3.303 \(\int \frac{x^6 (a+b \log (c x^n))}{(d+e x^2)^{5/2}} \, dx\)

Optimal. Leaf size=443 \[ \frac{5 b d^{3/2} n \sqrt{\frac{e x^2}{d}+1} \text{PolyLog}\left (2,e^{2 \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}\right )}{4 e^{7/2} \sqrt{d+e x^2}}-\frac{5 d^{3/2} \sqrt{\frac{e x^2}{d}+1} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^{7/2} \sqrt{d+e x^2}}-\frac{5 x^3 \left (a+b \log \left (c x^n\right )\right )}{3 e^2 \sqrt{d+e x^2}}+\frac{5 x \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{2 e^3}-\frac{x^5 \left (a+b \log \left (c x^n\right )\right )}{3 e \left (d+e x^2\right )^{3/2}}-\frac{5 b d^{3/2} n \sqrt{\frac{e x^2}{d}+1} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{4 e^{7/2} \sqrt{d+e x^2}}-\frac{31 b d^{3/2} n \sqrt{\frac{e x^2}{d}+1} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{12 e^{7/2} \sqrt{d+e x^2}}+\frac{5 b d^{3/2} n \sqrt{\frac{e x^2}{d}+1} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (1-e^{2 \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}\right )}{2 e^{7/2} \sqrt{d+e x^2}}-\frac{b n x \sqrt{d+e x^2}}{4 e^3}+\frac{b d n x}{3 e^3 \sqrt{d+e x^2}} \]

[Out]

(b*d*n*x)/(3*e^3*Sqrt[d + e*x^2]) - (b*n*x*Sqrt[d + e*x^2])/(4*e^3) - (31*b*d^(3/2)*n*Sqrt[1 + (e*x^2)/d]*ArcS
inh[(Sqrt[e]*x)/Sqrt[d]])/(12*e^(7/2)*Sqrt[d + e*x^2]) - (5*b*d^(3/2)*n*Sqrt[1 + (e*x^2)/d]*ArcSinh[(Sqrt[e]*x
)/Sqrt[d]]^2)/(4*e^(7/2)*Sqrt[d + e*x^2]) + (5*b*d^(3/2)*n*Sqrt[1 + (e*x^2)/d]*ArcSinh[(Sqrt[e]*x)/Sqrt[d]]*Lo
g[1 - E^(2*ArcSinh[(Sqrt[e]*x)/Sqrt[d]])])/(2*e^(7/2)*Sqrt[d + e*x^2]) - (x^5*(a + b*Log[c*x^n]))/(3*e*(d + e*
x^2)^(3/2)) - (5*x^3*(a + b*Log[c*x^n]))/(3*e^2*Sqrt[d + e*x^2]) + (5*x*Sqrt[d + e*x^2]*(a + b*Log[c*x^n]))/(2
*e^3) - (5*d^(3/2)*Sqrt[1 + (e*x^2)/d]*ArcSinh[(Sqrt[e]*x)/Sqrt[d]]*(a + b*Log[c*x^n]))/(2*e^(7/2)*Sqrt[d + e*
x^2]) + (5*b*d^(3/2)*n*Sqrt[1 + (e*x^2)/d]*PolyLog[2, E^(2*ArcSinh[(Sqrt[e]*x)/Sqrt[d]])])/(4*e^(7/2)*Sqrt[d +
e*x^2])

________________________________________________________________________________________

Rubi [A]  time = 0.539344, antiderivative size = 443, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 13, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.52, Rules used = {2341, 288, 321, 215, 2350, 21, 1157, 388, 5659, 3716, 2190, 2279, 2391} \[ \frac{5 b d^{3/2} n \sqrt{\frac{e x^2}{d}+1} \text{PolyLog}\left (2,e^{2 \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}\right )}{4 e^{7/2} \sqrt{d+e x^2}}-\frac{5 d^{3/2} \sqrt{\frac{e x^2}{d}+1} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^{7/2} \sqrt{d+e x^2}}-\frac{5 x^3 \left (a+b \log \left (c x^n\right )\right )}{3 e^2 \sqrt{d+e x^2}}+\frac{5 x \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{2 e^3}-\frac{x^5 \left (a+b \log \left (c x^n\right )\right )}{3 e \left (d+e x^2\right )^{3/2}}-\frac{5 b d^{3/2} n \sqrt{\frac{e x^2}{d}+1} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{4 e^{7/2} \sqrt{d+e x^2}}-\frac{31 b d^{3/2} n \sqrt{\frac{e x^2}{d}+1} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{12 e^{7/2} \sqrt{d+e x^2}}+\frac{5 b d^{3/2} n \sqrt{\frac{e x^2}{d}+1} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (1-e^{2 \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}\right )}{2 e^{7/2} \sqrt{d+e x^2}}-\frac{b n x \sqrt{d+e x^2}}{4 e^3}+\frac{b d n x}{3 e^3 \sqrt{d+e x^2}} \]

Antiderivative was successfully verified.

