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

Optimal. Leaf size=488 \[ -\frac{\sqrt{\frac{\pi }{2}} \sqrt{b} h \sqrt{p} \sqrt{q} (e+f x)^2 e^{-\frac{2 a}{b p q}} (f g-e h) \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac{2}{p q}} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt{b} \sqrt{p} \sqrt{q}}\right )}{2 f^3}-\frac{\sqrt{\pi } \sqrt{b} \sqrt{p} \sqrt{q} (e+f x) e^{-\frac{a}{b p q}} (f g-e h)^2 \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac{1}{p q}} \text{Erfi}\left (\frac{\sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt{b} \sqrt{p} \sqrt{q}}\right )}{2 f^3}-\frac{\sqrt{\frac{\pi }{3}} \sqrt{b} h^2 \sqrt{p} \sqrt{q} (e+f x)^3 e^{-\frac{3 a}{b p q}} \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac{3}{p q}} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt{b} \sqrt{p} \sqrt{q}}\right )}{6 f^3}+\frac{h (e+f x)^2 (f g-e h) \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{f^3}+\frac{(e+f x) (f g-e h)^2 \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{f^3}+\frac{h^2 (e+f x)^3 \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{3 f^3} \]

[Out]

-(Sqrt[b]*(f*g - e*h)^2*Sqrt[p]*Sqrt[Pi]*Sqrt[q]*(e + f*x)*Erfi[Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]]/(Sqrt[b]*
Sqrt[p]*Sqrt[q])])/(2*E^(a/(b*p*q))*f^3*(c*(d*(e + f*x)^p)^q)^(1/(p*q))) - (Sqrt[b]*h*(f*g - e*h)*Sqrt[p]*Sqrt
[Pi/2]*Sqrt[q]*(e + f*x)^2*Erfi[(Sqrt[2]*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]])/(Sqrt[b]*Sqrt[p]*Sqrt[q])])/(2*
E^((2*a)/(b*p*q))*f^3*(c*(d*(e + f*x)^p)^q)^(2/(p*q))) - (Sqrt[b]*h^2*Sqrt[p]*Sqrt[Pi/3]*Sqrt[q]*(e + f*x)^3*E
rfi[(Sqrt[3]*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]])/(Sqrt[b]*Sqrt[p]*Sqrt[q])])/(6*E^((3*a)/(b*p*q))*f^3*(c*(d*
(e + f*x)^p)^q)^(3/(p*q))) + ((f*g - e*h)^2*(e + f*x)*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]])/f^3 + (h*(f*g - e*
h)*(e + f*x)^2*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]])/f^3 + (h^2*(e + f*x)^3*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q
]])/(3*f^3)

________________________________________________________________________________________

Rubi [A]  time = 1.61476, antiderivative size = 488, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 10, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {2401, 2389, 2296, 2300, 2180, 2204, 2390, 2305, 2310, 2445} \[ -\frac{\sqrt{\frac{\pi }{2}} \sqrt{b} h \sqrt{p} \sqrt{q} (e+f x)^2 e^{-\frac{2 a}{b p q}} (f g-e h) \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac{2}{p q}} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt{b} \sqrt{p} \sqrt{q}}\right )}{2 f^3}-\frac{\sqrt{\pi } \sqrt{b} \sqrt{p} \sqrt{q} (e+f x) e^{-\frac{a}{b p q}} (f g-e h)^2 \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac{1}{p q}} \text{Erfi}\left (\frac{\sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt{b} \sqrt{p} \sqrt{q}}\right )}{2 f^3}-\frac{\sqrt{\frac{\pi }{3}} \sqrt{b} h^2 \sqrt{p} \sqrt{q} (e+f x)^3 e^{-\frac{3 a}{b p q}} \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac{3}{p q}} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt{b} \sqrt{p} \sqrt{q}}\right )}{6 f^3}+\frac{h (e+f x)^2 (f g-e h) \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{f^3}+\frac{(e+f x) (f g-e h)^2 \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{f^3}+\frac{h^2 (e+f x)^3 \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{3 f^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[b]*(f*g - e*h)^2*Sqrt[p]*Sqrt[Pi]*Sqrt[q]*(e + f*x)*Erfi[Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]]/(Sqrt[b]*
Sqrt[p]*Sqrt[q])])/(2*E^(a/(b*p*q))*f^3*(c*(d*(e + f*x)^p)^q)^(1/(p*q))) - (Sqrt[b]*h*(f*g - e*h)*Sqrt[p]*Sqrt
[Pi/2]*Sqrt[q]*(e + f*x)^2*Erfi[(Sqrt[2]*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]])/(Sqrt[b]*Sqrt[p]*Sqrt[q])])/(2*
E^((2*a)/(b*p*q))*f^3*(c*(d*(e + f*x)^p)^q)^(2/(p*q))) - (Sqrt[b]*h^2*Sqrt[p]*Sqrt[Pi/3]*Sqrt[q]*(e + f*x)^3*E
rfi[(Sqrt[3]*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]])/(Sqrt[b]*Sqrt[p]*Sqrt[q])])/(6*E^((3*a)/(b*p*q))*f^3*(c*(d*
(e + f*x)^p)^q)^(3/(p*q))) + ((f*g - e*h)^2*(e + f*x)*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]])/f^3 + (h*(f*g - e*
h)*(e + f*x)^2*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]])/f^3 + (h^2*(e + f*x)^3*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q
]])/(3*f^3)

