3.134 \(\int \frac{(f+g x)^2}{(a+b \log (c (d+e x)^n))^{5/2}} \, dx\)

Optimal. Leaf size=421 \[ \frac{16 \sqrt{2 \pi } g e^{-\frac{2 a}{b n}} (d+e x)^2 (e f-d g) \left (c (d+e x)^n\right )^{-2/n} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{3 b^{5/2} e^3 n^{5/2}}+\frac{4 \sqrt{\pi } e^{-\frac{a}{b n}} (d+e x) (e f-d g)^2 \left (c (d+e x)^n\right )^{-1/n} \text{Erfi}\left (\frac{\sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{3 b^{5/2} e^3 n^{5/2}}+\frac{4 \sqrt{3 \pi } g^2 e^{-\frac{3 a}{b n}} (d+e x)^3 \left (c (d+e x)^n\right )^{-3/n} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{b^{5/2} e^3 n^{5/2}}+\frac{8 (d+e x) (f+g x) (e f-d g)}{3 b^2 e^2 n^2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}-\frac{4 (d+e x) (f+g x)^2}{b^2 e n^2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}-\frac{2 (d+e x) (f+g x)^2}{3 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}} \]

[Out]

(4*(e*f - d*g)^2*Sqrt[Pi]*(d + e*x)*Erfi[Sqrt[a + b*Log[c*(d + e*x)^n]]/(Sqrt[b]*Sqrt[n])])/(3*b^(5/2)*e^3*E^(
a/(b*n))*n^(5/2)*(c*(d + e*x)^n)^n^(-1)) + (16*g*(e*f - d*g)*Sqrt[2*Pi]*(d + e*x)^2*Erfi[(Sqrt[2]*Sqrt[a + b*L
og[c*(d + e*x)^n]])/(Sqrt[b]*Sqrt[n])])/(3*b^(5/2)*e^3*E^((2*a)/(b*n))*n^(5/2)*(c*(d + e*x)^n)^(2/n)) + (4*g^2
*Sqrt[3*Pi]*(d + e*x)^3*Erfi[(Sqrt[3]*Sqrt[a + b*Log[c*(d + e*x)^n]])/(Sqrt[b]*Sqrt[n])])/(b^(5/2)*e^3*E^((3*a
)/(b*n))*n^(5/2)*(c*(d + e*x)^n)^(3/n)) - (2*(d + e*x)*(f + g*x)^2)/(3*b*e*n*(a + b*Log[c*(d + e*x)^n])^(3/2))
+ (8*(e*f - d*g)*(d + e*x)*(f + g*x))/(3*b^2*e^2*n^2*Sqrt[a + b*Log[c*(d + e*x)^n]]) - (4*(d + e*x)*(f + g*x)
^2)/(b^2*e*n^2*Sqrt[a + b*Log[c*(d + e*x)^n]])

________________________________________________________________________________________

Rubi [A]  time = 1.39878, antiderivative size = 421, normalized size of antiderivative = 1., number of steps used = 41, number of rules used = 8, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {2400, 2401, 2389, 2300, 2180, 2204, 2390, 2310} \[ \frac{16 \sqrt{2 \pi } g e^{-\frac{2 a}{b n}} (d+e x)^2 (e f-d g) \left (c (d+e x)^n\right )^{-2/n} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{3 b^{5/2} e^3 n^{5/2}}+\frac{4 \sqrt{\pi } e^{-\frac{a}{b n}} (d+e x) (e f-d g)^2 \left (c (d+e x)^n\right )^{-1/n} \text{Erfi}\left (\frac{\sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{3 b^{5/2} e^3 n^{5/2}}+\frac{4 \sqrt{3 \pi } g^2 e^{-\frac{3 a}{b n}} (d+e x)^3 \left (c (d+e x)^n\right )^{-3/n} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{b^{5/2} e^3 n^{5/2}}+\frac{8 (d+e x) (f+g x) (e f-d g)}{3 b^2 e^2 n^2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}-\frac{4 (d+e x) (f+g x)^2}{b^2 e n^2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}-\frac{2 (d+e x) (f+g x)^2}{3 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*(e*f - d*g)^2*Sqrt[Pi]*(d + e*x)*Erfi[Sqrt[a + b*Log[c*(d + e*x)^n]]/(Sqrt[b]*Sqrt[n])])/(3*b^(5/2)*e^3*E^(
a/(b*n))*n^(5/2)*(c*(d + e*x)^n)^n^(-1)) + (16*g*(e*f - d*g)*Sqrt[2*Pi]*(d + e*x)^2*Erfi[(Sqrt[2]*Sqrt[a + b*L
og[c*(d + e*x)^n]])/(Sqrt[b]*Sqrt[n])])/(3*b^(5/2)*e^3*E^((2*a)/(b*n))*n^(5/2)*(c*(d + e*x)^n)^(2/n)) + (4*g^2
*Sqrt[3*Pi]*(d + e*x)^3*Erfi[(Sqrt[3]*Sqrt[a + b*Log[c*(d + e*x)^n]])/(Sqrt[b]*Sqrt[n])])/(b^(5/2)*e^3*E^((3*a
)/(b*n))*n^(5/2)*(c*(d + e*x)^n)^(3/n)) - (2*(d + e*x)*(f + g*x)^2)/(3*b*e*n*(a + b*Log[c*(d + e*x)^n])^(3/2))
+ (8*(e*f - d*g)*(d + e*x)*(f + g*x))/(3*b^2*e^2*n^2*Sqrt[a + b*Log[c*(d + e*x)^n]]) - (4*(d + e*x)*(f + g*x)
^2)/(b^2*e*n^2*Sqrt[a + b*Log[c*(d + e*x)^n]])

