\(\int \frac {1}{(d x)^{3/2} (a+b \log (c x^n))} \, dx\) [105]

Optimal result

Integrand size = 20, antiderivative size = 67 \[ \int \frac {1}{(d x)^{3/2} \left (a+b \log \left (c x^n\right )\right )} \, dx=\frac {e^{\frac {a}{2 b n}} \left (c x^n\right )^{\left .\frac {1}{2}\right /n} \operatorname {ExpIntegralEi}\left (\frac {-a-b \log \left (c x^n\right )}{2 b n}\right )}{b d n \sqrt {d x}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.96, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2347, 2209} \[ \int \frac {1}{(d x)^{3/2} \left (a+b \log \left (c x^n\right )\right )} \, dx=\frac {e^{\frac {a}{2 b n}} \left (c x^n\right )^{\left .\frac {1}{2}\right /n} \operatorname {ExpIntegralEi}\left (-\frac {a+b \log \left (c x^n\right )}{2 b n}\right )}{b d n \sqrt {d x}} \]

[In]

Int[1/((d*x)^(3/2)*(a + b*Log[c*x^n])),x]

[Out]

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

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

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.93 \[ \int \frac {1}{(d x)^{3/2} \left (a+b \log \left (c x^n\right )\right )} \, dx=\frac {e^{\frac {a}{2 b n}} x \left (c x^n\right )^{\left .\frac {1}{2}\right /n} \operatorname {ExpIntegralEi}\left (-\frac {a+b \log \left (c x^n\right )}{2 b n}\right )}{b n (d x)^{3/2}} \]

[In]

Integrate[1/((d*x)^(3/2)*(a + b*Log[c*x^n])),x]

[Out]

(E^(a/(2*b*n))*x*(c*x^n)^(1/(2*n))*ExpIntegralEi[-1/2*(a + b*Log[c*x^n])/(b*n)])/(b*n*(d*x)^(3/2))

Maple [F]

\[\int \frac {1}{\left (d x \right )^{\frac {3}{2}} \left (a +b \ln \left (c \,x^{n}\right )\right )}d x\]

[In]

int(1/(d*x)^(3/2)/(a+b*ln(c*x^n)),x)

[Out]

int(1/(d*x)^(3/2)/(a+b*ln(c*x^n)),x)

Fricas [F]

\[ \int \frac {1}{(d x)^{3/2} \left (a+b \log \left (c x^n\right )\right )} \, dx=\int { \frac {1}{\left (d x\right )^{\frac {3}{2}} {\left (b \log \left (c x^{n}\right ) + a\right )}} \,d x } \]

[In]

integrate(1/(d*x)^(3/2)/(a+b*log(c*x^n)),x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int \frac {1}{(d x)^{3/2} \left (a+b \log \left (c x^n\right )\right )} \, dx=\int \frac {1}{\left (d x\right )^{\frac {3}{2}} \left (a + b \log {\left (c x^{n} \right )}\right )}\, dx \]

[In]

integrate(1/(d*x)**(3/2)/(a+b*ln(c*x**n)),x)

[Out]

Integral(1/((d*x)**(3/2)*(a + b*log(c*x**n))), x)

Maxima [F]

\[ \int \frac {1}{(d x)^{3/2} \left (a+b \log \left (c x^n\right )\right )} \, dx=\int { \frac {1}{\left (d x\right )^{\frac {3}{2}} {\left (b \log \left (c x^{n}\right ) + a\right )}} \,d x } \]

[In]

integrate(1/(d*x)^(3/2)/(a+b*log(c*x^n)),x, algorithm="maxima")

[Out]

-2*b*n*integrate(1/((b^2*d^(3/2)*log(c)^2 + b^2*d^(3/2)*log(x^n)^2 + 2*a*b*d^(3/2)*log(c) + a^2*d^(3/2) + 2*(b
^2*d^(3/2)*log(c) + a*b*d^(3/2))*log(x^n))*x^(3/2)), x) - 2/((b*d^(3/2)*log(c) + b*d^(3/2)*log(x^n) + a*d^(3/2
))*sqrt(x))

Giac [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.73 \[ \int \frac {1}{(d x)^{3/2} \left (a+b \log \left (c x^n\right )\right )} \, dx=\frac {c^{\frac {1}{2 \, n}} {\rm Ei}\left (-\frac {\log \left (c\right )}{2 \, n} - \frac {a}{2 \, b n} - \frac {1}{2} \, \log \left (x\right )\right ) e^{\left (\frac {a}{2 \, b n}\right )}}{b d^{\frac {3}{2}} n} \]

[In]

integrate(1/(d*x)^(3/2)/(a+b*log(c*x^n)),x, algorithm="giac")

[Out]

c^(1/2/n)*Ei(-1/2*log(c)/n - 1/2*a/(b*n) - 1/2*log(x))*e^(1/2*a/(b*n))/(b*d^(3/2)*n)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(d x)^{3/2} \left (a+b \log \left (c x^n\right )\right )} \, dx=\int \frac {1}{{\left (d\,x\right )}^{3/2}\,\left (a+b\,\ln \left (c\,x^n\right )\right )} \,d x \]

[In]

int(1/((d*x)^(3/2)*(a + b*log(c*x^n))),x)

[Out]

int(1/((d*x)^(3/2)*(a + b*log(c*x^n))), x)