3.133 \(\int \frac{(f+g x)^3}{(a+b \log (c (d+e x)^n))^{5/2}} \, dx\)

Optimal. Leaf size=520 \[ \frac{12 \sqrt{3 \pi } g^2 e^{-\frac{3 a}{b n}} (d+e x)^3 (e f-d g) \left (c (d+e x)^n\right )^{-3/n} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{b^{5/2} e^4 n^{5/2}}+\frac{8 \sqrt{2 \pi } g e^{-\frac{2 a}{b n}} (d+e x)^2 (e f-d g)^2 \left (c (d+e x)^n\right )^{-2/n} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{b^{5/2} e^4 n^{5/2}}+\frac{4 \sqrt{\pi } e^{-\frac{a}{b n}} (d+e x) (e f-d g)^3 \left (c (d+e x)^n\right )^{-1/n} \text{Erfi}\left (\frac{\sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{3 b^{5/2} e^4 n^{5/2}}+\frac{32 \sqrt{\pi } g^3 e^{-\frac{4 a}{b n}} (d+e x)^4 \left (c (d+e x)^n\right )^{-4/n} \text{Erfi}\left (\frac{2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{3 b^{5/2} e^4 n^{5/2}}+\frac{4 (d+e x) (f+g x)^2 (e f-d g)}{b^2 e^2 n^2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}-\frac{16 (d+e x) (f+g x)^3}{3 b^2 e n^2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}-\frac{2 (d+e x) (f+g x)^3}{3 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}} \]

[Out]

(4*(e*f - d*g)^3*Sqrt[Pi]*(d + e*x)*Erfi[Sqrt[a + b*Log[c*(d + e*x)^n]]/(Sqrt[b]*Sqrt[n])])/(3*b^(5/2)*e^4*E^(
a/(b*n))*n^(5/2)*(c*(d + e*x)^n)^n^(-1)) + (32*g^3*Sqrt[Pi]*(d + e*x)^4*Erfi[(2*Sqrt[a + b*Log[c*(d + e*x)^n]]
)/(Sqrt[b]*Sqrt[n])])/(3*b^(5/2)*e^4*E^((4*a)/(b*n))*n^(5/2)*(c*(d + e*x)^n)^(4/n)) + (8*g*(e*f - d*g)^2*Sqrt[
2*Pi]*(d + e*x)^2*Erfi[(Sqrt[2]*Sqrt[a + b*Log[c*(d + e*x)^n]])/(Sqrt[b]*Sqrt[n])])/(b^(5/2)*e^4*E^((2*a)/(b*n
))*n^(5/2)*(c*(d + e*x)^n)^(2/n)) + (12*g^2*(e*f - d*g)*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^4*E^((3*a)/(b*n))*n^(5/2)*(c*(d + e*x)^n)^(3/n)) - (2*(d + e*x)*
(f + g*x)^3)/(3*b*e*n*(a + b*Log[c*(d + e*x)^n])^(3/2)) + (4*(e*f - d*g)*(d + e*x)*(f + g*x)^2)/(b^2*e^2*n^2*S
qrt[a + b*Log[c*(d + e*x)^n]]) - (16*(d + e*x)*(f + g*x)^3)/(3*b^2*e*n^2*Sqrt[a + b*Log[c*(d + e*x)^n]])

________________________________________________________________________________________

Rubi [A]  time = 2.33263, antiderivative size = 520, normalized size of antiderivative = 1., number of steps used = 59, 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} \[ \frac{12 \sqrt{3 \pi } g^2 e^{-\frac{3 a}{b n}} (d+e x)^3 (e f-d g) \left (c (d+e x)^n\right )^{-3/n} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{b^{5/2} e^4 n^{5/2}}+\frac{8 \sqrt{2 \pi } g e^{-\frac{2 a}{b n}} (d+e x)^2 (e f-d g)^2 \left (c (d+e x)^n\right )^{-2/n} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{b^{5/2} e^4 n^{5/2}}+\frac{4 \sqrt{\pi } e^{-\frac{a}{b n}} (d+e x) (e f-d g)^3 \left (c (d+e x)^n\right )^{-1/n} \text{Erfi}\left (\frac{\sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{3 b^{5/2} e^4 n^{5/2}}+\frac{32 \sqrt{\pi } g^3 e^{-\frac{4 a}{b n}} (d+e x)^4 \left (c (d+e x)^n\right )^{-4/n} \text{Erfi}\left (\frac{2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{3 b^{5/2} e^4 n^{5/2}}+\frac{4 (d+e x) (f+g x)^2 (e f-d g)}{b^2 e^2 n^2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}-\frac{16 (d+e x) (f+g x)^3}{3 b^2 e n^2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}-\frac{2 (d+e x) (f+g x)^3}{3 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[(f + g*x)^3/(a + b*Log[c*(d + e*x)^n])^(5/2),x]