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]

Rubi steps

\begin{align*} \int \frac{(f+g x)^2}{\left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}} \, dx &=-\frac{2 (d+e x) (f+g x)^2}{3 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}+\frac{2 \int \frac{(f+g x)^2}{\left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}} \, dx}{b n}-\frac{(4 (e f-d g)) \int \frac{f+g x}{\left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}} \, dx}{3 b e n}\\ &=-\frac{2 (d+e x) (f+g x)^2}{3 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}+\frac{8 (e f-d g) (d+e x) (f+g x)}{3 b^2 e^2 n^2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}-\frac{4 (d+e x) (f+g x)^2}{b^2 e n^2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}+\frac{12 \int \frac{(f+g x)^2}{\sqrt{a+b \log \left (c (d+e x)^n\right )}} \, dx}{b^2 n^2}-\frac{(16 (e f-d g)) \int \frac{f+g x}{\sqrt{a+b \log \left (c (d+e x)^n\right )}} \, dx}{3 b^2 e n^2}-\frac{(8 (e f-d g)) \int \frac{f+g x}{\sqrt{a+b \log \left (c (d+e x)^n\right )}} \, dx}{b^2 e n^2}+\frac{\left (8 (e f-d g)^2\right ) \int \frac{1}{\sqrt{a+b \log \left (c (d+e x)^n\right )}} \, dx}{3 b^2 e^2 n^2}\\ &=-\frac{2 (d+e x) (f+g x)^2}{3 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}+\frac{8 (e f-d g) (d+e x) (f+g x)}{3 b^2 e^2 n^2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}-\frac{4 (d+e x) (f+g x)^2}{b^2 e n^2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}+\frac{12 \int \left (\frac{(e f-d g)^2}{e^2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}+\frac{2 g (e f-d g) (d+e x)}{e^2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}+\frac{g^2 (d+e x)^2}{e^2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}\right ) \, dx}{b^2 n^2}-\frac{(16 (e f-d g)) \int \left (\frac{e f-d g}{e \sqrt{a+b \log \left (c (d+e x)^n\right )}}+\frac{g (d+e x)}{e \sqrt{a+b \log \left (c (d+e x)^n\right )}}\right ) \, dx}{3 b^2 e n^2}-\frac{(8 (e f-d g)) \int \left (\frac{e f-d g}{e \sqrt{a+b \log \left (c (d+e x)^n\right )}}+\frac{g (d+e x)}{e \sqrt{a+b \log \left (c (d+e x)^n\right )}}\right ) \, dx}{b^2 e n^2}+\frac{\left (8 (e f-d g)^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b \log \left (c x^n\right )}} \, dx,x,d+e x\right )}{3 b^2 e^3 n^2}\\ &=-\frac{2 (d+e x) (f+g x)^2}{3 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}+\frac{8 (e f-d g) (d+e x) (f+g x)}{3 b^2 e^2 n^2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}-\frac{4 (d+e x) (f+g x)^2}{b^2 e n^2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}+\frac{\left (12 g^2\right ) \int \frac{(d+e x)^2}{\sqrt{a+b \log \left (c (d+e x)^n\right )}} \, dx}{b^2 e^2 n^2}-\frac{(16 g (e f-d g)) \int \frac{d+e x}{\sqrt{a+b \log \left (c (d+e x)^n\right )}} \, dx}{3 b^2 e^2 n^2}-\frac{(8 g (e f-d g)) \int \frac{d+e x}{\sqrt{a+b \log \left (c (d+e x)^n\right )}} \, dx}{b^2 e^2 n^2}+\frac{(24 g (e f-d g)) \int \frac{d+e x}{\sqrt{a+b \log \left (c (d+e x)^n\right )}} \, dx}{b^2 e^2 n^2}-\frac{\left (16 (e f-d g)^2\right ) \int \frac{1}{\sqrt{a+b \log \left (c (d+e x)^n\right )}} \, dx}{3 b^2 e^2 n^2}-\frac{\left (8 (e f-d g)^2\right ) \int \frac{1}{\sqrt{a+b \log \left (c (d+e x)^n\right )}} \, dx}{b^2 e^2 n^2}+\frac{\left (12 (e f-d g)^2\right ) \int \frac{1}{\sqrt{a+b \log \left (c (d+e x)^n\right )}} \, dx}{b^2 e^2 n^2}+\frac{\left (8 (e f-d g)^2 (d+e x) \left (c (d+e x)^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{x}{n}}}{\sqrt{a+b x}} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{3 b^2 e^3 n^3}\\ &=-\frac{2 (d+e x) (f+g x)^2}{3 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}+\frac{8 (e f-d g) (d+e x) (f+g x)}{3 b^2 e^2 n^2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}-\frac{4 (d+e x) (f+g x)^2}{b^2 e n^2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}+\frac{\left (12 g^2\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{a+b \log \left (c x^n\right )}} \, dx,x,d+e x\right )}{b^2 e^3 n^2}-\frac{(16 g (e f-d g)) \operatorname{Subst}\left (\int \frac{x}{\sqrt{a+b \log \left (c x^n\right )}} \, dx,x,d+e x\right )}{3 b^2 e^3 n^2}-\frac{(8 g (e f-d g)) \operatorname{Subst}\left (\int \frac{x}{\sqrt{a+b \log \left (c x^n\right )}} \, dx,x,d+e x\right )}{b^2 e^3 n^2}+\frac{(24 g (e f-d g)) \operatorname{Subst}\left (\int \frac{x}{\sqrt{a+b \log \left (c x^n\right )}} \, dx,x,d+e x\right )}{b^2 e^3 n^2}-\frac{\left (16 (e f-d g)^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b \log \left (c x^n\right )}} \, dx,x,d+e x\right )}{3 b^2 e^3 n^2}-\frac{\left (8 (e f-d g)^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b \log \left (c x^n\right )}} \, dx,x,d+e x\right )}{b^2 e^3 n^2}+\frac{\left (12 (e f-d g)^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b \log \left (c x^n\right )}} \, dx,x,d+e x\right )}{b^2 e^3 n^2}+\frac{\left (16 (e f-d g)^2 (d+e x) \left (c (d+e x)^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int e^{-\frac{a}{b n}+\frac{x^2}{b n}} \, dx,x,\sqrt{a+b \log \left (c (d+e x)^n\right )}\right )}{3 b^3 e^3 n^3}\\ &=\frac{8 e^{-\frac{a}{b n}} (e f-d g)^2 \sqrt{\pi } (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text{erfi}\left (\frac{\sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{3 b^{5/2} e^3 n^{5/2}}-\frac{2 (d+e x) (f+g x)^2}{3 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}+\frac{8 (e f-d g) (d+e x) (f+g x)}{3 b^2 e^2 n^2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}-\frac{4 (d+e x) (f+g x)^2}{b^2 e n^2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}+\frac{\left (12 g^2 (d+e x)^3 \left (c (d+e x)^n\right )^{-3/n}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{3 x}{n}}}{\sqrt{a+b x}} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{b^2 e^3 n^3}-\frac{\left (16 g (e f-d g) (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{2 x}{n}}}{\sqrt{a+b x}} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{3 b^2 e^3 n^3}-\frac{\left (8 g (e f-d g) (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{2 x}{n}}}{\sqrt{a+b x}} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{b^2 e^3 n^3}+\frac{\left (24 g (e f-d g) (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{2 x}{n}}}{\sqrt{a+b x}} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{b^2 e^3 n^3}-\frac{\left (16 (e f-d g)^2 (d+e x) \left (c (d+e x)^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{x}{n}}}{\sqrt{a+b x}} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{3 b^2 e^3 n^3}-\frac{\left (8 (e f-d g)^2 (d+e x) \left (c (d+e x)^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{x}{n}}}{\sqrt{a+b