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

Optimal. Leaf size=252 \[ -\frac{b e n \text{PolyLog}\left (2,1-\frac{2 \sqrt{d}}{\sqrt{d}-\sqrt{d+e x^2}}\right )}{4 \sqrt{d}}-\frac{\sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac{e \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 \sqrt{d}}-\frac{b n \sqrt{d+e x^2}}{4 x^2}+\frac{b e n \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )^2}{4 \sqrt{d}}-\frac{b e n \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )}{4 \sqrt{d}}-\frac{b e n \log \left (\frac{2 \sqrt{d}}{\sqrt{d}-\sqrt{d+e x^2}}\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )}{2 \sqrt{d}} \]

[Out]

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

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Rubi [A]  time = 0.375995, antiderivative size = 252, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 11, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.44, Rules used = {266, 47, 63, 208, 2350, 12, 14, 5984, 5918, 2402, 2315} \[ -\frac{b e n \text{PolyLog}\left (2,1-\frac{2 \sqrt{d}}{\sqrt{d}-\sqrt{d+e x^2}}\right )}{4 \sqrt{d}}-\frac{\sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac{e \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 \sqrt{d}}-\frac{b n \sqrt{d+e x^2}}{4 x^2}+\frac{b e n \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )^2}{4 \sqrt{d}}-\frac{b e n \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )}{4 \sqrt{d}}-\frac{b e n \log \left (\frac{2 \sqrt{d}}{\sqrt{d}-\sqrt{d+e x^2}}\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )}{2 \sqrt{d}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + 1)), x] - Dist[(d*n)/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/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 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 5984

Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(a + b*ArcTanh[c
*x])^(p + 1)/(b*e*(p + 1)), x] + Dist[1/(c*d), Int[(a + b*ArcTanh[c*x])^p/(1 - c*x), x], x] /; FreeQ[{a, b, c,
d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]

Rule 5918

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[((a + b*ArcTanh[c*x])^p*
Log[2/(1 + (e*x)/d)])/e, x] + Dist[(b*c*p)/e, Int[((a + b*ArcTanh[c*x])^(p - 1)*Log[2/(1 + (e*x)/d)])/(1 - c^2
*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 - e^2, 0]

Rule 2402

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> -Dist[e/g, Subst[Int[Log[2*d*x]/(1 - 2*
d*x), x], x, 1/(d + e*x)], x] /; FreeQ[{c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]

Rule 2315

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[PolyLog[2, 1 - c*x]/e, x] /; FreeQ[{c, d, e}, x] &
& EqQ[e + c*d, 0]

