3.304 \(\int \frac{x^4 (a+b \log (c x^n))}{(d+e x^2)^{5/2}} \, dx\)

Optimal. Leaf size=383 \[ -\frac{b \sqrt{d} n \sqrt{\frac{e x^2}{d}+1} \text{PolyLog}\left (2,e^{2 \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}\right )}{2 e^{5/2} \sqrt{d+e x^2}}-\frac{x \left (a+b \log \left (c x^n\right )\right )}{e^2 \sqrt{d+e x^2}}+\frac{\sqrt{d} \sqrt{\frac{e x^2}{d}+1} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^{5/2} \sqrt{d+e x^2}}-\frac{x^3 \left (a+b \log \left (c x^n\right )\right )}{3 e \left (d+e x^2\right )^{3/2}}-\frac{b n x}{3 e^2 \sqrt{d+e x^2}}+\frac{b \sqrt{d} n \sqrt{\frac{e x^2}{d}+1} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{2 e^{5/2} \sqrt{d+e x^2}}+\frac{4 b \sqrt{d} n \sqrt{\frac{e x^2}{d}+1} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{3 e^{5/2} \sqrt{d+e x^2}}-\frac{b \sqrt{d} n \sqrt{\frac{e x^2}{d}+1} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (1-e^{2 \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}\right )}{e^{5/2} \sqrt{d+e x^2}} \]

[Out]

-(b*n*x)/(3*e^2*Sqrt[d + e*x^2]) + (4*b*Sqrt[d]*n*Sqrt[1 + (e*x^2)/d]*ArcSinh[(Sqrt[e]*x)/Sqrt[d]])/(3*e^(5/2)
*Sqrt[d + e*x^2]) + (b*Sqrt[d]*n*Sqrt[1 + (e*x^2)/d]*ArcSinh[(Sqrt[e]*x)/Sqrt[d]]^2)/(2*e^(5/2)*Sqrt[d + e*x^2
]) - (b*Sqrt[d]*n*Sqrt[1 + (e*x^2)/d]*ArcSinh[(Sqrt[e]*x)/Sqrt[d]]*Log[1 - E^(2*ArcSinh[(Sqrt[e]*x)/Sqrt[d]])]
)/(e^(5/2)*Sqrt[d + e*x^2]) - (x^3*(a + b*Log[c*x^n]))/(3*e*(d + e*x^2)^(3/2)) - (x*(a + b*Log[c*x^n]))/(e^2*S
qrt[d + e*x^2]) + (Sqrt[d]*Sqrt[1 + (e*x^2)/d]*ArcSinh[(Sqrt[e]*x)/Sqrt[d]]*(a + b*Log[c*x^n]))/(e^(5/2)*Sqrt[
d + e*x^2]) - (b*Sqrt[d]*n*Sqrt[1 + (e*x^2)/d]*PolyLog[2, E^(2*ArcSinh[(Sqrt[e]*x)/Sqrt[d]])])/(2*e^(5/2)*Sqrt
[d + e*x^2])

________________________________________________________________________________________

Rubi [A]  time = 0.556751, antiderivative size = 383, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 11, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.44, Rules used = {2341, 288, 215, 2350, 21, 385, 5659, 3716, 2190, 2279, 2391} \[ -\frac{b \sqrt{d} n \sqrt{\frac{e x^2}{d}+1} \text{PolyLog}\left (2,e^{2 \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}\right )}{2 e^{5/2} \sqrt{d+e x^2}}-\frac{x \left (a+b \log \left (c x^n\right )\right )}{e^2 \sqrt{d+e x^2}}+\frac{\sqrt{d} \sqrt{\frac{e x^2}{d}+1} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^{5/2} \sqrt{d+e x^2}}-\frac{x^3 \left (a+b \log \left (c x^n\right )\right )}{3 e \left (d+e x^2\right )^{3/2}}-\frac{b n x}{3 e^2 \sqrt{d+e x^2}}+\frac{b \sqrt{d} n \sqrt{\frac{e x^2}{d}+1} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{2 e^{5/2} \sqrt{d+e x^2}}+\frac{4 b \sqrt{d} n \sqrt{\frac{e x^2}{d}+1} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{3 e^{5/2} \sqrt{d+e x^2}}-\frac{b \sqrt{d} n \sqrt{\frac{e x^2}{d}+1} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (1-e^{2 \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}\right )}{e^{5/2} \sqrt{d+e x^2}} \]