x}} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{b^2 e^3 n^3}+\frac{\left (12 (e f-d g)^2 (d+e x) \left (c (d+e x)^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{x}{n}}}{\sqrt{a+b x}} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{b^2 e^3 n^3}\\ &=\frac{8 e^{-\frac{a}{b n}} (e f-d g)^2 \sqrt{\pi } (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text{erfi}\left (\frac{\sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{3 b^{5/2} e^3 n^{5/2}}-\frac{2 (d+e x) (f+g x)^2}{3 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}+\frac{8 (e f-d g) (d+e x) (f+g x)}{3 b^2 e^2 n^2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}-\frac{4 (d+e x) (f+g x)^2}{b^2 e n^2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}+\frac{\left (24 g^2 (d+e x)^3 \left (c (d+e x)^n\right )^{-3/n}\right ) \operatorname{Subst}\left (\int e^{-\frac{3 a}{b n}+\frac{3 x^2}{b n}} \, dx,x,\sqrt{a+b \log \left (c (d+e x)^n\right )}\right )}{b^3 e^3 n^3}-\frac{\left (32 g (e f-d g) (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n}\right ) \operatorname{Subst}\left (\int e^{-\frac{2 a}{b n}+\frac{2 x^2}{b n}} \, dx,x,\sqrt{a+b \log \left (c (d+e x)^n\right )}\right )}{3 b^3 e^3 n^3}-\frac{\left (16 g (e f-d g) (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n}\right ) \operatorname{Subst}\left (\int e^{-\frac{2 a}{b n}+\frac{2 x^2}{b n}} \, dx,x,\sqrt{a+b \log \left (c (d+e x)^n\right )}\right )}{b^3 e^3 n^3}+\frac{\left (48 g (e f-d g) (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n}\right ) \operatorname{Subst}\left (\int e^{-\frac{2 a}{b n}+\frac{2 x^2}{b n}} \, dx,x,\sqrt{a+b \log \left (c (d+e x)^n\right )}\right )}{b^3 e^3 n^3}-\frac{\left (32 (e f-d g)^2 (d+e x) \left (c (d+e x)^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int e^{-\frac{a}{b n}+\frac{x^2}{b n}} \, dx,x,\sqrt{a+b \log \left (c (d+e x)^n\right )}\right )}{3 b^3 e^3 n^3}-\frac{\left (16 (e f-d g)^2 (d+e x) \left (c (d+e x)^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int e^{-\frac{a}{b n}+\frac{x^2}{b n}} \, dx,x,\sqrt{a+b \log \left (c (d+e x)^n\right )}\right )}{b^3 e^3 n^3}+\frac{\left (24 (e f-d g)^2 (d+e x) \left (c (d+e x)^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int e^{-\frac{a}{b n}+\frac{x^2}{b n}} \, dx,x,\sqrt{a+b \log \left (c (d+e x)^n\right )}\right )}{b^3 e^3 n^3}\\ &=\frac{4 e^{-\frac{a}{b n}} (e f-d g)^2 \sqrt{\pi } (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text{erfi}\left (\frac{\sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{3 b^{5/2} e^3 n^{5/2}}+\frac{16 e^{-\frac{2 a}{b n}} g (e f-d g) \sqrt{2 \pi } (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} \text{erfi}\left (\frac{\sqrt{2} \sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{3 b^{5/2} e^3 n^{5/2}}+\frac{4 e^{-\frac{3 a}{b n}} g^2 \sqrt{3 \pi } (d+e x)^3 \left (c (d+e x)^n\right )^{-3/n} \text{erfi}\left (\frac{\sqrt{3} \sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{b^{5/2} e^3 n^{5/2}}-\frac{2 (d+e x) (f+g x)^2}{3 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}+\frac{8 (e f-d g) (d+e x) (f+g x)}{3 b^2 e^2 n^2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}-\frac{4 (d+e x) (f+g x)^2}{b^2 e n^2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}\\ \end{align*}