[In]

Int[(x^6*(a + b*Log[c*x^n]))/(d + e*x^2)^(5/2),x]

[Out]

(b*d*n*x)/(3*e^3*Sqrt[d + e*x^2]) - (b*n*x*Sqrt[d + e*x^2])/(4*e^3) - (31*b*d^(3/2)*n*Sqrt[1 + (e*x^2)/d]*ArcS
inh[(Sqrt[e]*x)/Sqrt[d]])/(12*e^(7/2)*Sqrt[d + e*x^2]) - (5*b*d^(3/2)*n*Sqrt[1 + (e*x^2)/d]*ArcSinh[(Sqrt[e]*x
)/Sqrt[d]]^2)/(4*e^(7/2)*Sqrt[d + e*x^2]) + (5*b*d^(3/2)*n*Sqrt[1 + (e*x^2)/d]*ArcSinh[(Sqrt[e]*x)/Sqrt[d]]*Lo
g[1 - E^(2*ArcSinh[(Sqrt[e]*x)/Sqrt[d]])])/(2*e^(7/2)*Sqrt[d + e*x^2]) - (x^5*(a + b*Log[c*x^n]))/(3*e*(d + e*
x^2)^(3/2)) - (5*x^3*(a + b*Log[c*x^n]))/(3*e^2*Sqrt[d + e*x^2]) + (5*x*Sqrt[d + e*x^2]*(a + b*Log[c*x^n]))/(2
*e^3) - (5*d^(3/2)*Sqrt[1 + (e*x^2)/d]*ArcSinh[(Sqrt[e]*x)/Sqrt[d]]*(a + b*Log[c*x^n]))/(2*e^(7/2)*Sqrt[d + e*
x^2]) + (5*b*d^(3/2)*n*Sqrt[1 + (e*x^2)/d]*PolyLog[2, E^(2*ArcSinh[(Sqrt[e]*x)/Sqrt[d]])])/(4*e^(7/2)*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 321

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*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 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 1157

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> With[{Qx = PolynomialQ
uotient[(a + b*x^2 + c*x^4)^p, d + e*x^2, x], R = Coeff[PolynomialRemainder[(a + b*x^2 + c*x^4)^p, d + e*x^2,
x], x, 0]}, -Simp[(R*x*(d + e*x^2)^(q + 1))/(2*d*(q + 1)), x] + Dist[1/(2*d*(q + 1)), Int[(d + e*x^2)^(q + 1)*
ExpandToSum[2*d*(q + 1)*Qx + R*(2*q + 3), x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && N
eQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && LtQ[q, -1]

