3.2.28 \(\int \frac {(f+g x)^3}{(a+b \log (c (d+e x)^n))^{3/2}} \, dx\) [128]

Optimal. Leaf size=422 \[ \frac {2 e^{-\frac {a}{b n}} (e f-d g)^3 \sqrt {\pi } (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{b^{3/2} e^4 n^{3/2}}+\frac {4 e^{-\frac {4 a}{b n}} g^3 \sqrt {\pi } (d+e x)^4 \left (c (d+e x)^n\right )^{-4/n} \text {erfi}\left (\frac {2 \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{b^{3/2} e^4 n^{3/2}}+\frac {6 e^{-\frac {2 a}{b n}} g (e f-d g)^2 \sqrt {2 \pi } (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{b^{3/2} e^4 n^{3/2}}+\frac {6 e^{-\frac {3 a}{b n}} g^2 (e f-d g) \sqrt {3 \pi } (d+e x)^3 \left (c (d+e x)^n\right )^{-3/n} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{b^{3/2} e^4 n^{3/2}}-\frac {2 (d+e x) (f+g x)^3}{b e n \sqrt {a+b \log \left (c (d+e x)^n\right )}} \]

[Out]

2*(-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^(3/2)/e^4/exp(a/b/n)/n^(3/
2)/((c*(e*x+d)^n)^(1/n))+4*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^(3/2)/
e^4/exp(4*a/b/n)/n^(3/2)/((c*(e*x+d)^n)^(4/n))+6*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^(3/2)/e^4/exp(2*a/b/n)/n^(3/2)/((c*(e*x+d)^n)^(2/n))+6*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^(3/2)/e^4/exp(3*a/b/n)
/n^(3/2)/((c*(e*x+d)^n)^(3/n))-2*(e*x+d)*(g*x+f)^3/b/e/n/(a+b*ln(c*(e*x+d)^n))^(1/2)

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Rubi [A]
time = 0.96, antiderivative size = 422, normalized size of antiderivative = 1.00, number of steps used = 33, number of rules used = 8, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {2447, 2448, 2436, 2337, 2211, 2235, 2437, 2347} \begin {gather*} \frac {6 \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^{3/2} e^4 n^{3/2}}+\frac {6 \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^{3/2} e^4 n^{3/2}}+\frac {2 \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 )}{b^{3/2} e^4 n^{3/2}}+\frac {4 \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 )}{b^{3/2} e^4 n^{3/2}}-\frac {2 (d+e x) (f+g x)^3}{b e n \sqrt {a+b \log \left (c (d+e x)^n\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(e*f - d*g)^3*Sqrt[Pi]*(d + e*x)*Erfi[Sqrt[a + b*Log[c*(d + e*x)^n]]/(Sqrt[b]*Sqrt[n])])/(b^(3/2)*e^4*E^(a/
(b*n))*n^(3/2)*(c*(d + e*x)^n)^n^(-1)) + (4*g^3*Sqrt[Pi]*(d + e*x)^4*Erfi[(2*Sqrt[a + b*Log[c*(d + e*x)^n]])/(
Sqrt[b]*Sqrt[n])])/(b^(3/2)*e^4*E^((4*a)/(b*n))*n^(3/2)*(c*(d + e*x)^n)^(4/n)) + (6*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^(3/2)*e^4*E^((2*a)/(b*n))*n^
(3/2)*(c*(d + e*x)^n)^(2/n)) + (6*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^(3/2)*e^4*E^((3*a)/(b*n))*n^(3/2)*(c*(d + e*x)^n)^(3/n)) - (2*(d + e*x)*(f + g
*x)^3)/(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*} \int \frac {(f+g x)^3}{\left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}} \, dx &=-\frac {2 (d+e x) (f+g x)^3}{b e n \sqrt {a+b \log \left (c (d+e x)^n\right )}}+\frac {8 \int \frac {(f+g x)^3}{\sqrt {a+b \log \left (c (d+e x)^n\right )}} \, dx}{b n}-\frac {(6 (e f-d g)) \int \frac {(f+g x)^2}{\sqrt {a+b \log \left (c (d+e x)^n\right )}} \, dx}{b e n}\\ &=-\frac {2 (d+e x) (f+g x)^3}{b e n \sqrt {a+b \log \left (c (d+e x)^n\right )}}+\frac {8 \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}{b n}-\frac {(6 (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 e n}\\ &=-\frac {2 (d+e x) (f+g x)^3}{b e n \sqrt {a+b \log \left (c (d+e x)^n\right )}}+\frac {\left (8 g^3\right ) \int \frac {(d+e x)^3}{\sqrt {a+b \log \left (c (d+e x)^n\right )}} \, dx}{b e^3 n}-\frac {\left (6 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 e^3 n}+\frac {\left (24 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 e^3 n}-\frac {\left (12 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 e^3 n}+\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 e^3 n}-\frac {\left (6 (e f-d g)^3\right ) \int \frac {1}{\sqrt {a+b \log \left (c (d+e x)^n\right )}} \, dx}{b e^3 n}+\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 e^3 n}\\ &=-\frac {2 (d+e x) (f+g x)^3}{b e n \sqrt {a+b \log \left (c (d+e x)^n\right )}}+\frac {\left (8 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 )}{b e^4 n}-\frac {\left (6 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 e^4 n}+\frac {\left (24 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 e^4 n}-\frac {\left (12 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 e^4 n}+\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 e^4 n}-\frac {\left (6 (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 e^4 n}+\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 e^4 n}\\ &=-\frac {2 (d+e x) (f+g x)^3}{b e n \sqrt {a+b \log \left (c (d+e x)^n\right )}}+\frac {\left (8 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 )}{b e^4 n^2}-\frac {\left (6 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 e^4 n^2}+\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 \frac {e^{\frac {3 x}{n}}}{\sqrt {a+b x}} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{b e^4 n^2}-\frac {\left (12 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 e^4 n^2}+\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 e^4 n^2}-\frac {\left (6 (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 e^4 n^2}+\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 e^4 n^2}\\ &=-\frac {2 (d+e x) (f+g x)^3}{b e n \sqrt {a+b \log \left (c (d+e x)^n\right )}}+\frac {\left (16 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 )}{b^2 e^4 n^2}-\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 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^4 n^2}+\frac {\left (48 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^2 e^4 n^2}-\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 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^4 n^2}+\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^2 e^4 n^2}-\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 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^4 n^2}+\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^2 e^4 n^2}\\ &=\frac {2 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 )}{b^{3/2} e^4 n^{3/2}}+\frac {4 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 )}{b^{3/2} e^4 n^{3/2}}+\frac {6 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^{3/2} e^4 n^{3/2}}+\frac {6 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^{3/2} e^4 n^{3/2}}-\frac {2 (d+e x) (f+g x)^3}{b e n \sqrt {a+b \log \left (c (d+e x)^n\right )}}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(1667\) vs. \(2(422)=844\).