[Out]

(4*(e*f - d*g)^3*Sqrt[Pi]*(d + e*x)*Erfi[Sqrt[a + b*Log[c*(d + e*x)^n]]/(Sqrt[b]*Sqrt[n])])/(3*b^(5/2)*e^4*E^(
a/(b*n))*n^(5/2)*(c*(d + e*x)^n)^n^(-1)) + (32*g^3*Sqrt[Pi]*(d + e*x)^4*Erfi[(2*Sqrt[a + b*Log[c*(d + e*x)^n]]
)/(Sqrt[b]*Sqrt[n])])/(3*b^(5/2)*e^4*E^((4*a)/(b*n))*n^(5/2)*(c*(d + e*x)^n)^(4/n)) + (8*g*(e*f - d*g)^2*Sqrt[
2*Pi]*(d + e*x)^2*Erfi[(Sqrt[2]*Sqrt[a + b*Log[c*(d + e*x)^n]])/(Sqrt[b]*Sqrt[n])])/(b^(5/2)*e^4*E^((2*a)/(b*n
))*n^(5/2)*(c*(d + e*x)^n)^(2/n)) + (12*g^2*(e*f - d*g)*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^4*E^((3*a)/(b*n))*n^(5/2)*(c*(d + e*x)^n)^(3/n)) - (2*(d + e*x)*
(f + g*x)^3)/(3*b*e*n*(a + b*Log[c*(d + e*x)^n])^(3/2)) + (4*(e*f - d*g)*(d + e*x)*(f + g*x)^2)/(b^2*e^2*n^2*S
qrt[a + b*Log[c*(d + e*x)^n]]) - (16*(d + e*x)*(f + g*x)^3)/(3*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)^3}{\left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}} \, dx &=-\frac{2 (d+e x) (f+g x)^3}{3 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}+\frac{8 \int \frac{(f+g x)^3}{\left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}} \, dx}{3 b n}-\frac{(2 (e f-d g)) \int \frac{(f+g x)^2}{\left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}} \, dx}{b e n}\\ &=-\frac{2 (d+e x) (f+g x)^3}{3 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}+\frac{4 (e f-d g) (d+e x) (f+g x)^2}{b^2 e^2 n^2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}-\frac{16 (d+e x) (f+g x)^3}{3 b^2 e n^2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}+\frac{64 \int \frac{(f+g x)^3}{\sqrt{a+b \log \left (c (d+e x)^n\right )}} \, dx}{3 b^2 n^2}-\frac{(12 (e f-d g)) \int \frac{(f+g x)^2}{\sqrt{a+b \log \left (c (d+e x)^n\right )}} \, dx}{b^2 e n^2}-\frac{(16 (e f-d g)) \int \frac{(f+g x)^2}{\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{f+g x}{\sqrt{a+b \log \left (c (d+e x)^n\right )}} \, dx}{b^2 e^2 n^2}\\ &=-\frac{2 (d+e x) (f+g x)^3}{3 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}+\frac{4 (e f-d g) (d+e x) (f+g x)^2}{b^2 e^2 n^2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}-\frac{16 (d+e x) (f+g x)^3}{3 b^2 e n^2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}+\frac{64 \int \left (\frac{(e f-d g)^3}{e^3 \sqrt{a+b \log \left (c (d+e x)^n\right )}}+\frac{3 g (e f-d g)^2 (d+e x)}{e^3 \sqrt{a+b \log \left (c (d+e x)^n\right )}}+\frac{3 g^2 (e f-d g) (d+e x)^2}{e^3 \sqrt{a+b \log \left (c (d+e x)^n\right )}}+\frac{g^3 (d+e x)^3}{e^3 \sqrt{a+b \log \left (c (d+e x)^n\right )}}\right ) \, dx}{3 b^2 n^2}-\frac{(12 (e f-d g)) \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 e n^2}-\frac{(16 (e f-d g)) \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 e n^2}+\frac{\left (8 (e f-d g)^2\right ) \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^2 n^2}\\ &=-\frac{2 (d+e x) (f+g x)^3}{3 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}+\frac{4 (e f-d g) (d+e x) (f+g x)^2}{b^2 e^2 n^2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}-\frac{16 (d+e x) (f+g x)^3}{3 b^2 e n^2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}+\frac{\left (64 g^3\right ) \int \frac{(d+e x)^3}{\sqrt{a+b \log \left (c (d+e x)^n\right )}} \, dx}{3 b^2 e^3 n^2}-\frac{\left (12 g^2 (e f-d g)\right ) \int \frac{(d+e x)^2}{\sqrt{a+b \log \left (c (d+e x)^n\right )}} \, dx}{b^2 e^3 n^2}-\frac{\left (16 g^2 (e f-d