3.258 \(\int \sqrt{d+e x^2} (a+b \log (c x^n)) \, dx\)

Optimal. Leaf size=330 \[ -\frac{b d^{3/2} n \sqrt{\frac{e x^2}{d}+1} \text{PolyLog}\left (2,e^{2 \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}\right )}{4 \sqrt{e} \sqrt{d+e x^2}}+\frac{d^{3/2} \sqrt{\frac{e x^2}{d}+1} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 \sqrt{e} \sqrt{d+e x^2}}+\frac{1}{2} x \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )+\frac{b d^{3/2} n \sqrt{\frac{e x^2}{d}+1} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{4 \sqrt{e} \sqrt{d+e x^2}}-\frac{b d^{3/2} n \sqrt{\frac{e x^2}{d}+1} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (1-e^{2 \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}\right )}{2 \sqrt{e} \sqrt{d+e x^2}}-\frac{1}{4} b n x \sqrt{d+e x^2}-\frac{b d n \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{4 \sqrt{e}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.200677, antiderivative size = 330, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 11, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {2321, 195, 217, 206, 2327, 2325, 5659, 3716, 2190, 2279, 2391} \[ -\frac{b d^{3/2} n \sqrt{\frac{e x^2}{d}+1} \text{PolyLog}\left (2,e^{2 \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}\right )}{4 \sqrt{e} \sqrt{d+e x^2}}+\frac{d^{3/2} \sqrt{\frac{e x^2}{d}+1} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 \sqrt{e} \sqrt{d+e x^2}}+\frac{1}{2} x \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )+\frac{b d^{3/2} n \sqrt{\frac{e x^2}{d}+1} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{4 \sqrt{e} \sqrt{d+e x^2}}-\frac{b d^{3/2} n \sqrt{\frac{e x^2}{d}+1} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (1-e^{2 \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}\right )}{2 \sqrt{e} \sqrt{d+e x^2}}-\frac{1}{4} b n x \sqrt{d+e x^2}-\frac{b d n \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{4 \sqrt{e}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [C]  time = 0.354149, size = 237, normalized size = 0.72 \[ \frac{-2 b \sqrt{e} n x \sqrt{d+e x^2} \, _3F_2\left (\frac{1}{2},\frac{1}{2},\frac{1}{2};\frac{3}{2},\frac{3}{2};-\frac{e x^2}{d}\right )+\sqrt{\frac{e x^2}{d}+1} \left (\sqrt{e} x (2 a-b n) \sqrt{d+e x^2}+2 d \log \left (\sqrt{e} \sqrt{d+e x^2}+e x\right ) (a-b n \log (x))+2 b \log \left (c x^n\right ) \left (\sqrt{e} x \sqrt{d+e x^2}+d \log \left (\sqrt{e} \sqrt{d+e x^2}+e x\right )\right )\right )+b \sqrt{d} n (2 \log (x)-1) \sqrt{d+e x^2} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{4 \sqrt{e} \sqrt{\frac{e x^2}{d}+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [F]  time = 0.439, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \sqrt{e{x}^{2}+d}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*ln(c*x^n))*(e*x^2+d)^(1/2),x)

[Out]

int((a+b*ln(c*x^n))*(e*x^2+d)^(1/2),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((a+b*log(c*x^n))*(e*x^2+d)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))*(e*x^2+d)^(1/2),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*ln(c*x**n))*(e*x**2+d)**(1/2),x)

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))*(e*x^2+d)^(1/2),x, algorithm="giac")

[Out]

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