3.269 \(\int (d+e x^2)^{3/2} (a+b \log (c x^n)) \, dx\)

Optimal. Leaf size=378 \[ \frac {3 d^{5/2} \sqrt {\frac {e x^2}{d}+1} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{8 \sqrt {e} \sqrt {d+e x^2}}+\frac {3}{8} d x \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )+\frac {1}{4} x \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )-\frac {3 b d^{5/2} n \sqrt {\frac {e x^2}{d}+1} \text {Li}_2\left (e^{2 \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )}{16 \sqrt {e} \sqrt {d+e x^2}}+\frac {3 b d^{5/2} n \sqrt {\frac {e x^2}{d}+1} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{16 \sqrt {e} \sqrt {d+e x^2}}-\frac {3 b d^{5/2} n \sqrt {\frac {e x^2}{d}+1} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (1-e^{2 \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )}{8 \sqrt {e} \sqrt {d+e x^2}}-\frac {9 b d^2 n \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{32 \sqrt {e}}-\frac {9}{32} b d n x \sqrt {d+e x^2}-\frac {1}{16} b n x \left (d+e x^2\right )^{3/2} \]

[Out]

-1/16*b*n*x*(e*x^2+d)^(3/2)+1/4*x*(e*x^2+d)^(3/2)*(a+b*ln(c*x^n))-9/32*b*d^2*n*arctanh(x*e^(1/2)/(e*x^2+d)^(1/
2))/e^(1/2)-9/32*b*d*n*x*(e*x^2+d)^(1/2)+3/8*d*x*(a+b*ln(c*x^n))*(e*x^2+d)^(1/2)+3/16*b*d^(5/2)*n*arcsinh(x*e^
(1/2)/d^(1/2))^2*(1+e*x^2/d)^(1/2)/e^(1/2)/(e*x^2+d)^(1/2)-3/8*b*d^(5/2)*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)*(1+e*x^2/d)^(1/2)/e^(1/2)/(e*x^2+d)^(1/2)+3/8*d^(5/2)*arcsinh(x*e^(1/2)/
d^(1/2))*(a+b*ln(c*x^n))*(1+e*x^2/d)^(1/2)/e^(1/2)/(e*x^2+d)^(1/2)-3/16*b*d^(5/2)*n*polylog(2,(x*e^(1/2)/d^(1/
2)+(1+e*x^2/d)^(1/2))^2)*(1+e*x^2/d)^(1/2)/e^(1/2)/(e*x^2+d)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.27, antiderivative size = 378, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 11, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2321, 195, 217, 206, 2327, 2325, 5659, 3716, 2190, 2279, 2391} \[ -\frac {3 b d^{5/2} n \sqrt {\frac {e x^2}{d}+1} \text {PolyLog}\left (2,e^{2 \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )}{16 \sqrt {e} \sqrt {d+e x^2}}+\frac {3 d^{5/2} \sqrt {\frac {e x^2}{d}+1} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{8 \sqrt {e} \sqrt {d+e x^2}}+\frac {3}{8} d x \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )+\frac {1}{4} x \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )+\frac {3 b d^{5/2} n \sqrt {\frac {e x^2}{d}+1} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{16 \sqrt {e} \sqrt {d+e x^2}}-\frac {9 b d^2 n \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{32 \sqrt {e}}-\frac {3 b d^{5/2} n \sqrt {\frac {e x^2}{d}+1} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (1-e^{2 \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )}{8 \sqrt {e} \sqrt {d+e x^2}}-\frac {9}{32} b d n x \sqrt {d+e x^2}-\frac {1}{16} b n x \left (d+e x^2\right )^{3/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-9*b*d*n*x*Sqrt[d + e*x^2])/32 - (b*n*x*(d + e*x^2)^(3/2))/16 + (3*b*d^(5/2)*n*Sqrt[1 + (e*x^2)/d]*ArcSinh[(S
qrt[e]*x)/Sqrt[d]]^2)/(16*Sqrt[e]*Sqrt[d + e*x^2]) - (9*b*d^2*n*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(32*Sqrt
[e]) - (3*b*d^(5/2)*n*Sqrt[1 + (e*x^2)/d]*ArcSinh[(Sqrt[e]*x)/Sqrt[d]]*Log[1 - E^(2*ArcSinh[(Sqrt[e]*x)/Sqrt[d
]])])/(8*Sqrt[e]*Sqrt[d + e*x^2]) + (3*d*x*Sqrt[d + e*x^2]*(a + b*Log[c*x^n]))/8 + (x*(d + e*x^2)^(3/2)*(a + b
*Log[c*x^n]))/4 + (3*d^(5/2)*Sqrt[1 + (e*x^2)/d]*ArcSinh[(Sqrt[e]*x)/Sqrt[d]]*(a + b*Log[c*x^n]))/(8*Sqrt[e]*S
qrt[d + e*x^2]) - (3*b*d^(5/2)*n*Sqrt[1 + (e*x^2)/d]*PolyLog[2, E^(2*ArcSinh[(Sqrt[e]*x)/Sqrt[d]])])/(16*Sqrt[
e]*Sqrt[d + e*x^2])

