e+f x\right )}{5 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 {\left (\frac {f g-e h}{f}+\frac {h x}{f}\right )^{3/2} \left (a+b \log \left (c d^q x^{p q}\right )\right )}{x} \, dx,x,e+f x\right )}{5 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 (g+h x)^{5/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{25 h}+\frac {2 (g+h x)^{5/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{5 h}-\text {Subst}\left (\frac {(4 b (f g-e h) p q) \text {Subst}\left (\int \sqrt {\frac {f g-e h}{f}+\frac {h x}{f}} \left (a+b \log \left (c d^q x^{p q}\right )\right ) \, dx,x,e+f x\right )}{5 f^2},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 (f g-e h)^2 p q\right ) \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 )}{5 f^2 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 p^2 q^2\right ) \text {Subst}\left (\int \frac {\left (\frac {f g-e h}{f}+\frac {h x}{f}\right )^{5/2}}{x} \, dx,x,e+f x\right )}{25 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 (g+h x)^{5/2}}{125 h}-\frac {8 b (f g-e h) p q (g+h x)^{3/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{15 f h}-\frac {8 b p q (g+h x)^{5/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{25 h}+\frac {2 (g+h x)^{5/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{5 h}-\text {Subst}\left (\frac {\left (4 b (f g-e h)^2 p q\right ) \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 )}{5 f^3},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 (f g-e h)^3 p q\right ) \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 )}{5 f^3 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 {\left (\frac {f g-e h}{f}+\frac {h x}{f}\right )^{3/2}}{x} \, dx,x,e+f x\right )}{25 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 {\left (\frac {f g-e h}{f}+\frac {h x}{f}\right )^{3/2}}{x} \, dx,x,e+f x\right )}{15 f h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right ) \\ & = \frac {128 b^2 (f g-e h) p^2 q^2 (g+h x)^{3/2}}{225 f h}+\frac {16 b^2 p^2 q^2 (g+h x)^{5/2}}{125 h}-\frac {8 b (f g-e h)^2 p q \sqrt {g+h x} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{5 f^2 h}-\frac {8 b (f g-e h) p q (g+h x)^{3/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{15 f h}-\frac {8 b p q (g+h x)^{5/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{25 h}+\frac {8 b (f g-e h)^{5/2} 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 )}{5 f^{5/2} h}+\frac {2 (g+h x)^{5/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{5 h}+\text {Subst}\left (\frac {\left (8 b^2 (f g-e h)^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 )}{25 f^2 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)^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 )}{15 f^2 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)^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 )}{5 f^2 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)^3 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 )}{5 f^3 h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right ) \\ & = \frac {368 b^2 (f g-e h)^2 p^2 q^2 \sqrt {g+h x}}{75 f^2 h}+\frac {128 b^2 (f g-e h) p^2 q^2 (g+h x)^{3/2}}{225 f h}+\frac {16 b^2 p^2 q^2 (g+h x)^{5/2}}{125 h}-\frac {8 b (f g-e h)^2 p q \sqrt {g+h x} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{5 f^2 h}-\frac {8 b (f g-e h) p q (g+h x)^{3/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{15 f h}-\frac {8 b p q (g+h x)^{5/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{25 h}+\frac {8 b (f g-e h)^{5/2} 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 )}{5 f^{5/2} h}+\frac {2 (g+h x)^{5/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{5 h}-\text {Subst}\left (\frac {\left (8 b^2 (f g-e h)^{5/2} 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 )}{5 f^{5/2} 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)^3 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 )}{25 f^3 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)^3 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 )}{15 f^3 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)^3 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 )}{5 f^3 h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right ) \\ & = \frac {368 b^2 (f g-e h)^2 p^2 q^2 \sqrt {g+h x}}{75 f^2 h}+\frac {128 b^2 (f g-e h) p^2 q^2 (g+h x)^{3/2}}{225 f h}+\frac {16 b^2 p^2 q^2 (g+h x)^{5/2}}{125 h}-\frac {8 b (f g-e h)^2 p q \sqrt {g+h x} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{5 f^2 h}-\frac {8 b (f g-e h) p q (g+h x)^{3/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{15 f h}-\frac {8 b p q (g+h x)^{5/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{25 h}+\frac {8 b (f g-e h)^{5/2} 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 )}{5 f^{5/2} h}+\frac {2 (g+h x)^{5/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{5 h}-\text {Subst}\left (\frac {\left (16 b^2 (f g-e h)^{5/2} 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 )}{5 f^{3/2} 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)^3 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 )}{25 f^2 h^2},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)^3 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 )}{15 f^2 h^2},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)^3 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 )}{5 f^2 h^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right ) \\ & = \frac {368 b^2 (f g-e h)^2 p^2 q^2 \sqrt {g+h x}}{75 f^2 h}+\frac {128 b^2 (f g-e h) p^2 q^2 (g+h x)^{3/2}}{225 f h}+\frac {16 b^2 p^2 q^2 (g+h x)^{5/2}}{125 h}-\frac {368 b^2 (f g-e h)^{5/2} p^2 q^2 \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )}{75 f^{5/2} h}-\frac {8 b (f g-e h)^2 p q \sqrt {g+h x} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{5 f^2 h}-\frac {8 b (f g-e h) p q (g+h x)^{3/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{15 f h}-\frac {8 b p q (g+h x)^{5/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{25 h}+\frac {8 b (f g-e h)^{5/2} 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 )}{5 f^{5/2} h}+\frac {2 (g+h x)^{5/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{5 h}-\text {Subst}\left (\frac {\left (16 b^2 (f g-e h)^{5/2} 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 )}{5 f^{3/2} h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right ) \\ & = \frac {368 b^2 (f g-e h)^2 p^2 q^2 \sqrt {g+h x}}{75 f^2 h}+\frac {128 b^2 (f g-e h) p^2 q^2 (g+h x)^{3/2}}{225 f h}+\frac {16 b^2 p^2 q^2 (g+h x)^{5/2}}{125 h}-\frac {368 b^2 (f g-e h)^{5/2} p^2 q^2 \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )}{75 f^{5/2} h}-\frac {8 b^2 (f g-e h)^{5/2} p^2 q^2 \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )^2}{5 f^{5/2} h}-\frac {8 b (f g-e h)^2 p q \sqrt {g+h x} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{5 f^2 h}-\frac {8 b (f g-e h) p q (g+h x)^{3/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{15 f h}-\frac {8 b p q (g+h x)^{5/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{25 h}+\frac {8 b (f g-e h)^{5/2} 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 )}{5 f^{5/2} h}+\frac {2 (g+h x)^{5/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{5 h}+\text {Subst}\left (\frac {\left (16 b^2 (f g-e h)^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 )}{5 f^2 h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right ) \\ & = \frac {368 b^2 (f g-e h)^2 p^2 q^2 \sqrt {g+h x}}{75 f^2 h}+\frac {128 b^2 (f g-e h) p^2 q^2 (g+h x)^{3/2}}{225 f h}+\frac {16 b^2 p^2 q^2 (g+h x)^{5/2}}{125 h}-\frac {368 b^2 (f g-e h)^{5/2} p^2 q^2 \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )}{75 f^{5/2} h}-\frac {8 b^2 (f g-e h)^{5/2} p^2 q^2 \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )^2}{5 f^{5/2} h}-\frac {8 b (f g-e h)^2 p q \sqrt {g+h x} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{5 f^2 h}-\frac {8 b (f g-e h) p q (g+h x)^{3/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{15 f h}-\frac {8 b p q (g+h x)^{5/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{25 h}+\frac {8 b (f g-e h)^{5/2} 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 )}{5 f^{5/2} h}+\frac {2 (g+h x)^{5/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{5 h}+\frac {16 b^2 (f g-e h)^{5/2} 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 )}{5 f^{5/2} h}-\text {Subst}\left (\frac {\left (16 b^2 (f g-e h)^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 )}{5 f^2 h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right ) \\ & = \frac {368 b^2 (f g-e h)^2 p^2 q^2 \sqrt {g+h x}}{75 f^2 h}+\frac {128 b^2 (f g-e h) p^2 q^2 (g+h x)^{3/2}}{225 f h}+\frac {16 b^2 p^2 q^2 (g+h x)^{5/2}}{125 h}-\frac {368 b^2 (f g-e h)^{5/2} p^2 q^2 \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )}{75 f^{5/2} h}-\frac {8 b^2 (f g-e h)^{5/2} p^2 q^2 \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )^2}{5 f^{5/2} h}-\frac {8 b (f g-e h)^2 p q \sqrt {g+h x} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{5 f^2 h}-\frac {8 b (f g-e h) p q (g+h x)^{3/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{15 f h}-\frac {8 b p q (g+h x)^{5/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{25 h}+\frac {8 b (f g-e h)^{5/2} 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 )}{5 f^{5/2} h}+\frac {2 (g+h x)^{5/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{5 h}+\frac {16 b^2 (f g-e h)^{5/2} 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 )}{5 f^{5/2} h}+\text {Subst}\left (\frac {\left (16 b^2 (f g-e h)^{5/2} 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 )}{5 f^{5/2} h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right ) \\ & = \frac {368 b^2 (f g-e h)^2 p^2 q^2 \sqrt {g+h x}}{75 f^2 h}+\frac {128 b^2 (f g-e h) p^2 q^2 (g+h x)^{3/2}}{225 f h}+\frac {16 b^2 p^2 q^2 (g+h x)^{5/2}}{125 h}-\frac {368 b^2 (f g-e h)^{5/2} p^2 q^2 \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )}{75 f^{5/2} h}-\frac {8 b^2 (f g-e h)^{5/2} p^2 q^2 \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )^2}{5 f^{5/2} h}-\frac {8 b (f g-e h)^2 p q \sqrt {g+h x} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{5 f^2 h}-\frac {8 b (f g-e h) p q (g+h x)^{3/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{15 f h}-\frac {8 b p q (g+h x)^{5/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{25 h}+\frac {8 b (f g-e h)^{5/2} 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 )}{5 f^{5/2} h}+\frac {2 (g+h x)^{5/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{5 h}+\frac {16 b^2 (f g-e h)^{5/2} 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 )}{5 f^{5/2} h}+\frac {8 b^2 (f g-e h)^{5/2} 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 )}{5 f^{5/2} h} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(4354\) vs. \(2(635)=1270\).

