3.2.98 \(\int \frac {(f+g x)^{5/2} (a+b \log (c (d+e x)^n))}{d+e x} \, dx\) [198]

Optimal. Leaf size=485 \[ -\frac {92 b (e f-d g)^2 n \sqrt {f+g x}}{15 e^3}-\frac {32 b (e f-d g) n (f+g x)^{3/2}}{45 e^2}-\frac {4 b n (f+g x)^{5/2}}{25 e}+\frac {92 b (e f-d g)^{5/2} n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{15 e^{7/2}}+\frac {2 b (e f-d g)^{5/2} n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )^2}{e^{7/2}}+\frac {2 (e f-d g)^2 \sqrt {f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right )}{e^3}+\frac {2 (e f-d g) (f+g x)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 e^2}+\frac {2 (f+g x)^{5/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{5 e}-\frac {2 (e f-d g)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{e^{7/2}}-\frac {4 b (e f-d g)^{5/2} n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \log \left (\frac {2}{1-\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}}\right )}{e^{7/2}}-\frac {2 b (e f-d g)^{5/2} n \text {Li}_2\left (1-\frac {2}{1-\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}}\right )}{e^{7/2}} \]

[Out]

-32/45*b*(-d*g+e*f)*n*(g*x+f)^(3/2)/e^2-4/25*b*n*(g*x+f)^(5/2)/e+92/15*b*(-d*g+e*f)^(5/2)*n*arctanh(e^(1/2)*(g
*x+f)^(1/2)/(-d*g+e*f)^(1/2))/e^(7/2)+2*b*(-d*g+e*f)^(5/2)*n*arctanh(e^(1/2)*(g*x+f)^(1/2)/(-d*g+e*f)^(1/2))^2
/e^(7/2)+2/3*(-d*g+e*f)*(g*x+f)^(3/2)*(a+b*ln(c*(e*x+d)^n))/e^2+2/5*(g*x+f)^(5/2)*(a+b*ln(c*(e*x+d)^n))/e-2*(-
d*g+e*f)^(5/2)*arctanh(e^(1/2)*(g*x+f)^(1/2)/(-d*g+e*f)^(1/2))*(a+b*ln(c*(e*x+d)^n))/e^(7/2)-4*b*(-d*g+e*f)^(5
/2)*n*arctanh(e^(1/2)*(g*x+f)^(1/2)/(-d*g+e*f)^(1/2))*ln(2/(1-e^(1/2)*(g*x+f)^(1/2)/(-d*g+e*f)^(1/2)))/e^(7/2)
-2*b*(-d*g+e*f)^(5/2)*n*polylog(2,1-2/(1-e^(1/2)*(g*x+f)^(1/2)/(-d*g+e*f)^(1/2)))/e^(7/2)-92/15*b*(-d*g+e*f)^2
*n*(g*x+f)^(1/2)/e^3+2*(-d*g+e*f)^2*(a+b*ln(c*(e*x+d)^n))*(g*x+f)^(1/2)/e^3

