\(\int \frac {(f+g x)^3}{(a+b \log (c (d+e x)^n))^{5/2}} \, dx\) [133]

Optimal result

Integrand size = 26, antiderivative size = 520 \[ \int \frac {(f+g x)^3}{\left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}} \, dx=\frac {4 e^{-\frac {a}{b n}} (e f-d g)^3 \sqrt {\pi } (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{3 b^{5/2} e^4 n^{5/2}}+\frac {32 e^{-\frac {4 a}{b n}} g^3 \sqrt {\pi } (d+e x)^4 \left (c (d+e x)^n\right )^{-4/n} \text {erfi}\left (\frac {2 \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{3 b^{5/2} e^4 n^{5/2}}+\frac {8 e^{-\frac {2 a}{b n}} g (e f-d g)^2 \sqrt {2 \pi } (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{b^{5/2} e^4 n^{5/2}}+\frac {12 e^{-\frac {3 a}{b n}} g^2 (e f-d g) \sqrt {3 \pi } (d+e x)^3 \left (c (d+e x)^n\right )^{-3/n} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{b^{5/2} e^4 n^{5/2}}-\frac {2 (d+e x) (f+g x)^3}{3 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}+\frac {4 (e f-d g) (d+e x) (f+g x)^2}{b^2 e^2 n^2 \sqrt {a+b \log \left (c (d+e x)^n\right )}}-\frac {16 (d+e x) (f+g x)^3}{3 b^2 e n^2 \sqrt {a+b \log \left (c (d+e x)^n\right )}} \]

[Out]

-2/3*(e*x+d)*(g*x+f)^3/b/e/n/(a+b*ln(c*(e*x+d)^n))^(3/2)+4/3*(-d*g+e*f)^3*(e*x+d)*erfi((a+b*ln(c*(e*x+d)^n))^(
1/2)/b^(1/2)/n^(1/2))*Pi^(1/2)/b^(5/2)/e^4/exp(a/b/n)/n^(5/2)/((c*(e*x+d)^n)^(1/n))+32/3*g^3*(e*x+d)^4*erfi(2*
(a+b*ln(c*(e*x+d)^n))^(1/2)/b^(1/2)/n^(1/2))*Pi^(1/2)/b^(5/2)/e^4/exp(4*a/b/n)/n^(5/2)/((c*(e*x+d)^n)^(4/n))+8
*g*(-d*g+e*f)^2*(e*x+d)^2*erfi(2^(1/2)*(a+b*ln(c*(e*x+d)^n))^(1/2)/b^(1/2)/n^(1/2))*2^(1/2)*Pi^(1/2)/b^(5/2)/e
^4/exp(2*a/b/n)/n^(5/2)/((c*(e*x+d)^n)^(2/n))+12*g^2*(-d*g+e*f)*(e*x+d)^3*erfi(3^(1/2)*(a+b*ln(c*(e*x+d)^n))^(
1/2)/b^(1/2)/n^(1/2))*3^(1/2)*Pi^(1/2)/b^(5/2)/e^4/exp(3*a/b/n)/n^(5/2)/((c*(e*x+d)^n)^(3/n))+4*(-d*g+e*f)*(e*
x+d)*(g*x+f)^2/b^2/e^2/n^2/(a+b*ln(c*(e*x+d)^n))^(1/2)-16/3*(e*x+d)*(g*x+f)^3/b^2/e/n^2/(a+b*ln(c*(e*x+d)^n))^
(1/2)

Rubi [A] (verified)