Rule 195

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 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 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 2321

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

Rule 2325

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(ArcSinh[(Rt[e, 2]*x)/S
qrt[d]]*(a + b*Log[c*x^n]))/Rt[e, 2], x] - Dist[(b*n)/Rt[e, 2], Int[ArcSinh[(Rt[e, 2]*x)/Sqrt[d]]/x, x], x] /;
FreeQ[{a, b, c, d, e, n}, x] && GtQ[d, 0] && PosQ[e]

Rule 2327

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Dist[Sqrt[1 + (e*x^2)/d]/Sqr
t[d + e*x^2], Int[(a + b*Log[c*x^n])/Sqrt[1 + (e*x^2)/d], x], x] /; FreeQ[{a, b, c, d, e, n}, x] &&  !GtQ[d, 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]

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 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]

Rubi steps

\begin {align*} \int \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac {1}{4} x \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )+\frac {1}{4} (3 d) \int \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right ) \, dx-\frac {1}{4} (b n) \int \left (d+e x^2\right )^{3/2} \, dx\\ &=-\frac {1}{16} b n x \left (d+e x^2\right )^{3/2}+\frac {3}{8} d x \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )+\frac {1}{4} x \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )+\frac {1}{8} \left (3 d^2\right ) \int \frac {a+b \log \left (c x^n\right )}{\sqrt {d+e x^2}} \, dx-\frac {1}{16} (3 b d n) \int \sqrt {d+e x^2} \, dx-\frac {1}{8} (3 b d n) \int \sqrt {d+e x^2} \, dx\\ &=-\frac {9}{32} b d n x \sqrt {d+e x^2}-\frac {1}{16} b n x \left (d+e x^2\right )^{3/2}+\frac {3}{8} d x \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )+\frac {1}{4} x \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )-\frac {1}{32} \left (3 b d^2 n\right ) \int \frac {1}{\sqrt {d+e x^2}} \, dx-\frac {1}{16} \left (3 b d^2 n\right ) \int \frac {1}{\sqrt {d+e x^2}} \, dx+\frac {\left (3 d^2 \sqrt {1+\frac {e x^2}{d}}\right ) \int \frac {a+b \log \left (c x^n\right )}{\sqrt {1+\frac {e x^2}{d}}} \, dx}{8 \sqrt {d+e x^2}}\\ &=-\frac {9}{32} b d n x \sqrt {d+e x^2}-\frac {1}{16} b n x \left (d+e x^2\right )^{3/2}+\frac {3}{8} d x \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )+\frac {1}{4} x \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )+\frac {3 d^{5/2} \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 )}{8 \sqrt {e} \sqrt {d+e x^2}}-\frac {1}{32} \left (3 b d^2 n\right ) \operatorname {Subst}\left (\int \frac {1}{1-e x^2} \, dx,x,\frac {x}{\sqrt {d+e x^2}}\right )-\frac {1}{16} \left (3 b d^2 n\right ) \operatorname {Subst}\left (\int \frac {1}{1-e x^2} \, dx,x,\frac {x}{\sqrt {d+e x^2}}\right )-\frac {\left (3 b d^{5/2} n \sqrt {1+\frac {e x^2}{d}}\right ) \int \frac {\sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{x} \, dx}{8 \sqrt {e} \sqrt {d+e x^2}}\\ &=-\frac {9}{32} b d n x \sqrt {d+e x^2}-\frac {1}{16} b n x \left (d+e x^2\right )^{3/2}-\frac {9 b d^2 n \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{32 \sqrt {e}}+\frac {3}{8} d x \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )+\frac {1}{4} x \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )+\frac {3 d^{5/2} \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 )}{8 \sqrt {e} \sqrt {d+e x^2}}-\frac {\left (3 b d^{5/2} 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 )}{8 \sqrt {e} \sqrt {d+e x^2}}\\ &=-\frac {9}{32} b d n x \sqrt {d+e x^2}-\frac {1}{16} b n x \left (d+e x^2\right )^{3/2}+\frac {3 b d^{5/2} n \sqrt {1+\frac {e x^2}{d}} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{16 \sqrt {e} \sqrt {d+e x^2}}-\frac {9 b d^2 n \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{32 \sqrt {e}}+\frac {3}{8} d x \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )+\frac {1}{4} x \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )+\frac {3 d^{5/2} \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 )}{8 \sqrt {e} \sqrt {d+e x^2}}+\frac {\left (3 b d^{5/2} 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 )}{4 \sqrt {e} \sqrt {d+e x^2}}\\ &=-\frac {9}{32} b d n x \sqrt {d+e x^2}-\frac {1}{16} b n x \left (d+e x^2\right )^{3/2}+\frac {3 b d^{5/2} n \sqrt {1+\frac {e x^2}{d}} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{16 \sqrt {e} \sqrt {d+e x^2}}-\frac {9 b d^2 n \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{32 \sqrt {e}}-\frac {3 b d^{5/2} 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 )}{8 \sqrt {e} \sqrt {d+e x^2}}+\frac {3}{8} d x \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )+\frac {1}{4} x \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )+\frac {3 d^{5/2} \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 )}{8 \sqrt {e} \sqrt {d+e x^2}}+\frac {\left (3 b d^{5/2} 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 )}{8 \sqrt {e} \sqrt {d+e x^2}}\\ &=-\frac {9}{32} b d n x \sqrt {d+e x^2}-\frac {1}{16} b n x \left (d+e x^2\right )^{3/2}+\frac {3 b d^{5/2} n \sqrt {1+\frac {e x^2}{d}} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{16 \sqrt {e} \sqrt {d+e x^2}}-\frac {9 b d^2 n \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{32 \sqrt {e}}-\frac {3 b d^{5/2} 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 )}{8 \sqrt {e} \sqrt {d+e x^2}}+\frac {3}{8} d x \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )+\frac {1}{4} x \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )+\frac {3 d^{5/2} \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 )}{8 \sqrt {e} \sqrt {d+e x^2}}+\frac {\left (3 b d^{5/2} 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 )}{16 \sqrt {e} \sqrt {d+e x^2}}\\ &=-\frac {9}{32} b d n x \sqrt {d+e x^2}-\frac {1}{16} b n x \left (d+e x^2\right )^{3/2}+\frac {3 b d^{5/2} n \sqrt {1+\frac {e x^2}{d}} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{16 \sqrt {e} \sqrt {d+e x^2}}-\frac {9 b d^2 n \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{32 \sqrt {e}}-\frac {3 b d^{5/2} 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 )}{8 \sqrt {e} \sqrt {d+e x^2}}+\frac {3}{8} d x \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )+\frac {1}{4} x \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )+\frac {3 d^{5/2} \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 )}{8 \sqrt {e} \sqrt {d+e x^2}}-\frac {3 b d^{5/2} 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 )}{16 \sqrt {e} \sqrt {d+e x^2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 1.03, size = 314, normalized size = 0.83 \[ \frac {9 \left (-4 b d \sqrt {e} n x \sqrt {d+e x^2} \, _3F_2\left (\frac {1}{2},\frac {1}{2},\frac {1}{2};\frac {3}{2},\frac {3}{2};-\frac {e x^2}{d}\right )+\sqrt {\frac {e x^2}{d}+1} \left (3 d^2 \log \left (\sqrt {e} \sqrt {d+e x^2}+e x\right ) (a-b n \log (x))+\sqrt {e} x \sqrt {d+e x^2} \left (5 a d+2 a e x^2-2 b d n\right )+b \log \left (c x^n\right ) \left (3 d^2 \log \left (\sqrt {e} \sqrt {d+e x^2}+e x\right )+\sqrt {e} x \sqrt {d+e x^2} \left (5 d+2 e x^2\right )\right )\right )+b d^{3/2} n (3 \log (x)-2) \sqrt {d+e x^2} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right )-8 b e^{3/2} n x^3 \sqrt {d+e x^2} \, _3F_2\left (-\frac {1}{2},\frac {3}{2},\frac {3}{2};\frac {5}{2},\frac {5}{2};-\frac {e x^2}{d}\right )}{72 \sqrt {e} \sqrt {\frac {e x^2}{d}+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-8*b*e^(3/2)*n*x^3*Sqrt[d + e*x^2]*HypergeometricPFQ[{-1/2, 3/2, 3/2}, {5/2, 5/2}, -((e*x^2)/d)] + 9*(-4*b*d*
Sqrt[e]*n*x*Sqrt[d + e*x^2]*HypergeometricPFQ[{1/2, 1/2, 1/2}, {3/2, 3/2}, -((e*x^2)/d)] + b*d^(3/2)*n*Sqrt[d
+ e*x^2]*ArcSinh[(Sqrt[e]*x)/Sqrt[d]]*(-2 + 3*Log[x]) + Sqrt[1 + (e*x^2)/d]*(Sqrt[e]*x*Sqrt[d + e*x^2]*(5*a*d
- 2*b*d*n + 2*a*e*x^2) + 3*d^2*(a - b*n*Log[x])*Log[e*x + Sqrt[e]*Sqrt[d + e*x^2]] + b*Log[c*x^n]*(Sqrt[e]*x*S
qrt[d + e*x^2]*(5*d + 2*e*x^2) + 3*d^2*Log[e*x + Sqrt[e]*Sqrt[d + e*x^2]]))))/(72*Sqrt[e]*Sqrt[1 + (e*x^2)/d])