________________________________________________________________________________________

Rubi [A]
time = 1.45, antiderivative size = 485, normalized size of antiderivative = 1.00, number of steps used = 27, number of rules used = 14, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.452, Rules used = {2458, 2388, 65, 214, 2390, 12, 1601, 6873, 6131, 6055, 2449, 2352, 2356, 52} \begin {gather*} -\frac {2 b n (e f-d g)^{5/2} \text {PolyLog}\left (2,1-\frac {2}{1-\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}}\right )}{e^{7/2}}-\frac {2 (e f-d g)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{e^{7/2}}+\frac {2 \sqrt {f+g x} (e f-d g)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{e^3}+\frac {2 (f+g x)^{3/2} (e f-d g) \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 e^2}+\frac {2 (f+g x)^{5/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{5 e}+\frac {2 b n (e f-d g)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )^2}{e^{7/2}}+\frac {92 b n (e f-d g)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{15 e^{7/2}}-\frac {4 b n (e f-d g)^{5/2} \log \left (\frac {2}{1-\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}}\right ) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{e^{7/2}}-\frac {92 b n \sqrt {f+g x} (e f-d g)^2}{15 e^3}-\frac {32 b n (f+g x)^{3/2} (e f-d g)}{45 e^2}-\frac {4 b n (f+g x)^{5/2}}{25 e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-92*b*(e*f - d*g)^2*n*Sqrt[f + g*x])/(15*e^3) - (32*b*(e*f - d*g)*n*(f + g*x)^(3/2))/(45*e^2) - (4*b*n*(f + g
*x)^(5/2))/(25*e) + (92*b*(e*f - d*g)^(5/2)*n*ArcTanh[(Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g]])/(15*e^(7/2)) +
(2*b*(e*f - d*g)^(5/2)*n*ArcTanh[(Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g]]^2)/e^(7/2) + (2*(e*f - d*g)^2*Sqrt[
f + g*x]*(a + b*Log[c*(d + e*x)^n]))/e^3 + (2*(e*f - d*g)*(f + g*x)^(3/2)*(a + b*Log[c*(d + e*x)^n]))/(3*e^2)
+ (2*(f + g*x)^(5/2)*(a + b*Log[c*(d + e*x)^n]))/(5*e) - (2*(e*f - d*g)^(5/2)*ArcTanh[(Sqrt[e]*Sqrt[f + g*x])/
Sqrt[e*f - d*g]]*(a + b*Log[c*(d + e*x)^n]))/e^(7/2) - (4*b*(e*f - d*g)^(5/2)*n*ArcTanh[(Sqrt[e]*Sqrt[f + g*x]
)/Sqrt[e*f - d*g]]*Log[2/(1 - (Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g])])/e^(7/2) - (2*b*(e*f - d*g)^(5/2)*n*Po
lyLog[2, 1 - 2/(1 - (Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g])])/e^(7/2)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + n + 1))), x] + Dist[n*((b*c - a*d)/(b*(m + n + 1))), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rule 1601

Int[(Pp_)/(Qq_), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x]}, Simp[Coeff[Pp, x, p]*(Log[RemoveConte
nt[Qq, x]]/(q*Coeff[Qq, x, q])), x] /; EqQ[p, q - 1] && EqQ[Pp, Simplify[(Coeff[Pp, x, p]/(q*Coeff[Qq, x, q]))
*D[Qq, x]]]] /; PolyQ[Pp, x] && PolyQ[Qq, x]

Rule 2352

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLog[2, 1 - c*x], x] /; FreeQ[{c, d, e
}, x] && EqQ[e + c*d, 0]

Rule 2356

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[(d + e*x)^(q + 1)
*((a + b*Log[c*x^n])^p/(e*(q + 1))), x] - Dist[b*n*(p/(e*(q + 1))), Int[((d + e*x)^(q + 1)*(a + b*Log[c*x^n])^
(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && GtQ[p, 0] && NeQ[q, -1] && (EqQ[p, 1] || (Integers
Q[2*p, 2*q] &&  !IGtQ[q, 0]) || (EqQ[p, 2] && NeQ[q, 1]))

Rule 2388

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_.))/(x_), x_Symbol] :> Dist[d, Int[(d
+ e*x)^(q - 1)*((a + b*Log[c*x^n])^p/x), x], x] + Dist[e, Int[(d + e*x)^(q - 1)*(a + b*Log[c*x^n])^p, x], x] /
; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && GtQ[q, 0] && IntegerQ[2*q]

Rule 2390

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^(r_.))^(q_.))/(x_), x_Symbol] :> With[{u = IntHi
de[(d + e*x^r)^q/x, x]}, Simp[u*(a + b*Log[c*x^n]), x] - Dist[b*n, Int[Dist[1/x, u, x], x], x]] /; FreeQ[{a, b
, c, d, e, n, r}, x] && IntegerQ[q - 1/2]

Rule 2449

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> Dist[-e/g, Subst[Int[Log[2*d*x]/(1 - 2*
d*x), x], x, 1/(d + e*x)], x] /; FreeQ[{c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]