Time = 1.62 (sec) , antiderivative size = 520, normalized size of antiderivative = 1.00, number of steps used = 59, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {2447, 2448, 2436, 2337, 2211, 2235, 2437, 2347} \[ \int \frac {(f+g x)^3}{\left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}} \, dx=\frac {12 \sqrt {3 \pi } g^2 e^{-\frac {3 a}{b n}} (d+e x)^3 (e f-d g) \left (c (d+e x)^n\right )^{-3/n} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{b^{5/2} e^4 n^{5/2}}+\frac {8 \sqrt {2 \pi } g e^{-\frac {2 a}{b n}} (d+e x)^2 (e f-d g)^2 \left (c (d+e x)^n\right )^{-2/n} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{b^{5/2} e^4 n^{5/2}}+\frac {4 \sqrt {\pi } e^{-\frac {a}{b n}} (d+e x) (e f-d g)^3 \left (c (d+e x)^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{3 b^{5/2} e^4 n^{5/2}}+\frac {32 \sqrt {\pi } g^3 e^{-\frac {4 a}{b n}} (d+e x)^4 \left (c (d+e x)^n\right )^{-4/n} \text {erfi}\left (\frac {2 \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{3 b^{5/2} e^4 n^{5/2}}+\frac {4 (d+e x) (f+g x)^2 (e f-d g)}{b^2 e^2 n^2 \sqrt {a+b \log \left (c (d+e x)^n\right )}}-\frac {16 (d+e x) (f+g x)^3}{3 b^2 e n^2 \sqrt {a+b \log \left (c (d+e x)^n\right )}}-\frac {2 (d+e x) (f+g x)^3}{3 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}} \]

[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 2211

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] &&  !TrueQ[$UseGamma]

Rule 2235

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 2337

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 2347

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)/n)*x)*(a + b*x)^p, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, d, m, n, p}
, x]

Rule 2436

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 2437

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 2447

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 2448

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]

Rubi steps \begin{align*} \text {integral}& = -\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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1523\) vs. \(2(520)=1040\).

Time = 6.12 (sec) , antiderivative size = 1523, normalized size of antiderivative = 2.93 \[ \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) \left (-24 d e^2 e^{-\frac {a}{b n}} f^2 g \sqrt {\pi } \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 )-6 d^2 e e^{-\frac {a}{b n}} f g^2 \sqrt {\pi } \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 )-2 d^3 e^{-\frac {a}{b n}} g^3 \sqrt {\pi } \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 )+16 e^{-\frac {4 a}{b n}} g^3 \sqrt {\pi } (d+e x)^3 \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 )+12 e^2 e^{-\frac {2 a}{b n}} f^2 g \sqrt {2 \pi } (d+e x) \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 )+30 d e e^{-\frac {2 a}{b n}} f g^2 \sqrt {2 \pi } (d+e x) \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 )+6 d^2 e^{-\frac {2 a}{b n}} g^3 \sqrt {2 \pi } (d+e x) \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 )+16 d e^{-\frac {3 a}{b n}} g^3 \sqrt {\pi } (d+e x) \left (c (d+e x)^n\right )^{-3/n} \left (3 \sqrt {2} d e^{\frac {a}{b n}} \left (c (d+e x)^n\right )^{\frac {1}{n}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )-2 \sqrt {3} (d+e x) \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )\right )-14 d e^{-\frac {3 a}{b n}} g^3 \sqrt {\pi } (d+e x) \left (c (d+e x)^n\right )^{-3/n} \left (3 \sqrt {2} d e^{\frac {a}{b n}} \left (c (d+e x)^n\right )^{\frac {1}{n}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )-\sqrt {3} (d+e x) \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )\right )+18 e e^{-\frac {3 a}{b n}} f g^2 \sqrt {\pi } (d+e x) \left (c (d+e x)^n\right )^{-3/n} \left (-3 \sqrt {2} d e^{\frac {a}{b n}} \left (c (d+e x)^n\right )^{\frac {1}{n}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )+\sqrt {3} (d+e x) \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )\right )-\frac {b^{3/2} e^3 n^{3/2} (f+g x)^3}{\left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}-\frac {2 a \sqrt {b} e^2 \sqrt {n} (f+g x)^2 (3 d g+e (f+4 g x))}{\left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}-\frac {2 b^{3/2} e^2 \sqrt {n} (f+g x)^2 (3 d g+e (f+4 g x)) \log \left (c (d+e x)^n\right )}{\left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}+\frac {2 \sqrt {b} e^3 e^{-\frac {a}{b n}} f^3 \sqrt {n} \left (c (d+e x)^n\right )^{-1/n} \Gamma \left (\frac {1}{2},-\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right ) \sqrt {-\frac {a+b \log \left (c (d+e x)^n\right )}{b n}}}{\sqrt {a+b \log \left (c (d+e x)^n\right )}}+\frac {18 \sqrt {b} d e^2 e^{-\frac {a}{b n}} f^2 g \sqrt {n} \left (c (d+e x)^n\right )^{-1/n} \Gamma \left (\frac {1}{2},-\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right ) \sqrt {-\frac {a+b \log \left (c (d+e x)^n\right )}{b n}}}{\sqrt {a+b \log \left (c (d+e x)^n\right )}}+\frac {12 \sqrt {b} d^2 e e^{-\frac {a}{b n}} f g^2 \sqrt {n} \left (c (d+e x)^n\right )^{-1/n} \Gamma \left (\frac {1}{2},-\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right ) \sqrt {-\frac {a+b \log \left (c (d+e x)^n\right )}{b n}}}{\sqrt {a+b \log \left (c (d+e x)^n\right )}}\right )}{3 b^{5/2} e^4 n^{5/2}} \]

