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

Optimal. Leaf size=76 \[ \frac{3 x^3 e^{-\frac{3 a}{b n}} \left (c x^n\right )^{-3/n} \text{Ei}\left (\frac{3 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{b^2 n^2}-\frac{x^3}{b n \left (a+b \log \left (c x^n\right )\right )} \]

[Out]

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

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Rubi [A]  time = 0.0793871, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {2306, 2310, 2178} \[ \frac{3 x^3 e^{-\frac{3 a}{b n}} \left (c x^n\right )^{-3/n} \text{Ei}\left (\frac{3 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{b^2 n^2}-\frac{x^3}{b n \left (a+b \log \left (c x^n\right )\right )} \]

Antiderivative was successfully verified.

[In]

Int[x^2/(a + b*Log[c*x^n])^2,x]

[Out]

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

Mathematica [A]  time = 0.120686, size = 70, normalized size = 0.92 \[ \frac{x^3 \left (3 e^{-\frac{3 a}{b n}} \left (c x^n\right )^{-3/n} \text{Ei}\left (\frac{3 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )-\frac{b n}{a+b \log \left (c x^n\right )}\right )}{b^2 n^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/(a + b*Log[c*x^n])^2,x]

[Out]

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

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Maple [F]  time = 0.691, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{2}}{ \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 0.888405, size = 257, normalized size = 3.38 \begin{align*} -\frac{{\left (b n x^{3} e^{\left (\frac{3 \,{\left (b \log \left (c\right ) + a\right )}}{b n}\right )} - 3 \,{\left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )} \logintegral \left (x^{3} e^{\left (\frac{3 \,{\left (b \log \left (c\right ) + a\right )}}{b n}\right )}\right )\right )} e^{\left (-\frac{3 \,{\left (b \log \left (c\right ) + a\right )}}{b n}\right )}}{b^{3} n^{3} \log \left (x\right ) + b^{3} n^{2} \log \left (c\right ) + a b^{2} n^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(b*n*x^3*e^(3*(b*log(c) + a)/(b*n)) - 3*(b*n*log(x) + b*log(c) + a)*log_integral(x^3*e^(3*(b*log(c) + a)/(b*n
))))*e^(-3*(b*log(c) + a)/(b*n))/(b^3*n^3*log(x) + b^3*n^2*log(c) + a*b^2*n^2)

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

[Out]

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

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Giac [B]  time = 1.33935, size = 352, normalized size = 4.63 \begin{align*} -\frac{b n x^{3}}{b^{3} n^{3} \log \left (x\right ) + b^{3} n^{2} \log \left (c\right ) + a b^{2} n^{2}} + \frac{3 \, b n{\rm Ei}\left (\frac{3 \, \log \left (c\right )}{n} + \frac{3 \, a}{b n} + 3 \, \log \left (x\right )\right ) e^{\left (-\frac{3 \, a}{b n}\right )} \log \left (x\right )}{{\left (b^{3} n^{3} \log \left (x\right ) + b^{3} n^{2} \log \left (c\right ) + a b^{2} n^{2}\right )} c^{\frac{3}{n}}} + \frac{3 \, b{\rm Ei}\left (\frac{3 \, \log \left (c\right )}{n} + \frac{3 \, a}{b n} + 3 \, \log \left (x\right )\right ) e^{\left (-\frac{3 \, a}{b n}\right )} \log \left (c\right )}{{\left (b^{3} n^{3} \log \left (x\right ) + b^{3} n^{2} \log \left (c\right ) + a b^{2} n^{2}\right )} c^{\frac{3}{n}}} + \frac{3 \, a{\rm Ei}\left (\frac{3 \, \log \left (c\right )}{n} + \frac{3 \, a}{b n} + 3 \, \log \left (x\right )\right ) e^{\left (-\frac{3 \, a}{b n}\right )}}{{\left (b^{3} n^{3} \log \left (x\right ) + b^{3} n^{2} \log \left (c\right ) + a b^{2} n^{2}\right )} c^{\frac{3}{n}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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