Antiderivative was successfully verified.

[In]

Int[(x^4*(a + b*Log[c*x^n]))/(d + e*x^2)^(5/2),x]

[Out]

-(b*n*x)/(3*e^2*Sqrt[d + e*x^2]) + (4*b*Sqrt[d]*n*Sqrt[1 + (e*x^2)/d]*ArcSinh[(Sqrt[e]*x)/Sqrt[d]])/(3*e^(5/2)
*Sqrt[d + e*x^2]) + (b*Sqrt[d]*n*Sqrt[1 + (e*x^2)/d]*ArcSinh[(Sqrt[e]*x)/Sqrt[d]]^2)/(2*e^(5/2)*Sqrt[d + e*x^2
]) - (b*Sqrt[d]*n*Sqrt[1 + (e*x^2)/d]*ArcSinh[(Sqrt[e]*x)/Sqrt[d]]*Log[1 - E^(2*ArcSinh[(Sqrt[e]*x)/Sqrt[d]])]
)/(e^(5/2)*Sqrt[d + e*x^2]) - (x^3*(a + b*Log[c*x^n]))/(3*e*(d + e*x^2)^(3/2)) - (x*(a + b*Log[c*x^n]))/(e^2*S
qrt[d + e*x^2]) + (Sqrt[d]*Sqrt[1 + (e*x^2)/d]*ArcSinh[(Sqrt[e]*x)/Sqrt[d]]*(a + b*Log[c*x^n]))/(e^(5/2)*Sqrt[
d + e*x^2]) - (b*Sqrt[d]*n*Sqrt[1 + (e*x^2)/d]*PolyLog[2, E^(2*ArcSinh[(Sqrt[e]*x)/Sqrt[d]])])/(2*e^(5/2)*Sqrt
[d + e*x^2])

Rule 2341

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

Rule 288

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^
n)^(p + 1))/(b*n*(p + 1)), x] - Dist[(c^n*(m - n + 1))/(b*n*(p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, p, x]

Rule 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rule 2350

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> Wit
h[{u = IntHide[(f*x)^m*(d + e*x^r)^q, x]}, Dist[a + b*Log[c*x^n], u, x] - Dist[b*n, Int[SimplifyIntegrand[u/x,
x], x], x] /; ((EqQ[r, 1] || EqQ[r, 2]) && IntegerQ[m] && IntegerQ[q - 1/2]) || InverseFunctionFreeQ[u, x]] /
; FreeQ[{a, b, c, d, e, f, m, n, q, r}, x] && IntegerQ[2*q] && ((IntegerQ[m] && IntegerQ[r]) || IGtQ[q, 0])

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 385

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> -Simp[((b*c - a*d)*x*(a + b*x^n)^(p +
1))/(a*b*n*(p + 1)), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(a*b*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /
; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/n + p, 0])

Rule 5659

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Subst[Int[(a + b*x)^n/Tanh[x], x], x, ArcSinh
[c*x]] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0]

Rule 3716

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)], x_Symbol] :> -Simp[(I*(c
+ d*x)^(m + 1))/(d*(m + 1)), x] + Dist[2*I, Int[((c + d*x)^m*E^(2*(-(I*e) + f*fz*x)))/(E^(2*I*k*Pi)*(1 + E^(2*
(-(I*e) + f*fz*x))/E^(2*I*k*Pi))), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 2190

