\(\int \frac {(d x^m+e \log ^{-1+q}(c x^n)) (a x^m+b \log ^q(c x^n))^3}{x} \, dx\) [31]

Optimal result

Integrand size = 40, antiderivative size = 331 \[ \int \frac {\left (d x^m+e \log ^{-1+q}\left (c x^n\right )\right ) \left (a x^m+b \log ^q\left (c x^n\right )\right )^3}{x} \, dx=-\frac {a^3 (a e m-b d n q) x^{4 m}}{4 b m n q}-\frac {b^2 (a e m-b d n q) x^m \left (c x^n\right )^{-\frac {m}{n}} \Gamma \left (1+3 q,-\frac {m \log \left (c x^n\right )}{n}\right ) \log ^{3 q}\left (c x^n\right ) \left (-\frac {m \log \left (c x^n\right )}{n}\right )^{-3 q}}{m n q}-\frac {3\ 2^{-1-2 q} a b (a e m-b d n q) x^{2 m} \left (c x^n\right )^{-\frac {2 m}{n}} \Gamma \left (1+2 q,-\frac {2 m \log \left (c x^n\right )}{n}\right ) \log ^{2 q}\left (c x^n\right ) \left (-\frac {m \log \left (c x^n\right )}{n}\right )^{-2 q}}{m n q}-\frac {3^{-q} a^2 (a e m-b d n q) x^{3 m} \left (c x^n\right )^{-\frac {3 m}{n}} \Gamma \left (1+q,-\frac {3 m \log \left (c x^n\right )}{n}\right ) \log ^q\left (c x^n\right ) \left (-\frac {m \log \left (c x^n\right )}{n}\right )^{-q}}{m n q}+\frac {e \left (a x^m+b \log ^q\left (c x^n\right )\right )^4}{4 b n q} \] Output:

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

Mathematica [A] (verified)

Time = 1.84 (sec) , antiderivative size = 445, normalized size of antiderivative = 1.34 \[ \int \frac {\left (d x^m+e \log ^{-1+q}\left (c x^n\right )\right ) \left (a x^m+b \log ^q\left (c x^n\right )\right )^3}{x} \, dx=\frac {3^{-q} 4^{-1-q} \left (c x^n\right )^{-\frac {3 m}{n}} \left (-\frac {m \log \left (c x^n\right )}{n}\right )^{-3 q} \left (-12^{1+q} a b^2 e m q x^m \left (c x^n\right )^{\frac {2 m}{n}} \Gamma \left (3 q,-\frac {m \log \left (c x^n\right )}{n}\right ) \log ^{3 q}\left (c x^n\right )+3^q 4^{1+q} b^3 d n q x^m \left (c x^n\right )^{\frac {2 m}{n}} \Gamma \left (1+3 q,-\frac {m \log \left (c x^n\right )}{n}\right ) \log ^{3 q}\left (c x^n\right )+\left (-\frac {m \log \left (c x^n\right )}{n}\right )^q \left (-4 3^{1+q} a^2 b e m q x^{2 m} \left (c x^n\right )^{m/n} \Gamma \left (2 q,-\frac {2 m \log \left (c x^n\right )}{n}\right ) \log ^{2 q}\left (c x^n\right )+2\ 3^{1+q} a b^2 d n q x^{2 m} \left (c x^n\right )^{m/n} \Gamma \left (1+2 q,-\frac {2 m \log \left (c x^n\right )}{n}\right ) \log ^{2 q}\left (c x^n\right )+4^q \left (-\frac {m \log \left (c x^n\right )}{n}\right )^q \left (-4 a^3 e m q x^{3 m} \Gamma \left (q,-\frac {3 m \log \left (c x^n\right )}{n}\right ) \log ^q\left (c x^n\right )+4 a^2 b d n q x^{3 m} \Gamma \left (1+q,-\frac {3 m \log \left (c x^n\right )}{n}\right ) \log ^q\left (c x^n\right )+3^q \left (c x^n\right )^{\frac {3 m}{n}} \left (-\frac {m \log \left (c x^n\right )}{n}\right )^q \left (a^3 d n q x^{4 m}+b^3 e m \log ^{4 q}\left (c x^n\right )\right )\right )\right )\right )}{m n q} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.84 (sec) , antiderivative size = 279, normalized size of antiderivative = 0.84, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.075, Rules used = {3025, 7293, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 3025

\(\displaystyle \left (d-\frac {a e m}{b n q}\right ) \int x^{m-1} \left (a x^m+b \log ^q\left (c x^n\right )\right )^3dx+\frac {e \left (a x^m+b \log ^q\left (c x^n\right )\right )^4}{4 b n q}\)

\(\Big \downarrow \) 7293

\(\displaystyle \left (d-\frac {a e m}{b n q}\right ) \int \left (b^3 \log ^{3 q}\left (c x^n\right ) x^{m-1}+3 a b^2 \log ^{2 q}\left (c x^n\right ) x^{2 m-1}+3 a^2 b \log ^q\left (c x^n\right ) x^{3 m-1}+a^3 x^{4 m-1}\right )dx+\frac {e \left (a x^m+b \log ^q\left (c x^n\right )\right )^4}{4 b n q}\)