Rule 2458

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_.)*(x_))^(q_.)*((h_.) + (i_.)*(x_))
^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[(g*(x/e))^q*((e*h - d*i)/e + i*(x/e))^r*(a + b*Log[c*x^n])^p, x], x,
d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - d*g, 0] && (IGtQ[p, 0] || IGtQ[
r, 0]) && IntegerQ[2*r]

Rule 6055

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcTanh[c*x])^p)
*(Log[2/(1 + e*(x/d))]/e), x] + Dist[b*c*(p/e), Int[(a + b*ArcTanh[c*x])^(p - 1)*(Log[2/(1 + e*(x/d))]/(1 - c^
2*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 - e^2, 0]

Rule 6131

Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(a + b*ArcTanh[c
*x])^(p + 1)/(b*e*(p + 1)), x] + Dist[1/(c*d), Int[(a + b*ArcTanh[c*x])^p/(1 - c*x), x], x] /; FreeQ[{a, b, c,
d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]

Rule 6873

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =!= u]

Rubi steps

\begin {align*} \int \frac {(f+g x)^{5/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{d+e x} \, dx &=\frac {\text {Subst}\left (\int \frac {\left (\frac {e f-d g}{e}+\frac {g x}{e}\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{x} \, dx,x,d+e x\right )}{e}\\ &=\frac {g \text {Subst}\left (\int \left (\frac {e f-d g}{e}+\frac {g x}{e}\right )^{3/2} \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e x\right )}{e^2}+\frac {(e f-d g) \text {Subst}\left (\int \frac {\left (\frac {e f-d g}{e}+\frac {g x}{e}\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x} \, dx,x,d+e x\right )}{e^2}\\ &=\frac {2 (f+g x)^{5/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{5 e}+\frac {(g (e f-d g)) \text {Subst}\left (\int \sqrt {\frac {e f-d g}{e}+\frac {g x}{e}} \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e x\right )}{e^3}+\frac {(e f-d g)^2 \text {Subst}\left (\int \frac {\sqrt {\frac {e f-d g}{e}+\frac {g x}{e}} \left (a+b \log \left (c x^n\right )\right )}{x} \, dx,x,d+e x\right )}{e^3}-\frac {(2 b n) \text {Subst}\left (\int \frac {\left (\frac {e f-d g}{e}+\frac {g x}{e}\right )^{5/2}}{x} \, dx,x,d+e x\right )}{5 e}\\ &=-\frac {4 b n (f+g x)^{5/2}}{25 e}+\frac {2 (e f-d g) (f+g x)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 e^2}+\frac {2 (f+g x)^{5/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{5 e}+\frac {\left (g (e f-d g)^2\right ) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{\sqrt {\frac {e f-d g}{e}+\frac {g x}{e}}} \, dx,x,d+e x\right )}{e^4}+\frac {(e f-d g)^3 \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x \sqrt {\frac {e f-d g}{e}+\frac {g x}{e}}} \, dx,x,d+e x\right )}{e^4}-\frac {(2 b (e f-d g) n) \text {Subst}\left (\int \frac {\left (\frac {e f-d g}{e}+\frac {g x}{e}\right )^{3/2}}{x} \, dx,x,d+e x\right )}{5 e^2}-\frac {(2 b (e f-d g) n) \text {Subst}\left (\int \frac {\left (\frac {e f-d g}{e}+\frac {g x}{e}\right )^{3/2}}{x} \, dx,x,d+e x\right )}{3 