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

Optimal. Leaf size=404 \[ \frac{4 \sqrt{2 \pi } h (e+f x)^2 e^{-\frac{2 a}{b p q}} (f g-e h) \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac{2}{p q}} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt{b} \sqrt{p} \sqrt{q}}\right )}{b^{3/2} f^3 p^{3/2} q^{3/2}}+\frac{2 \sqrt{\pi } (e+f x) e^{-\frac{a}{b p q}} (f g-e h)^2 \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac{1}{p q}} \text{Erfi}\left (\frac{\sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt{b} \sqrt{p} \sqrt{q}}\right )}{b^{3/2} f^3 p^{3/2} q^{3/2}}+\frac{2 \sqrt{3 \pi } h^2 (e+f x)^3 e^{-\frac{3 a}{b p q}} \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac{3}{p q}} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt{b} \sqrt{p} \sqrt{q}}\right )}{b^{3/2} f^3 p^{3/2} q^{3/2}}-\frac{2 (e+f x) (g+h x)^2}{b f p q \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}} \]

[Out]

(2*(f*g - e*h)^2*Sqrt[Pi]*(e + f*x)*Erfi[Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]]/(Sqrt[b]*Sqrt[p]*Sqrt[q])])/(b^(
3/2)*E^(a/(b*p*q))*f^3*p^(3/2)*q^(3/2)*(c*(d*(e + f*x)^p)^q)^(1/(p*q))) + (4*h*(f*g - e*h)*Sqrt[2*Pi]*(e + f*x
)^2*Erfi[(Sqrt[2]*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]])/(Sqrt[b]*Sqrt[p]*Sqrt[q])])/(b^(3/2)*E^((2*a)/(b*p*q))
*f^3*p^(3/2)*q^(3/2)*(c*(d*(e + f*x)^p)^q)^(2/(p*q))) + (2*h^2*Sqrt[3*Pi]*(e + f*x)^3*Erfi[(Sqrt[3]*Sqrt[a + b
*Log[c*(d*(e + f*x)^p)^q]])/(Sqrt[b]*Sqrt[p]*Sqrt[q])])/(b^(3/2)*E^((3*a)/(b*p*q))*f^3*p^(3/2)*q^(3/2)*(c*(d*(
e + f*x)^p)^q)^(3/(p*q))) - (2*(e + f*x)*(g + h*x)^2)/(b*f*p*q*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]])

________________________________________________________________________________________

Rubi [A]  time = 2.24988, antiderivative size = 404, normalized size of antiderivative = 1., number of steps used = 26, number of rules used = 9, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {2400, 2401, 2389, 2300, 2180, 2204, 2390, 2310, 2445} \[ \frac{4 \sqrt{2 \pi } h (e+f x)^2 e^{-\frac{2 a}{b p q}} (f g-e h) \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac{2}{p q}} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt{b} \sqrt{p} \sqrt{q}}\right )}{b^{3/2} f^3 p^{3/2} q^{3/2}}+\frac{2 \sqrt{\pi } (e+f x) e^{-\frac{a}{b p q}} (f g-e h)^2 \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac{1}{p q}} \text{Erfi}\left (\frac{\sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt{b} \sqrt{p} \sqrt{q}}\right )}{b^{3/2} f^3 p^{3/2} q^{3/2}}+\frac{2 \sqrt{3 \pi } h^2 (e+f x)^3 e^{-\frac{3 a}{b p q}} \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac{3}{p q}} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt{b} \sqrt{p} \sqrt{q}}\right )}{b^{3/2} f^3 p^{3/2} q^{3/2}}-\frac{2 (e+f x) (g+h x)^2}{b f p q \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(f*g - e*h)^2*Sqrt[Pi]*(e + f*x)*Erfi[Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]]/(Sqrt[b]*Sqrt[p]*Sqrt[q])])/(b^(
3/2)*E^(a/(b*p*q))*f^3*p^(3/2)*q^(3/2)*(c*(d*(e + f*x)^p)^q)^(1/(p*q))) + (4*h*(f*g - e*h)*Sqrt[2*Pi]*(e + f*x
)^2*Erfi[(Sqrt[2]*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]])/(Sqrt[b]*Sqrt[p]*Sqrt[q])])/(b^(3/2)*E^((2*a)/(b*p*q))
*f^3*p^(3/2)*q^(3/2)*(c*(d*(e + f*x)^p)^q)^(2/(p*q))) + (2*h^2*Sqrt[3*Pi]*(e + f*x)^3*Erfi[(Sqrt[3]*Sqrt[a + b
*Log[c*(d*(e + f*x)^p)^q]])/(Sqrt[b]*Sqrt[p]*Sqrt[q])])/(b^(3/2)*E^((3*a)/(b*p*q))*f^3*p^(3/2)*q^(3/2)*(c*(d*(
e + f*x)^p)^q)^(3/(p*q))) - (2*(e + f*x)*(g + h*x)^2)/(b*f*p*q*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]])