g)\right ) \int \frac{(d+e x)^2}{\sqrt{a+b \log \left (c (d+e x)^n\right )}} \, dx}{b^2 e^3 n^2}+\frac{\left (64 g^2 (e f-d g)\right ) \int \frac{(d+e x)^2}{\sqrt{a+b \log \left (c (d+e x)^n\right )}} \, dx}{b^2 e^3 n^2}+\frac{\left (8 g (e f-d g)^2\right ) \int \frac{d+e x}{\sqrt{a+b \log \left (c (d+e x)^n\right )}} \, dx}{b^2 e^3 n^2}-\frac{\left (24 g (e f-d g)^2\right ) \int \frac{d+e x}{\sqrt{a+b \log \left (c (d+e x)^n\right )}} \, dx}{b^2 e^3 n^2}-\frac{\left (32 g (e f-d g)^2\right ) \int \frac{d+e x}{\sqrt{a+b \log \left (c (d+e x)^n\right )}} \, dx}{b^2 e^3 n^2}+\frac{\left (64 g (e f-d g)^2\right ) \int \frac{d+e x}{\sqrt{a+b \log \left (c (d+e x)^n\right )}} \, dx}{b^2 e^3 n^2}+\frac{\left (8 (e f-d g)^3\right ) \int \frac{1}{\sqrt{a+b \log \left (c (d+e x)^n\right )}} \, dx}{b^2 e^3 n^2}-\frac{\left (12 (e f-d g)^3\right ) \int \frac{1}{\sqrt{a+b \log \left (c (d+e x)^n\right )}} \, dx}{b^2 e^3 n^2}-\frac{\left (16 (e f-d g)^3\right ) \int \frac{1}{\sqrt{a+b \log \left (c (d+e x)^n\right )}} \, dx}{b^2 e^3 n^2}+\frac{\left (64 (e f-d g)^3\right ) \int \frac{1}{\sqrt{a+b \log \left (c (d+e x)^n\right )}} \, dx}{3 b^2 e^3 n^2}\\ &=-\frac{2 (d+e x) (f+g x)^3}{3 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}+\frac{4 (e f-d g) (d+e x) (f+g x)^2}{b^2 e^2 n^2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}-\frac{16 (d+e x) (f+g x)^3}{3 b^2 e n^2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}+\frac{\left (64 g^3\right ) \operatorname{Subst}\left (\int \frac{x^3}{\sqrt{a+b \log \left (c x^n\right )}} \, dx,x,d+e x\right )}{3 b^2 e^4 n^2}-\frac{\left (12 g^2 (e f-d g)\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^4 n^2}-\frac{\left (16 g^2 (e f-d g)\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^4 n^2}+\frac{\left (64 g^2 (e f-d g)\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^4 n^2}+\frac{\left (8 g (e f-d g)^2\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{a+b \log \left (c x^n\right )}} \, dx,x,d+e x\right )}{b^2 e^4 n^2}-\frac{\left (24 g (e f-d g)^2\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{a+b \log \left (c x^n\right )}} \, dx,x,d+e x\right )}{b^2 e^4 n^2}-\frac{\left (32 g (e f-d g)^2\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{a+b \log \left (c x^n\right )}} \, dx,x,d+e x\right )}{b^2 e^4 n^2}+\frac{\left (64 g (e f-d g)^2\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{a+b \log \left (c x^n\right )}} \, dx,x,d+e x\right )}{b^2 e^4 n^2}+\frac{\left (8 (e f-d g)^3\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^4 n^2}-\frac{\left (12 (e f-d g)^3\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^4 n^2}-\frac{\left (16 (e f-d g)^3\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^4 n^2}+\frac{\left (64 (e f-d g)^3\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^4 n^2}\\ &=-\frac{2 (d+e x) (f+g x)^3}{3 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}+\frac{4 (e f-d g) (d+e x) (f+g x)^2}{b^2 e^2 n^2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}-\frac{16 (d+e x) (f+g x)^3}{3 b^2 e n^2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}+\frac{\left (64 g^3 (d+e x)^4 \left (c (d+e x)^n\right )^{-4/n}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{4 x}{n}}}{\sqrt{a+b x}} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{3 b^2 e^4 n^3}-\frac{\left (12 g^2 (e f-d g) (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^4 n^3}-\frac{\left (16 g^2 (e f-d g) (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^4 n^3}+\frac{\left (64 g^2 (e f-d g) (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^4 n^3}+\frac{\left (8 g (e f-d g)^2 (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^4 n^3}-\frac{\left (24 g (e f-d g)^2 (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^4 n^3}-\frac{\left (32 g (e f-d g)^2 (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^4 n^3}+\frac{\left (64 g (e f-d g)^2 (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^4 n^3}+\frac{\left (8 (e f-d g)^3 (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^4 n^3}-\frac{\left (12 (e f-d g)^3 (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^4 n^3}-\frac{\left (16 (e f-d g)^3 (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^4 n^3}+\frac{\left (64 (e f-d g)^3 (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^4 n^3}\\ &=-\frac{2 (d+e x) (f+g x)^3}{3 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}+\frac{4 (e f-d g) (d+e x) (f+g x)^2}{b^2 e^2 n^2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}-\frac{16 (d+e x) (f+g x)^3}{3 b^2 e n^2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}+\frac{\left (128 g^3 (d+e x)^4 \left (c (d+e x)^n\right )^{-4/n}\right ) \operatorname{Subst}\left (\int e^{-\frac{4 a}{b n}+\frac{4 x^2}{b n}} \, dx,x,\sqrt{a+b \log \left (c (d+e x)^n\right )}\right )}{3 b^3 e^4 n^3}-\frac{\left (24 g^2 (e f-d g) (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^4 n^3}-\frac{\left (32 g^2 (e f-d g) (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^4 n^3}+\frac{\left (128 g^2 (e f-d g) (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^4 n^3}+\frac{\left (16 g (e f-d g)^2 (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^4 n^3}-\frac{\left (48 g (e f-d g)^2 (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^4 n^3}-\frac{\left (64 g (e f-d g)^2 (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^4 n^3}+\frac{\left (128 g (e f-d g)^2 (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^4 n^3}+\frac{\left (16 (e f-d g)^3 (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^4 n^3}-\frac{\left (24 (e f-d g)^3 (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^4 n^3}-\frac{\left (32 (e f-d g)^3 (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^4 n^3}+\frac{\left (128 (e f-d g)^3 (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^4 n^3}\\ &=\frac{4 e^{-\frac{a}{b n}} (e f-d g)^3 \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^4 n^{5/2}}+\frac{32 e^{-\frac{4 a}{b n}} g^3 \sqrt{\pi } (d+e x)^4 \left (c (d+e x)^n\right )^{-4/n} \text{erfi}\left (\frac{2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{3 b^{5/2} e^4 n^{5/2}}+\frac{8 e^{-\frac{2 a}{b n}} g (e f-d g)^2 \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 )}{b^{5/2} e^4 n^{5/2}}+\frac{12 e^{-\frac{3 a}{b n}} g^2 (e f-d g) \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^4 n^{5/2}}-\frac{2 (d+e x) (f+g x)^3}{3 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}+\frac{4 (e f-d g) (d+e x) (f+g x)^2}{b^2 e^2 n^2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}-\frac{16 (d+e x) (f+g x)^3}{3 b^2 e n^2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}\\ \end{align*}

