3.479 \(\int \frac{g+h x}{(a+b \log (c (d (e+f x)^p)^q))^{5/2}} \, dx\)

Optimal. Leaf size=380 \[ \frac{4 \sqrt{\pi } (e+f x) e^{-\frac{a}{b p q}} (f g-e h) \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac{1}{p q}} \text{Erfi}\left (\frac{\sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt{b} \sqrt{p} \sqrt{q}}\right )}{3 b^{5/2} f^2 p^{5/2} q^{5/2}}+\frac{8 \sqrt{2 \pi } h (e+f x)^2 e^{-\frac{2 a}{b p q}} \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac{2}{p q}} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt{b} \sqrt{p} \sqrt{q}}\right )}{3 b^{5/2} f^2 p^{5/2} q^{5/2}}+\frac{4 (e+f x) (f g-e h)}{3 b^2 f^2 p^2 q^2 \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}-\frac{8 (e+f x) (g+h x)}{3 b^2 f p^2 q^2 \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}-\frac{2 (e+f x) (g+h x)}{3 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}} \]

[Out]

(4*(f*g - e*h)*Sqrt[Pi]*(e + f*x)*Erfi[Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]]/(Sqrt[b]*Sqrt[p]*Sqrt[q])])/(3*b^(
5/2)*E^(a/(b*p*q))*f^2*p^(5/2)*q^(5/2)*(c*(d*(e + f*x)^p)^q)^(1/(p*q))) + (8*h*Sqrt[2*Pi]*(e + f*x)^2*Erfi[(Sq
rt[2]*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]])/(Sqrt[b]*Sqrt[p]*Sqrt[q])])/(3*b^(5/2)*E^((2*a)/(b*p*q))*f^2*p^(5/
2)*q^(5/2)*(c*(d*(e + f*x)^p)^q)^(2/(p*q))) - (2*(e + f*x)*(g + h*x))/(3*b*f*p*q*(a + b*Log[c*(d*(e + f*x)^p)^
q])^(3/2)) + (4*(f*g - e*h)*(e + f*x))/(3*b^2*f^2*p^2*q^2*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]]) - (8*(e + f*x)
*(g + h*x))/(3*b^2*f*p^2*q^2*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]])

________________________________________________________________________________________

Rubi [A]  time = 1.59795, antiderivative size = 380, normalized size of antiderivative = 1., number of steps used = 22, number of rules used = 10, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357, Rules used = {2400, 2401, 2389, 2300, 2180, 2204, 2390, 2310, 2297, 2445} \[ \frac{4 \sqrt{\pi } (e+f x) e^{-\frac{a}{b p q}} (f g-e h) \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac{1}{p q}} \text{Erfi}\left (\frac{\sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt{b} \sqrt{p} \sqrt{q}}\right )}{3 b^{5/2} f^2 p^{5/2} q^{5/2}}+\frac{8 \sqrt{2 \pi } h (e+f x)^2 e^{-\frac{2 a}{b p q}} \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac{2}{p q}} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt{b} \sqrt{p} \sqrt{q}}\right )}{3 b^{5/2} f^2 p^{5/2} q^{5/2}}+\frac{4 (e+f x) (f g-e h)}{3 b^2 f^2 p^2 q^2 \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}-\frac{8 (e+f x) (g+h x)}{3 b^2 f p^2 q^2 \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}-\frac{2 (e+f x) (g+h x)}{3 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[(g + h*x)/(a + b*Log[c*(d*(e + f*x)^p)^q])^(5/2),x]

[Out]

(4*(f*g - e*h)*Sqrt[Pi]*(e + f*x)*Erfi[Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]]/(Sqrt[b]*Sqrt[p]*Sqrt[q])])/(3*b^(
5/2)*E^(a/(b*p*q))*f^2*p^(5/2)*q^(5/2)*(c*(d*(e + f*x)^p)^q)^(1/(p*q))) + (8*h*Sqrt[2*Pi]*(e + f*x)^2*Erfi[(Sq
rt[2]*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]])/(Sqrt[b]*Sqrt[p]*Sqrt[q])])/(3*b^(5/2)*E^((2*a)/(b*p*q))*f^2*p^(5/
2)*q^(5/2)*(c*(d*(e + f*x)^p)^q)^(2/(p*q))) - (2*(e + f*x)*(g + h*x))/(3*b*f*p*q*(a + b*Log[c*(d*(e + f*x)^p)^
q])^(3/2)) + (4*(f*g - e*h)*(e + f*x))/(3*b^2*f^2*p^2*q^2*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]]) - (8*(e + f*x)
*(g + h*x))/(3*b^2*f*p^2*q^2*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 2297

