3.81 \(\int \frac{x^3}{(a+b \log (c x^n))^3} \, dx\)

Optimal. Leaf size=101 \[ \frac{8 x^4 e^{-\frac{4 a}{b n}} \left (c x^n\right )^{-4/n} \text{Ei}\left (\frac{4 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{b^3 n^3}-\frac{2 x^4}{b^2 n^2 \left (a+b \log \left (c x^n\right )\right )}-\frac{x^4}{2 b n \left (a+b \log \left (c x^n\right )\right )^2} \]

[Out]

(8*x^4*ExpIntegralEi[(4*(a + b*Log[c*x^n]))/(b*n)])/(b^3*E^((4*a)/(b*n))*n^3*(c*x^n)^(4/n)) - x^4/(2*b*n*(a +
b*Log[c*x^n])^2) - (2*x^4)/(b^2*n^2*(a + b*Log[c*x^n]))

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Rubi [A]  time = 0.10761, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {2306, 2310, 2178} \[ \frac{8 x^4 e^{-\frac{4 a}{b n}} \left (c x^n\right )^{-4/n} \text{Ei}\left (\frac{4 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{b^3 n^3}-\frac{2 x^4}{b^2 n^2 \left (a+b \log \left (c x^n\right )\right )}-\frac{x^4}{2 b n \left (a+b \log \left (c x^n\right )\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[x^3/(a + b*Log[c*x^n])^3,x]

[Out]

(8*x^4*ExpIntegralEi[(4*(a + b*Log[c*x^n]))/(b*n)])/(b^3*E^((4*a)/(b*n))*n^3*(c*x^n)^(4/n)) - x^4/(2*b*n*(a +
b*Log[c*x^n])^2) - (2*x^4)/(b^2*n^2*(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^3}{\left (a+b \log \left (c x^n\right )\right )^3} \, dx &=-\frac{x^4}{2 b n \left (a+b \log \left (c x^n\right )\right )^2}+\frac{2 \int \frac{x^3}{\left (a+b \log \left (c x^n\right )\right )^2} \, dx}{b n}\\ &=-\frac{x^4}{2 b n \left (a+b \log \left (c x^n\right )\right )^2}-\frac{2 x^4}{b^2 n^2 \left (a+b \log \left (c x^n\right )\right )}+\frac{8 \int \frac{x^3}{a+b \log \left (c x^n\right )} \, dx}{b^2 n^2}\\ &=-\frac{x^4}{2 b n \left (a+b \log \left (c x^n\right )\right )^2}-\frac{2 x^4}{b^2 n^2 \left (a+b \log \left (c x^n\right )\right )}+\frac{\left (8 x^4 \left (c x^n\right )^{-4/n}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{4 x}{n}}}{a+b x} \, dx,x,\log \left (c x^n\right )\right )}{b^2 n^3}\\ &=\frac{8 e^{-\frac{4 a}{b n}} x^4 \left (c x^n\right )^{-4/n} \text{Ei}\left (\frac{4 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{b^3 n^3}-\frac{x^4}{2 b n \left (a+b \log \left (c x^n\right )\right )^2}-\frac{2 x^4}{b^2 n^2 \left (a+b \log \left (c x^n\right )\right )}\\ \end{align*}

Mathematica [A]  time = 0.135277, size = 89, normalized size = 0.88 \[ \frac{x^4 \left (16 e^{-\frac{4 a}{b n}} \left (c x^n\right )^{-4/n} \text{Ei}\left (\frac{4 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )-\frac{b n \left (4 a+4 b \log \left (c x^n\right )+b n\right )}{\left (a+b \log \left (c x^n\right )\right )^2}\right )}{2 b^3 n^3} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3/(a + b*Log[c*x^n])^3,x]

[Out]

(x^4*((16*ExpIntegralEi[(4*(a + b*Log[c*x^n]))/(b*n)])/(E^((4*a)/(b*n))*(c*x^n)^(4/n)) - (b*n*(4*a + b*n + 4*b
*Log[c*x^n]))/(a + b*Log[c*x^n])^2))/(2*b^3*n^3)

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Maple [F]  time = 0.705, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{3}}{ \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{4 \, b x^{4} \log \left (x^{n}\right ) +{\left (b{\left (n + 4 \, \log \left (c\right )\right )} + 4 \, a\right )} x^{4}}{2 \,{\left (b^{4} n^{2} \log \left (c\right )^{2} + b^{4} n^{2} \log \left (x^{n}\right )^{2} + 2 \, a b^{3} n^{2} \log \left (c\right ) + a^{2} b^{2} n^{2} + 2 \,{\left (b^{4} n^{2} \log \left (c\right ) + a b^{3} n^{2}\right )} \log \left (x^{n}\right )\right )}} + 8 \, \int \frac{x^{3}}{b^{3} n^{2} \log \left (c\right ) + b^{3} n^{2} \log \left (x^{n}\right ) + a b^{2} n^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(4*b*x^4*log(x^n) + (b*(n + 4*log(c)) + 4*a)*x^4)/(b^4*n^2*log(c)^2 + b^4*n^2*log(x^n)^2 + 2*a*b^3*n^2*lo
g(c) + a^2*b^2*n^2 + 2*(b^4*n^2*log(c) + a*b^3*n^2)*log(x^n)) + 8*integrate(x^3/(b^3*n^2*log(c) + b^3*n^2*log(
x^n) + a*b^2*n^2), x)