Time = 18.56 (sec) , antiderivative size = 4354, normalized size of antiderivative = 6.86 \[ \int (g+h x)^{3/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \, dx=\text {Result too large to show} \]

[In]

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

[Out]

(4*b*g*p*q*((6*(f*g - e*h)^(3/2)*ArcTanh[(Sqrt[f]*Sqrt[(f*g - e*h + h*(e + f*x))/f])/Sqrt[f*g - e*h]])/Sqrt[f]
- Sqrt[(f*g - e*h + h*(e + f*x))/f]*(h*(e + f*x)*(2 - 3*Log[e + f*x]) + (f*g - e*h)*(8 - 3*Log[e + f*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])
])))/(9*f*h) - (4*b*p*q*(30*(f*g - e*h)^(3/2)*(2*f*g + 3*e*h)*ArcTanh[(Sqrt[f]*Sqrt[(f*g - e*h + h*(e + f*x))/
f])/Sqrt[f*g - e*h]] + Sqrt[f]*Sqrt[(f*g - e*h + h*(e + f*x))/f]*(9*h^2*(e + f*x)^2*(2 - 5*Log[e + f*x]) + (f*
g - e*h)*(3*e*h*(-46 + 15*Log[e + f*x]) + 2*f*g*(-31 + 15*Log[e + f*x])) + h*(e + f*x)*(f*g*(16 - 15*Log[e + f
*x]) + 6*e*h*(-11 + 15*Log[e + f*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])])))/(225*f^(5/2)*h) + Sqrt[g + h*x]*((2*g^2*(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)/(5*h) + (4*g*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)/5 + (2*h*x^2*(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)/5) + (2*b^2*p^2*q^2*Sqrt[(f*g - e*h + h*(e + f*x))/f]*(3720*f^3*g^3*Sqrt[h]*Sq
rt[f*g - e*h]*Sqrt[e + f*x]*Sqrt[(f*g + f*h*x)/(h*(e + f*x))]*ArcSin[Sqrt[-(f*g) + e*h]/(Sqrt[h]*Sqrt[e + f*x]
)] + 840*e*f^2*g^2*h^(3/2)*Sqrt[f*g - e*h]*Sqrt[e + f*x]*Sqrt[(f*g + f*h*x)/(h*(e + f*x))]*ArcSin[Sqrt[-(f*g)
+ e*h]/(Sqrt[h]*Sqrt[e + f*x])] - 12840*e^2*f*g*h^(5/2)*Sqrt[f*g - e*h]*Sqrt[e + f*x]*Sqrt[(f*g + f*h*x)/(h*(e
+ f*x))]*ArcSin[Sqrt[-(f*g) + e*h]/(Sqrt[h]*Sqrt[e + f*x])] + 8280*e^3*h^(7/2)*Sqrt[f*g - e*h]*Sqrt[e + f*x]*
Sqrt[(f*g + f*h*x)/(h*(e + f*x))]*ArcSin[Sqrt[-(f*g) + e*h]/(Sqrt[h]*Sqrt[e + f*x])] + Sqrt[-(f*g - e*h)^2]*(f
*g + f*h*x)*(27*h^2*(e + f*x)^2*(8 - 20*Log[e + f*x] + 25*Log[e + f*x]^2) - (f*g - e*h)*(3*e*h*(3152 - 1380*Lo
g[e + f*x] + 225*Log[e + f*x]^2) + 2*f*g*(1772 - 930*Log[e + f*x] + 225*Log[e + f*x]^2)) + h*(e + f*x)*(f*g*(3
92 - 480*Log[e + f*x] + 225*Log[e + f*x]^2) - 6*e*h*(232 - 330*Log[e + f*x] + 225*Log[e + f*x]^2))) - 450*f^3*