Rule 2400

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

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 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 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 \frac{(g+h x)^2}{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}} \, dx &=\operatorname{Subst}\left (\int \frac{(g+h x)^2}{\left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^{3/2}} \, dx,c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac{2 (e+f x) (g+h x)^2}{b f p q \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}+\operatorname{Subst}\left (\frac{6 \int \frac{(g+h x)^2}{\sqrt{a+b \log \left (c d^q (e+f x)^{p q}\right )}} \, dx}{b p q},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname{Subst}\left (\frac{(4 (f g-e h)) \int \frac{g+h x}{\sqrt{a+b \log \left (c d^q (e+f x)^{p q}\right )}} \, dx}{b f p q},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac{2 (e+f x) (g+h x)^2}{b f p q \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}+\operatorname{Subst}\left (\frac{6 \int \left (\frac{(f g-e h)^2}{f^2 \sqrt{a+b \log \left (c d^q (e+f x)^{p q}\right )}}+\frac{2 h (f g-e h) (e+f x)}{f^2 \sqrt{a+b \log \left (c d^q (e+f x)^{p q}\right )}}+\frac{h^2 (e+f x)^2}{f^2 \sqrt{a+b \log \left (c d^q (e+f x)^{p q}\right )}}\right ) \, dx}{b p q},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname{Subst}\left (\frac{(4 (f g-e h)) \int \left (\frac{f g-e h}{f \sqrt{a+b \log \left (c d^q (e+f x)^{p q}\right )}}+\frac{h (e+f x)}{f \sqrt{a+b \log \left (c d^q (e+f x)^{p q}\right )}}\right ) \, dx}{b f p q},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac{2 (e+f x) (g+h x)^2}{b f p q \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}+\operatorname{Subst}\left (\frac{\left (6 h^2\right ) \int \frac{(e+f x)^2}{\sqrt{a+b \log \left (c d^q (e+f x)^{p q}\right )}} \, dx}{b f^2 p q},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname{Subst}\left (\frac{(4 h (f g-e h)) \int \frac{e+f x}{\sqrt{a+b \log \left (c d^q (e+f x)^{p q}\right )}} \, dx}{b f^2 p q},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname{Subst}\left (\frac{(12 h (f g-e h)) \int \frac{e+f x}{\sqrt{a+b \log \left (c d^q (e+f x)^{p q}\right )}} \, dx}{b f^2 p q},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname{Subst}\left (\frac{\left (4 (f g-e h)^2\right ) \int \frac{1}{\sqrt{a+b \log \left (c d^q (e+f x)^{p q}\right )}} \, dx}{b f^2 p q},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname{Subst}\left (\frac{\left (6 (f g-e h)^2\right ) \int \frac{1}{\sqrt{a+b \log \left (c d^q (e+f x)^{p q}\right )}} \, dx}{b f^2 p q},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac{2 (e+f x) (g+h x)^2}{b f p q \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}+\operatorname{Subst}\left (\frac{\left (6 h^2\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 )}{b f^3 p q},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname{Subst}\left (\frac{(4 h (f g-e h)) \operatorname{Subst}\left (\int \frac{x}{\sqrt{a+b \log \left (c d^q x^{p q}\right )}} \, dx,x,e+f x\right )}{b f^3 p q},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname{Subst}\left (\frac{(12 h (f g-e h)) \operatorname{Subst}\left (\int \frac{x}{\sqrt{a+b \log \left (c d^q x^{p q}\right )}} \, dx,x,e+f x\right )}{b f^3 p q},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname{Subst}\left (\frac{\left (4 (f g-e h)^2\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 )}{b f^3 p q},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname{Subst}\left (\frac{\left (6 (f g-e h)^2\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 )}{b f^3 p q},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac{2 (e+f x) (g+h x)^2}{b f p q \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}+\operatorname{Subst}\left (\frac{\left (6 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 )}{b f^3 p^2 q^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname{Subst}\left (\frac{\left (4 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 )}{b f^3 p^2 q^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname{Subst}\left (\frac{\left (12 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 )}{b f^3 p^2 q^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname{Subst}\left (\frac{\left (4 (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 )}{b f^3 p^2 q^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname{Subst}\left (\frac{\left (6 (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 )}{b f^3 p^2 q^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac{2 (e+f x) (g+h x)^2}{b f p q \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}+\operatorname{Subst}\left (\frac{\left (12 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 )}{b^2 f^3 p^2 q^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname{Subst}\left (\frac{\left (8 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 )}{b^2 f^3 p^2 q^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname{Subst}\left (\frac{\left (24 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 )}{b^2 f^3 p^2 q^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname{Subst}\left (\frac{\left (8 (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 )}{b^2 f^3 p^2 q^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname{Subst}\left (\frac{\left (12 (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 )}{b^2 f^3 p^2 q^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{2 e^{-\frac{a}{b p q}} (f g-e h)^2 \sqrt{\pi } (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 )}{b^{3/2} f^3 p^{3/2} q^{3/2}}+\frac{4 e^{-\frac{2 a}{b p q}} h (f g-e h) \sqrt{2 \pi } (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 )}{b^{3/2} f^3 p^{3/2} q^{3/2}}+\frac{2 e^{-\frac{3 a}{b p q}} h^2 \sqrt{3 \pi } (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 )}{b^{3/2} f^3 p^{3/2} q^{3/2}}-\frac{2 (e+f x) (g+h x)^2}{b f p q \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}\\ \end{align*}