Rule 2401

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

Rule 2389

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.), x_Symbol] :> Dist[1/e, Subst[Int[(a + b*Log[c*
x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, n, p}, x]

Rule 2296

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*Log[c*x^n])^p, x] - Dist[b*n*p, In
t[(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, n}, x] && GtQ[p, 0] && IntegerQ[2*p]

Rule 2300

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_), x_Symbol] :> Dist[x/(n*(c*x^n)^(1/n)), Subst[Int[E^(x/n)*(a +
b*x)^p, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

Rule 2180

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[F^(g*(e - (c*
f)/d) + (f*g*x^2)/d), x], x, Sqrt[c + d*x]], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !$UseGamma === True

Rule 2204

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[(F^a*Sqrt[Pi]*Erfi[(c + d*x)*Rt[b*Log[F], 2
]])/(2*d*Rt[b*Log[F], 2]), x] /; FreeQ[{F, a, b, c, d}, x] && PosQ[b]

Rule 2390

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

Rule 2305

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Lo
g[c*x^n])^p)/(d*(m + 1)), x] - Dist[(b*n*p)/(m + 1), Int[(d*x)^m*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{
a, b, c, d, m, n}, x] && NeQ[m, -1] && GtQ[p, 0]

Rule 2310

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_)*((d_.)*(x_))^(m_.), x_Symbol] :> Dist[(d*x)^(m + 1)/(d*n*(c*x^n
)^((m + 1)/n)), Subst[Int[E^(((m + 1)*x)/n)*(a + b*x)^p, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, d, m, n, p}
, x]

