\(\int \frac {(f+g x)^2}{(a+b \log (c (d+e x)^n))^{3/2}} \, dx\) [129]

Optimal result

Integrand size = 26, antiderivative size = 325 \[ \int \frac {(f+g x)^2}{\left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}} \, dx=\frac {2 e^{-\frac {a}{b n}} (e f-d g)^2 \sqrt {\pi } (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{b^{3/2} e^3 n^{3/2}}+\frac {4 e^{-\frac {2 a}{b n}} g (e f-d g) \sqrt {2 \pi } (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{b^{3/2} e^3 n^{3/2}}+\frac {2 e^{-\frac {3 a}{b n}} g^2 \sqrt {3 \pi } (d+e x)^3 \left (c (d+e x)^n\right )^{-3/n} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{b^{3/2} e^3 n^{3/2}}-\frac {2 (d+e x) (f+g x)^2}{b e n \sqrt {a+b \log \left (c (d+e x)^n\right )}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.61 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.00, number of steps used = 25, 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} \[ \int \frac {(f+g x)^2}{\left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}} \, dx=\frac {4 \sqrt {2 \pi } g e^{-\frac {2 a}{b n}} (d+e x)^2 (e f-d g) \left (c (d+e x)^n\right )^{-2/n} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{b^{3/2} e^3 n^{3/2}}+\frac {2 \sqrt {\pi } e^{-\frac {a}{b n}} (d+e x) (e f-d g)^2 \left (c (d+e x)^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{b^{3/2} e^3 n^{3/2}}+\frac {2 \sqrt {3 \pi } g^2 e^{-\frac {3 a}{b n}} (d+e x)^3 \left (c (d+e x)^n\right )^{-3/n} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{b^{3/2} e^3 n^{3/2}}-\frac {2 (d+e x) (f+g x)^2}{b e n \sqrt {a+b \log \left (c (d+e x)^n\right )}} \]

[In]

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

[Out]

