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

Optimal. Leaf size=64 \[ \frac{e^{\frac{a}{2 b n}} \left (c x^n\right )^{\left .\frac{1}{2}\right /n} \text{Ei}\left (-\frac{a+b \log \left (c x^n\right )}{2 b n}\right )}{b d n \sqrt{d x}} \]

[Out]

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

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Rubi [A]  time = 0.0607804, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {2310, 2178} \[ \frac{e^{\frac{a}{2 b n}} \left (c x^n\right )^{\left .\frac{1}{2}\right /n} \text{Ei}\left (-\frac{a+b \log \left (c x^n\right )}{2 b n}\right )}{b d n \sqrt{d x}} \]

Antiderivative was successfully verified.

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

Rule 2310

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)*x)/n)*(a + b*x)^p, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, d, m, n, p}
, x]

Rule 2178

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

Rubi steps

\begin{align*} \int \frac{1}{(d x)^{3/2} \left (a+b \log \left (c x^n\right )\right )} \, dx &=\frac{\left (c x^n\right )^{\left .\frac{1}{2}\right /n} \operatorname{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]  time = 0.0712396, size = 62, normalized size = 0.97 \[ \frac{x e^{\frac{a}{2 b n}} \left (c x^n\right )^{\left .\frac{1}{2}\right /n} \text{Ei}\left (-\frac{a+b \log \left (c x^n\right )}{2 b n}\right )}{b n (d x)^{3/2}} \]

Antiderivative was successfully verified.

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

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Maple [F]  time = 0.109, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{a+b\ln \left ( c{x}^{n} \right ) } \left ( dx \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -2 \, b n \int \frac{1}{{\left (b^{2} d^{\frac{3}{2}} \log \left (c\right )^{2} + b^{2} d^{\frac{3}{2}} \log \left (x^{n}\right )^{2} + 2 \, a b d^{\frac{3}{2}} \log \left (c\right ) + a^{2} d^{\frac{3}{2}} + 2 \,{\left (b^{2} d^{\frac{3}{2}} \log \left (c\right ) + a b d^{\frac{3}{2}}\right )} \log \left (x^{n}\right )\right )} x^{\frac{3}{2}}}\,{d x} - \frac{2}{{\left (b d^{\frac{3}{2}} \log \left (c\right ) + b d^{\frac{3}{2}} \log \left (x^{n}\right ) + a d^{\frac{3}{2}}\right )} \sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (d x\right )^{\frac{3}{2}} \left (a + b \log{\left (c x^{n} \right )}\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [A]  time = 1.28626, size = 66, normalized size = 1.03 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)