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Fricas [B]  time = 0.827485, size = 520, normalized size = 5.15 \begin{align*} -\frac{{\left ({\left (4 \, b^{2} n^{2} x^{4} \log \left (x\right ) + 4 \, b^{2} n x^{4} \log \left (c\right ) +{\left (b^{2} n^{2} + 4 \, a b n\right )} x^{4}\right )} e^{\left (\frac{4 \,{\left (b \log \left (c\right ) + a\right )}}{b n}\right )} - 16 \,{\left (b^{2} n^{2} \log \left (x\right )^{2} + b^{2} \log \left (c\right )^{2} + 2 \, a b \log \left (c\right ) + a^{2} + 2 \,{\left (b^{2} n \log \left (c\right ) + a b n\right )} \log \left (x\right )\right )} \logintegral \left (x^{4} e^{\left (\frac{4 \,{\left (b \log \left (c\right ) + a\right )}}{b n}\right )}\right )\right )} e^{\left (-\frac{4 \,{\left (b \log \left (c\right ) + a\right )}}{b n}\right )}}{2 \,{\left (b^{5} n^{5} \log \left (x\right )^{2} + b^{5} n^{3} \log \left (c\right )^{2} + 2 \, a b^{4} n^{3} \log \left (c\right ) + a^{2} b^{3} n^{3} + 2 \,{\left (b^{5} n^{4} \log \left (c\right ) + a b^{4} n^{4}\right )} \log \left (x\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*((4*b^2*n^2*x^4*log(x) + 4*b^2*n*x^4*log(c) + (b^2*n^2 + 4*a*b*n)*x^4)*e^(4*(b*log(c) + a)/(b*n)) - 16*(b
^2*n^2*log(x)^2 + b^2*log(c)^2 + 2*a*b*log(c) + a^2 + 2*(b^2*n*log(c) + a*b*n)*log(x))*log_integral(x^4*e^(4*(
b*log(c) + a)/(b*n))))*e^(-4*(b*log(c) + a)/(b*n))/(b^5*n^5*log(x)^2 + b^5*n^3*log(c)^2 + 2*a*b^4*n^3*log(c) +
a^2*b^3*n^3 + 2*(b^5*n^4*log(c) + a*b^4*n^4)*log(x))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.44135, size = 1389, normalized size = 13.75 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*b^2*n^2*x^4*log(x)/(b^5*n^5*log(x)^2 + 2*b^5*n^4*log(c)*log(x) + b^5*n^3*log(c)^2 + 2*a*b^4*n^4*log(x) + 2*
a*b^4*n^3*log(c) + a^2*b^3*n^3) - 1/2*b^2*n^2*x^4/(b^5*n^5*log(x)^2 + 2*b^5*n^4*log(c)*log(x) + b^5*n^3*log(c)
^2 + 2*a*b^4*n^4*log(x) + 2*a*b^4*n^3*log(c) + a^2*b^3*n^3) - 2*b^2*n*x^4*log(c)/(b^5*n^5*log(x)^2 + 2*b^5*n^4
*log(c)*log(x) + b^5*n^3*log(c)^2 + 2*a*b^4*n^4*log(x) + 2*a*b^4*n^3*log(c) + a^2*b^3*n^3) - 2*a*b*n*x^4/(b^5*
n^5*log(x)^2 + 2*b^5*n^4*log(c)*log(x) + b^5*n^3*log(c)^2 + 2*a*b^4*n^4*log(x) + 2*a*b^4*n^3*log(c) + a^2*b^3*
n^3) + 8*b^2*n^2*Ei(4*log(c)/n + 4*a/(b*n) + 4*log(x))*e^(-4*a/(b*n))*log(x)^2/((b^5*n^5*log(x)^2 + 2*b^5*n^4*
log(c)*log(x) + b^5*n^3*log(c)^2 + 2*a*b^4*n^4*log(x) + 2*a*b^4*n^3*log(c) + a^2*b^3*n^3)*c^(4/n)) + 16*b^2*n*
Ei(4*log(c)/n + 4*a/(b*n) + 4*log(x))*e^(-4*a/(b*n))*log(c)*log(x)/((b^5*n^5*log(x)^2 + 2*b^5*n^4*log(c)*log(x
) + b^5*n^3*log(c)^2 + 2*a*b^4*n^4*log(x) + 2*a*b^4*n^3*log(c) + a^2*b^3*n^3)*c^(4/n)) + 8*b^2*Ei(4*log(c)/n +
4*a/(b*n) + 4*log(x))*e^(-4*a/(b*n))*log(c)^2/((b^5*n^5*log(x)^2 + 2*b^5*n^4*log(c)*log(x) + b^5*n^3*log(c)^2
+ 2*a*b^4*n^4*log(x) + 2*a*b^4*n^3*log(c) + a^2*b^3*n^3)*c^(4/n)) + 16*a*b*n*Ei(4*log(c)/n + 4*a/(b*n) + 4*lo
g(x))*e^(-4*a/(b*n))*log(x)/((b^5*n^5*log(x)^2 + 2*b^5*n^4*log(c)*log(x) + b^5*n^3*log(c)^2 + 2*a*b^4*n^4*log(
x) + 2*a*b^4*n^3*log(c) + a^2*b^3*n^3)*c^(4/n)) + 16*a*b*Ei(4*log(c)/n + 4*a/(b*n) + 4*log(x))*e^(-4*a/(b*n))*
log(c)/((b^5*n^5*log(x)^2 + 2*b^5*n^4*log(c)*log(x) + b^5*n^3*log(c)^2 + 2*a*b^4*n^4*log(x) + 2*a*b^4*n^3*log(
c) + a^2*b^3*n^3)*c^(4/n)) + 8*a^2*Ei(4*log(c)/n + 4*a/(b*n) + 4*log(x))*e^(-4*a/(b*n))/((b^5*n^5*log(x)^2 + 2
*b^5*n^4*log(c)*log(x) + b^5*n^3*log(c)^2 + 2*a*b^4*n^4*log(x) + 2*a*b^4*n^3*log(c) + a^2*b^3*n^3)*c^(4/n))