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

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}{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{5/2}} \, dx &=\operatorname{Subst}\left (\int \frac{g+h x}{\left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^{5/2}} \, 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)}{3 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}}+\operatorname{Subst}\left (\frac{4 \int \frac{g+h x}{\left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^{3/2}} \, dx}{3 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{(2 (f g-e h)) \int \frac{1}{\left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^{3/2}} \, dx}{3 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)}{3 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}}-\frac{8 (e+f x) (g+h x)}{3 b^2 f p^2 q^2 \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}+\operatorname{Subst}\left (\frac{16 \int \frac{g+h x}{\sqrt{a+b \log \left (c d^q (e+f x)^{p q}\right )}} \, dx}{3 b^2 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{(8 (f g-e h)) \int \frac{1}{\sqrt{a+b \log \left (c d^q (e+f x)^{p q}\right )}} \, dx}{3 b^2 f 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{(2 (f g-e h)) \operatorname{Subst}\left (\int \frac{1}{\left (a+b \log \left (c d^q x^{p q}\right )\right )^{3/2}} \, dx,x,e+f x\right )}{3 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)}{3 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}}+\frac{4 (f g-e h) (e+f x)}{3 b^2 f^2 p^2 q^2 \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}-\frac{8 (e+f x) (g+h x)}{3 b^2 f p^2 q^2 \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}+\operatorname{Subst}\left (\frac{16 \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}{3 b^2 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{(4 (f g-e h)) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b \log \left (c d^q x^{p q}\right )}} \, dx,x,e+f x\right )}{3 b^2 f^2 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{(8 (f g-e h)) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b \log \left (c d^q x^{p q}\right )}} \, dx,x,e+f x\right )}{3 b^2 f^2 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)}{3 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}}+\frac{4 (f g-e h) (e+f x)}{3 b^2 f^2 p^2 q^2 \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}-\frac{8 (e+f x) (g+h x)}{3 b^2 f p^2 q^2 \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}+\operatorname{Subst}\left (\frac{(16 h) \int \frac{e+f x}{\sqrt{a+b \log \left (c d^q (e+f x)^{p q}\right )}} \, dx}{3 b^2 f 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{(16 (f g-e h)) \int \frac{1}{\sqrt{a+b \log \left (c d^q (e+f x)^{p q}\right )}} \, dx}{3 b^2 f 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) (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 )}{3 b^2 f^2 p^3 q^3},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) (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 )}{3 b^2 f^2 p^3 q^3},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)}{3 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}}+\frac{4 (f g-e h) (e+f x)}{3 b^2 f^2 p^2 q^2 \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}-\frac{8 (e+f x) (g+h x)}{3 b^2 f p^2 q^2 \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}+\operatorname{Subst}\left (\frac{(16 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 )}{3 b^2 f^2 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{(16 (f g-e h)) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b \log \left (c d^q x^{p q}\right )}} \, dx,x,e+f x\right )}{3 b^2 f^2 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) (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 )}{3 b^3 f^2 p^3 q^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname{Subst}\left (\frac{\left (16 (f g-e h) (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 )}{3 b^3 f^2 p^3 q^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac{4 e^{-\frac{a}{b p q}} (f g-e h) \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^{5/2} f^2 p^{5/2} q^{5/2}}-\frac{2 (e+f x) (g+h x)}{3 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}}+\frac{4 (f g-e h) (e+f x)}{3 b^2 f^2 p^2 q^2 \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}-\frac{8 (e+f x) (g+h x)}{3 b^2 f p^2 q^2 \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}+\operatorname{Subst}\left (\frac{\left (16 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 )}{3 b^2 f^2 p^3 q^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname{Subst}\left (\frac{\left (16 (f g-e h) (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 )}{3 b^2 f^2 p^3 q^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac{4 e^{-\frac{a}{b p q}} (f g-e h) \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^{5/2} f^2 p^{5/2} q^{5/2}}-\frac{2 (e+f x) (g+h x)}{3 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}}+\frac{4 (f g-e h) (e+f x)}{3 b^2 f^2 p^2 q^2 \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}-\frac{8 (e+f x) (g+h x)}{3 b^2 f p^2 q^2 \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}+\operatorname{Subst}\left (\frac{\left (32 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 )}{3 b^3 f^2 p^3 q^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname{Subst}\left (\frac{\left (32 (f g-e h) (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 )}{3 b^3 f^2 p^3 q^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{4 e^{-\frac{a}{b p q}} (f g-e h) \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 )}{3 b^{5/2} f^2 p^{5/2} q^{5/2}}+\frac{8 e^{-\frac{2 a}{b p q}} 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 )}{3 b^{5/2} f^2 p^{5/2} q^{5/2}}-\frac{2 (e+f x) (g+h x)}{3 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}}+\frac{4 (f g-e h) (e+f x)}{3 b^2 f^2 p^2 q^2 \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}-\frac{8 (e+f x) (g+h x)}{3 b^2 f p^2 q^2 \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}\\ \end{align*}