[In]

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

[Out]

(2*(d + e*x)*((-24*d*e^2*f^2*g*Sqrt[Pi]*Erfi[Sqrt[a + b*Log[c*(d + e*x)^n]]/(Sqrt[b]*Sqrt[n])])/(E^(a/(b*n))*(
c*(d + e*x)^n)^n^(-1)) - (6*d^2*e*f*g^2*Sqrt[Pi]*Erfi[Sqrt[a + b*Log[c*(d + e*x)^n]]/(Sqrt[b]*Sqrt[n])])/(E^(a
/(b*n))*(c*(d + e*x)^n)^n^(-1)) - (2*d^3*g^3*Sqrt[Pi]*Erfi[Sqrt[a + b*Log[c*(d + e*x)^n]]/(Sqrt[b]*Sqrt[n])])/
(E^(a/(b*n))*(c*(d + e*x)^n)^n^(-1)) + (16*g^3*Sqrt[Pi]*(d + e*x)^3*Erfi[(2*Sqrt[a + b*Log[c*(d + e*x)^n]])/(S
qrt[b]*Sqrt[n])])/(E^((4*a)/(b*n))*(c*(d + e*x)^n)^(4/n)) + (12*e^2*f^2*g*Sqrt[2*Pi]*(d + e*x)*Erfi[(Sqrt[2]*S
qrt[a + b*Log[c*(d + e*x)^n]])/(Sqrt[b]*Sqrt[n])])/(E^((2*a)/(b*n))*(c*(d + e*x)^n)^(2/n)) + (30*d*e*f*g^2*Sqr
t[2*Pi]*(d + e*x)*Erfi[(Sqrt[2]*Sqrt[a + b*Log[c*(d + e*x)^n]])/(Sqrt[b]*Sqrt[n])])/(E^((2*a)/(b*n))*(c*(d + e
*x)^n)^(2/n)) + (6*d^2*g^3*Sqrt[2*Pi]*(d + e*x)*Erfi[(Sqrt[2]*Sqrt[a + b*Log[c*(d + e*x)^n]])/(Sqrt[b]*Sqrt[n]
)])/(E^((2*a)/(b*n))*(c*(d + e*x)^n)^(2/n)) + (16*d*g^3*Sqrt[Pi]*(d + e*x)*(3*Sqrt[2]*d*E^(a/(b*n))*(c*(d + e*
x)^n)^n^(-1)*Erfi[(Sqrt[2]*Sqrt[a + b*Log[c*(d + e*x)^n]])/(Sqrt[b]*Sqrt[n])] - 2*Sqrt[3]*(d + e*x)*Erfi[(Sqrt
[3]*Sqrt[a + b*Log[c*(d + e*x)^n]])/(Sqrt[b]*Sqrt[n])]))/(E^((3*a)/(b*n))*(c*(d + e*x)^n)^(3/n)) - (14*d*g^3*S
qrt[Pi]*(d + e*x)*(3*Sqrt[2]*d*E^(a/(b*n))*(c*(d + e*x)^n)^n^(-1)*Erfi[(Sqrt[2]*Sqrt[a + b*Log[c*(d + e*x)^n]]
)/(Sqrt[b]*Sqrt[n])] - Sqrt[3]*(d + e*x)*Erfi[(Sqrt[3]*Sqrt[a + b*Log[c*(d + e*x)^n]])/(Sqrt[b]*Sqrt[n])]))/(E
^((3*a)/(b*n))*(c*(d + e*x)^n)^(3/n)) + (18*e*f*g^2*Sqrt[Pi]*(d + e*x)*(-3*Sqrt[2]*d*E^(a/(b*n))*(c*(d + e*x)^
n)^n^(-1)*Erfi[(Sqrt[2]*Sqrt[a + b*Log[c*(d + e*x)^n]])/(Sqrt[b]*Sqrt[n])] + Sqrt[3]*(d + e*x)*Erfi[(Sqrt[3]*S
qrt[a + b*Log[c*(d + e*x)^n]])/(Sqrt[b]*Sqrt[n])]))/(E^((3*a)/(b*n))*(c*(d + e*x)^n)^(3/n)) - (b^(3/2)*e^3*n^(
