\(\int \frac {x^2}{(a+b \log (c x^n))^3} \, dx\) [82]

Optimal result

Integrand size = 16, antiderivative size = 105 \[ \int \frac {x^2}{\left (a+b \log \left (c x^n\right )\right )^3} \, dx=\frac {9 e^{-\frac {3 a}{b n}} x^3 \left (c x^n\right )^{-3/n} \operatorname {ExpIntegralEi}\left (\frac {3 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{2 b^3 n^3}-\frac {x^3}{2 b n \left (a+b \log \left (c x^n\right )\right )^2}-\frac {3 x^3}{2 b^2 n^2 \left (a+b \log \left (c x^n\right )\right )} \]

[Out]

9/2*x^3*Ei(3*(a+b*ln(c*x^n))/b/n)/b^3/exp(3*a/b/n)/n^3/((c*x^n)^(3/n))-1/2*x^3/b/n/(a+b*ln(c*x^n))^2-3/2*x^3/b
^2/n^2/(a+b*ln(c*x^n))

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {2343, 2347, 2209} \[ \int \frac {x^2}{\left (a+b \log \left (c x^n\right )\right )^3} \, dx=\frac {9 x^3 e^{-\frac {3 a}{b n}} \left (c x^n\right )^{-3/n} \operatorname {ExpIntegralEi}\left (\frac {3 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{2 b^3 n^3}-\frac {3 x^3}{2 b^2 n^2 \left (a+b \log \left (c x^n\right )\right )}-\frac {x^3}{2 b n \left (a+b \log \left (c x^n\right )\right )^2} \]

[In]

Int[x^2/(a + b*Log[c*x^n])^3,x]

[Out]

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

Rule 2209

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

Rule 2343

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 2347

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

Rubi steps \begin{align*} \text {integral}& = -\frac {x^3}{2 b n \left (a+b \log \left (c x^n\right )\right )^2}+\frac {3 \int \frac {x^2}{\left (a+b \log \left (c x^n\right )\right )^2} \, dx}{2 b n} \\ & = -\frac {x^3}{2 b n \left (a+b \log \left (c x^n\right )\right )^2}-\frac {3 x^3}{2 b^2 n^2 \left (a+b \log \left (c x^n\right )\right )}+\frac {9 \int \frac {x^2}{a+b \log \left (c x^n\right )} \, dx}{2 b^2 n^2} \\ & = -\frac {x^3}{2 b n \left (a+b \log \left (c x^n\right )\right )^2}-\frac {3 x^3}{2 b^2 n^2 \left (a+b \log \left (c x^n\right )\right )}+\frac {\left (9 x^3 \left (c x^n\right )^{-3/n}\right ) \text {Subst}\left (\int \frac {e^{\frac {3 x}{n}}}{a+b x} \, dx,x,\log \left (c x^n\right )\right )}{2 b^2 n^3} \\ & = \frac {9 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 )}{2 b^3 n^3}-\frac {x^3}{2 b n \left (a+b \log \left (c x^n\right )\right )^2}-\frac {3 x^3}{2 b^2 n^2 \left (a+b \log \left (c x^n\right )\right )} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.85 \[ \int \frac {x^2}{\left (a+b \log \left (c x^n\right )\right )^3} \, dx=\frac {x^3 \left (9 e^{-\frac {3 a}{b n}} \left (c x^n\right )^{-3/n} \operatorname {ExpIntegralEi}\left (\frac {3 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )-\frac {b n \left (3 a+b n+3 b \log \left (c x^n\right )\right )}{\left (a+b \log \left (c x^n\right )\right )^2}\right )}{2 b^3 n^3} \]

[In]

Integrate[x^2/(a + b*Log[c*x^n])^3,x]

[Out]

(x^3*((9*ExpIntegralEi[(3*(a + b*Log[c*x^n]))/(b*n)])/(E^((3*a)/(b*n))*(c*x^n)^(3/n)) - (b*n*(3*a + b*n + 3*b*
Log[c*x^n]))/(a + b*Log[c*x^n])^2))/(2*b^3*n^3)

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.36 (sec) , antiderivative size = 477, normalized size of antiderivative = 4.54

method	result	size
risch	\(-\frac {2 b n \,x^{3}-3 i \pi b \,x^{3} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+3 i \pi b \,x^{3} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+3 i \pi b \,x^{3} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-3 i \pi b \,x^{3} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+6 \ln \left (c \right ) b \,x^{3}+6 b \,x^{3} \ln \left (x^{n}\right )+6 x^{3} a}{{\left (-i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+2 b \ln \left (c \right )+2 \ln \left (x^{n}\right ) b +2 a \right )}^{2} b^{2} n^{2}}-\frac {9 x^{3} c^{-\frac {3}{n}} \left (x^{n}\right )^{-\frac {3}{n}} {\mathrm e}^{-\frac {3 \left (-i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+2 a \right )}{2 b n}} \operatorname {Ei}_{1}\left (-3 \ln \left (x \right )-\frac {3 \left (-i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+2 b \ln \left (c \right )+2 b \left (\ln \left (x^{n}\right )-n \ln \left (x \right )\right )+2 a \right )}{2 b n}\right )}{2 b^{3} n^{3}}\)	\(477\)

[In]

int(x^2/(a+b*ln(c*x^n))^3,x,method=_RETURNVERBOSE)

[Out]