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), x]
- Dist[(d*m)/(b*f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*Log[1 + (b*(F^(g*(e + f*x)))^n)/a], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2279

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2391

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

Rubi steps

\begin{align*} \int \frac{x^4 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^{5/2}} \, dx &=\frac{\sqrt{1+\frac{e x^2}{d}} \int \frac{x^4 \left (a+b \log \left (c x^n\right )\right )}{\left (1+\frac{e x^2}{d}\right )^{5/2}} \, dx}{d^2 \sqrt{d+e x^2}}\\ &=-\frac{x^3 \left (a+b \log \left (c x^n\right )\right )}{3 e \left (d+e x^2\right )^{3/2}}-\frac{x \left (a+b \log \left (c x^n\right )\right )}{e^2 \sqrt{d+e x^2}}+\frac{\sqrt{d} \sqrt{1+\frac{e x^2}{d}} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^{5/2} \sqrt{d+e x^2}}-\frac{\left (b n \sqrt{1+\frac{e x^2}{d}}\right ) \int \left (-\frac{d^3 \left (3 d+4 e x^2\right ) \sqrt{1+\frac{e x^2}{d}}}{3 e^2 \left (d+e x^2\right )^2}+\frac{d^{5/2} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{e^{5/2} x}\right ) \, dx}{d^2 \sqrt{d+e x^2}}\\ &=-\frac{x^3 \left (a+b \log \left (c x^n\right )\right )}{3 e \left (d+e x^2\right )^{3/2}}-\frac{x \left (a+b \log \left (c x^n\right )\right )}{e^2 \sqrt{d+e x^2}}+\frac{\sqrt{d} \sqrt{1+\frac{e x^2}{d}} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^{5/2} \sqrt{d+e x^2}}-\frac{\left (b \sqrt{d} n \sqrt{1+\frac{e x^2}{d}}\right ) \int \frac{\sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{x} \, dx}{e^{5/2} \sqrt{d+e x^2}}+\frac{\left (b d n \sqrt{1+\frac{e x^2}{d}}\right ) \int \frac{\left (3 d+4 e x^2\right ) \sqrt{1+\frac{e x^2}{d}}}{\left (d+e x^2\right )^2} \, dx}{3 e^2 \sqrt{d+e x^2}}\\ &=-\frac{x^3 \left (a+b \log \left (c x^n\right )\right )}{3 e \left (d+e x^2\right )^{3/2}}-\frac{x \left (a+b \log \left (c x^n\right )\right )}{e^2 \sqrt{d+e x^2}}+\frac{\sqrt{d} \sqrt{1+\frac{e x^2}{d}} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^{5/2} \sqrt{d+e x^2}}-\frac{\left (b \sqrt{d} n \sqrt{1+\frac{e x^2}{d}}\right ) \operatorname{Subst}\left (\int x \coth (x) \, dx,x,\sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )\right )}{e^{5/2} \sqrt{d+e x^2}}+\frac{\left (b n \sqrt{1+\frac{e x^2}{d}}\right ) \int \frac{3 d+4 e x^2}{\left (1+\frac{e x^2}{d}\right )^{3/2}} \, dx}{3 d e^2 \sqrt{d+e x^2}}\\ &=-\frac{b n x}{3 e^2 \sqrt{d+e x^2}}+\frac{b \sqrt{d} n \sqrt{1+\frac{e x^2}{d}} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{2 e^{5/2} \sqrt{d+e x^2}}-\frac{x^3 \left (a+b \log \left (c x^n\right )\right )}{3 e \left (d+e x^2\right )^{3/2}}-\frac{x \left (a+b \log \left (c x^n\right )\right )}{e^2 \sqrt{d+e x^2}}+\frac{\sqrt{d} \sqrt{1+\frac{e x^2}{d}} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^{5/2} \sqrt{d+e x^2}}+\frac{\left (2 b \sqrt{d} n \sqrt{1+\frac{e x^2}{d}}\right ) \operatorname{Subst}\left (\int \frac{e^{2 x} x}{1-e^{2 