Integrand size = 40, antiderivative size = 235 \[ \int \frac {\left (d x^m+e \log ^{-1+q}\left (c x^n\right )\right ) \left (a x^m+b \log ^q\left (c x^n\right )\right )^2}{x} \, dx=-\frac {a^2 (a e m-b d n q) x^{3 m}}{3 b m n q}-\frac {b (a e m-b d n q) x^m \left (c x^n\right )^{-\frac {m}{n}} \Gamma \left (1+2 q,-\frac {m \log \left (c x^n\right )}{n}\right ) \log ^{2 q}\left (c x^n\right ) \left (-\frac {m \log \left (c x^n\right )}{n}\right )^{-2 q}}{m n q}-\frac {2^{-q} a (a e m-b d n q) x^{2 m} \left (c x^n\right )^{-\frac {2 m}{n}} \Gamma \left (1+q,-\frac {2 m \log \left (c x^n\right )}{n}\right ) \log ^q\left (c x^n\right ) \left (-\frac {m \log \left (c x^n\right )}{n}\right )^{-q}}{m n q}+\frac {e \left (a x^m+b \log ^q\left (c x^n\right )\right )^3}{3 b n q} \]
-1/3*a^2*(-b*d*n*q+a*e*m)*x^(3*m)/b/m/n/q-b*(-b*d*n*q+a*e*m)*x^m*GAMMA(1+2 *q,-m*ln(c*x^n)/n)*ln(c*x^n)^(2*q)/m/n/q/((c*x^n)^(m/n))/((-m*ln(c*x^n)/n) ^(2*q))-a*(-b*d*n*q+a*e*m)*x^(2*m)*GAMMA(1+q,-2*m*ln(c*x^n)/n)*ln(c*x^n)^q /(2^q)/m/n/q/((c*x^n)^(2*m/n))/((-m*ln(c*x^n)/n)^q)+1/3*e*(a*x^m+b*ln(c*x^ n)^q)^3/b/n/q
Time = 0.67 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.27 \[ \int \frac {\left (d x^m+e \log ^{-1+q}\left (c x^n\right )\right ) \left (a x^m+b \log ^q\left (c x^n\right )\right )^2}{x} \, dx=\frac {2^{-q} \left (c x^n\right )^{-\frac {2 m}{n}} \left (-\frac {m \log \left (c x^n\right )}{n}\right )^{-2 q} \left (-3 2^{1+q} a b e m q x^m \left (c x^n\right )^{m/n} \Gamma \left (2 q,-\frac {m \log \left (c x^n\right )}{n}\right ) \log ^{2 q}\left (c x^n\right )+3\ 2^q b^2 d n q x^m \left (c x^n\right )^{m/n} \Gamma \left (1+2 q,-\frac {m \log \left (c x^n\right )}{n}\right ) \log ^{2 q}\left (c x^n\right )+\left (-\frac {m \log \left (c x^n\right )}{n}\right )^q \left (-3 a^2 e m q x^{2 m} \Gamma \left (q,-\frac {2 m \log \left (c x^n\right )}{n}\right ) \log ^q\left (c x^n\right )+3 a b d n q x^{2 m} \Gamma \left (1+q,-\frac {2 m \log \left (c x^n\right )}{n}\right ) \log ^q\left (c x^n\right )+2^q \left (c x^n\right )^{\frac {2 m}{n}} \left (-\frac {m \log \left (c x^n\right )}{n}\right )^q \left (a^2 d n q x^{3 m}+b^2 e m \log ^{3 q}\left (c x^n\right )\right )\right )\right )}{3 m n q} \]
(-3*2^(1 + q)*a*b*e*m*q*x^m*(c*x^n)^(m/n)*Gamma[2*q, -((m*Log[c*x^n])/n)]* Log[c*x^n]^(2*q) + 3*2^q*b^2*d*n*q*x^m*(c*x^n)^(m/n)*Gamma[1 + 2*q, -((m*L og[c*x^n])/n)]*Log[c*x^n]^(2*q) + (-((m*Log[c*x^n])/n))^q*(-3*a^2*e*m*q*x^ (2*m)*Gamma[q, (-2*m*Log[c*x^n])/n]*Log[c*x^n]^q + 3*a*b*d*n*q*x^(2*m)*Gam ma[1 + q, (-2*m*Log[c*x^n])/n]*Log[c*x^n]^q + 2^q*(c*x^n)^((2*m)/n)*(-((m* Log[c*x^n])/n))^q*(a^2*d*n*q*x^(3*m) + b^2*e*m*Log[c*x^n]^(3*q))))/(3*2^q* m*n*q*(c*x^n)^((2*m)/n)*(-((m*Log[c*x^n])/n))^(2*q))
Time = 0.63 (sec) , antiderivative size = 200, normalized size of antiderivative = 0.85, 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.
