\(\int \frac {\log ^{-1+q}(c x^n) (a x^m+b \log ^q(c x^n))^3}{x} \, dx\) [2]

Optimal result

Integrand size = 32, antiderivative size = 231 \[ \int \frac {\log ^{-1+q}\left (c x^n\right ) \left (a x^m+b \log ^q\left (c x^n\right )\right )^3}{x} \, dx=\frac {b^3 \log ^{4 q}\left (c x^n\right )}{4 n q}-\frac {3 a b^2 x^m \left (c x^n\right )^{-\frac {m}{n}} \Gamma \left (3 q,-\frac {m \log \left (c x^n\right )}{n}\right ) \log ^{3 q}\left (c x^n\right ) \left (-\frac {m \log \left (c x^n\right )}{n}\right )^{-3 q}}{n}-\frac {3\ 4^{-q} a^2 b x^{2 m} \left (c x^n\right )^{-\frac {2 m}{n}} \Gamma \left (2 q,-\frac {2 m \log \left (c x^n\right )}{n}\right ) \log ^{2 q}\left (c x^n\right ) \left (-\frac {m \log \left (c x^n\right )}{n}\right )^{-2 q}}{n}-\frac {3^{-q} a^3 x^{3 m} \left (c x^n\right )^{-\frac {3 m}{n}} \Gamma \left (q,-\frac {3 m \log \left (c x^n\right )}{n}\right ) \log ^q\left (c x^n\right ) \left (-\frac {m \log \left (c x^n\right )}{n}\right )^{-q}}{n} \]

[Out]

1/4*b^3*ln(c*x^n)^(4*q)/n/q-3*a*b^2*x^m*GAMMA(3*q,-m*ln(c*x^n)/n)*ln(c*x^n)^(3*q)/n/((c*x^n)^(m/n))/((-m*ln(c*
x^n)/n)^(3*q))-3*a^2*b*x^(2*m)*GAMMA(2*q,-2*m*ln(c*x^n)/n)*ln(c*x^n)^(2*q)/(4^q)/n/((c*x^n)^(2*m/n))/((-m*ln(c
*x^n)/n)^(2*q))-a^3*x^(3*m)*GAMMA(q,-3*m*ln(c*x^n)/n)*ln(c*x^n)^q/(3^q)/n/((c*x^n)^(3*m/n))/((-m*ln(c*x^n)/n)^
q)

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {2619, 2347, 2212, 2339, 30} \[ \int \frac {\log ^{-1+q}\left (c x^n\right ) \left (a x^m+b \log ^q\left (c x^n\right )\right )^3}{x} \, dx=-\frac {a^3 3^{-q} x^{3 m} \left (c x^n\right )^{-\frac {3 m}{n}} \log ^q\left (c x^n\right ) \left (-\frac {m \log \left (c x^n\right )}{n}\right )^{-q} \Gamma \left (q,-\frac {3 m \log \left (c x^n\right )}{n}\right )}{n}-\frac {3 a^2 b 4^{-q} x^{2 m} \left (c x^n\right )^{-\frac {2 m}{n}} \log ^{2 q}\left (c x^n\right ) \left (-\frac {m \log \left (c x^n\right )}{n}\right )^{-2 q} \Gamma \left (2 q,-\frac {2 m \log \left (c x^n\right )}{n}\right )}{n}-\frac {3 a b^2 x^m \left (c x^n\right )^{-\frac {m}{n}} \log ^{3 q}\left (c x^n\right ) \left (-\frac {m \log \left (c x^n\right )}{n}\right )^{-3 q} \Gamma \left (3 q,-\frac {m \log \left (c x^n\right )}{n}\right )}{n}+\frac {b^3 \log ^{4 q}\left (c x^n\right )}{4 n q} \]

[In]

Int[(Log[c*x^n]^(-1 + q)*(a*x^m + b*Log[c*x^n]^q)^3)/x,x]

[Out]

(b^3*Log[c*x^n]^(4*q))/(4*n*q) - (3*a*b^2*x^m*Gamma[3*q, -((m*Log[c*x^n])/n)]*Log[c*x^n]^(3*q))/(n*(c*x^n)^(m/
n)*(-((m*Log[c*x^n])/n))^(3*q)) - (3*a^2*b*x^(2*m)*Gamma[2*q, (-2*m*Log[c*x^n])/n]*Log[c*x^n]^(2*q))/(4^q*n*(c
*x^n)^((2*m)/n)*(-((m*Log[c*x^n])/n))^(2*q)) - (a^3*x^(3*m)*Gamma[q, (-3*m*Log[c*x^n])/n]*Log[c*x^n]^q)/(3^q*n
*(c*x^n)^((3*m)/n)*(-((m*Log[c*x^n])/n))^q)