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 (g+h x)^2 \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )} \, dx &=\operatorname{Subst}\left (\int (g+h x)^2 \sqrt{a+b \log \left (c d^q (e+f x)^{p q}\right )} \, dx,c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\operatorname{Subst}\left (\int \left (\frac{(f g-e h)^2 \sqrt{a+b \log \left (c d^q (e+f x)^{p q}\right )}}{f^2}+\frac{2 h (f g-e h) (e+f x) \sqrt{a+b \log \left (c d^q (e+f x)^{p q}\right )}}{f^2}+\frac{h^2 (e+f x)^2 \sqrt{a+b \log \left (c d^q (e+f x)^{p q}\right )}}{f^2}\right ) \, dx,c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\operatorname{Subst}\left (\frac{h^2 \int (e+f x)^2 \sqrt{a+b \log \left (c d^q (e+f x)^{p q}\right )} \, dx}{f^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname{Subst}\left (\frac{(2 h (f g-e h)) \int (e+f x) \sqrt{a+b \log \left (c d^q (e+f x)^{p q}\right )} \, dx}{f^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname{Subst}\left (\frac{(f g-e h)^2 \int \sqrt{a+b \log \left (c d^q (e+f x)^{p q}\right )} \, dx}{f^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\operatorname{Subst}\left (\frac{h^2 \operatorname{Subst}\left (\int x^2 \sqrt{a+b \log \left (c d^q x^{p q}\right )} \, dx,x,e+f x\right )}{f^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname{Subst}\left (\frac{(2 h (f g-e h)) \operatorname{Subst}\left (\int x \sqrt{a+b \log \left (c d^q x^{p q}\right )} \, dx,x,e+f x\right )}{f^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname{Subst}\left (\frac{(f g-e h)^2 \operatorname{Subst}\left (\int \sqrt{a+b \log \left (c d^q x^{p q}\right )} \, dx,x,e+f x\right )}{f^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{(f g-e h)^2 (e+f x) \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{f^3}+\frac{h (f g-e h) (e+f x)^2 \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{f^3}+\frac{h^2 (e+f x)^3 \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{3 f^3}-\operatorname{Subst}\left (\frac{\left (b h^2 p q\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{a+b \log \left (c d^q x^{p q}\right )}} \, dx,x,e+f x\right )}{6 f^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname{Subst}\left (\frac{(b h (f g-e h) p q) \operatorname{Subst}\left (\int \frac{x}{\sqrt{a+b \log \left (c d^q x^{p q}\right )}} \, dx,x,e+f x\right )}{2 f^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname{Subst}\left (\frac{\left (b (f g-e h)^2 p q\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b \log \left (c d^q x^{p q}\right )}} \, dx,x,e+f x\right )}{2 f^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{(f g-e h)^2 (e+f x) \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{f^3}+\frac{h (f g-e h) (e+f x)^2 \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{f^3}+\frac{h^2 (e+f x)^3 \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{3 f^3}-\operatorname{Subst}\left (\frac{\left (b h^2 (e+f x)^3 \left (c d^q (e+f x)^{p q}\right )^{-\frac{3}{p q}}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{3 x}{p q}}}{\sqrt{a+b x}} \, dx,x,\log \left (c d^q (e+f x)^{p q}\right )\right )}{6 f^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname{Subst}\left (\frac{\left (b h (f g-e h) (e+f x)^2 \left (c d^q (e+f x)^{p q}\right )^{-\frac{2}{p q}}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{2 x}{p q}}}{\sqrt{a+b x}} \, dx,x,\log \left (c d^q (e+f x)^{p q}\right )\right )}{2 f^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname{Subst}\left (\frac{\left (b (f g-e h)^2 (e+f x) \left (c d^q (e+f x)^{p q}\right )^{-\frac{1}{p q}}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{x}{p q}}}{\sqrt{a+b x}} \, dx,x,\log \left (c d^q (e+f x)^{p q}\right )\right )}{2 f^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{(f g-e h)^2 (e+f x) \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{f^3}+\frac{h (f g-e h) (e+f x)^2 \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{f^3}+\frac{h^2 (e+f x)^3 \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{3 f^3}-\operatorname{Subst}\left (\frac{\left (h^2 (e+f x)^3 \left (c d^q (e+f x)^{p q}\right )^{-\frac{3}{p q}}\right ) \operatorname{Subst}\left (\int e^{-\frac{3 a}{b p q}+\frac{3 x^2}{b p q}} \, dx,x,\sqrt{a+b \log \left (c d^q (e+f x)^{p q}\right )}\right )}{3 f^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname{Subst}\left (\frac{\left (h (f g-e h) (e+f x)^2 \left (c d^q (e+f x)^{p q}\right )^{-\frac{2}{p q}}\right ) \operatorname{Subst}\left (\int e^{-\frac{2 a}{b p q}+\frac{2 x^2}{b p q}} \, dx,x,\sqrt{a+b \log \left (c d^q (e+f x)^{p q}\right )}\right )}{f^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname{Subst}\left (\frac{\left ((f g-e h)^2 (e+f x) \left (c d^q (e+f x)^{p q}\right )^{-\frac{1}{p q}}\right ) \operatorname{Subst}\left (\int e^{-\frac{a}{b p q}+\frac{x^2}{b p q}} \, dx,x,\sqrt{a+b \log \left (c d^q (e+f x)^{p q}\right )}\right )}{f^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac{\sqrt{b} e^{-\frac{a}{b p q}} (f g-e h)^2 \sqrt{p} \sqrt{\pi } \sqrt{q} (e+f x) \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac{1}{p q}} \text{erfi}\left (\frac{\sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt{b} \sqrt{p} \sqrt{q}}\right )}{2 f^3}-\frac{\sqrt{b} e^{-\frac{2 a}{b p q}} h (f g-e h) \sqrt{p} \sqrt{\frac{\pi }{2}} \sqrt{q} (e+f x)^2 \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac{2}{p q}} \text{erfi}\left (\frac{\sqrt{2} \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt{b} \sqrt{p} \sqrt{q}}\right )}{2 f^3}-\frac{\sqrt{b} e^{-\frac{3 a}{b p q}} h^2 \sqrt{p} \sqrt{\frac{\pi }{3}} \sqrt{q} (e+f x)^3 \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac{3}{p q}} \text{erfi}\left (\frac{\sqrt{3} \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt{b} \sqrt{p} \sqrt{q}}\right )}{6 f^3}+\frac{(f g-e h)^2 (e+f x) \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{f^3}+\frac{h (f g-e h) (e+f x)^2 \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{f^3}+\frac{h^2 (e+f x)^3 \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{3 f^3}\\ \end{align*}