Mathematica [A]  time = 2.30621, size = 491, normalized size = 1.29 \[ -\frac{2 (e+f x) e^{-\frac{2 a}{b p q}} \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac{2}{p q}} \left (\sqrt{b} \sqrt{p} \sqrt{q} e^{\frac{a}{b p q}} \left (c \left (d (e+f x)^p\right )^q\right )^{\frac{1}{p q}} \left (2 b p q (3 e h+f g) \left (-\frac{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{b p q}\right )^{3/2} \text{Gamma}\left (\frac{1}{2},-\frac{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{b p q}\right )+e^{\frac{a}{b p q}} \left (c \left (d (e+f x)^p\right )^q\right )^{\frac{1}{p q}} \left (2 a (e h+f g+2 f h x)+2 b (e h+f (g+2 h x)) \log \left (c \left (d (e+f x)^p\right )^q\right )+b f p q (g+h x)\right )\right )+8 \sqrt{\pi } e h e^{\frac{a}{b p q}} \left (c \left (d (e+f x)^p\right )^q\right )^{\frac{1}{p q}} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2} \text{Erfi}\left (\frac{\sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt{b} \sqrt{p} \sqrt{q}}\right )-4 \sqrt{2 \pi } h (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt{b} \sqrt{p} \sqrt{q}}\right )\right )}{3 b^{5/2} f^2 p^{5/2} q^{5/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(g + h*x)/(a + b*Log[c*(d*(e + f*x)^p)^q])^(5/2),x]

[Out]

(-2*(e + f*x)*(8*e*E^(a/(b*p*q))*h*Sqrt[Pi]*(c*(d*(e + f*x)^p)^q)^(1/(p*q))*Erfi[Sqrt[a + b*Log[c*(d*(e + f*x)
^p)^q]]/(Sqrt[b]*Sqrt[p]*Sqrt[q])]*(a + b*Log[c*(d*(e + f*x)^p)^q])^(3/2) - 4*h*Sqrt[2*Pi]*(e + f*x)*Erfi[(Sqr
t[2]*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]])/(Sqrt[b]*Sqrt[p]*Sqrt[q])]*(a + b*Log[c*(d*(e + f*x)^p)^q])^(3/2) +
Sqrt[b]*E^(a/(b*p*q))*Sqrt[p]*Sqrt[q]*(c*(d*(e + f*x)^p)^q)^(1/(p*q))*(2*b*(f*g + 3*e*h)*p*q*Gamma[1/2, -((a
+ b*Log[c*(d*(e + f*x)^p)^q])/(b*p*q))]*(-((a + b*Log[c*(d*(e + f*x)^p)^q])/(b*p*q)))^(3/2) + E^(a/(b*p*q))*(c
*(d*(e + f*x)^p)^q)^(1/(p*q))*(b*f*p*q*(g + h*x) + 2*a*(f*g + e*h + 2*f*h*x) + 2*b*(e*h + f*(g + 2*h*x))*Log[c
*(d*(e + f*x)^p)^q]))))/(3*b^(5/2)*E^((2*a)/(b*p*q))*f^2*p^(5/2)*q^(5/2)*(c*(d*(e + f*x)^p)^q)^(2/(p*q))*(a +
b*Log[c*(d*(e + f*x)^p)^q])^(3/2))

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Maple [F]  time = 0.273, size = 0, normalized size = 0. \begin{align*} \int{(hx+g) \left ( a+b\ln \left ( c \left ( d \left ( fx+e \right ) ^{p} \right ) ^{q} \right ) \right ) ^{-{\frac{5}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((h*x+g)/(a+b*ln(c*(d*(f*x+e)^p)^q))^(5/2),x)

[Out]

int((h*x+g)/(a+b*ln(c*(d*(f*x+e)^p)^q))^(5/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{h x + g}{{\left (b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((h*x+g)/(a+b*log(c*(d*(f*x+e)^p)^q))^(5/2),x, algorithm="maxima")

[Out]

integrate((h*x + g)/(b*log(((f*x + e)^p*d)^q*c) + a)^(5/2), 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)/(a+b*log(c*(d*(f*x+e)^p)^q))^(5/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((h*x+g)/(a+b*ln(c*(d*(f*x+e)**p)**q))**(5/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{h x + g}{{\left (b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((h*x+g)/(a+b*log(c*(d*(f*x+e)^p)^q))^(5/2),x, algorithm="giac")

[Out]

integrate((h*x + g)/(b*log(((f*x + e)^p*d)^q*c) + a)^(5/2), x)