x}} \, dx,x,\sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )\right )}{e^{5/2} \sqrt{d+e x^2}}+\frac{\left (4 b n \sqrt{1+\frac{e x^2}{d}}\right ) \int \frac{1}{\sqrt{1+\frac{e x^2}{d}}} \, dx}{3 e^2 \sqrt{d+e x^2}}\\ &=-\frac{b n x}{3 e^2 \sqrt{d+e x^2}}+\frac{4 b \sqrt{d} n \sqrt{1+\frac{e x^2}{d}} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{3 e^{5/2} \sqrt{d+e x^2}}+\frac{b \sqrt{d} n \sqrt{1+\frac{e x^2}{d}} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{2 e^{5/2} \sqrt{d+e x^2}}-\frac{b \sqrt{d} n \sqrt{1+\frac{e x^2}{d}} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (1-e^{2 \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}\right )}{e^{5/2} \sqrt{d+e x^2}}-\frac{x^3 \left (a+b \log \left (c x^n\right )\right )}{3 e \left (d+e x^2\right )^{3/2}}-\frac{x \left (a+b \log \left (c x^n\right )\right )}{e^2 \sqrt{d+e x^2}}+\frac{\sqrt{d} \sqrt{1+\frac{e x^2}{d}} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^{5/2} \sqrt{d+e x^2}}+\frac{\left (b \sqrt{d} n \sqrt{1+\frac{e x^2}{d}}\right ) \operatorname{Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )\right )}{e^{5/2} \sqrt{d+e x^2}}\\ &=-\frac{b n x}{3 e^2 \sqrt{d+e x^2}}+\frac{4 b \sqrt{d} n \sqrt{1+\frac{e x^2}{d}} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{3 e^{5/2} \sqrt{d+e x^2}}+\frac{b \sqrt{d} n \sqrt{1+\frac{e x^2}{d}} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{2 e^{5/2} \sqrt{d+e x^2}}-\frac{b \sqrt{d} n \sqrt{1+\frac{e x^2}{d}} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (1-e^{2 \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}\right )}{e^{5/2} \sqrt{d+e x^2}}-\frac{x^3 \left (a+b \log \left (c x^n\right )\right )}{3 e \left (d+e x^2\right )^{3/2}}-\frac{x \left (a+b \log \left (c x^n\right )\right )}{e^2 \sqrt{d+e x^2}}+\frac{\sqrt{d} \sqrt{1+\frac{e x^2}{d}} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^{5/2} \sqrt{d+e x^2}}+\frac{\left (b \sqrt{d} n \sqrt{1+\frac{e x^2}{d}}\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}\right )}{2 e^{5/2} \sqrt{d+e x^2}}\\ &=-\frac{b n x}{3 e^2 \sqrt{d+e x^2}}+\frac{4 b \sqrt{d} n \sqrt{1+\frac{e x^2}{d}} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{3 e^{5/2} \sqrt{d+e x^2}}+\frac{b \sqrt{d} n \sqrt{1+\frac{e x^2}{d}} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{2 e^{5/2} \sqrt{d+e x^2}}-\frac{b \sqrt{d} n \sqrt{1+\frac{e x^2}{d}} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (1-e^{2 \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}\right )}{e^{5/2} \sqrt{d+e x^2}}-\frac{x^3 \left (a+b \log \left (c x^n\right )\right )}{3 e \left (d+e x^2\right )^{3/2}}-\frac{x \left (a+b \log \left (c x^n\right )\right )}{e^2 \sqrt{d+e x^2}}+\frac{\sqrt{d} \sqrt{1+\frac{e x^2}{d}} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^{5/2} \sqrt{d+e x^2}}-\frac{b \sqrt{d} n \sqrt{1+\frac{e x^2}{d}} \text{Li}_2\left (e^{2 \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}\right )}{2 e^{5/2} \sqrt{d+e x^2}}\\ \end{align*}