Mathematica [B]  time = 2.48928, size = 1040, normalized size = 2.57 \[ \frac{2 \left (e^{-\frac{3 a}{b p q}} h^2 \sqrt{3 \pi } (e+f x)^3 \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 ) \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )} \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac{3}{p q}}-2 e e^{-\frac{2 a}{b p q}} h^2 \sqrt{2 \pi } (e+f x)^2 \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 ) \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )} \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac{2}{p q}}+2 e^{-\frac{2 a}{b p q}} f g h \sqrt{2 \pi } (e+f x)^2 \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 ) \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )} \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac{2}{p q}}+e^2 e^{-\frac{a}{b p q}} h^2 \sqrt{\pi } (e+f x) \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 ) \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )} \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac{1}{p q}}-4 e e^{-\frac{a}{b p q}} f g h \sqrt{\pi } (e+f x) \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 ) \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )} \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac{1}{p q}}+\sqrt{b} e^{-\frac{a}{b p q}} f^2 g^2 \sqrt{p} \sqrt{q} (e+f x) \text{Gamma}\left (\frac{1}{2},-\frac{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{b p q}\right ) \sqrt{-\frac{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{b p q}} \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac{1}{p q}}+2 \sqrt{b} e e^{-\frac{a}{b p q}} f g h \sqrt{p} \sqrt{q} (e+f x) \text{Gamma}\left (\frac{1}{2},-\frac{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{b p q}\right ) \sqrt{-\frac{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{b p q}} \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac{1}{p q}}-\sqrt{b} f^3 h^2 \sqrt{p} \sqrt{q} x^3-\sqrt{b} e f^2 h^2 \sqrt{p} \sqrt{q} x^2-2 \sqrt{b} f^3 g h \sqrt{p} \sqrt{q} x^2-\sqrt{b} f^3 g^2 \sqrt{p} \sqrt{q} x-2 \sqrt{b} e f^2 g h \sqrt{p} \sqrt{q} x-\sqrt{b} e f^2 g^2 \sqrt{p} \sqrt{q}\right )}{b^{3/2} f^3 p^{3/2} q^{3/2} \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(-(Sqrt[b]*e*f^2*g^2*Sqrt[p]*Sqrt[q]) - Sqrt[b]*f^3*g^2*Sqrt[p]*Sqrt[q]*x - 2*Sqrt[b]*e*f^2*g*h*Sqrt[p]*Sqr
t[q]*x - 2*Sqrt[b]*f^3*g*h*Sqrt[p]*Sqrt[q]*x^2 - Sqrt[b]*e*f^2*h^2*Sqrt[p]*Sqrt[q]*x^2 - Sqrt[b]*f^3*h^2*Sqrt[