Rule 388

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(d*x*(a + b*x^n)^(p + 1))/(b*(n*
(p + 1) + 1)), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(b*(n*(p + 1) + 1)), Int[(a + b*x^n)^p, x], x] /; FreeQ[{
a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && NeQ[n*(p + 1) + 1, 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^6 \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^6 \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^5 \left (a+b \log \left (c x^n\right )\right )}{3 e \left (d+e x^2\right )^{3/2}}-\frac{5 x^3 \left (a+b \log \left (c x^n\right )\right )}{3 e^2 \sqrt{d+e x^2}}+\frac{5 x \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{2 e^3}-\frac{5 d^{3/2} \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 )}{2 e^{7/2} \sqrt{d+e x^2}}-\frac{\left (b n \sqrt{1+\frac{e x^2}{d}}\right ) \int \left (\frac{d^3 \sqrt{1+\frac{e x^2}{d}} \left (15 d^2+20 d e x^2+3 e^2 x^4\right )}{6 e^3 \left (d+e x^2\right )^2}-\frac{5 d^{7/2} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 e^{7/2} x}\right ) \, dx}{d^2 \sqrt{d+e x^2}}\\ &=-\frac{x^5 \left (a+b \log \left (c x^n\right )\right )}{3 e \left (d+e x^2\right )^{3/2}}-\frac{5 x^3 \left (a+b \log \left (c x^n\right )\right )}{3 e^2 \sqrt{d+e x^2}}+\frac{5 x \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{2 e^3}-\frac{5 d^{3/2} \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 )}{2 e^{7/2} \sqrt{d+e x^2}}+\frac{\left (5 b d^{3/2} n \sqrt{1+\frac{e x^2}{d}}\right ) \int \frac{\sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{x} \, dx}{2 e^{7/2} \sqrt{d+e x^2}}-\frac{\left (b d n \sqrt{1+\frac{e x^2}{d}}\right ) \int \frac{\sqrt{1+\frac{e x^2}{d}} \left (15 d^2+20 d e x^2+3 e^2 x^4\right )}{\left (d+e x^2\right )^2} \, dx}{6 e^3 \sqrt{d+e x^2}}\\ &=-\frac{x^5 \left (a+b \log \left (c x^n\right )\right )}{3 e \left (d+e x^2\right )^{3/2}}-\frac{5 x^3 \left (a+b \log \left (c x^n\right )\right )}{3 e^2 \sqrt{d+e x^2}}+\frac{5 x \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{2 e^3}-\frac{5 d^{3/2} \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 )}{2 e^{7/2} \sqrt{d+e x^2}}+\frac{\left (5 b d^{3/2} 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 )}{2 e^{7/2} \sqrt{d+e x^2}}-\frac{\left (b n \sqrt{1+\frac{e x^2}{d}}\right ) \int \frac{15 d^2+20 d e x^2+3 e^2 x^4}{\left (1+\frac{e x^2}{d}\right )^{3/2}} \, dx}{6 d e^3 \sqrt{d+e x^2}}\\ &=\frac{b d n x}{3 e^3 \sqrt{d+e x^2}}-\frac{5 b d^{3/2} n \sqrt{1+\frac{e x^2}{d}} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{4 e^{7/2} \sqrt{d+e x^2}}-\frac{x^5 \left (a+b \log \left (c x^n\right )\right )}{3 e \left (d+e x^2\right )^{3/2}}-\frac{5 x^3 \left (a+b \log \left (c x^n\right )\right )}{3 e^2 \sqrt{d+e x^2}}+\frac{5 x \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{2 e^3}-\frac{5 d^{3/2} \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 )}{2 e^{7/2} \sqrt{d+e x^2}}-\frac{\left (5 b d^{3/2} 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^{7/2} \sqrt{d+e x^2}}+\frac{\left (b n \sqrt{1+\frac{e x^2}{d}}\right ) \int \frac{-17 d^2-3 d e x^2}{\sqrt{1+\frac{e x^2}{d}}} \, dx}{6 d e^3 \sqrt{d+e x^2}}\\ &=\frac{b d n x}{3 e^3 \sqrt{d+e x^2}}-\frac{b n x \sqrt{d+e x^2}}{4 e^3}-\frac{5 b d^{3/2} n \sqrt{1+\frac{e x^2}{d}} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{4 e^{7/2} \sqrt{d+e x^2}}+\frac{5 b d^{3/2} 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 )}{2 e^{7/2} \sqrt{d+e x^2}}-\frac{x^5 \left (a+b \log \left (c x^n\right )\right )}{3 e \left (d+e x^2\right )^{3/2}}-\frac{5 x^3 \left (a+b \log \left (c x^n\right )\right )}{3 e^2 \sqrt{d+e x^2}}+\frac{5 x \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{2 e^3}-\frac{5 d^{3/2} \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 )}{2 e^{7/2} \sqrt{d+e x^2}}-\frac{\left (5 b d^{3/2} 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 )}{2 e^{7/2} \sqrt{d+e x^2}}-\frac{\left (31 b d n \sqrt{1+\frac{e x^2}{d}}\right ) \int \frac{1}{\sqrt{1+\frac{e x^2}{d}}} \, dx}{12 e^3 \sqrt{d+e x^2}}\\ &=\frac{b d n x}{3 e^3 \sqrt{d+e x^2}}-\frac{b n x \sqrt{d+e x^2}}{4 e^3}-\frac{31 b d^{3/2} n \sqrt{1+\frac{e x^2}{d}} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{12 e^{7/2} \sqrt{d+e x^2}}-\frac{5 b d^{3/2} n \sqrt{1+\frac{e x^2}{d}} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{4 e^{7/2} \sqrt{d+e x^2}}+\frac{5 b d^{3/2} 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 )}{2 e^{7/2} \sqrt{d+e x^2}}-\frac{x^5 \left (a+b \log \left (c x^n\right )\right )}{3 e \left (d+e x^2\right )^{3/2}}-\frac{5 x^3 \left (a+b \log \left (c x^n\right )\right )}{3 e^2 \sqrt{d+e x^2}}+\frac{5 x \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{2 e^3}-\frac{5 d^{3/2} \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 )}{2 e^{7/2} \sqrt{d+e x^2}}-\frac{\left (5 b d^{3/2} 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 )}{4 e^{7/2} \sqrt{d+e x^2}}\\ &=\frac{b d n x}{3 e^3 \sqrt{d+e x^2}}-\frac{b n x \sqrt{d+e x^2}}{4 e^3}-\frac{31 b d^{3/2} n \sqrt{1+\frac{e x^2}{d}} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{12 e^{7/2} \sqrt{d+e x^2}}-\frac{5 b d^{3/2} n \sqrt{1+\frac{e x^2}{d}} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{4 e^{7/2} \sqrt{d+e x^2}}+\frac{5 b d^{3/2} 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 )}{2 e^{7/2} \sqrt{d+e x^2}}-\frac{x^5 \left (a+b \log \left (c x^n\right )\right )}{3 e \left (d+e x^2\right )^{3/2}}-\frac{5 x^3 \left (a+b \log \left (c x^n\right )\right )}{3 e^2 \sqrt{d+e x^2}}+\frac{5 x \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{2 e^3}-\frac{5 d^{3/2} \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 )}{2 e^{7/2} \sqrt{d+e x^2}}+\frac{5 b d^{3/2} 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 )}{4 e^{7/2} \sqrt{d+e x^2}}\\ \end{align*}

