\(\int \frac {(d x)^{3/2}}{a+b \log (c x^n)} \, dx\) [102]

Optimal result

Integrand size = 20, antiderivative size = 64 \[ \int \frac {(d x)^{3/2}}{a+b \log \left (c x^n\right )} \, dx=\frac {e^{-\frac {5 a}{2 b n}} (d x)^{5/2} \left (c x^n\right )^{\left .-\frac {5}{2}\right /n} \operatorname {ExpIntegralEi}\left (\frac {5 \left (a+b \log \left (c x^n\right )\right )}{2 b n}\right )}{b d n} \]

[Out]

(d*x)^(5/2)*Ei(5/2*(a+b*ln(c*x^n))/b/n)/b/d/exp(5/2*a/b/n)/n/((c*x^n)^(5/2/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} \[ \int \frac {(d x)^{3/2}}{a+b \log \left (c x^n\right )} \, dx=\frac {(d x)^{5/2} e^{-\frac {5 a}{2 b n}} \left (c x^n\right )^{\left .-\frac {5}{2}\right /n} \operatorname {ExpIntegralEi}\left (\frac {5 \left (a+b \log \left (c x^n\right )\right )}{2 b n}\right )}{b d n} \]

[In]

Int[(d*x)^(3/2)/(a + b*Log[c*x^n]),x]

[Out]

((d*x)^(5/2)*ExpIntegralEi[(5*(a + b*Log[c*x^n]))/(2*b*n)])/(b*d*E^((5*a)/(2*b*n))*n*(c*x^n)^(5/(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 ((d x)^{5/2} \left (c x^n\right )^{\left .-\frac {5}{2}\right /n}\right ) \text {Subst}\left (\int \frac {e^{\frac {5 x}{2 n}}}{a+b x} \, dx,x,\log \left (c x^n\right )\right )}{d n} \\ & = \frac {e^{-\frac {5 a}{2 b n}} (d x)^{5/2} \left (c x^n\right )^{\left .-\frac {5}{2}\right /n} \text {Ei}\left (\frac {5 \left (a+b \log \left (c x^n\right )\right )}{2 b n}\right )}{b d n} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.97 \[ \int \frac {(d x)^{3/2}}{a+b \log \left (c x^n\right )} \, dx=\frac {e^{-\frac {5 a}{2 b n}} x (d x)^{3/2} \left (c x^n\right )^{\left .-\frac {5}{2}\right /n} \operatorname {ExpIntegralEi}\left (\frac {5 \left (a+b \log \left (c x^n\right )\right )}{2 b n}\right )}{b n} \]

[In]

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

[Out]

(x*(d*x)^(3/2)*ExpIntegralEi[(5*(a + b*Log[c*x^n]))/(2*b*n)])/(b*E^((5*a)/(2*b*n))*n*(c*x^n)^(5/(2*n)))

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

2*b*d^(3/2)*n*integrate(1/5*x^(3/2)/(b^2*log(c)^2 + b^2*log(x^n)^2 + 2*a*b*log(c) + a^2 + 2*(b^2*log(c) + a*b)
*log(x^n)), x) + 2/5*d^(3/2)*x^(5/2)/(b*log(c) + b*log(x^n) + a)

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int((d*x)^(3/2)/(a + b*log(c*x^n)),x)

[Out]

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