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

Optimal. Leaf size=221 \[ -\frac{b e n \text{PolyLog}\left (2,1-\frac{2 \sqrt{d}}{\sqrt{d}-\sqrt{d+e x}}\right )}{\sqrt{d}}-\frac{\sqrt{d+e x} \left (a+b \log \left (c x^n\right )\right )}{x}-\frac{e \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt{d}}-\frac{b n \sqrt{d+e x}}{x}+\frac{b e n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )^2}{\sqrt{d}}-\frac{b e n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{\sqrt{d}}-\frac{2 b e n \log \left (\frac{2 \sqrt{d}}{\sqrt{d}-\sqrt{d+e x}}\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{\sqrt{d}} \]

[Out]

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

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Rubi [A]  time = 0.277843, antiderivative size = 221, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 9, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.391, Rules used = {47, 63, 208, 2350, 14, 5984, 5918, 2402, 2315} \[ -\frac{b e n \text{PolyLog}\left (2,1-\frac{2 \sqrt{d}}{\sqrt{d}-\sqrt{d+e x}}\right )}{\sqrt{d}}-\frac{\sqrt{d+e x} \left (a+b \log \left (c x^n\right )\right )}{x}-\frac{e \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt{d}}-\frac{b n \sqrt{d+e x}}{x}+\frac{b e n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )^2}{\sqrt{d}}-\frac{b e n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{\sqrt{d}}-\frac{2 b e n \log \left (\frac{2 \sqrt{d}}{\sqrt{d}-\sqrt{d+e x}}\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{\sqrt{d}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.326947, size = 392, normalized size = 1.77 \[ -\frac{2 b e n x \text{PolyLog}\left (2,\frac{1}{2}-\frac{\sqrt{d+e x}}{2 \sqrt{d}}\right )-2 b e n x \text{PolyLog}\left (2,\frac{1}{2} \left (\frac{\sqrt{d+e x}}{\sqrt{d}}+1\right )\right )+4 a \sqrt{d} \sqrt{d+e x}-2 a e x \log \left (\sqrt{d}-\sqrt{d+e x}\right )+2 a e x \log \left (\sqrt{d+e x}+\sqrt{d}\right )-2 b e x \log \left (c x^n\right ) \log \left (\sqrt{d}-\sqrt{d+e x}\right )+4 b \sqrt{d} \sqrt{d+e x} \log \left (c x^n\right )+2 b e x \log \left (c x^n\right ) \log \left (\sqrt{d+e x}+\sqrt{d}\right )+4 b \sqrt{d} n \sqrt{d+e x}+b e n x \log ^2\left (\sqrt{d}-\sqrt{d+e x}\right )-b e n x \log ^2\left (\sqrt{d+e x}+\sqrt{d}\right )+2 b e n x \log \left (\frac{1}{2} \left (\frac{\sqrt{d+e x}}{\sqrt{d}}+1\right )\right ) \log \left (\sqrt{d}-\sqrt{d+e x}\right )-2 b e n x \log \left (\sqrt{d+e x}+\sqrt{d}\right ) \log \left (\frac{1}{2}-\frac{\sqrt{d+e x}}{2 \sqrt{d}}\right )+4 b e n x \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{4 \sqrt{d} x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.515, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b\ln \left ( c{x}^{n} \right ) }{{x}^{2}}\sqrt{ex+d}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((a+b*ln(c*x^n))*(e*x+d)^(1/2)/x^2,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+d)^(1/2)/x^2,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 + d} b \log \left (c x^{n}\right ) + \sqrt{e x + d} a}{x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((sqrt(e*x + d)*b*log(c*x^n) + sqrt(e*x + d)*a)/x^2, 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}}{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+d)**(1/2)/x**2,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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