\(\int \frac {1}{\sqrt {d x} (a+b \log (c x^n))} \, dx\) [104]

Optimal result

Integrand size = 20, antiderivative size = 64 \[ \int \frac {1}{\sqrt {d x} \left (a+b \log \left (c x^n\right )\right )} \, dx=\frac {e^{-\frac {a}{2 b n}} \sqrt {d x} \left (c x^n\right )^{\left .-\frac {1}{2}\right /n} \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c x^n\right )}{2 b n}\right )}{b d n} \]

[Out]

Ei(1/2*(a+b*ln(c*x^n))/b/n)*(d*x)^(1/2)/b/d/exp(1/2*a/b/n)/n/((c*x^n)^(1/2/n))

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2347, 2209} \[ \int \frac {1}{\sqrt {d x} \left (a+b \log \left (c x^n\right )\right )} \, dx=\frac {\sqrt {d x} e^{-\frac {a}{2 b n}} \left (c x^n\right )^{\left .-\frac {1}{2}\right /n} \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c x^n\right )}{2 b n}\right )}{b d n} \]

[In]

Int[1/(Sqrt[d*x]*(a + b*Log[c*x^n])),x]

[Out]

(Sqrt[d*x]*ExpIntegralEi[(a + b*Log[c*x^n])/(2*b*n)])/(b*d*E^(a/(2*b*n))*n*(c*x^n)^(1/(2*n)))

Rule 2209

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - c*(f/d)))/d)*ExpInteg
ralEi[f*g*(c + d*x)*(Log[F]/d)], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !TrueQ[$UseGamma]

Rule 2347

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_)*((d_.)*(x_))^(m_.), x_Symbol] :> Dist[(d*x)^(m + 1)/(d*n*(c*x^n
)^((m + 1)/n)), Subst[Int[E^(((m + 1)/n)*x)*(a + b*x)^p, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, d, m, n, p}
, x]

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

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.97 \[ \int \frac {1}{\sqrt {d x} \left (a+b \log \left (c x^n\right )\right )} \, dx=\frac {e^{-\frac {a}{2 b n}} x \left (c x^n\right )^{\left .-\frac {1}{2}\right /n} \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c x^n\right )}{2 b n}\right )}{b n \sqrt {d x}} \]

[In]

Integrate[1/(Sqrt[d*x]*(a + b*Log[c*x^n])),x]

[Out]

(x*ExpIntegralEi[(a + b*Log[c*x^n])/(2*b*n)])/(b*E^(a/(2*b*n))*n*Sqrt[d*x]*(c*x^n)^(1/(2*n)))

Maple [F]

\[\int \frac {1}{\sqrt {d x}\, \left (a +b \ln \left (c \,x^{n}\right )\right )}d x\]

[In]

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

[Out]

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

Fricas [F]

\[ \int \frac {1}{\sqrt {d x} \left (a+b \log \left (c x^n\right )\right )} \, dx=\int { \frac {1}{\sqrt {d x} {\left (b \log \left (c x^{n}\right ) + a\right )}} \,d x } \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {1}{\sqrt {d x} \left (a+b \log \left (c x^n\right )\right )} \, dx=\int \frac {1}{\sqrt {d x} \left (a + b \log {\left (c x^{n} \right )}\right )}\, dx \]

[In]

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

[Out]

Integral(1/(sqrt(d*x)*(a + b*log(c*x**n))), x)

Maxima [F]

\[ \int \frac {1}{\sqrt {d x} \left (a+b \log \left (c x^n\right )\right )} \, dx=\int { \frac {1}{\sqrt {d x} {\left (b \log \left (c x^{n}\right ) + a\right )}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {1}{\sqrt {d x} \left (a+b \log \left (c x^n\right )\right )} \, dx=\int { \frac {1}{\sqrt {d x} {\left (b \log \left (c x^{n}\right ) + a\right )}} \,d x } \]

[In]

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

[Out]

integrate(1/(sqrt(d*x)*(b*log(c*x^n) + a)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {d x} \left (a+b \log \left (c x^n\right )\right )} \, dx=\int \frac {1}{\sqrt {d\,x}\,\left (a+b\,\ln \left (c\,x^n\right )\right )} \,d x \]

[In]

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

[Out]

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