Mathematica [A]  time = 0.676829, size = 458, normalized size = 0.94 \[ \frac{(e+f x) \left (9 \sqrt{2 \pi } \sqrt{b} h \sqrt{p} \sqrt{q} (e+f x) e^{-\frac{2 a}{b p q}} (e h-f g) \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac{2}{p q}} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt{b} \sqrt{p} \sqrt{q}}\right )-18 \sqrt{\pi } \sqrt{b} \sqrt{p} \sqrt{q} e^{-\frac{a}{b p q}} (f g-e h)^2 \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac{1}{p q}} \text{Erfi}\left (\frac{\sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt{b} \sqrt{p} \sqrt{q}}\right )-2 \sqrt{3 \pi } \sqrt{b} h^2 \sqrt{p} \sqrt{q} (e+f x)^2 e^{-\frac{3 a}{b p q}} \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac{3}{p q}} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt{b} \sqrt{p} \sqrt{q}}\right )+36 (f g-e h)^2 \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}+36 h (e+f x) (f g-e h) \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}+12 h^2 (e+f x)^2 \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}\right )}{36 f^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((e + f*x)*((-18*Sqrt[b]*(f*g - e*h)^2*Sqrt[p]*Sqrt[Pi]*Sqrt[q]*Erfi[Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]]/(Sqr
t[b]*Sqrt[p]*Sqrt[q])])/(E^(a/(b*p*q))*(c*(d*(e + f*x)^p)^q)^(1/(p*q))) + (9*Sqrt[b]*h*(-(f*g) + e*h)*Sqrt[p]*
Sqrt[2*Pi]*Sqrt[q]*(e + f*x)*Erfi[(Sqrt[2]*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]])/(Sqrt[b]*Sqrt[p]*Sqrt[q])])/(
E^((2*a)/(b*p*q))*(c*(d*(e + f*x)^p)^q)^(2/(p*q))) - (2*Sqrt[b]*h^2*Sqrt[p]*Sqrt[3*Pi]*Sqrt[q]*(e + f*x)^2*Erf
i[(Sqrt[3]*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]])/(Sqrt[b]*Sqrt[p]*Sqrt[q])])/(E^((3*a)/(b*p*q))*(c*(d*(e + f*x
)^p)^q)^(3/(p*q))) + 36*(f*g - e*h)^2*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]] + 36*h*(f*g - e*h)*(e + f*x)*Sqrt[a
+ b*Log[c*(d*(e + f*x)^p)^q]] + 12*h^2*(e + f*x)^2*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]]))/(36*f^3)

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Maple [F]  time = 0.888, size = 0, normalized size = 0. \begin{align*} \int \left ( hx+g \right ) ^{2}\sqrt{a+b\ln \left ( c \left ( d \left ( fx+e \right ) ^{p} \right ) ^{q} \right ) }\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: UnboundLocalError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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