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2212

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[(-F^(g*(e - c*(f/d))))*((c
+ d*x)^FracPart[m]/(d*((-f)*g*(Log[F]/d))^(IntPart[m] + 1)*((-f)*g*Log[F]*((c + d*x)/d))^FracPart[m]))*Gamma[m
+ 1, ((-f)*g*(Log[F]/d))*(c + d*x)], x] /; FreeQ[{F, c, d, e, f, g, m}, x] &&  !IntegerQ[m]

Rule 2339

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*L
og[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

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]

Rule 2619

Int[(Log[(c_.)*(x_)^(n_.)]^(r_.)*(Log[(c_.)*(x_)^(n_.)]^(q_)*(b_.) + (a_.)*(x_)^(m_.))^(p_.))/(x_), x_Symbol]
:> Int[ExpandIntegrand[Log[c*x^n]^r/x, (a*x^m + b*Log[c*x^n]^q)^p, x], x] /; FreeQ[{a, b, c, m, n, p, q, r}, x
] && EqQ[r, q - 1] && IGtQ[p, 0]

Rubi steps \begin{align*} \text {integral}& = \int \left (a^3 x^{-1+3 m} \log ^{-1+q}\left (c x^n\right )+3 a^2 b x^{-1+2 m} \log ^{-1+2 q}\left (c x^n\right )+3 a b^2 x^{-1+m} \log ^{-1+3 q}\left (c x^n\right )+\frac {b^3 \log ^{-1+4 q}\left (c x^n\right )}{x}\right ) \, dx \\ & = a^3 \int x^{-1+3 m} \log ^{-1+q}\left (c x^n\right ) \, dx+\left (3 a^2 b\right ) \int x^{-1+2 m} \log ^{-1+2 q}\left (c x^n\right ) \, dx+\left (3 a b^2\right ) \int x^{-1+m} \log ^{-1+3 q}\left (c x^n\right ) \, dx+b^3 \int \frac {\log ^{-1+4 q}\left (c x^n\right )}{x} \, dx \\ & = \frac {b^3 \text {Subst}\left (\int x^{-1+4 q} \, dx,x,\log \left (c x^n\right )\right )}{n}+\frac {\left (a^3 x^{3 m} \left (c x^n\right )^{-\frac {3 m}{n}}\right ) \text {Subst}\left (\int e^{\frac {3 m x}{n}} x^{-1+q} \, dx,x,\log \left (c x^n\right )\right )}{n}+\frac {\left (3 a^2 b x^{2 m} \left (c x^n\right )^{-\frac {2 m}{n}}\right ) \text {Subst}\left (\int e^{\frac {2 m x}{n}} x^{-1+2 q} \, dx,x,\log \left (c x^n\right )\right )}{n}+\frac {\left (3 a b^2 x^m \left (c x^n\right )^{-\frac {m}{n}}\right ) \text {Subst}\left (\int e^{\frac {m x}{n}} x^{-1+3 q} \, dx,x,\log \left (c x^n\right )\right )}{n} \\ & = \frac {b^3 \log ^{4 q}\left (c x^n\right )}{4 n q}-\frac {3 a b^2 x^m \left (c x^n\right )^{-\frac {m}{n}} \Gamma \left (3 q,-\frac {m \log \left (c x^n\right )}{n}\right ) \log ^{3 q}\left (c x^n\right ) \left (-\frac {m \log \left (c x^n\right )}{n}\right )^{-3 q}}{n}-\frac {3\ 4^{-q} a^2 b x^{2 m} \left (c x^n\right )^{-\frac {2 m}{n}} \Gamma \left (2 q,-\frac {2 m \log \left (c x^n\right )}{n}\right ) \log ^{2 q}\left (c x^n\right ) \left (-\frac {m \log \left (c x^n\right )}{n}\right )^{-2 q}}{n}-\frac {3^{-q} a^3 x^{3 m} \left (c x^n\right )^{-\frac {3 m}{n}} \Gamma \left (q,-\frac {3 m \log \left (c x^n\right )}{n}\right ) \log ^q\left (c x^n\right ) \left (-\frac {m \log \left (c x^n\right )}{n}\right )^{-q}}{n} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.47 (sec) , antiderivative size = 223, normalized size of antiderivative = 0.97 \[ \int \frac {\log ^{-1+q}\left (c x^n\right ) \left (a x^m+b \log ^q\left (c x^n\right )\right )^3}{x} \, dx=\frac {\log ^q\left (c x^n\right ) \left (\frac {b^3 \log ^{3 q}\left (c x^n\right )}{q}-12 a b^2 x^m \left (c x^n\right )^{-\frac {m}{n}} \Gamma \left (3 q,-\frac {m \log \left (c x^n\right )}{n}\right ) \log ^{2 q}\left (c x^n\right ) \left (-\frac {m \log \left (c x^n\right )}{n}\right )^{-3 q}-3\ 4^{1-q} a^2 b x^{2 m} \left (c x^n\right )^{-\frac {2 m}{n}} \Gamma \left (2 q,-\frac {2 m \log \left (c x^n\right )}{n}\right ) \log ^q\left (c x^n\right ) \left (-\frac {m \log \left (c x^n\right )}{n}\right )^{-2 q}-4\ 3^{-q} a^3 x^{3 m} \left (c x^n\right )^{-\frac {3 m}{n}} \Gamma \left (q,-\frac {3 m \log \left (c x^n\right )}{n}\right ) \left (-\frac {m \log \left (c x^n\right )}{n}\right )^{-q}\right )}{4 n} \]