3/2)*(f + g*x)^3)/(a + b*Log[c*(d + e*x)^n])^(3/2) - (2*a*Sqrt[b]*e^2*Sqrt[n]*(f + g*x)^2*(3*d*g + e*(f + 4*g*
x)))/(a + b*Log[c*(d + e*x)^n])^(3/2) - (2*b^(3/2)*e^2*Sqrt[n]*(f + g*x)^2*(3*d*g + e*(f + 4*g*x))*Log[c*(d +
e*x)^n])/(a + b*Log[c*(d + e*x)^n])^(3/2) + (2*Sqrt[b]*e^3*f^3*Sqrt[n]*Gamma[1/2, -((a + b*Log[c*(d + e*x)^n])
/(b*n))]*Sqrt[-((a + b*Log[c*(d + e*x)^n])/(b*n))])/(E^(a/(b*n))*(c*(d + e*x)^n)^n^(-1)*Sqrt[a + b*Log[c*(d +
e*x)^n]]) + (18*Sqrt[b]*d*e^2*f^2*g*Sqrt[n]*Gamma[1/2, -((a + b*Log[c*(d + e*x)^n])/(b*n))]*Sqrt[-((a + b*Log[
c*(d + e*x)^n])/(b*n))])/(E^(a/(b*n))*(c*(d + e*x)^n)^n^(-1)*Sqrt[a + b*Log[c*(d + e*x)^n]]) + (12*Sqrt[b]*d^2
*e*f*g^2*Sqrt[n]*Gamma[1/2, -((a + b*Log[c*(d + e*x)^n])/(b*n))]*Sqrt[-((a + b*Log[c*(d + e*x)^n])/(b*n))])/(E
^(a/(b*n))*(c*(d + e*x)^n)^n^(-1)*Sqrt[a + b*Log[c*(d + e*x)^n]])))/(3*b^(5/2)*e^4*n^(5/2))

Maple [F]

\[\int \frac {\left (g x +f \right )^{3}}{{\left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )}^{\frac {5}{2}}}d x\]

[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)

Fricas [F(-2)]

Exception generated. \[ \int \frac {(f+g x)^3}{\left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

Sympy [F]

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

[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)

Maxima [F]

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

[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)

Giac [F]

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

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {(f+g x)^3}{\left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}} \, dx=\int \frac {{\left (f+g\,x\right )}^3}{{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^{5/2}} \,d x \]

[In]

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

[Out]

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