e^2}\\ &=-\frac {32 b (e f-d g) n (f+g x)^{3/2}}{45 e^2}-\frac {4 b n (f+g x)^{5/2}}{25 e}+\frac {2 (e f-d g)^2 \sqrt {f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right )}{e^3}+\frac {2 (e f-d g) (f+g x)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 e^2}+\frac {2 (f+g x)^{5/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{5 e}-\frac {2 (e f-d g)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{e^{7/2}}-\frac {\left (2 b (e f-d g)^2 n\right ) \text {Subst}\left (\int \frac {\sqrt {\frac {e f-d g}{e}+\frac {g x}{e}}}{x} \, dx,x,d+e x\right )}{5 e^3}-\frac {\left (2 b (e f-d g)^2 n\right ) \text {Subst}\left (\int \frac {\sqrt {\frac {e f-d g}{e}+\frac {g x}{e}}}{x} \, dx,x,d+e x\right )}{3 e^3}-\frac {\left (2 b (e f-d g)^2 n\right ) \text {Subst}\left (\int \frac {\sqrt {\frac {e f-d g}{e}+\frac {g x}{e}}}{x} \, dx,x,d+e x\right )}{e^3}-\frac {\left (b (e f-d g)^3 n\right ) \text {Subst}\left (\int -\frac {2 \sqrt {e} \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g x}{e}}}{\sqrt {e f-d g}}\right )}{\sqrt {e f-d g} x} \, dx,x,d+e x\right )}{e^4}\\ &=-\frac {92 b (e f-d g)^2 n \sqrt {f+g x}}{15 e^3}-\frac {32 b (e f-d g) n (f+g x)^{3/2}}{45 e^2}-\frac {4 b n (f+g x)^{5/2}}{25 e}+\frac {2 (e f-d g)^2 \sqrt {f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right )}{e^3}+\frac {2 (e f-d g) (f+g x)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 e^2}+\frac {2 (f+g x)^{5/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{5 e}-\frac {2 (e f-d g)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{e^{7/2}}+\frac {\left (2 b (e f-d g)^{5/2} n\right ) \text {Subst}\left (\int \frac {\tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g x}{e}}}{\sqrt {e f-d g}}\right )}{x} \, dx,x,d+e x\right )}{e^{7/2}}-\frac {\left (2 b (e f-d g)^3 n\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {\frac {e f-d g}{e}+\frac {g x}{e}}} \, dx,x,d+e x\right )}{5 e^4}-\frac {\left (2 b (e f-d g)^3 n\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {\frac {e f-d g}{e}+\frac {g x}{e}}} \, dx,x,d+e x\right )}{3 e^4}-\frac {\left (2 b (e f-d g)^3 n\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {\frac {e f-d g}{e}+\frac {g x}{e}}} \, dx,x,d+e x\right )}{e^4}\\ &=-\frac {92 b (e f-d g)^2 n \sqrt {f+g x}}{15 e^3}-\frac {32 b (e f-d g) n (f+g x)^{3/2}}{45 e^2}-\frac {4 b n (f+g x)^{5/2}}{25 e}+\frac {2 (e f-d g)^2 \sqrt {f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right )}{e^3}+\frac {2 (e f-d g) (f+g x)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 e^2}+\frac {2 (f+g x)^{5/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{5 e}-\frac {2 (e f-d g)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{e^{7/2}}+\frac {\left (4 b (e f-d g)^{5/2} n\right ) \text {Subst}\left (\int \frac {x \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {e f-d g}}\right )}{d g+e \left (-f+x^2\right )} \, dx,x,\sqrt {f+g x}\right )}{e^{5/2}}-\frac {\left (4 b (e f-d g)^3 n\right ) \text {Subst}\left (\int \frac {1}{-\frac {e f-d g}{g}+\frac {e x^2}{g}} \, dx,x,\sqrt {f+g x}\right )}{5 e^3 g}-\frac {\left (4 b (e f-d g)^3 n\right ) \text {Subst}\left (\int \frac {1}{-\frac {e f-d g}{g}+\frac {e x^2}{g}} \, dx,x,\sqrt {f+g x}\right )}{3 e^3 g}-\frac {\left (4 b (e f-d g)^3 n\right ) \text {Subst}\left (\int \frac {1}{-\frac {e f-d g}{g}+\frac {e x^2}{g}} \, dx,x,\sqrt {f+g x}\right )}{e^3 g}\\ &=-\frac {92 b (e f-d g)^2 n \sqrt {f+g x}}{15 e^3}-\frac {32 b (e f-d g) n (f+g x)^{3/2}}{45 e^2}-\frac {4 b n (f+g x)^{5/2}}{25 e}+\frac {92 b (e f-d g)^{5/2} n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{15 e^{7/2}}+\frac {2 (e f-d g)^2 \sqrt {f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right )}{e^3}+\frac {2 (e f-d g) (f+g x)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 e^2}+\frac {2 (f+g x)^{5/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{5 e}-\frac {2 (e f-d g)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{e^{7/2}}+\frac {\left (4 b (e f-d g)^{5/2} n\right ) \text {Subst}\left (\int \frac {x \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {e f-d g}}\right )}{-e f+d g+e x^2} \, dx,x,\sqrt {f+g x}\right )}{e^{5/2}}\\ &=-\frac {92 b (e f-d g)^2 n \sqrt {f+g x}}{15 e^3}-\frac {32 b (e f-d g) n (f+g x)^{3/2}}{45 e^2}-\frac {4 b n (f+g x)^{5/2}}{25 e}+\frac {92 b (e f-d g)^{5/2} n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{15 e^{7/2}}+\frac {2 b (e f-d g)^{5/2} n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )^2}{e^{7/2}}+\frac {2 (e f-d g)^2 \sqrt {f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right )}{e^3}+\frac {2 (e f-d g) (f+g x)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 e^2}+\frac {2 (f+g x)^{5/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{5 e}-\frac {2 (e f-d g)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{e^{7/2}}-\frac {\left (4 b (e f-d g)^2 n\right ) \text {Subst}\left (\int \frac {\tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {e f-d g}}\right )}{1-\frac {\sqrt {e} x}{\sqrt {e f-d g}}} \, dx,x,\sqrt {f+g x}\right )}{e^3}\\ &=-\frac {92 b (e f-d g)^2 n \sqrt {f+g x}}{15 e^3}-\frac {32 b (e f-d g) n (f+g x)^{3/2}}{45 e^2}-\frac {4 b n (f+g x)^{5/2}}{25 e}+\frac {92 b (e f-d g)^{5/2} n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{15 e^{7/2}}+\frac {2 b (e f-d g)^{5/2} n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )^2}{e^{7/2}}+\frac {2 (e f-d g)^2 \sqrt {f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right )}{e^3}+\frac {2 (e f-d g) (f+g x)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 e^2}+\frac {2 (f+g x)^{5/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{5 e}-\frac {2 (e f-d g)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{e^{7/2}}-\frac {4 b (e f-d