p]*Sqrt[q]*x^3 - (4*e*f*g*h*Sqrt[Pi]*(e + f*x)*Erfi[Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]]/(Sqrt[b]*Sqrt[p]*Sqrt
[q])]*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]])/(E^(a/(b*p*q))*(c*(d*(e + f*x)^p)^q)^(1/(p*q))) + (e^2*h^2*Sqrt[Pi
]*(e + f*x)*Erfi[Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]]/(Sqrt[b]*Sqrt[p]*Sqrt[q])]*Sqrt[a + b*Log[c*(d*(e + f*x)
^p)^q]])/(E^(a/(b*p*q))*(c*(d*(e + f*x)^p)^q)^(1/(p*q))) + (2*f*g*h*Sqrt[2*Pi]*(e + f*x)^2*Erfi[(Sqrt[2]*Sqrt[
a + b*Log[c*(d*(e + f*x)^p)^q]])/(Sqrt[b]*Sqrt[p]*Sqrt[q])]*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]])/(E^((2*a)/(b
*p*q))*(c*(d*(e + f*x)^p)^q)^(2/(p*q))) - (2*e*h^2*Sqrt[2*Pi]*(e + f*x)^2*Erfi[(Sqrt[2]*Sqrt[a + b*Log[c*(d*(e
+ f*x)^p)^q]])/(Sqrt[b]*Sqrt[p]*Sqrt[q])]*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]])/(E^((2*a)/(b*p*q))*(c*(d*(e +
f*x)^p)^q)^(2/(p*q))) + (h^2*Sqrt[3*Pi]*(e + f*x)^3*Erfi[(Sqrt[3]*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]])/(Sqrt
[b]*Sqrt[p]*Sqrt[q])]*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]])/(E^((3*a)/(b*p*q))*(c*(d*(e + f*x)^p)^q)^(3/(p*q))
) + (Sqrt[b]*f^2*g^2*Sqrt[p]*Sqrt[q]*(e + f*x)*Gamma[1/2, -((a + b*Log[c*(d*(e + f*x)^p)^q])/(b*p*q))]*Sqrt[-(
(a + b*Log[c*(d*(e + f*x)^p)^q])/(b*p*q))])/(E^(a/(b*p*q))*(c*(d*(e + f*x)^p)^q)^(1/(p*q))) + (2*Sqrt[b]*e*f*g
*h*Sqrt[p]*Sqrt[q]*(e + f*x)*Gamma[1/2, -((a + b*Log[c*(d*(e + f*x)^p)^q])/(b*p*q))]*Sqrt[-((a + b*Log[c*(d*(e
+ f*x)^p)^q])/(b*p*q))])/(E^(a/(b*p*q))*(c*(d*(e + f*x)^p)^q)^(1/(p*q)))))/(b^(3/2)*f^3*p^(3/2)*q^(3/2)*Sqrt[
a + b*Log[c*(d*(e + f*x)^p)^q]])

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Maple [F]  time = 0.887, size = 0, normalized size = 0. \begin{align*} \int{ \left ( hx+g \right ) ^{2} \left ( a+b\ln \left ( c \left ( d \left ( fx+e \right ) ^{p} \right ) ^{q} \right ) \right ) ^{-{\frac{3}{2}}}}\, 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))^(3/2),x)

[Out]

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

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

[Out]

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

[Out]

Exception raised: UnboundLocalError

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

[Out]

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

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

[Out]

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