\(\Big \downarrow \) 2009

\(\displaystyle \left (d-\frac {a e m}{b n q}\right ) \left (\frac {a^3 x^{4 m}}{4 m}+\frac {a^2 b 3^{-q} x^{3 m} \left (c x^n\right )^{-\frac {3 m}{n}} \log ^q\left (c x^n\right ) \left (-\frac {m \log \left (c x^n\right )}{n}\right )^{-q} \Gamma \left (q+1,-\frac {3 m \log \left (c x^n\right )}{n}\right )}{m}+\frac {3 a b^2 2^{-2 q-1} x^{2 m} \left (c x^n\right )^{-\frac {2 m}{n}} \log ^{2 q}\left (c x^n\right ) \left (-\frac {m \log \left (c x^n\right )}{n}\right )^{-2 q} \Gamma \left (2 q+1,-\frac {2 m \log \left (c x^n\right )}{n}\right )}{m}+\frac {b^3 x^m \left (c x^n\right )^{-\frac {m}{n}} \log ^{3 q}\left (c x^n\right ) \left (-\frac {m \log \left (c x^n\right )}{n}\right )^{-3 q} \Gamma \left (3 q+1,-\frac {m \log \left (c x^n\right )}{n}\right )}{m}\right )+\frac {e \left (a x^m+b \log ^q\left (c x^n\right )\right )^4}{4 b n q}\)

Input:

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

Output:

(e*(a*x^m + b*Log[c*x^n]^q)^4)/(4*b*n*q) + (d - (a*e*m)/(b*n*q))*((a^3*x^( 
4*m))/(4*m) + (b^3*x^m*Gamma[1 + 3*q, -((m*Log[c*x^n])/n)]*Log[c*x^n]^(3*q 
))/(m*(c*x^n)^(m/n)*(-((m*Log[c*x^n])/n))^(3*q)) + (3*2^(-1 - 2*q)*a*b^2*x 
^(2*m)*Gamma[1 + 2*q, (-2*m*Log[c*x^n])/n]*Log[c*x^n]^(2*q))/(m*(c*x^n)^(( 
2*m)/n)*(-((m*Log[c*x^n])/n))^(2*q)) + (a^2*b*x^(3*m)*Gamma[1 + q, (-3*m*L 
og[c*x^n])/n]*Log[c*x^n]^q)/(3^q*m*(c*x^n)^((3*m)/n)*(-((m*Log[c*x^n])/n)) 
^q))
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3025

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

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral((a^3*e*x^(3*m)*log(c*x^n)^(q - 1) + a^3*d*x^(4*m) + (b^3*d*x^m + 
b^3*e*log(c*x^n)^(q - 1))*log(c*x^n)^(3*q) + 3*(a*b^2*e*x^m*log(c*x^n)^(q 
- 1) + a*b^2*d*x^(2*m))*log(c*x^n)^(2*q) + 3*(a^2*b*e*x^(2*m)*log(c*x^n)^( 
q - 1) + a^2*b*d*x^(3*m))*log(c*x^n)^q)/x, x)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

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

Output:

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]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (d x^m+e \log ^{-1+q}\left (c x^n\right )\right ) \left (a x^m+b \log ^q\left (c x^n\right )\right )^3}{x} \, dx=\frac {\mathrm {log}\left (x^{n} c \right )^{4 q} b^{3} e m +4 x^{m} \mathrm {log}\left (x^{n} c \right )^{3 q} b^{3} d n q +6 x^{2 m} \mathrm {log}\left (x^{n} c \right )^{2 q} a \,b^{2} d n q +4 x^{3 m} \mathrm {log}\left (x^{n} c \right )^{q} a^{2} b d n q +x^{4 m} a^{3} d n q +4 \left (\int \frac {x^{3 m} \mathrm {log}\left (x^{n} c \right )^{q}}{\mathrm {log}\left (x^{n} c \right ) x}d x \right ) a^{3} e m n q -4 \left (\int \frac {x^{3 m} \mathrm {log}\left (x^{n} c \right )^{q}}{\mathrm {log}\left (x^{n} c \right ) x}d x \right ) a^{2} b d \,n^{2} q^{2}+12 \left (\int \frac {x^{2 m} \mathrm {log}\left (x^{n} c \right )^{2 q}}{\mathrm {log}\left (x^{n} c \right ) x}d x \right ) a^{2} b e m n q -12 \left (\int \frac {x^{2 m} \mathrm {log}\left (x^{n} c \right )^{2 q}}{\mathrm {log}\left (x^{n} c \right ) x}d x \right ) a \,b^{2} d \,n^{2} q^{2}+12 \left (\int \frac {x^{m} \mathrm {log}\left (x^{n} c \right )^{3 q}}{\mathrm {log}\left (x^{n} c \right ) x}d x \right ) a \,b^{2} e m n q -12 \left (\int \frac {x^{m} \mathrm {log}\left (x^{n} c \right )^{3 q}}{\mathrm {log}\left (x^{n} c \right ) x}d x \right ) b^{3} d \,n^{2} q^{2}}{4 m n q} \] Input:

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

Output:

(log(x**n*c)**(4*q)*b**3*e*m + 4*x**m*log(x**n*c)**(3*q)*b**3*d*n*q + 6*x* 
*(2*m)*log(x**n*c)**(2*q)*a*b**2*d*n*q + 4*x**(3*m)*log(x**n*c)**q*a**2*b* 
d*n*q + x**(4*m)*a**3*d*n*q + 4*int((x**(3*m)*log(x**n*c)**q)/(log(x**n*c) 
*x),x)*a**3*e*m*n*q - 4*int((x**(3*m)*log(x**n*c)**q)/(log(x**n*c)*x),x)*a 
**2*b*d*n**2*q**2 + 12*int((x**(2*m)*log(x**n*c)**(2*q))/(log(x**n*c)*x),x 
)*a**2*b*e*m*n*q - 12*int((x**(2*m)*log(x**n*c)**(2*q))/(log(x**n*c)*x),x) 
*a*b**2*d*n**2*q**2 + 12*int((x**m*log(x**n*c)**(3*q))/(log(x**n*c)*x),x)* 
a*b**2*e*m*n*q - 12*int((x**m*log(x**n*c)**(3*q))/(log(x**n*c)*x),x)*b**3* 
d*n**2*q**2)/(4*m*n*q)