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

Optimal. Leaf size=98 \[ -\frac{e^{\frac{a}{2 b n}} \left (c x^n\right )^{\left .\frac{1}{2}\right /n} \text{Ei}\left (-\frac{a+b \log \left (c x^n\right )}{2 b n}\right )}{2 b^2 d n^2 \sqrt{d x}}-\frac{1}{b d n \sqrt{d x} \left (a+b \log \left (c x^n\right )\right )} \]

[Out]

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

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Rubi [A]  time = 0.0925152, 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} \[ -\frac{e^{\frac{a}{2 b n}} \left (c x^n\right )^{\left .\frac{1}{2}\right /n} \text{Ei}\left (-\frac{a+b \log \left (c x^n\right )}{2 b n}\right )}{2 b^2 d n^2 \sqrt{d x}}-\frac{1}{b d n \sqrt{d x} \left (a+b \log \left (c x^n\right )\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(E^(a/(2*b*n))*(c*x^n)^(1/(2*n))*ExpIntegralEi[-(a + b*Log[c*x^n])/(2*b*n)])/(2*b^2*d*n^2*Sqrt[d*x]) - 1/(b*d
*n*Sqrt[d*x]*(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{1}{(d x)^{3/2} \left (a+b \log \left (c x^n\right )\right )^2} \, dx &=-\frac{1}{b d n \sqrt{d x} \left (a+b \log \left (c x^n\right )\right )}-\frac{\int \frac{1}{(d x)^{3/2} \left (a+b \log \left (c x^n\right )\right )} \, dx}{2 b n}\\ &=-\frac{1}{b d n \sqrt{d x} \left (a+b \log \left (c x^n\right )\right )}-\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 )}{2 b d n^2 \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 )}{2 b^2 d n^2 \sqrt{d x}}-\frac{1}{b d n \sqrt{d x} \left (a+b \log \left (c x^n\right )\right )}\\ \end{align*}

Mathematica [A]  time = 0.120076, size = 93, normalized size = 0.95 \[ -\frac{x \left (e^{\frac{a}{2 b n}} \left (c x^n\right )^{\left .\frac{1}{2}\right /n} \left (a+b \log \left (c x^n\right )\right ) \text{Ei}\left (-\frac{a+b \log \left (c x^n\right )}{2 b n}\right )+2 b n\right )}{2 b^2 n^2 (d x)^{3/2} \left (a+b \log \left (c x^n\right )\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 4.557, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{2}} \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))^2,x)

[Out]

int(1/(d*x)^(3/2)/(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 n \int \frac{1}{{\left (b^{3} d^{\frac{3}{2}} \log \left (c\right )^{3} + b^{3} d^{\frac{3}{2}} \log \left (x^{n}\right )^{3} + 3 \, a b^{2} d^{\frac{3}{2}} \log \left (c\right )^{2} + 3 \, a^{2} b d^{\frac{3}{2}} \log \left (c\right ) + a^{3} d^{\frac{3}{2}} + 3 \,{\left (b^{3} d^{\frac{3}{2}} \log \left (c\right ) + a b^{2} d^{\frac{3}{2}}\right )} \log \left (x^{n}\right )^{2} + 3 \,{\left (b^{3} d^{\frac{3}{2}} \log \left (c\right )^{2} + 2 \, a b^{2} d^{\frac{3}{2}} \log \left (c\right ) + a^{2} b d^{\frac{3}{2}}\right )} \log \left (x^{n}\right )\right )} x^{\frac{3}{2}}}\,{d x} - \frac{2}{{\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 )} \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))^2,x, algorithm="maxima")

[Out]

-4*b*n*integrate(1/((b^3*d^(3/2)*log(c)^3 + b^3*d^(3/2)*log(x^n)^3 + 3*a*b^2*d^(3/2)*log(c)^2 + 3*a^2*b*d^(3/2
)*log(c) + a^3*d^(3/2) + 3*(b^3*d^(3/2)*log(c) + a*b^2*d^(3/2))*log(x^n)^2 + 3*(b^3*d^(3/2)*log(c)^2 + 2*a*b^2
*d^(3/2)*log(c) + a^2*b*d^(3/2))*log(x^n))*x^(3/2)), x) - 2/((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))*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^{2} d^{2} x^{2} \log \left (c x^{n}\right )^{2} + 2 \, a b d^{2} x^{2} \log \left (c x^{n}\right ) + a^{2} 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))^2,x, algorithm="fricas")

[Out]

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

[Out]

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

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Giac [B]  time = 1.31324, size = 379, normalized size = 3.87 \begin{align*} -\frac{\frac{b c^{\frac{1}{2 \, n}} \sqrt{d} 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 )} \log \left (x\right )}{b^{3} d n^{3} \log \left (x\right ) + b^{3} d n^{2} \log \left (c\right ) + a b^{2} d n^{2}} + \frac{b c^{\frac{1}{2 \, n}} \sqrt{d}{\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 )} \log \left (c\right )}{b^{3} d n^{3} \log \left (x\right ) + b^{3} d n^{2} \log \left (c\right ) + a b^{2} d n^{2}} + \frac{a c^{\frac{1}{2 \, n}} \sqrt{d}{\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^{3} d n^{3} \log \left (x\right ) + b^{3} d n^{2} \log \left (c\right ) + a b^{2} d n^{2}} + \frac{2 \, b \sqrt{d} n}{{\left (b^{3} d n^{3} \log \left (x\right ) + b^{3} d n^{2} \log \left (c\right ) + a b^{2} d n^{2}\right )} \sqrt{x}}}{2 \, d} \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))^2,x, algorithm="giac")

[Out]

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