3.3.70 \(\int \frac {(d+e x^2)^{3/2} (a+b \log (c x^n))}{x^2} \, dx\) [270]

Optimal. Leaf size=400 \[ -\frac {b d n \sqrt {d+e x^2}}{x}-\frac {1}{4} b e n x \sqrt {d+e x^2}+\frac {3 b \sqrt {d} \sqrt {e} n \sqrt {d+e x^2} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{4 \sqrt {1+\frac {e x^2}{d}}}+\frac {3 b \sqrt {d} \sqrt {e} n \sqrt {d+e x^2} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{4 \sqrt {1+\frac {e x^2}{d}}}-\frac {3 b \sqrt {d} \sqrt {e} n \sqrt {d+e x^2} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (1-e^{2 \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )}{2 \sqrt {1+\frac {e x^2}{d}}}+\frac {3}{2} e x \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x}+\frac {3 \sqrt {d} \sqrt {e} \sqrt {d+e x^2} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 \sqrt {1+\frac {e x^2}{d}}}-\frac {3 b \sqrt {d} \sqrt {e} n \sqrt {d+e x^2} \text {Li}_2\left (e^{2 \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )}{4 \sqrt {1+\frac {e x^2}{d}}} \]

[Out]

-(e*x^2+d)^(3/2)*(a+b*ln(c*x^n))/x-b*d*n*(e*x^2+d)^(1/2)/x-1/4*b*e*n*x*(e*x^2+d)^(1/2)+3/2*e*x*(a+b*ln(c*x^n))
*(e*x^2+d)^(1/2)+3/4*b*n*arcsinh(x*e^(1/2)/d^(1/2))*d^(1/2)*e^(1/2)*(e*x^2+d)^(1/2)/(1+e*x^2/d)^(1/2)+3/4*b*n*
arcsinh(x*e^(1/2)/d^(1/2))^2*d^(1/2)*e^(1/2)*(e*x^2+d)^(1/2)/(1+e*x^2/d)^(1/2)-3/2*b*n*arcsinh(x*e^(1/2)/d^(1/
2))*ln(1-(x*e^(1/2)/d^(1/2)+(1+e*x^2/d)^(1/2))^2)*d^(1/2)*e^(1/2)*(e*x^2+d)^(1/2)/(1+e*x^2/d)^(1/2)+3/2*arcsin
h(x*e^(1/2)/d^(1/2))*(a+b*ln(c*x^n))*d^(1/2)*e^(1/2)*(e*x^2+d)^(1/2)/(1+e*x^2/d)^(1/2)-3/4*b*n*polylog(2,(x*e^
(1/2)/d^(1/2)+(1+e*x^2/d)^(1/2))^2)*d^(1/2)*e^(1/2)*(e*x^2+d)^(1/2)/(1+e*x^2/d)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.32, antiderivative size = 400, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 12, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.480, Rules used = {2386, 283, 201, 221, 2392, 12, 14, 5775, 3797, 2221, 2317, 2438} \begin {gather*} -\frac {3 b \sqrt {d} \sqrt {e} n \sqrt {d+e x^2} \text {PolyLog}\left (2,e^{2 \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )}{4 \sqrt {\frac {e x^2}{d}+1}}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x}+\frac {3}{2} e x \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )+\frac {3 \sqrt {d} \sqrt {e} \sqrt {d+e x^2} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 \sqrt {\frac {e x^2}{d}+1}}-\frac {1}{4} b e n x \sqrt {d+e x^2}-\frac {b d n \sqrt {d+e x^2}}{x}+\frac {3 b \sqrt {d} \sqrt {e} n \sqrt {d+e x^2} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{4 \sqrt {\frac {e x^2}{d}+1}}+\frac {3 b \sqrt {d} \sqrt {e} n \sqrt {d+e x^2} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{4 \sqrt {\frac {e x^2}{d}+1}}-\frac {3 b \sqrt {d} \sqrt {e} n \sqrt {d+e x^2} \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 \sqrt {\frac {e x^2}{d}+1}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 12

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

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 201

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^p/(n*p + 1)), x] + Dist[a*n*(p/(n*p + 1)),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 221

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 283

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

Rule 2221

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/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], 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 2317

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 2386

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/d)*x^2)^FracPart[q]), Int[x^m*(1 + (e/d)*x^2)^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 2392

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 2438

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]

Rule 3797

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))/(1 + E^(2*((-I)*e + f*
fz*x))/E^(2*I*k*Pi))))/E^(2*I*k*Pi), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 5775

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Dist[1/b, Subst[Int[x^n*Coth[-a/b + x/b], x],
x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0]

Rubi steps

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

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 0.64, size = 329, normalized size = 0.82 \begin {gather*} -\frac {b \sqrt {d} n \sqrt {d+e x^2} \left (\sqrt {d} \, _3F_2\left (-\frac {1}{2},-\frac {1}{2},-\frac {1}{2};\frac {1}{2},\frac {1}{2};-\frac {e x^2}{d}\right )+\left (\sqrt {d} \sqrt {1+\frac {e x^2}{d}}-\sqrt {e} x \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right ) \log (x)\right )}{x \sqrt {1+\frac {e x^2}{d}}}+\frac {b \sqrt {e} n \sqrt {d+e x^2} \left (-2 \sqrt {e} x \, _3F_2\left (\frac {1}{2},\frac {1}{2},\frac {1}{2};\frac {3}{2},\frac {3}{2};-\frac {e x^2}{d}\right )+\left (\sqrt {e} x \sqrt {1+\frac {e x^2}{d}}+\sqrt {d} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right ) (-1+2 \log (x))\right )}{4 \sqrt {1+\frac {e x^2}{d}}}-\frac {\left (2 d-e x^2\right ) \sqrt {d+e x^2} \left (a-b n \log (x)+b \log \left (c x^n\right )\right )}{2 x}+\frac {3}{2} d \sqrt {e} \left (a-b n \log (x)+b \log \left (c x^n\right )\right ) \log \left (e x+\sqrt {e} \sqrt {d+e x^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (e \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \ln \left (c \,x^{n}\right )\right )}{x^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

1/2*(3*d*arcsinh(x*e^(1/2)/sqrt(d))*e^(1/2) + 3*sqrt(x^2*e + d)*x*e - 2*(x^2*e + d)^(3/2)/x)*a + b*integrate((
x^2*e*log(c) + d*log(c) + (x^2*e + d)*log(x^n))*sqrt(x^2*e + d)/x^2, x)

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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