Mathematica [B]  time = 7.00728, size = 2647, normalized size = 5.09 \[ \text{Result too large to show} \]

Antiderivative was successfully verified.

[In]

Integrate[(f + g*x)^3/(a + b*Log[c*(d + e*x)^n])^(5/2),x]

[Out]

(8*f^2*g*Sqrt[Pi]*(-2*d*E^((a + b*(-(n*Log[d + e*x]) + Log[c*(d + e*x)^n]))/(b*n))*Erfi[Sqrt[a + b*n*Log[d + e
*x] + b*(-(n*Log[d + e*x]) + Log[c*(d + e*x)^n])]/(Sqrt[b]*Sqrt[n])] + Sqrt[2]*Erfi[(Sqrt[2]*Sqrt[a + b*n*Log[
d + e*x] + b*(-(n*Log[d + e*x]) + Log[c*(d + e*x)^n])])/(Sqrt[b]*Sqrt[n])])*Sqrt[a + b*Log[c*(d + e*x)^n]])/(b
^(5/2)*e^2*E^((2*(a + b*(-(n*Log[d + e*x]) + Log[c*(d + e*x)^n])))/(b*n))*n^(5/2)*Sqrt[a + b*n*Log[d + e*x] +
b*(-(n*Log[d + e*x]) + Log[c*(d + e*x)^n])]) + (20*d*f*g^2*Sqrt[Pi]*(-2*d*E^((a + b*(-(n*Log[d + e*x]) + Log[c
*(d + e*x)^n]))/(b*n))*Erfi[Sqrt[a + b*n*Log[d + e*x] + b*(-(n*Log[d + e*x]) + Log[c*(d + e*x)^n])]/(Sqrt[b]*S
qrt[n])] + Sqrt[2]*Erfi[(Sqrt[2]*Sqrt[a + b*n*Log[d + e*x] + b*(-(n*Log[d + e*x]) + Log[c*(d + e*x)^n])])/(Sqr
t[b]*Sqrt[n])])*Sqrt[a + b*Log[c*(d + e*x)^n]])/(b^(5/2)*e^3*E^((2*(a + b*(-(n*Log[d + e*x]) + Log[c*(d + e*x)
^n])))/(b*n))*n^(5/2)*Sqrt[a + b*n*Log[d + e*x] + b*(-(n*Log[d + e*x]) + Log[c*(d + e*x)^n])]) + (4*d^2*g^3*Sq
rt[Pi]*(-2*d*E^((a + b*(-(n*Log[d + e*x]) + Log[c*(d + e*x)^n]))/(b*n))*Erfi[Sqrt[a + b*n*Log[d + e*x] + b*(-(
n*Log[d + e*x]) + Log[c*(d + e*x)^n])]/(Sqrt[b]*Sqrt[n])] + Sqrt[2]*Erfi[(Sqrt[2]*Sqrt[a + b*n*Log[d + e*x] +
b*(-(n*Log[d + e*x]) + Log[c*(d + e*x)^n])])/(Sqrt[b]*Sqrt[n])])*Sqrt[a + b*Log[c*(d + e*x)^n]])/(b^(5/2)*e^4*
E^((2*(a + b*(-(n*Log[d + e*x]) + Log[c*(d + e*x)^n])))/(b*n))*n^(5/2)*Sqrt[a + b*n*Log[d + e*x] + b*(-(n*Log[
d + e*x]) + Log[c*(d + e*x)^n])]) + (12*f*g^2*Sqrt[Pi]*(3*d^2*E^((2*(a + b*(-(n*Log[d + e*x]) + Log[c*(d + e*x
)^n])))/(b*n))*Erfi[Sqrt[a + b*n*Log[d + e*x] + b*(-(n*Log[d + e*x]) + Log[c*(d + e*x)^n])]/(Sqrt[b]*Sqrt[n])]
- 3*Sqrt[2]*d*E^((a + b*(-(n*Log[d + e*x]) + Log[c*(d + e*x)^n]))/(b*n))*Erfi[(Sqrt[2]*Sqrt[a + b*n*Log[d + e
*x] + b*(-(n*Log[d + e*x]) + Log[c*(d + e*x)^n])])/(Sqrt[b]*Sqrt[n])] + Sqrt[3]*Erfi[(Sqrt[3]*Sqrt[a + b*n*Log