g)^{5/2} n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \log \left (\frac {2}{1-\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}}\right )}{e^{7/2}}+\frac {\left (4 b (e f-d g)^2 n\right ) \text {Subst}\left (\int \frac {\log \left (\frac {2}{1-\frac {\sqrt {e} x}{\sqrt {e f-d g}}}\right )}{1-\frac {e x^2}{e f-d g}} \, dx,x,\sqrt {f+g x}\right )}{e^3}\\ &=-\frac {92 b (e f-d g)^2 n \sqrt {f+g x}}{15 e^3}-\frac {32 b (e f-d g) n (f+g x)^{3/2}}{45 e^2}-\frac {4 b n (f+g x)^{5/2}}{25 e}+\frac {92 b (e f-d g)^{5/2} n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{15 e^{7/2}}+\frac {2 b (e f-d g)^{5/2} n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )^2}{e^{7/2}}+\frac {2 (e f-d g)^2 \sqrt {f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right )}{e^3}+\frac {2 (e f-d g) (f+g x)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 e^2}+\frac {2 (f+g x)^{5/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{5 e}-\frac {2 (e f-d g)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{e^{7/2}}-\frac {4 b (e f-d g)^{5/2} n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \log \left (\frac {2}{1-\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}}\right )}{e^{7/2}}-\frac {\left (4 b (e f-d g)^{5/2} n\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}}\right )}{e^{7/2}}\\ &=-\frac {92 b (e f-d g)^2 n \sqrt {f+g x}}{15 e^3}-\frac {32 b (e f-d g) n (f+g x)^{3/2}}{45 e^2}-\frac {4 b n (f+g x)^{5/2}}{25 e}+\frac {92 b (e f-d g)^{5/2} n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{15 e^{7/2}}+\frac {2 b (e f-d g)^{5/2} n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )^2}{e^{7/2}}+\frac {2 (e f-d g)^2 \sqrt {f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right )}{e^3}+\frac {2 (e f-d g) (f+g x)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 e^2}+\frac {2 (f+g x)^{5/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{5 e}-\frac {2 (e f-d g)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{e^{7/2}}-\frac {4 b (e f-d g)^{5/2} n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \log \left (\frac {2}{1-\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}}\right )}{e^{7/2}}-\frac {2 b (e f-d g)^{5/2} n \text {Li}_2\left (1-\frac {2}{1-\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}}\right )}{e^{7/2}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 14.54, size = 1400, normalized size = 2.89 \begin {gather*} -\frac {2 b f^2 n (f+g x)^{3/2} \left (2 \sqrt {g} \sqrt {d+e x} \, _3F_2\left (-\frac {1}{2},-\frac {1}{2},-\frac {1}{2};\frac {1}{2},\frac {1}{2};\frac {-e f+d g}{g (d+e x)}\right )+\left (-\sqrt {g} \sqrt {d+e x} \sqrt {\frac {e (f+g x)}{g (d+e x)}}+\sqrt {e f-d g} \sinh ^{-1}\left (\frac {\sqrt {e f-d g}}{\sqrt {g} \sqrt {d+e x}}\right )\right ) \log (d+e x)\right )}{g^{3/2} (d+e x)^{3/2} \left (\frac {e (f+g x)}{g (d+e x)}\right )^{3/2}}+\frac {2 b f n \sqrt {f+g x} \left (12 