Mathematica [C]  time = 0.862396, size = 244, normalized size = 0.64 \[ -\frac{b n \sqrt{\frac{e x^2}{d}+1} \left (3 e^{5/2} x^5 \left (d+e x^2\right )^2 \, _3F_2\left (\frac{5}{2},\frac{5}{2},\frac{5}{2};\frac{7}{2},\frac{7}{2};-\frac{e x^2}{d}\right )+25 d^3 \sqrt{e} x \log (x) \left (3 d+4 e x^2\right ) \sqrt{\frac{e x^2}{d}+1}-75 d^{5/2} \log (x) \left (d+e x^2\right )^2 \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )\right )}{75 d^2 e^{5/2} \left (d+e x^2\right )^{5/2}}-\frac{x \left (3 d+4 e x^2\right ) \left (a+b \log \left (c x^n\right )-b n \log (x)\right )}{3 e^2 \left (d+e x^2\right )^{3/2}}+\frac{\log \left (\sqrt{e} \sqrt{d+e x^2}+e x\right ) \left (a+b \log \left (c x^n\right )-b n \log (x)\right )}{e^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^4*(a + b*Log[c*x^n]))/(d + e*x^2)^(5/2),x]

[Out]

-(b*n*Sqrt[1 + (e*x^2)/d]*(3*e^(5/2)*x^5*(d + e*x^2)^2*HypergeometricPFQ[{5/2, 5/2, 5/2}, {7/2, 7/2}, -((e*x^2
)/d)] + 25*d^3*Sqrt[e]*x*(3*d + 4*e*x^2)*Sqrt[1 + (e*x^2)/d]*Log[x] - 75*d^(5/2)*(d + e*x^2)^2*ArcSinh[(Sqrt[e
]*x)/Sqrt[d]]*Log[x]))/(75*d^2*e^(5/2)*(d + e*x^2)^(5/2)) - (x*(3*d + 4*e*x^2)*(a - b*n*Log[x] + b*Log[c*x^n])
)/(3*e^2*(d + e*x^2)^(3/2)) + ((a - b*n*Log[x] + b*Log[c*x^n])*Log[e*x + Sqrt[e]*Sqrt[d + e*x^2]])/e^(5/2)

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Maple [F]  time = 0.406, size = 0, normalized size = 0. \begin{align*} \int{{x}^{4} \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \left ( e{x}^{2}+d \right ) ^{-{\frac{5}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(a+b*ln(c*x^n))/(e*x^2+d)^(5/2),x)

[Out]

int(x^4*(a+b*ln(c*x^n))/(e*x^2+d)^(5/2),x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(a+b*log(c*x^n))/(e*x^2+d)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{e x^{2} + d} b x^{4} \log \left (c x^{n}\right ) + \sqrt{e x^{2} + d} a x^{4}}{e^{3} x^{6} + 3 \, d e^{2} x^{4} + 3 \, d^{2} e x^{2} + d^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(a+b*log(c*x^n))/(e*x^2+d)^(5/2),x, algorithm="fricas")

[Out]

integral((sqrt(e*x^2 + d)*b*x^4*log(c*x^n) + sqrt(e*x^2 + d)*a*x^4)/(e^3*x^6 + 3*d*e^2*x^4 + 3*d^2*e*x^2 + d^3
), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(a+b*ln(c*x**n))/(e*x**2+d)**(5/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \log \left (c x^{n}\right ) + a\right )} x^{4}}{{\left (e x^{2} + d\right )}^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(a+b*log(c*x^n))/(e*x^2+d)^(5/2),x, algorithm="giac")

[Out]

integrate((b*log(c*x^n) + a)*x^4/(e*x^2 + d)^(5/2), x)