\(\displaystyle \int \frac {\left (a x^m+b \log ^q\left (c x^n\right )\right )^2 \left (e \log ^{q-1}\left (c x^n\right )+d x^m\right )}{x} \, dx\) |
\(\Big \downarrow \) 3025 |
\(\displaystyle \left (d-\frac {a e m}{b n q}\right ) \int x^{m-1} \left (a x^m+b \log ^q\left (c x^n\right )\right )^2dx+\frac {e \left (a x^m+b \log ^q\left (c x^n\right )\right )^3}{3 b n q}\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \left (d-\frac {a e m}{b n q}\right ) \int \left (b^2 \log ^{2 q}\left (c x^n\right ) x^{m-1}+2 a b \log ^q\left (c x^n\right ) x^{2 m-1}+a^2 x^{3 m-1}\right )dx+\frac {e \left (a x^m+b \log ^q\left (c x^n\right )\right )^3}{3 b n q}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \left (d-\frac {a e m}{b n q}\right ) \left (\frac {a^2 x^{3 m}}{3 m}+\frac {a b 2^{-q} x^{2 m} \left (c x^n\right )^{-\frac {2 m}{n}} \log ^q\left (c x^n\right ) \left (-\frac {m \log \left (c x^n\right )}{n}\right )^{-q} \Gamma \left (q+1,-\frac {2 m \log \left (c x^n\right )}{n}\right )}{m}+\frac {b^2 x^m \left (c x^n\right )^{-\frac {m}{n}} \log ^{2 q}\left (c x^n\right ) \left (-\frac {m \log \left (c x^n\right )}{n}\right )^{-2 q} \Gamma \left (2 q+1,-\frac {m \log \left (c x^n\right )}{n}\right )}{m}\right )+\frac {e \left (a x^m+b \log ^q\left (c x^n\right )\right )^3}{3 b n q}\) |
(e*(a*x^m + b*Log[c*x^n]^q)^3)/(3*b*n*q) + (d - (a*e*m)/(b*n*q))*((a^2*x^( 3*m))/(3*m) + (b^2*x^m*Gamma[1 + 2*q, -((m*Log[c*x^n])/n)]*Log[c*x^n]^(2*q ))/(m*(c*x^n)^(m/n)*(-((m*Log[c*x^n])/n))^(2*q)) + (a*b*x^(2*m)*Gamma[1 + q, (-2*m*Log[c*x^n])/n]*Log[c*x^n]^q)/(2^q*m*(c*x^n)^((2*m)/n)*(-((m*Log[c *x^n])/n))^q))
3.1.32.3.1 Defintions of rubi rules used
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]
\[\int \frac {\left (d \,x^{m}+e \ln \left (c \,x^{n}\right )^{-1+q}\right ) \left (a \,x^{m}+b \ln \left (c \,x^{n}\right )^{q}\right )^{2}}{x}d x\]
\[ \int \frac {\left (d x^m+e \log ^{-1+q}\left (c x^n\right )\right ) \left (a x^m+b \log ^q\left (c x^n\right )\right )^2}{x} \, dx=\int { \frac {{\left (a x^{m} + b \log \left (c x^{n}\right )^{q}\right )}^{2} {\left (d x^{m} + e \log \left (c x^{n}\right )^{q - 1}\right )}}{x} \,d x } \]
integral((a^2*e*x^(2*m)*log(c*x^n)^(q - 1) + a^2*d*x^(3*m) + (b^2*d*x^m + b^2*e*log(c*x^n)^(q - 1))*log(c*x^n)^(2*q) + 2*(a*b*e*x^m*log(c*x^n)^(q - 1) + a*b*d*x^(2*m))*log(c*x^n)^q)/x, x)
\[ \int \frac {\left (d x^m+e \log ^{-1+q}\left (c x^n\right )\right ) \left (a x^m+b \log ^q\left (c x^n\right )\right )^2}{x} \, dx=\int \frac {\left (a x^{m} + b \log {\left (c x^{n} \right )}^{q}\right )^{2} \left (d x^{m} + e \log {\left (c x^{n} \right )}^{q - 1}\right )}{x}\, dx \]
Exception generated. \[ \int \frac {\left (d x^m+e \log ^{-1+q}\left (c x^n\right )\right ) \left (a x^m+b \log ^q\left (c x^n\right )\right )^2}{x} \, dx=\text {Exception raised: RuntimeError} \]
Exception raised: RuntimeError >> ECL says: In function CAR, the value of the first argument is 0which is not of the expected type LIST
\[ \int \frac {\left (d x^m+e \log ^{-1+q}\left (c x^n\right )\right ) \left (a x^m+b \log ^q\left (c x^n\right )\right )^2}{x} \, dx=\int { \frac {{\left (a x^{m} + b \log \left (c x^{n}\right )^{q}\right )}^{2} {\left (d x^{m} + e \log \left (c x^{n}\right )^{q - 1}\right )}}{x} \,d x } \]
Timed out. \[ \int \frac {\left (d x^m+e \log ^{-1+q}\left (c x^n\right )\right ) \left (a x^m+b \log ^q\left (c x^n\right )\right )^2}{x} \, dx=\int \frac {{\left (a\,x^m+b\,{\ln \left (c\,x^n\right )}^q\right )}^2\,\left (d\,x^m+e\,{\ln \left (c\,x^n\right )}^{q-1}\right )}{x} \,d x \]