d g \sqrt {d+e x} \sqrt {\frac {e (f+g x)}{e f-d g}} \, _3F_2\left (-\frac {1}{2},-\frac {1}{2},-\frac {1}{2};\frac {1}{2},\frac {1}{2};\frac {-e f+d g}{g (d+e x)}\right )-3 g (d+e x)^{3/2} \sqrt {\frac {e (f+g x)}{g (d+e x)}} \, _3F_2\left (-\frac {1}{2},1,1;2,2;\frac {g (d+e x)}{-e f+d g}\right )+2 \left (\sqrt {d+e x} \sqrt {\frac {e (f+g x)}{g (d+e x)}} \left (d g-3 d g \sqrt {\frac {e (f+g x)}{e f-d g}}+e g x \sqrt {\frac {e (f+g x)}{e f-d g}}+e f \left (-1+\sqrt {\frac {e (f+g x)}{e f-d g}}\right )\right )+3 d \sqrt {g} \sqrt {e f-d g} \sqrt {\frac {e (f+g x)}{e f-d g}} \sinh ^{-1}\left (\frac {\sqrt {e f-d g}}{\sqrt {g} \sqrt {d+e x}}\right )\right ) \log (d+e x)\right )}{3 e^2 \sqrt {d+e x} \sqrt {\frac {e (f+g x)}{e f-d g}} \sqrt {\frac {e (f+g x)}{g (d+e x)}}}+\frac {b g^2 n \left (-\frac {2 d (d+e x) \sqrt {f+g x} \left (-\, _3F_2\left (-\frac {1}{2},1,1;2,2;\frac {g (d+e x)}{-e f+d g}\right )+\frac {2 (e f-d g) \left (-1+\left (\frac {e (f+g x)}{e f-d g}\right )^{3/2}\right ) \log (d+e x)}{3 g (d+e x)}\right )}{\sqrt {\frac {e (f+g x)}{e f-d g}}}-\frac {2 \sqrt {f+g x} \left (4 d e f g-2 d^2 g^2+2 e^2 \left (2 f g x \sqrt {\frac {e (f+g x)}{e f-d g}}+g^2 x^2 \sqrt {\frac {e (f+g x)}{e f-d g}}+f^2 \left (-1+\sqrt {\frac {e (f+g x)}{e f-d g}}\right )\right )+5 g (-e f+d g) (d+e x) \, _3F_2\left (-\frac {3}{2},1,1;2,2;\frac {g (d+e x)}{-e f+d g}\right )+\left (-2 d^2 g^2+e^2 \left (-f g x \sqrt {\frac {e (f+g x)}{e f-d g}}-3 g^2 x^2 \sqrt {\frac {e (f+g x)}{e f-d g}}+2 f^2 \left (-1+\sqrt {\frac {e (f+g x)}{e f-d g}}\right )\right )-d e g \left (5 g x \sqrt {\frac {e (f+g x)}{e f-d g}}+f \left (-4+5 \sqrt {\frac {e (f+g x)}{e f-d g}}\right )\right )\right ) \log (d+e x)\right )}{15 g^2 \sqrt {\frac {e (f+g x)}{e f-d g}}}-\frac {2 d^2 \sqrt {f+g x} \left (2 \sqrt {g} \sqrt {d+e x} \, _3F_2\left (-\frac {1}{2},-\frac {1}{2},-\frac {1}{2};\frac {1}{2},\frac {1}{2};\frac {-e f+d g}{g (d+e x)}\right )+\left (-\sqrt {g} \sqrt {d+e x} \sqrt {\frac {e (f+g x)}{g (d+e x)}}+\sqrt {e f-d g} \sinh ^{-1}\left (\frac {\sqrt {e f-d g}}{\sqrt {g} \sqrt {d+e x}}\right )\right ) \log (d+e x)\right )}{\sqrt {g} \sqrt {d+e x} \sqrt {\frac {e (f+g x)}{g (d+e x)}}}\right )}{e^3}+\frac {2 \sqrt {f+g x} \left (15 d^2 g^2-5 d e g (7 f+g x)+e^2 \left (23 f^2+11 f g x+3 g^2 x^2\right )\right ) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )}{15 e^3}-\frac {2 (e f-d g)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )}{e^{7/2}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-2*b*f^2*n*(f + g*x)^(3/2)*(2*Sqrt[g]*Sqrt[d + e*x]*HypergeometricPFQ[{-1/2, -1/2, -1/2}, {1/2, 1/2}, (-(e*f)
+ d*g)/(g*(d + e*x))] + (-(Sqrt[g]*Sqrt[d + e*x]*Sqrt[(e*(f + g*x))/(g*(d + e*x))]) + Sqrt[e*f - d*g]*ArcSinh
[Sqrt[e*f - d*g]/(Sqrt[g]*Sqrt[d + e*x])])*Log[d + e*x]))/(g^(3/2)*(d + e*x)^(3/2)*((e*(f + g*x))/(g*(d + e*x)