-(2*b*n*x^3-3*I*Pi*b*x^3*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+3*I*Pi*b*x^3*csgn(I*c)*csgn(I*c*x^n)^2+3*I*Pi*b*x
^3*csgn(I*x^n)*csgn(I*c*x^n)^2-3*I*Pi*b*x^3*csgn(I*c*x^n)^3+6*ln(c)*b*x^3+6*b*x^3*ln(x^n)+6*x^3*a)/(-I*b*Pi*cs
gn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+I*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2+I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-I*b*Pi*c
sgn(I*c*x^n)^3+2*b*ln(c)+2*ln(x^n)*b+2*a)^2/b^2/n^2-9/2/b^3/n^3*x^3*c^(-3/n)*(x^n)^(-3/n)*exp(-3/2*(-I*b*Pi*cs
gn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+I*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2+I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-I*b*Pi*c
sgn(I*c*x^n)^3+2*a)/b/n)*Ei(1,-3*ln(x)-3/2*(-I*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+I*b*Pi*csgn(I*c)*csgn(
I*c*x^n)^2+I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-I*b*Pi*csgn(I*c*x^n)^3+2*b*ln(c)+2*b*(ln(x^n)-n*ln(x))+2*a)/b/n)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 211 vs. \(2 (100) = 200\).

Time = 0.30 (sec) , antiderivative size = 211, normalized size of antiderivative = 2.01 \[ \int \frac {x^2}{\left (a+b \log \left (c x^n\right )\right )^3} \, dx=-\frac {{\left ({\left (3 \, b^{2} n^{2} x^{3} \log \left (x\right ) + 3 \, b^{2} n x^{3} \log \left (c\right ) + {\left (b^{2} n^{2} + 3 \, a b n\right )} x^{3}\right )} e^{\left (\frac {3 \, {\left (b \log \left (c\right ) + a\right )}}{b n}\right )} - 9 \, {\left (b^{2} n^{2} \log \left (x\right )^{2} + b^{2} \log \left (c\right )^{2} + 2 \, a b \log \left (c\right ) + a^{2} + 2 \, {\left (b^{2} n \log \left (c\right ) + a b n\right )} \log \left (x\right )\right )} \operatorname {log\_integral}\left (x^{3} e^{\left (\frac {3 \, {\left (b \log \left (c\right ) + a\right )}}{b n}\right )}\right )\right )} e^{\left (-\frac {3 \, {\left (b \log \left (c\right ) + a\right )}}{b n}\right )}}{2 \, {\left (b^{5} n^{5} \log \left (x\right )^{2} + b^{5} n^{3} \log \left (c\right )^{2} + 2 \, a b^{4} n^{3} \log \left (c\right ) + a^{2} b^{3} n^{3} + 2 \, {\left (b^{5} n^{4} \log \left (c\right ) + a b^{4} n^{4}\right )} \log \left (x\right )\right )}} \]

[In]

integrate(x^2/(a+b*log(c*x^n))^3,x, algorithm="fricas")

[Out]

-1/2*((3*b^2*n^2*x^3*log(x) + 3*b^2*n*x^3*log(c) + (b^2*n^2 + 3*a*b*n)*x^3)*e^(3*(b*log(c) + a)/(b*n)) - 9*(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^3*e^(3*(b
*log(c) + a)/(b*n))))*e^(-3*(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))

Sympy [F]

\[ \int \frac {x^2}{\left (a+b \log \left (c x^n\right )\right )^3} \, dx=\int \frac {x^{2}}{\left (a + b \log {\left (c x^{n} \right )}\right )^{3}}\, dx \]

[In]

integrate(x**2/(a+b*ln(c*x**n))**3,x)

[Out]

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

Maxima [F]

\[ \int \frac {x^2}{\left (a+b \log \left (c x^n\right )\right )^3} \, dx=\int { \frac {x^{2}}{{\left (b \log \left (c x^{n}\right ) + a\right )}^{3}} \,d x } \]

[In]

integrate(x^2/(a+b*log(c*x^n))^3,x, algorithm="maxima")

[Out]

-1/2*(3*b*x^3*log(x^n) + (b*(n + 3*log(c)) + 3*a)*x^3)/(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)) + 9*integrate(1/2*x^2/(b^3*n^2*log(c) + b^3*n^2*
log(x^n) + a*b^2*n^2), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1029 vs. \(2 (100) = 200\).

Time = 0.36 (sec) , antiderivative size = 1029, normalized size of antiderivative = 9.80 \[ \int \frac {x^2}{\left (a+b \log \left (c x^n\right )\right )^3} \, dx=\text {Too large to display} \]

[In]

integrate(x^2/(a+b*log(c*x^n))^3,x, algorithm="giac")

[Out]

-3/2*b^2*n^2*x^3*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^3/(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) - 3/2*b^2*n*x^3*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) + 9/2*b^2*n^2*E
i(3*log(c)/n + 3*a/(b*n) + 3*log(x))*e^(-3*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^(3/n)) - 3/2*a*b*n*x^3/(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) + 9
*b^2*n*Ei(3*log(c)/n + 3*a/(b*n) + 3*log(x))*e^(-3*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^(3/n)) + 9/2*b^2*Ei(3*l
og(c)/n + 3*a/(b*n) + 3*log(x))*e^(-3*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^(3/n)) + 9*a*b*n*Ei(3*log(c)/n + 3*a/(b*n
) + 3*log(x))*e^(-3*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^(3/n)) + 9*a*b*Ei(3*log(c)/n + 3*a/(b*n) + 3*log(x))*e^(-3*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^(3/n)) + 9/2*a^2*Ei(3*log(c)/n + 3*a/(b*n) + 3*log(x))*e^(-3*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^
(3/n))

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2}{\left (a+b \log \left (c x^n\right )\right )^3} \, dx=\int \frac {x^2}{{\left (a+b\,\ln \left (c\,x^n\right )\right )}^3} \,d x \]

[In]

int(x^2/(a + b*log(c*x^n))^3,x)

[Out]

int(x^2/(a + b*log(c*x^n))^3, x)