(2*(e*f - d*g)^2*Sqrt[Pi]*(d + e*x)*Erfi[Sqrt[a + b*Log[c*(d + e*x)^n]]/(Sqrt[b]*Sqrt[n])])/(b^(3/2)*e^3*E^(a/
(b*n))*n^(3/2)*(c*(d + e*x)^n)^n^(-1)) + (4*g*(e*f - d*g)*Sqrt[2*Pi]*(d + e*x)^2*Erfi[(Sqrt[2]*Sqrt[a + b*Log[
c*(d + e*x)^n]])/(Sqrt[b]*Sqrt[n])])/(b^(3/2)*e^3*E^((2*a)/(b*n))*n^(3/2)*(c*(d + e*x)^n)^(2/n)) + (2*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^(3/2)*e^3*E^((3*a)/(b*
n))*n^(3/2)*(c*(d + e*x)^n)^(3/n)) - (2*(d + e*x)*(f + g*x)^2)/(b*e*n*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)^2}{b e n \sqrt {a+b \log \left (c (d+e x)^n\right )}}+\frac {6 \int \frac {(f+g x)^2}{\sqrt {a+b \log \left (c (d+e x)^n\right )}} \, dx}{b n}-\frac {(4 (e f-d g)) \int \frac {f+g x}{\sqrt {a+b \log \left (c (d+e x)^n\right )}} \, dx}{b e n} \\ & = -\frac {2 (d+e x) (f+g x)^2}{b e n \sqrt {a+b \log \left (c (d+e x)^n\right )}}+\frac {6 \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 n}-\frac {(4 (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 e n} \\ & = -\frac {2 (d+e x) (f+g x)^2}{b e n \sqrt {a+b \log \left (c (d+e x)^n\right )}}+\frac {\left (6 g^2\right ) \int \frac {(d+e x)^2}{\sqrt {a+b \log \left (c (d+e x)^n\right )}} \, dx}{b e^2 n}-\frac {(4 g (e f-d g)) \int \frac {d+e x}{\sqrt {a+b \log \left (c (d+e x)^n\right )}} \, dx}{b e^2 n}+\frac {(12 g (e f-d g)) \int \frac {d+e x}{\sqrt {a+b \log \left (c (d+e x)^n\right )}} \, dx}{b e^2 n}-\frac {\left (4 (e f-d g)^2\right ) \int \frac {1}{\sqrt {a+b \log \left (c (d+e x)^n\right )}} \, dx}{b e^2 n}+\frac {\left (6 (e f-d g)^2\right ) \int \frac {1}{\sqrt {a+b \log \left (c (d+e x)^n\right )}} \, dx}{b e^2 n} \\ & = -\frac {2 (d+e x) (f+g x)^2}{b e n \sqrt {a+b \log \left (c (d+e x)^n\right )}}+\frac {\left (6 g^2\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {a+b \log \left (c x^n\right )}} \, dx,x,d+e x\right )}{b e^3 n}-\frac {(4 g (e f-d g)) \text {Subst}\left (\int \frac {x}{\sqrt {a+b \log \left (c x^n\right )}} \, dx,x,d+e x\right )}{b e^3 n}+\frac {(12 g (e f-d g)) \text {Subst}\left (\int \frac {x}{\sqrt {a+b \log \left (c x^n\right )}} \, dx,x,d+e x\right )}{b e^3 n}-\frac {\left (4 (e f-d g)^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b \log \left (c x^n\right )}} \, dx,x,d+e x\right )}{b e^3 n}+\frac {\left (6 (e f-d g)^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b \log \left (c x^n\right )}} \, dx,x,d+e x\right )}{b e^3 n} \\ & = -\frac {2 (d+e x) (f+g x)^2}{b e n \sqrt {a+b \log \left (c (d+e x)^n\right )}}+\frac {\left (6 g^2 (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 e^3 n^2}-\frac {\left (4 g (e f-d g) (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 e^3 n^2}+\frac {\left (12 g (e f-d g) (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 e^3 n^2}-\frac {\left (4 (e f-d g)^2 (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 e^3 n^2}+\frac {\left (6 (e f-d g)^2 (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 e^3 n^2} \\ & = -\frac {2 (d+e x) (f+g x)^2}{b e n \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 ) \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^2 e^3 n^2}-\frac {\left (8 g (e f-d g) (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^2 e^3 n^2}+\frac {\left (24 g (e f-d g) (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^2 e^3 n^2}-\frac {\left (8 (e f-d g)^2 (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^2 e^3 n^2}+\frac {\left (12 (e f-d g)^2 (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^2 e^3 n^2} \\ & = \frac {2 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 )}{b^{3/2} e^3 n^{3/2}}+\frac {4 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 )}{b^{3/2} e^3 n^{3/2}}+\frac {2 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^{3/2} e^3 n^{3/2}}-\frac {2 (d+e x) (f+g x)^2}{b e n \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. \(828\) vs. \(2(325)=650\).

Time = 0.70 (sec) , antiderivative size = 828, normalized size of antiderivative = 2.55 \[ \int \frac {(f+g x)^2}{\left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}} \, dx=\frac {2 \left (-\sqrt {b} d e^2 f^2 \sqrt {n}-\sqrt {b} e^3 f^2 \sqrt {n} x-2 \sqrt {b} d e^2 f g \sqrt {n} x-2 \sqrt {b} e^3 f g \sqrt {n} x^2-\sqrt {b} d e^2 g^2 \sqrt {n} x^2-\sqrt {b} e^3 g^2 \sqrt {n} x^3-4 d e e^{-\frac {a}{b n}} f g \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 ) \sqrt {a+b \log \left (c (d+e x)^n\right )}+d^2 e^{-\frac {a}{b n}} 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 ) \sqrt {a+b \log \left (c (d+e x)^n\right )}+2 e e^{-\frac {2 a}{b n}} f 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 ) \sqrt {a+b \log \left (c (d+e x)^n\right )}-2 d e^{-\frac {2 a}{b n}} 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 ) \sqrt {a+b \log \left (c (d+e x)^n\right )}+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 ) \sqrt {a+b \log \left (c (d+e x)^n\right )}+\sqrt {b} e^2 e^{-\frac {a}{b n}} f^2 \sqrt {n} (d+e x) \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}}+2 \sqrt {b} d e e^{-\frac {a}{b n}} f g \sqrt {n} (d+e x) \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}}\right )}{b^{3/2} e^3 n^{3/2} \sqrt {a+b \log \left (c (d+e x)^n\right )}} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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