))^(3/2)) + (2*b*f*n*Sqrt[f + g*x]*(12*d*g*Sqrt[d + e*x]*Sqrt[(e*(f + g*x))/(e*f - d*g)]*HypergeometricPFQ[{-1
/2, -1/2, -1/2}, {1/2, 1/2}, (-(e*f) + d*g)/(g*(d + e*x))] - 3*g*(d + e*x)^(3/2)*Sqrt[(e*(f + g*x))/(g*(d + e*
x))]*HypergeometricPFQ[{-1/2, 1, 1}, {2, 2}, (g*(d + e*x))/(-(e*f) + d*g)] + 2*(Sqrt[d + e*x]*Sqrt[(e*(f + g*x
))/(g*(d + e*x))]*(d*g - 3*d*g*Sqrt[(e*(f + g*x))/(e*f - d*g)] + e*g*x*Sqrt[(e*(f + g*x))/(e*f - d*g)] + e*f*(
-1 + Sqrt[(e*(f + g*x))/(e*f - d*g)])) + 3*d*Sqrt[g]*Sqrt[e*f - d*g]*Sqrt[(e*(f + g*x))/(e*f - d*g)]*ArcSinh[S
qrt[e*f - d*g]/(Sqrt[g]*Sqrt[d + e*x])])*Log[d + e*x]))/(3*e^2*Sqrt[d + e*x]*Sqrt[(e*(f + g*x))/(e*f - d*g)]*S
qrt[(e*(f + g*x))/(g*(d + e*x))]) + (b*g^2*n*((-2*d*(d + e*x)*Sqrt[f + g*x]*(-HypergeometricPFQ[{-1/2, 1, 1},
{2, 2}, (g*(d + e*x))/(-(e*f) + d*g)] + (2*(e*f - d*g)*(-1 + ((e*(f + g*x))/(e*f - d*g))^(3/2))*Log[d + e*x])/
(3*g*(d + e*x))))/Sqrt[(e*(f + g*x))/(e*f - d*g)] - (2*Sqrt[f + g*x]*(4*d*e*f*g - 2*d^2*g^2 + 2*e^2*(2*f*g*x*S
qrt[(e*(f + g*x))/(e*f - d*g)] + g^2*x^2*Sqrt[(e*(f + g*x))/(e*f - d*g)] + f^2*(-1 + Sqrt[(e*(f + g*x))/(e*f -
d*g)])) + 5*g*(-(e*f) + d*g)*(d + e*x)*HypergeometricPFQ[{-3/2, 1, 1}, {2, 2}, (g*(d + e*x))/(-(e*f) + d*g)]
+ (-2*d^2*g^2 + e^2*(-(f*g*x*Sqrt[(e*(f + g*x))/(e*f - d*g)]) - 3*g^2*x^2*Sqrt[(e*(f + g*x))/(e*f - d*g)] + 2*
f^2*(-1 + Sqrt[(e*(f + g*x))/(e*f - d*g)])) - d*e*g*(5*g*x*Sqrt[(e*(f + g*x))/(e*f - d*g)] + f*(-4 + 5*Sqrt[(e
*(f + g*x))/(e*f - d*g)])))*Log[d + e*x]))/(15*g^2*Sqrt[(e*(f + g*x))/(e*f - d*g)]) - (2*d^2*Sqrt[f + g*x]*(2*
Sqrt[g]*Sqrt[d + e*x]*HypergeometricPFQ[{-1/2, -1/2, -1/2}, {1/2, 1/2}, (-(e*f) + d*g)/(g*(d + e*x))] + (-(Sqr
t[g]*Sqrt[d + e*x]*Sqrt[(e*(f + g*x))/(g*(d + e*x))]) + Sqrt[e*f - d*g]*ArcSinh[Sqrt[e*f - d*g]/(Sqrt[g]*Sqrt[
d + e*x])])*Log[d + e*x]))/(Sqrt[g]*Sqrt[d + e*x]*Sqrt[(e*(f + g*x))/(g*(d + e*x))])))/e^3 + (2*Sqrt[f + g*x]*
(15*d^2*g^2 - 5*d*e*g*(7*f + g*x) + e^2*(23*f^2 + 11*f*g*x + 3*g^2*x^2))*(a - b*n*Log[d + e*x] + b*Log[c*(d +
e*x)^n]))/(15*e^3) - (2*(e*f - d*g)^(5/2)*ArcTanh[(Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g]]*(a - b*n*Log[d + e*
x] + b*Log[c*(d + e*x)^n]))/e^(7/2)

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Maple [F]
time = 0.31, size = 0, normalized size = 0.00 \[\int \frac {\left (g x +f \right )^{\frac {5}{2}} \left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )}{e x +d}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*%e^2*f-4*%e*d*g>0)', see `as
sume?` for m

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Fricas [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)^(5/2)*(a+b*log(c*(e*x+d)^n))/(e*x+d),x, algorithm="fricas")

[Out]

integral(((b*g^2*x^2 + 2*b*f*g*x + b*f^2)*sqrt(g*x + f)*log((x*e + d)^n*c) + (a*g^2*x^2 + 2*a*f*g*x + a*f^2)*s
qrt(g*x + f))/(x*e + d), x)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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