Mathematica [A]  time = 4.52702, size = 527, normalized size = 1.25 \[ -\frac{2 e^{-\frac{3 a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-3/n} \left (\sqrt{b} \sqrt{n} e^{\frac{2 a}{b n}} \left (c (d+e x)^n\right )^{2/n} \left (2 b n \left (2 d^2 g^2+6 d e f g+e^2 f^2\right ) \left (-\frac{a+b \log \left (c (d+e x)^n\right )}{b n}\right )^{3/2} \text{Gamma}\left (\frac{1}{2},-\frac{a+b \log \left (c (d+e x)^n\right )}{b n}\right )+e e^{\frac{a}{b n}} (f+g x) \left (c (d+e x)^n\right )^{\frac{1}{n}} \left (2 a (2 d g+e f+3 e g x)+2 b (2 d g+e (f+3 g x)) \log \left (c (d+e x)^n\right )+b e n (f+g x)\right )\right )+2 \sqrt{\pi } d g e^{\frac{2 a}{b n}} (d g+8 e f) \left (c (d+e x)^n\right )^{2/n} \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2} \text{Erfi}\left (\frac{\sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )+8 \sqrt{2 \pi } g e^{\frac{a}{b n}} (d+e x) (d g-e f) \left (c (d+e x)^n\right )^{\frac{1}{n}} \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )-6 \sqrt{3 \pi } g^2 (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )\right )}{3 b^{5/2} e^3 n^{5/2} \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(d + e*x)*(2*d*E^((2*a)/(b*n))*g*(8*e*f + d*g)*Sqrt[Pi]*(c*(d + e*x)^n)^(2/n)*Erfi[Sqrt[a + b*Log[c*(d + e
*x)^n]]/(Sqrt[b]*Sqrt[n])]*(a + b*Log[c*(d + e*x)^n])^(3/2) + 8*E^(a/(b*n))*g*(-(e*f) + d*g)*Sqrt[2*Pi]*(d + e
*x)*(c*(d + e*x)^n)^n^(-1)*Erfi[(Sqrt[2]*Sqrt[a + b*Log[c*(d + e*x)^n]])/(Sqrt[b]*Sqrt[n])]*(a + b*Log[c*(d +
e*x)^n])^(3/2) - 6*g^2*Sqrt[3*Pi]*(d + e*x)^2*Erfi[(Sqrt[3]*Sqrt[a + b*Log[c*(d + e*x)^n]])/(Sqrt[b]*Sqrt[n])]
*(a + b*Log[c*(d + e*x)^n])^(3/2) + Sqrt[b]*E^((2*a)/(b*n))*Sqrt[n]*(c*(d + e*x)^n)^(2/n)*(2*b*(e^2*f^2 + 6*d*
e*f*g + 2*d^2*g^2)*n*Gamma[1/2, -((a + b*Log[c*(d + e*x)^n])/(b*n))]*(-((a + b*Log[c*(d + e*x)^n])/(b*n)))^(3/
2) + e*E^(a/(b*n))*(c*(d + e*x)^n)^n^(-1)*(f + g*x)*(b*e*n*(f + g*x) + 2*a*(e*f + 2*d*g + 3*e*g*x) + 2*b*(2*d*
g + e*(f + 3*g*x))*Log[c*(d + e*x)^n]))))/(3*b^(5/2)*e^3*E^((3*a)/(b*n))*n^(5/2)*(c*(d + e*x)^n)^(3/n)*(a + b*
Log[c*(d + e*x)^n])^(3/2))

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Maple [F]  time = 0.681, size = 0, normalized size = 0. \begin{align*} \int{ \left ( gx+f \right ) ^{2} \left ( a+b\ln \left ( c \left ( ex+d \right ) ^{n} \right ) \right ) ^{-{\frac{5}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((g*x + f)^2/(b*log((e*x + d)^n*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((g*x+f)^2/(a+b*log(c*(e*x+d)^n))^(5/2),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 (f + g x\right )^{2}}{\left (a + b \log{\left (c \left (d + e x\right )^{n} \right )}\right )^{\frac{5}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((f + g*x)**2/(a + b*log(c*(d + e*x)**n))**(5/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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