[d + e*x] + b*(-(n*Log[d + e*x]) + Log[c*(d + e*x)^n])])/(Sqrt[b]*Sqrt[n])])*Sqrt[a + b*Log[c*(d + e*x)^n]])/(
b^(5/2)*e^3*E^((3*(a + b*(-(n*Log[d + e*x]) + Log[c*(d + e*x)^n])))/(b*n))*n^(5/2)*Sqrt[a + b*n*Log[d + e*x] +
b*(-(n*Log[d + e*x]) + Log[c*(d + e*x)^n])]) + (28*d*g^3*Sqrt[Pi]*(3*d^2*E^((2*(a + b*(-(n*Log[d + e*x]) + Lo
g[c*(d + e*x)^n])))/(b*n))*Erfi[Sqrt[a + b*n*Log[d + e*x] + b*(-(n*Log[d + e*x]) + Log[c*(d + e*x)^n])]/(Sqrt[
b]*Sqrt[n])] - 3*Sqrt[2]*d*E^((a + b*(-(n*Log[d + e*x]) + Log[c*(d + e*x)^n]))/(b*n))*Erfi[(Sqrt[2]*Sqrt[a + b
*n*Log[d + e*x] + b*(-(n*Log[d + e*x]) + Log[c*(d + e*x)^n])])/(Sqrt[b]*Sqrt[n])] + Sqrt[3]*Erfi[(Sqrt[3]*Sqrt
[a + b*n*Log[d + e*x] + b*(-(n*Log[d + e*x]) + Log[c*(d + e*x)^n])])/(Sqrt[b]*Sqrt[n])])*Sqrt[a + b*Log[c*(d +
e*x)^n]])/(3*b^(5/2)*e^4*E^((3*(a + b*(-(n*Log[d + e*x]) + Log[c*(d + e*x)^n])))/(b*n))*n^(5/2)*Sqrt[a + b*n*
Log[d + e*x] + b*(-(n*Log[d + e*x]) + Log[c*(d + e*x)^n])]) + (32*g^3*Sqrt[Pi]*(-2*d^3*E^((3*(a + b*(-(n*Log[d
+ e*x]) + Log[c*(d + e*x)^n])))/(b*n))*Erfi[Sqrt[a + b*n*Log[d + e*x] + b*(-(n*Log[d + e*x]) + Log[c*(d + e*x
)^n])]/(Sqrt[b]*Sqrt[n])] + Erfi[(2*Sqrt[a + b*n*Log[d + e*x] + b*(-(n*Log[d + e*x]) + Log[c*(d + e*x)^n])])/(
Sqrt[b]*Sqrt[n])] + d*E^((a + b*(-(n*Log[d + e*x]) + Log[c*(d + e*x)^n]))/(b*n))*(3*Sqrt[2]*d*E^((a + b*(-(n*L
og[d + e*x]) + Log[c*(d + e*x)^n]))/(b*n))*Erfi[(Sqrt[2]*Sqrt[a + b*n*Log[d + e*x] + b*(-(n*Log[d + e*x]) + Lo
g[c*(d + e*x)^n])])/(Sqrt[b]*Sqrt[n])] - 2*Sqrt[3]*Erfi[(Sqrt[3]*Sqrt[a + b*n*Log[d + e*x] + b*(-(n*Log[d + e*
x]) + Log[c*(d + e*x)^n])])/(Sqrt[b]*Sqrt[n])]))*Sqrt[a + b*Log[c*(d + e*x)^n]])/(3*b^(5/2)*e^4*E^((4*(a + b*(
-(n*Log[d + e*x]) + Log[c*(d + e*x)^n])))/(b*n))*n^(5/2)*Sqrt[a + b*n*Log[d + e*x] + b*(-(n*Log[d + e*x]) + Lo
g[c*(d + e*x)^n])]) + (4*f^3*Gamma[1/2, -((a + b*n*Log[d + e*x] + b*(-(n*Log[d + e*x]) + Log[c*(d + e*x)^n]))/
(b*n))]*Sqrt[-((a + b*n*Log[d + e*x] + b*(-(n*Log[d + e*x]) + Log[c*(d + e*x)^n]))/(b*n))])/(3*b^2*e*E^((a + b