[In]

Integrate[(Log[c*x^n]^(-1 + q)*(a*x^m + b*Log[c*x^n]^q)^3)/x,x]

[Out]

(Log[c*x^n]^q*((b^3*Log[c*x^n]^(3*q))/q - (12*a*b^2*x^m*Gamma[3*q, -((m*Log[c*x^n])/n)]*Log[c*x^n]^(2*q))/((c*
x^n)^(m/n)*(-((m*Log[c*x^n])/n))^(3*q)) - (3*4^(1 - q)*a^2*b*x^(2*m)*Gamma[2*q, (-2*m*Log[c*x^n])/n]*Log[c*x^n
]^q)/((c*x^n)^((2*m)/n)*(-((m*Log[c*x^n])/n))^(2*q)) - (4*a^3*x^(3*m)*Gamma[q, (-3*m*Log[c*x^n])/n])/(3^q*(c*x
^n)^((3*m)/n)*(-((m*Log[c*x^n])/n))^q)))/(4*n)

Maple [F]

\[\int \frac {\ln \left (c \,x^{n}\right )^{-1+q} \left (a \,x^{m}+b \ln \left (c \,x^{n}\right )^{q}\right )^{3}}{x}d x\]

[In]

int(ln(c*x^n)^(-1+q)*(a*x^m+b*ln(c*x^n)^q)^3/x,x)

[Out]

int(ln(c*x^n)^(-1+q)*(a*x^m+b*ln(c*x^n)^q)^3/x,x)

Fricas [F]

\[ \int \frac {\log ^{-1+q}\left (c x^n\right ) \left (a x^m+b \log ^q\left (c x^n\right )\right )^3}{x} \, dx=\int { \frac {{\left (a x^{m} + b \log \left (c x^{n}\right )^{q}\right )}^{3} \log \left (c x^{n}\right )^{q - 1}}{x} \,d x } \]

[In]

integrate(log(c*x^n)^(-1+q)*(a*x^m+b*log(c*x^n)^q)^3/x,x, algorithm="fricas")

[Out]

integral((3*a*b^2*x^m*log(c*x^n)^(2*q)*log(c*x^n)^(q - 1) + 3*a^2*b*x^(2*m)*log(c*x^n)^(q - 1)*log(c*x^n)^q +
a^3*x^(3*m)*log(c*x^n)^(q - 1) + b^3*log(c*x^n)^(3*q)*log(c*x^n)^(q - 1))/x, x)

Sympy [F(-1)]

Timed out. \[ \int \frac {\log ^{-1+q}\left (c x^n\right ) \left (a x^m+b \log ^q\left (c x^n\right )\right )^3}{x} \, dx=\text {Timed out} \]

[In]

integrate(ln(c*x**n)**(-1+q)*(a*x**m+b*ln(c*x**n)**q)**3/x,x)

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {\log ^{-1+q}\left (c x^n\right ) \left (a x^m+b \log ^q\left (c x^n\right )\right )^3}{x} \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate(log(c*x^n)^(-1+q)*(a*x^m+b*log(c*x^n)^q)^3/x,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: In function CAR, the value of the first argument is  0which is not
of the expected type LIST

Giac [F]

\[ \int \frac {\log ^{-1+q}\left (c x^n\right ) \left (a x^m+b \log ^q\left (c x^n\right )\right )^3}{x} \, dx=\int { \frac {{\left (a x^{m} + b \log \left (c x^{n}\right )^{q}\right )}^{3} \log \left (c x^{n}\right )^{q - 1}}{x} \,d x } \]

[In]

integrate(log(c*x^n)^(-1+q)*(a*x^m+b*log(c*x^n)^q)^3/x,x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\log ^{-1+q}\left (c x^n\right ) \left (a x^m+b \log ^q\left (c x^n\right )\right )^3}{x} \, dx=\int \frac {{\ln \left (c\,x^n\right )}^{q-1}\,{\left (a\,x^m+b\,{\ln \left (c\,x^n\right )}^q\right )}^3}{x} \,d x \]

[In]

int((log(c*x^n)^(q - 1)*(a*x^m + b*log(c*x^n)^q)^3)/x,x)

[Out]

int((log(c*x^n)^(q - 1)*(a*x^m + b*log(c*x^n)^q)^3)/x, x)