3.109 \(\int \frac{\sqrt{d x}}{(a+b \log (c x^n))^2} \, dx\)

Optimal. Leaf size=98 \[ \frac{3 (d x)^{3/2} e^{-\frac{3 a}{2 b n}} \left (c x^n\right )^{\left .-\frac{3}{2}\right /n} \text{Ei}\left (\frac{3 \left (a+b \log \left (c x^n\right )\right )}{2 b n}\right )}{2 b^2 d n^2}-\frac{(d x)^{3/2}}{b d n \left (a+b \log \left (c x^n\right )\right )} \]

[Out]

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

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Rubi [A]  time = 0.0870387, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {2306, 2310, 2178} \[ \frac{3 (d x)^{3/2} e^{-\frac{3 a}{2 b n}} \left (c x^n\right )^{\left .-\frac{3}{2}\right /n} \text{Ei}\left (\frac{3 \left (a+b \log \left (c x^n\right )\right )}{2 b n}\right )}{2 b^2 d n^2}-\frac{(d x)^{3/2}}{b d n \left (a+b \log \left (c x^n\right )\right )} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[d*x]/(a + b*Log[c*x^n])^2,x]

[Out]

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

Rule 2306

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

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{\sqrt{d x}}{\left (a+b \log \left (c x^n\right )\right )^2} \, dx &=-\frac{(d x)^{3/2}}{b d n \left (a+b \log \left (c x^n\right )\right )}+\frac{3 \int \frac{\sqrt{d x}}{a+b \log \left (c x^n\right )} \, dx}{2 b n}\\ &=-\frac{(d x)^{3/2}}{b d n \left (a+b \log \left (c x^n\right )\right )}+\frac{\left (3 (d x)^{3/2} \left (c x^n\right )^{\left .-\frac{3}{2}\right /n}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{3 x}{2 n}}}{a+b x} \, dx,x,\log \left (c x^n\right )\right )}{2 b d n^2}\\ &=\frac{3 e^{-\frac{3 a}{2 b n}} (d x)^{3/2} \left (c x^n\right )^{\left .-\frac{3}{2}\right /n} \text{Ei}\left (\frac{3 \left (a+b \log \left (c x^n\right )\right )}{2 b n}\right )}{2 b^2 d n^2}-\frac{(d x)^{3/2}}{b d n \left (a+b \log \left (c x^n\right )\right )}\\ \end{align*}

Mathematica [A]  time = 0.132666, size = 84, normalized size = 0.86 \[ \frac{x \sqrt{d x} \left (3 e^{-\frac{3 a}{2 b n}} \left (c x^n\right )^{\left .-\frac{3}{2}\right /n} \text{Ei}\left (\frac{3 \left (a+b \log \left (c x^n\right )\right )}{2 b n}\right )-\frac{2 b n}{a+b \log \left (c x^n\right )}\right )}{2 b^2 n^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 4.658, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{2}}\sqrt{dx}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} 4 \, b \sqrt{d} n \int \frac{\sqrt{x}}{3 \,{\left (b^{3} \log \left (c\right )^{3} + b^{3} \log \left (x^{n}\right )^{3} + 3 \, a b^{2} \log \left (c\right )^{2} + 3 \, a^{2} b \log \left (c\right ) + a^{3} + 3 \,{\left (b^{3} \log \left (c\right ) + a b^{2}\right )} \log \left (x^{n}\right )^{2} + 3 \,{\left (b^{3} \log \left (c\right )^{2} + 2 \, a b^{2} \log \left (c\right ) + a^{2} b\right )} \log \left (x^{n}\right )\right )}}\,{d x} + \frac{2 \, \sqrt{d} x^{\frac{3}{2}}}{3 \,{\left (b^{2} \log \left (c\right )^{2} + b^{2} \log \left (x^{n}\right )^{2} + 2 \, a b \log \left (c\right ) + a^{2} + 2 \,{\left (b^{2} \log \left (c\right ) + a b\right )} \log \left (x^{n}\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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