*(-(n*Log[d + e*x]) + Log[c*(d + e*x)^n]))/(b*n))*n^2*Sqrt[a + b*n*Log[d + e*x] + b*(-(n*Log[d + e*x]) + Log[c
*(d + e*x)^n])]) + (12*d*f^2*g*Gamma[1/2, -((a + b*n*Log[d + e*x] + b*(-(n*Log[d + e*x]) + Log[c*(d + e*x)^n])
)/(b*n))]*Sqrt[-((a + b*n*Log[d + e*x] + b*(-(n*Log[d + e*x]) + Log[c*(d + e*x)^n]))/(b*n))])/(b^2*e^2*E^((a +
b*(-(n*Log[d + e*x]) + Log[c*(d + e*x)^n]))/(b*n))*n^2*Sqrt[a + b*n*Log[d + e*x] + b*(-(n*Log[d + e*x]) + Log
[c*(d + e*x)^n])]) + (8*d^2*f*g^2*Gamma[1/2, -((a + b*n*Log[d + e*x] + b*(-(n*Log[d + e*x]) + Log[c*(d + e*x)^
n]))/(b*n))]*Sqrt[-((a + b*n*Log[d + e*x] + b*(-(n*Log[d + e*x]) + Log[c*(d + e*x)^n]))/(b*n))])/(b^2*e^3*E^((
a + b*(-(n*Log[d + e*x]) + Log[c*(d + e*x)^n]))/(b*n))*n^2*Sqrt[a + b*n*Log[d + e*x] + b*(-(n*Log[d + e*x]) +
Log[c*(d + e*x)^n])]) + Sqrt[a + b*n*Log[d + e*x] + b*(-(n*Log[d + e*x]) + Log[c*(d + e*x)^n])]*((-2*(d + e*x)
*(f + g*x)^3)/(3*b*e*n*(a + b*n*Log[d + e*x] + b*(-(n*Log[d + e*x]) + Log[c*(d + e*x)^n]))^2) - (4*(d + e*x)*(
f + g*x)^2*(e*f + 3*d*g + 4*e*g*x))/(3*b^2*e^2*n^2*(a + b*n*Log[d + e*x] + b*(-(n*Log[d + e*x]) + Log[c*(d + e
*x)^n]))))

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Maple [F]  time = 0.781, size = 0, normalized size = 0. \begin{align*} \int{ \left ( gx+f \right ) ^{3} \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)^3/(a+b*ln(c*(e*x+d)^n))^(5/2),x)

[Out]

int((g*x+f)^3/(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 )}^{3}}{{\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)^3/(a+b*log(c*(e*x+d)^n))^(5/2),x, algorithm="maxima")

[Out]

integrate((g*x + f)^3/(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)^3/(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 )^{3}}{\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)**3/(a+b*ln(c*(e*x+d)**n))**(5/2),x)

[Out]

Integral((f + g*x)**3/(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 )}^{3}}{{\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)^3/(a+b*log(c*(e*x+d)^n))^(5/2),x, algorithm="giac")

[Out]

integrate((g*x + f)^3/(b*log((e*x + d)^n*c) + a)^(5/2), x)