________________________________________________________________________________________

fricas [F]  time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b e x^{2} + b d\right )} \sqrt {e x^{2} + d} \log \left (c x^{n}\right ) + {\left (a e x^{2} + a d\right )} \sqrt {e x^{2} + d}, x\right ) \]

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, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

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

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, algorithm="giac")

[Out]

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

________________________________________________________________________________________

maple [F]  time = 0.34, size = 0, normalized size = 0.00 \[ \int \left (e \,x^{2}+d \right )^{\frac {3}{2}} \left (b \ln \left (c \,x^{n}\right )+a \right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{8} \, {\left (2 \, {\left (e x^{2} + d\right )}^{\frac {3}{2}} x + 3 \, \sqrt {e x^{2} + d} d x + \frac {3 \, d^{2} \operatorname {arsinh}\left (\frac {e x}{\sqrt {d e}}\right )}{\sqrt {e}}\right )} a + b \int {\left (e x^{2} \log \relax (c) + d \log \relax (c) + {\left (e x^{2} + d\right )} \log \left (x^{n}\right )\right )} \sqrt {e x^{2} + d}\,{d x} \]

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, algorithm="maxima")

[Out]

1/8*(2*(e*x^2 + d)^(3/2)*x + 3*sqrt(e*x^2 + d)*d*x + 3*d^2*arcsinh(e*x/sqrt(d*e))/sqrt(e))*a + b*integrate((e*
x^2*log(c) + d*log(c) + (e*x^2 + d)*log(x^n))*sqrt(e*x^2 + d), x)

________________________________________________________________________________________

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

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)

[Out]

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

________________________________________________________________________________________

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

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)

[Out]

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

________________________________________________________________________________________