time = 5.48, size = 1667, normalized size = 3.95 \begin {gather*} \frac {2 (d+e x) \left (-3 e e^{-\frac {3 a}{b n}} f g^2 \sqrt {\pi } \left (c (d+e x)^n\right )^{-3/n} \left (\sqrt {3} (d+e x)^2-3 \sqrt {2} d e^{\frac {a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{\frac {1}{n}}+3 d^2 e^{\frac {2 a}{b n}} \left (c (d+e x)^n\right )^{2/n}-3 d^2 e^{\frac {2 a}{b n}} \left (c (d+e x)^n\right )^{2/n} \text {erf}\left (\sqrt {-\frac {a+b \log \left (c (d+e x)^n\right )}{b n}}\right )+3 \sqrt {2} d e^{\frac {a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{\frac {1}{n}} \text {erf}\left (\sqrt {2} \sqrt {-\frac {a+b \log \left (c (d+e x)^n\right )}{b n}}\right )-\sqrt {3} (d+e x)^2 \text {erf}\left (\sqrt {3} \sqrt {-\frac {a+b \log \left (c (d+e x)^n\right )}{b n}}\right )\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-d e^{-\frac {3 a}{b n}} g^3 \sqrt {\pi } \left (c (d+e x)^n\right )^{-3/n} \left (\sqrt {3} (d+e x)^2-3 \sqrt {2} d e^{\frac {a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{\frac {1}{n}}+3 d^2 e^{\frac {2 a}{b n}} \left (c (d+e x)^n\right )^{2/n}-3 d^2 e^{\frac {2 a}{b n}} \left (c (d+e x)^n\right )^{2/n} \text {erf}\left (\sqrt {-\frac {a+b \log \left (c (d+e x)^n\right )}{b n}}\right )+3 \sqrt {2} d e^{\frac {a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{\frac {1}{n}} \text {erf}\left (\sqrt {2} \sqrt {-\frac {a+b \log \left (c (d+e x)^n\right )}{b n}}\right )-\sqrt {3} (d+e x)^2 \text {erf}\left (\sqrt {3} \sqrt {-\frac {a+b \log \left (c (d+e x)^n\right )}{b n}}\right )\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-2 e^{-\frac {4 a}{b n}} g^3 \sqrt {\pi } \left (c (d+e x)^n\right )^{-4/n} \left ((d+e x)^3-2 \sqrt {3} d e^{\frac {a}{b n}} (d+e x)^2 \left (c (d+e x)^n\right )^{\frac {1}{n}}+3 \sqrt {2} d^2 e^{\frac {2 a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{2/n}-2 d^3 e^{\frac {3 a}{b n}} \left (c (d+e x)^n\right )^{3/n}+2 d^3 e^{\frac {3 a}{b n}} \left (c (d+e x)^n\right )^{3/n} \text {erf}\left (\sqrt {-\frac {a+b \log \left (c (d+e x)^n\right )}{b n}}\right )-(d+e x)^3 \text {erf}\left (2 \sqrt {-\frac {a+b \log \left (c (d+e x)^n\right )}{b n}}\right )-3 \sqrt {2} d^2 e^{\frac {2 a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{2/n} \text {erf}\left (\sqrt {2} \sqrt {-\frac {a+b \log \left (c (d+e x)^n\right )}{b n}}\right )+2 \sqrt {3} d e^{\frac {a}{b n}} (d+e x)^2 \left (c (d+e x)^n\right )^{\frac {1}{n}} \text {erf}\left (\sqrt {3} \sqrt {-\frac {a+b \log \left (c (d+e x)^n\right )}{b n}}\right )\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-e^3 e^{-\frac {a}{b n}} f^3 \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 ) \left (a+b \log \left (c (d+e x)^n\right )\right )-3 d e^2 e^{-\frac {a}{b n}} f^2 g \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 ) \left (a+b \log \left (c (d+e x)^n\right )\right )-b e^3 n (f+g x)^3 \sqrt {-\frac {a+b \log \left (c (d+e x)^n\right )}{b n}}-3 \sqrt {b} d e e^{-\frac {2 a}{b n}} f g^2 \sqrt {n} \sqrt {\pi } \left (c (d+e x)^n\right )^{-2/n} \left (2 d e^{\frac {a}{b n}} \left (c (d+e x)^n\right )^{\frac {1}{n}} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )-\sqrt {2} (d+e x) \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )\right ) \sqrt {a+b \log \left (c (d+e x)^n\right )} \sqrt {-\frac {a+b \log \left (c (d+e x)^n\right )}{b n}}+3 \sqrt {b} e^2 e^{-\frac {2 a}{b n}} f^2 g \sqrt {n} \sqrt {\pi } \left (c (d+e x)^n\right )^{-2/n} \left (-2 d e^{\frac {a}{b n}} \left (c (d+e x)^n\right )^{\frac {1}{n}} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )+\sqrt {2} (d+e x) \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )\right ) \sqrt {a+b \log \left (c (d+e x)^n\right )} \sqrt {-\frac {a+b \log \left (c (d+e x)^n\right )}{b n}}\right )}{b^2 e^4 n^2 \sqrt {a+b \log \left (c (d+e x)^n\right )} \sqrt {-\frac {a+b \log \left (c (d+e x)^n\right )}{b n}}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(d + e*x)*((-3*e*f*g^2*Sqrt[Pi]*(Sqrt[3]*(d + e*x)^2 - 3*Sqrt[2]*d*E^(a/(b*n))*(d + e*x)*(c*(d + e*x)^n)^n^
