\(\int \frac {a+b \log (c x^n)}{x (d+e x)^{3/2}} \, dx\) [155]

Optimal result

Integrand size = 23, antiderivative size = 201 \[ \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^{3/2}} \, dx=\frac {4 b n \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{d^{3/2}}+\frac {2 b n \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )^2}{d^{3/2}}+\frac {2 \left (a+b \log \left (c x^n\right )\right )}{d \sqrt {d+e x}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^{3/2}}-\frac {4 b n \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}\right )}{d^{3/2}}-\frac {2 b n \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}\right )}{d^{3/2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {2389, 65, 214, 2390, 12, 6131, 6055, 2449, 2352, 2356} \[ \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^{3/2}} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^{3/2}}+\frac {2 \left (a+b \log \left (c x^n\right )\right )}{d \sqrt {d+e x}}+\frac {2 b n \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )^2}{d^{3/2}}+\frac {4 b n \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{d^{3/2}}-\frac {4 b n \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}\right )}{d^{3/2}}-\frac {2 b n \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}\right )}{d^{3/2}} \]

[In]

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

[Out]

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

Rule 12

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

Rule 65

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 214

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

Rule 2352

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

Rule 2356

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[(d + e*x)^(q + 1)
*((a + b*Log[c*x^n])^p/(e*(q + 1))), x] - Dist[b*n*(p/(e*(q + 1))), Int[((d + e*x)^(q + 1)*(a + b*Log[c*x^n])^
(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && GtQ[p, 0] && NeQ[q, -1] && (EqQ[p, 1] || (Integers
Q[2*p, 2*q] &&  !IGtQ[q, 0]) || (EqQ[p, 2] && NeQ[q, 1]))

Rule 2389

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_))/(x_), x_Symbol] :> Dist[1/d, Int[(d
+ e*x)^(q + 1)*((a + b*Log[c*x^n])^p/x), x], x] - Dist[e/d, Int[(d + e*x)^q*(a + b*Log[c*x^n])^p, x], x] /; F
reeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && LtQ[q, -1] && IntegerQ[2*q]

Rule 2390

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^(r_.))^(q_.))/(x_), x_Symbol] :> With[{u = IntHi
de[(d + e*x^r)^q/x, x]}, Simp[u*(a + b*Log[c*x^n]), x] - Dist[b*n, Int[Dist[1/x, u, x], x], x]] /; FreeQ[{a, b
, c, d, e, n, r}, x] && IntegerQ[q - 1/2]

Rule 2449

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 6055

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 6131

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]

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

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.47 \[ \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^{3/2}} \, dx=\frac {8 b n \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )+\frac {4 \sqrt {d} \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d+e x}}+2 \left (a+b \log \left (c x^n\right )\right ) \log \left (\sqrt {d}-\sqrt {d+e x}\right )-2 \left (a+b \log \left (c x^n\right )\right ) \log \left (\sqrt {d}+\sqrt {d+e x}\right )-b n \left (\log \left (\sqrt {d}-\sqrt {d+e x}\right ) \left (\log \left (\sqrt {d}-\sqrt {d+e x}\right )+2 \log \left (\frac {1}{2} \left (1+\frac {\sqrt {d+e x}}{\sqrt {d}}\right )\right )\right )+2 \operatorname {PolyLog}\left (2,\frac {1}{2}-\frac {\sqrt {d+e x}}{2 \sqrt {d}}\right )\right )+b n \left (\log \left (\sqrt {d}+\sqrt {d+e x}\right ) \left (\log \left (\sqrt {d}+\sqrt {d+e x}\right )+2 \log \left (\frac {1}{2}-\frac {\sqrt {d+e x}}{2 \sqrt {d}}\right )\right )+2 \operatorname {PolyLog}\left (2,\frac {1}{2} \left (1+\frac {\sqrt {d+e x}}{\sqrt {d}}\right )\right )\right )}{2 d^{3/2}} \]

[In]

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

[Out]

(8*b*n*ArcTanh[Sqrt[d + e*x]/Sqrt[d]] + (4*Sqrt[d]*(a + b*Log[c*x^n]))/Sqrt[d + e*x] + 2*(a + b*Log[c*x^n])*Lo
g[Sqrt[d] - Sqrt[d + e*x]] - 2*(a + b*Log[c*x^n])*Log[Sqrt[d] + Sqrt[d + e*x]] - b*n*(Log[Sqrt[d] - Sqrt[d + e
*x]]*(Log[Sqrt[d] - Sqrt[d + e*x]] + 2*Log[(1 + Sqrt[d + e*x]/Sqrt[d])/2]) + 2*PolyLog[2, 1/2 - Sqrt[d + e*x]/
(2*Sqrt[d])]) + b*n*(Log[Sqrt[d] + Sqrt[d + e*x]]*(Log[Sqrt[d] + Sqrt[d + e*x]] + 2*Log[1/2 - Sqrt[d + e*x]/(2
*Sqrt[d])]) + 2*PolyLog[2, (1 + Sqrt[d + e*x]/Sqrt[d])/2]))/(2*d^(3/2))

Maple [F]

\[\int \frac {a +b \ln \left (c \,x^{n}\right )}{x \left (e x +d \right )^{\frac {3}{2}}}d x\]

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^{3/2}} \, dx=\int \frac {a+b\,\ln \left (c\,x^n\right )}{x\,{\left (d+e\,x\right )}^{3/2}} \,d x \]

[In]

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

[Out]

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