Rubi steps

\begin{align*} \int \frac{\sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx &=-\frac{\sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac{e \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 \sqrt{d}}-(b n) \int \frac{-\sqrt{d+e x^2}-\frac{e x^2 \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )}{\sqrt{d}}}{2 x^3} \, dx\\ &=-\frac{\sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac{e \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 \sqrt{d}}-\frac{1}{2} (b n) \int \frac{-\sqrt{d+e x^2}-\frac{e x^2 \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )}{\sqrt{d}}}{x^3} \, dx\\ &=-\frac{\sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac{e \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 \sqrt{d}}-\frac{1}{2} (b n) \int \left (-\frac{\sqrt{d+e x^2}}{x^3}-\frac{e \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )}{\sqrt{d} x}\right ) \, dx\\ &=-\frac{\sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac{e \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 \sqrt{d}}+\frac{1}{2} (b n) \int \frac{\sqrt{d+e x^2}}{x^3} \, dx+\frac{(b e n) \int \frac{\tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )}{x} \, dx}{2 \sqrt{d}}\\ &=-\frac{\sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac{e \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 \sqrt{d}}+\frac{1}{4} (b n) \operatorname{Subst}\left (\int \frac{\sqrt{d+e x}}{x^2} \, dx,x,x^2\right )+\frac{(b e n) \operatorname{Subst}\left (\int \frac{\tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{x} \, dx,x,x^2\right )}{4 \sqrt{d}}\\ &=-\frac{b n \sqrt{d+e x^2}}{4 x^2}-\frac{\sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac{e \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 \sqrt{d}}+\frac{1}{8} (b e n) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{d+e x}} \, dx,x,x^2\right )+\frac{(b e n) \operatorname{Subst}\left (\int \frac{x \tanh ^{-1}\left (\frac{x}{\sqrt{d}}\right )}{-d+x^2} \, dx,x,\sqrt{d+e x^2}\right )}{2 \sqrt{d}}\\ &=-\frac{b n \sqrt{d+e x^2}}{4 x^2}+\frac{b e n \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )^2}{4 \sqrt{d}}-\frac{\sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac{e \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 \sqrt{d}}+\frac{1}{4} (b n) \operatorname{Subst}\left (\int \frac{1}{-\frac{d}{e}+\frac{x^2}{e}} \, dx,x,\sqrt{d+e x^2}\right )-\frac{(b e n) \operatorname{Subst}\left (\int \frac{\tanh ^{-1}\left (\frac{x}{\sqrt{d}}\right )}{1-\frac{x}{\sqrt{d}}} \, dx,x,\sqrt{d+e x^2}\right )}{2 d}\\ &=-\frac{b n \sqrt{d+e x^2}}{4 x^2}-\frac{b e n \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )}{4 \sqrt{d}}+\frac{b e n \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )^2}{4 \sqrt{d}}-\frac{\sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac{e \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 \sqrt{d}}-\frac{b e n \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right ) \log \left (\frac{2 \sqrt{d}}{\sqrt{d}-\sqrt{d+e x^2}}\right )}{2 \sqrt{d}}+\frac{(b e n) \operatorname{Subst}\left (\int \frac{\log \left (\frac{2}{1-\frac{x}{\sqrt{d}}}\right )}{1-\frac{x^2}{d}} \, dx,x,\sqrt{d+e x^2}\right )}{2 d}\\ &=-\frac{b n \sqrt{d+e x^2}}{4 x^2}-\frac{b e n \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )}{4 \sqrt{d}}+\frac{b e n \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )^2}{4 \sqrt{d}}-\frac{\sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac{e \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 \sqrt{d}}-\frac{b e n \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right ) \log \left (\frac{2 \sqrt{d}}{\sqrt{d}-\sqrt{d+e x^2}}\right )}{2 \sqrt{d}}-\frac{(b e n) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1-\frac{\sqrt{d+e x^2}}{\sqrt{d}}}\right )}{2 \sqrt{d}}\\ &=-\frac{b n \sqrt{d+e x^2}}{4 x^2}-\frac{b e n \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )}{4 \sqrt{d}}+\frac{b e n \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )^2}{4 \sqrt{d}}-\frac{\sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac{e \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 \sqrt{d}}-\frac{b e n \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right ) \log \left (\frac{2 \sqrt{d}}{\sqrt{d}-\sqrt{d+e x^2}}\right )}{2 \sqrt{d}}-\frac{b e n \text{Li}_2\left (1-\frac{2}{1-\frac{\sqrt{d+e x^2}}{\sqrt{d}}}\right )}{4 \sqrt{d}}\\ \end{align*}

Mathematica [C]  time = 0.487486, size = 303, normalized size = 1.2 \[ \frac{-2 b \sqrt{d} n \sqrt{d+e x^2} \, _3F_2\left (\frac{1}{2},\frac{1}{2},\frac{1}{2};\frac{3}{2},\frac{3}{2};-\frac{d}{e x^2}\right )+\sqrt{\frac{d}{e x^2}+1} \left (2 e x^2 \log (x) \left (a+b \log \left (c x^n\right )+b n \log \left (\sqrt{d} \sqrt{d+e x^2}+d\right )\right )-2 a \sqrt{d} \sqrt{d+e x^2}-2 a e x^2 \log \left (\sqrt{d} \sqrt{d+e x^2}+d\right )-2 b \log \left (c x^n\right ) \left (\sqrt{d} \sqrt{d+e x^2}+e x^2 \log \left (\sqrt{d} \sqrt{d+e x^2}+d\right )\right )-b \sqrt{d} n \sqrt{d+e x^2}-2 b e n x^2 \log ^2(x)\right )-b \sqrt{e} n x (2 \log (x)+1) \sqrt{d+e x^2} \sinh ^{-1}\left (\frac{\sqrt{d}}{\sqrt{e} x}\right )}{4 \sqrt{d} x^2 \sqrt{\frac{d}{e x^2}+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.449, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b\ln \left ( c{x}^{n} \right ) }{{x}^{3}}\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^3,x)

[Out]

int((a+b*ln(c*x^n))*(e*x^2+d)^(1/2)/x^3,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((a+b*log(c*x^n))*(e*x^2+d)^(1/2)/x^3,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 \log \left (c x^{n}\right ) + \sqrt{e x^{2} + d} a}{x^{3}}, 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^3,x, algorithm="fricas")

[Out]

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

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{e x^{2} + d}{\left (b \log \left (c x^{n}\right ) + a\right )}}{x^{3}}\,{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^3,x, algorithm="giac")

[Out]

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