Mathematica [C]  time = 0.287677, size = 199, normalized size = 0.45 \[ \frac{b n x^7 \sqrt{\frac{e x^2}{d}+1} \left (5 \, _3F_2\left (\frac{7}{2},\frac{7}{2},\frac{7}{2};\frac{9}{2},\frac{9}{2};-\frac{e x^2}{d}\right )+7 (2 \log (x)-1) \, _2F_1\left (\frac{5}{2},\frac{7}{2};\frac{9}{2};-\frac{e x^2}{d}\right )\right )}{98 d^2 \sqrt{d+e x^2}}+\frac{x \left (15 d^2+20 d e x^2+3 e^2 x^4\right ) \left (a+b \log \left (c x^n\right )-b n \log (x)\right )}{6 e^3 \left (d+e x^2\right )^{3/2}}-\frac{5 d \log \left (\sqrt{e} \sqrt{d+e x^2}+e x\right ) \left (a+b \log \left (c x^n\right )-b n \log (x)\right )}{2 e^{7/2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(x^6*(a + b*Log[c*x^n]))/(d + e*x^2)^(5/2),x]

[Out]

(b*n*x^7*Sqrt[1 + (e*x^2)/d]*(5*HypergeometricPFQ[{7/2, 7/2, 7/2}, {9/2, 9/2}, -((e*x^2)/d)] + 7*Hypergeometri
c2F1[5/2, 7/2, 9/2, -((e*x^2)/d)]*(-1 + 2*Log[x])))/(98*d^2*Sqrt[d + e*x^2]) + (x*(15*d^2 + 20*d*e*x^2 + 3*e^2
*x^4)*(a - b*n*Log[x] + b*Log[c*x^n]))/(6*e^3*(d + e*x^2)^(3/2)) - (5*d*(a - b*n*Log[x] + b*Log[c*x^n])*Log[e*
x + Sqrt[e]*Sqrt[d + e*x^2]])/(2*e^(7/2))

________________________________________________________________________________________

Maple [F]  time = 0.408, size = 0, normalized size = 0. \begin{align*} \int{{x}^{6} \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^6*(a+b*ln(c*x^n))/(e*x^2+d)^(5/2),x)

[Out]

int(x^6*(a+b*ln(c*x^n))/(e*x^2+d)^(5/2),x)

________________________________________________________________________________________

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^6*(a+b*log(c*x^n))/(e*x^2+d)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{e x^{2} + d} b x^{6} \log \left (c x^{n}\right ) + \sqrt{e x^{2} + d} a x^{6}}{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^6*(a+b*log(c*x^n))/(e*x^2+d)^(5/2),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \log \left (c x^{n}\right ) + a\right )} x^{6}}{{\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^6*(a+b*log(c*x^n))/(e*x^2+d)^(5/2),x, algorithm="giac")

[Out]

integrate((b*log(c*x^n) + a)*x^6/(e*x^2 + d)^(5/2), x)