(-1) + 3*d^2*E^((2*a)/(b*n))*(c*(d + e*x)^n)^(2/n) - 3*d^2*E^((2*a)/(b*n))*(c*(d + e*x)^n)^(2/n)*Erf[Sqrt[-((a
+ b*Log[c*(d + e*x)^n])/(b*n))]] + 3*Sqrt[2]*d*E^(a/(b*n))*(d + e*x)*(c*(d + e*x)^n)^n^(-1)*Erf[Sqrt[2]*Sqrt[
-((a + b*Log[c*(d + e*x)^n])/(b*n))]] - Sqrt[3]*(d + e*x)^2*Erf[Sqrt[3]*Sqrt[-((a + b*Log[c*(d + e*x)^n])/(b*n
))]])*(a + b*Log[c*(d + e*x)^n]))/(E^((3*a)/(b*n))*(c*(d + e*x)^n)^(3/n)) - (d*g^3*Sqrt[Pi]*(Sqrt[3]*(d + e*x)
^2 - 3*Sqrt[2]*d*E^(a/(b*n))*(d + e*x)*(c*(d + e*x)^n)^n^(-1) + 3*d^2*E^((2*a)/(b*n))*(c*(d + e*x)^n)^(2/n) -
3*d^2*E^((2*a)/(b*n))*(c*(d + e*x)^n)^(2/n)*Erf[Sqrt[-((a + b*Log[c*(d + e*x)^n])/(b*n))]] + 3*Sqrt[2]*d*E^(a/
(b*n))*(d + e*x)*(c*(d + e*x)^n)^n^(-1)*Erf[Sqrt[2]*Sqrt[-((a + b*Log[c*(d + e*x)^n])/(b*n))]] - Sqrt[3]*(d +
e*x)^2*Erf[Sqrt[3]*Sqrt[-((a + b*Log[c*(d + e*x)^n])/(b*n))]])*(a + b*Log[c*(d + e*x)^n]))/(E^((3*a)/(b*n))*(c
*(d + e*x)^n)^(3/n)) - (2*g^3*Sqrt[Pi]*((d + e*x)^3 - 2*Sqrt[3]*d*E^(a/(b*n))*(d + e*x)^2*(c*(d + e*x)^n)^n^(-
1) + 3*Sqrt[2]*d^2*E^((2*a)/(b*n))*(d + e*x)*(c*(d + e*x)^n)^(2/n) - 2*d^3*E^((3*a)/(b*n))*(c*(d + e*x)^n)^(3/
n) + 2*d^3*E^((3*a)/(b*n))*(c*(d + e*x)^n)^(3/n)*Erf[Sqrt[-((a + b*Log[c*(d + e*x)^n])/(b*n))]] - (d + e*x)^3*
Erf[2*Sqrt[-((a + b*Log[c*(d + e*x)^n])/(b*n))]] - 3*Sqrt[2]*d^2*E^((2*a)/(b*n))*(d + e*x)*(c*(d + e*x)^n)^(2/
n)*Erf[Sqrt[2]*Sqrt[-((a + b*Log[c*(d + e*x)^n])/(b*n))]] + 2*Sqrt[3]*d*E^(a/(b*n))*(d + e*x)^2*(c*(d + e*x)^n
)^n^(-1)*Erf[Sqrt[3]*Sqrt[-((a + b*Log[c*(d + e*x)^n])/(b*n))]])*(a + b*Log[c*(d + e*x)^n]))/(E^((4*a)/(b*n))*
(c*(d + e*x)^n)^(4/n)) - (e^3*f^3*Gamma[1/2, -((a + b*Log[c*(d + e*x)^n])/(b*n))]*(a + b*Log[c*(d + e*x)^n]))/
(E^(a/(b*n))*(c*(d + e*x)^n)^n^(-1)) - (3*d*e^2*f^2*g*Gamma[1/2, -((a + b*Log[c*(d + e*x)^n])/(b*n))]*(a + b*L
og[c*(d + e*x)^n]))/(E^(a/(b*n))*(c*(d + e*x)^n)^n^(-1)) - b*e^3*n*(f + g*x)^3*Sqrt[-((a + b*Log[c*(d + e*x)^n
])/(b*n))] - (3*Sqrt[b]*d*e*f*g^2*Sqrt[n]*Sqrt[Pi]*(2*d*E^(a/(b*n))*(c*(d + e*x)^n)^n^(-1)*Erfi[Sqrt[a + b*Log
[c*(d + e*x)^n]]/(Sqrt[b]*Sqrt[n])] - Sqrt[2]*(d + e*x)*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]]*Sqrt[-((a + b*Log[c*(d + e*x)^n])/(b*n))])/(E^((2*a)/(b*n))*(c*(d +
e*x)^n)^(2/n)) + (3*Sqrt[b]*e^2*f^2*g*Sqrt[n]*Sqrt[Pi]*(-2*d*E^(a/(b*n))*(c*(d + e*x)^n)^n^(-1)*Erfi[Sqrt[a +
b*Log[c*(d + e*x)^n]]/(Sqrt[b]*Sqrt[n])] + Sqrt[2]*(d + e*x)*Erfi[(Sqrt[2]*Sqrt[a + b*Log[c*(d + e*x)^n]])/(S
qrt[b]*Sqrt[n])])*Sqrt[a + b*Log[c*(d + e*x)^n]]*Sqrt[-((a + b*Log[c*(d + e*x)^n])/(b*n))])/(E^((2*a)/(b*n))*(
c*(d + e*x)^n)^(2/n))))/(b^2*e^4*n^2*Sqrt[a + b*Log[c*(d + e*x)^n]]*Sqrt[-((a + b*Log[c*(d + e*x)^n])/(b*n))])

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Maple [F]
time = 0.17, size = 0, normalized size = 0.00 \[\int \frac {\left (g x +f \right )^{3}}{\left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )^{\frac {3}{2}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)^3/(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)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (f + g x\right )^{3}}{\left (a + b \log {\left (c \left (d + e x\right )^{n} \right )}\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (f+g\,x\right )}^3}{{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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