g^3*Sqrt[-(f*g) + e*h]*(4*Sqrt[f*g + f*h*x]*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 - Sqrt[1 + (h*(e + f*x))/(f*g - e*h)])/
2])) + 225*e*f^2*g^2*h*Sqrt[-(f*g) + e*h]*(4*Sqrt[f*g + f*h*x]*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 - Sqrt[1 + (h*(e + f
*x))/(f*g - e*h)])/2])) + 900*e^2*f*g*h^2*Sqrt[-(f*g) + e*h]*(4*Sqrt[f*g + f*h*x]*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 -
Sqrt[1 + (h*(e + f*x))/(f*g - e*h)])/2])) - 675*e^3*h^3*Sqrt[-(f*g) + e*h]*(4*Sqrt[f*g + f*h*x]*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 - Sqrt[1 + (h*(e + f*x))/(f*g - e*h)])/2]))))/(3375*f^2*h*Sqrt[-(f*g - e*h)^2]*(f*g + f*h*x)) +
(b^2*g*p^2*q^2*((-64*f^2*g^2*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])])/(9*Sqrt[h]*Sqrt[-(f*g) + e*h]*(f*g - e*h + h*(e + f*x))
) + (128*e*f*g*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]*Arc
Sin[Sqrt[-(f*g) + e*h]/(Sqrt[h]*Sqrt[e + f*x])])/(9*Sqrt[-(f*g) + e*h]*(f*g - e*h + h*(e + f*x))) - (64*e^2*h^
(3/2)*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])])/(9*Sqrt[-(f*g) + e*h]*(f*g - e*h + h*(e + f*x))) + Sqrt[(f*g - e*h + h*(e + f*
x))/f]*((-2*(-(f*g) + e*h)*(104 - 48*Log[e + f*x] + 9*Log[e + f*x]^2))/(27*h) + (2*(e + f*x)*(8 - 12*Log[e + f
*x] + 9*Log[e + f*x]^2))/27) + (4*e*f*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 - Sqrt[1 + (h*(e + f*x))/(f*g - e*h)])/2]))/Sqrt[f*g - e*h + h*(e + f*x)]))/(3*Sqrt[f*g - e*h + h*(e + f*x)
]) - (2*f^2*g^2*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)]*(Lo
g[-((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)]))/(3*h*Sqrt[f*g - e*h + h*(e + f*x)]) - (2*e^2*h*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]))/S
qrt[f*g - e*h + h*(e + f*x)]))/(3*Sqrt[f*g - e*h + h*(e + f*x)])))/f

Maple [F]

\[\int \left (h x +g \right )^{\frac {3}{2}} {\left (a +b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )\right )}^{2}d x\]

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F(-2)]

Exception generated. \[ \int (g+h x)^{3/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \, dx=\text {Exception raised: ValueError} \]

[In]

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

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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