Integrand size = 27, antiderivative size = 480 \[ \int (f x)^m \left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )^p \, dx=\frac {d^3 e^{-\frac {a (1+m)}{b n}} (f x)^{1+m} \left (c x^n\right )^{-\frac {1+m}{n}} \Gamma \left (1+p,-\frac {(1+m) \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \left (a+b \log \left (c x^n\right )\right )^p \left (-\frac {(1+m) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p}}{f (1+m)}+\frac {3 d^2 e e^{-\frac {a (1+m+r)}{b n}} x^{1+r} (f x)^m \left (c x^n\right )^{-\frac {1+m+r}{n}} \Gamma \left (1+p,-\frac {(1+m+r) \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \left (a+b \log \left (c x^n\right )\right )^p \left (-\frac {(1+m+r) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p}}{1+m+r}+\frac {3 d e^2 e^{-\frac {a (1+m+2 r)}{b n}} x^{1+2 r} (f x)^m \left (c x^n\right )^{-\frac {1+m+2 r}{n}} \Gamma \left (1+p,-\frac {(1+m+2 r) \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \left (a+b \log \left (c x^n\right )\right )^p \left (-\frac {(1+m+2 r) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p}}{1+m+2 r}+\frac {e^3 e^{-\frac {a (1+m+3 r)}{b n}} x^{1+3 r} (f x)^m \left (c x^n\right )^{-\frac {1+m+3 r}{n}} \Gamma \left (1+p,-\frac {(1+m+3 r) \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \left (a+b \log \left (c x^n\right )\right )^p \left (-\frac {(1+m+3 r) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p}}{1+m+3 r} \]
d^3*(f*x)^(1+m)*GAMMA(p+1,-(1+m)*(a+b*ln(c*x^n))/b/n)*(a+b*ln(c*x^n))^p/ex p(a*(1+m)/b/n)/f/(1+m)/((c*x^n)^((1+m)/n))/((-(1+m)*(a+b*ln(c*x^n))/b/n)^p )+3*d^2*e*x^(1+r)*(f*x)^m*GAMMA(p+1,-(1+m+r)*(a+b*ln(c*x^n))/b/n)*(a+b*ln( c*x^n))^p/exp(a*(1+m+r)/b/n)/(1+m+r)/((c*x^n)^((1+m+r)/n))/((-(1+m+r)*(a+b *ln(c*x^n))/b/n)^p)+3*d*e^2*x^(1+2*r)*(f*x)^m*GAMMA(p+1,-(1+m+2*r)*(a+b*ln (c*x^n))/b/n)*(a+b*ln(c*x^n))^p/exp(a*(1+m+2*r)/b/n)/(1+m+2*r)/((c*x^n)^(( 1+m+2*r)/n))/((-(1+m+2*r)*(a+b*ln(c*x^n))/b/n)^p)+e^3*x^(1+3*r)*(f*x)^m*GA MMA(p+1,-(1+m+3*r)*(a+b*ln(c*x^n))/b/n)*(a+b*ln(c*x^n))^p/exp(a*(1+m+3*r)/ b/n)/(1+m+3*r)/((c*x^n)^((1+m+3*r)/n))/((-(1+m+3*r)*(a+b*ln(c*x^n))/b/n)^p )
Time = 1.52 (sec) , antiderivative size = 408, normalized size of antiderivative = 0.85 \[ \int (f x)^m \left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )^p \, dx=x^{-m} (f x)^m \left (a+b \log \left (c x^n\right )\right )^p \left (\frac {d^3 e^{-\frac {(1+m) \left (a-b n \log (x)+b \log \left (c x^n\right )\right )}{b n}} \Gamma \left (1+p,-\frac {(1+m) \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \left (-\frac {(1+m) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p}}{1+m}+e \left (\frac {3 d^2 e^{-\frac {(1+m+r) \left (a-b n \log (x)+b \log \left (c x^n\right )\right )}{b n}} \Gamma \left (1+p,-\frac {(1+m+r) \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \left (-\frac {(1+m+r) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p}}{1+m+r}+e \left (\frac {3 d e^{-\frac {(1+m+2 r) \left (a-b n \log (x)+b \log \left (c x^n\right )\right )}{b n}} \Gamma \left (1+p,-\frac {(1+m+2 r) \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \left (-\frac {(1+m+2 r) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p}}{1+m+2 r}+\frac {e e^{-\frac {(1+m+3 r) \left (a-b n \log (x)+b \log \left (c x^n\right )\right )}{b n}} \Gamma \left (1+p,-\frac {(1+m+3 r) \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \left (-\frac {(1+m+3 r) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p}}{1+m+3 r}\right )\right )\right ) \]
((f*x)^m*(a + b*Log[c*x^n])^p*((d^3*Gamma[1 + p, -(((1 + m)*(a + b*Log[c*x ^n]))/(b*n))])/(E^(((1 + m)*(a - b*n*Log[x] + b*Log[c*x^n]))/(b*n))*(1 + m )*(-(((1 + m)*(a + b*Log[c*x^n]))/(b*n)))^p) + e*((3*d^2*Gamma[1 + p, -((( 1 + m + r)*(a + b*Log[c*x^n]))/(b*n))])/(E^(((1 + m + r)*(a - b*n*Log[x] + b*Log[c*x^n]))/(b*n))*(1 + m + r)*(-(((1 + m + r)*(a + b*Log[c*x^n]))/(b* n)))^p) + e*((3*d*Gamma[1 + p, -(((1 + m + 2*r)*(a + b*Log[c*x^n]))/(b*n)) ])/(E^(((1 + m + 2*r)*(a - b*n*Log[x] + b*Log[c*x^n]))/(b*n))*(1 + m + 2*r )*(-(((1 + m + 2*r)*(a + b*Log[c*x^n]))/(b*n)))^p) + (e*Gamma[1 + p, -(((1 + m + 3*r)*(a + b*Log[c*x^n]))/(b*n))])/(E^(((1 + m + 3*r)*(a - b*n*Log[x ] + b*Log[c*x^n]))/(b*n))*(1 + m + 3*r)*(-(((1 + m + 3*r)*(a + b*Log[c*x^n ]))/(b*n)))^p)))))/x^m
Time = 0.97 (sec) , antiderivative size = 480, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {2795, 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.
| \(\displaystyle \int (f x)^m \left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )^p \, dx\) |
| \(\Big \downarrow \) 2795 |
| \(\displaystyle \int \left (d^3 (f x)^m \left (a+b \log \left (c x^n\right )\right )^p+3 d^2 e x^r (f x)^m \left (a+b \log \left (c x^n\right )\right )^p+3 d e^2 x^{2 r} (f x)^m \left (a+b \log \left (c x^n\right )\right )^p+e^3 x^{3 r} (f x)^m \left (a+b \log \left (c x^n\right )\right )^p\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {d^3 (f x)^{m+1} e^{-\frac {a (m+1)}{b n}} \left (c x^n\right )^{-\frac {m+1}{n}} \left (a+b \log \left (c x^n\right )\right )^p \left (-\frac {(m+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p} \Gamma \left (p+1,-\frac {(m+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{f (m+1)}+\frac {3 d^2 e x^{r+1} (f x)^m e^{-\frac {a (m+r+1)}{b n}} \left (c x^n\right )^{-\frac {m+r+1}{n}} \left (a+b \log \left (c x^n\right )\right )^p \left (-\frac {(m+r+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p} \Gamma \left (p+1,-\frac {(m+r+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{m+r+1}+\frac {3 d e^2 x^{2 r+1} (f x)^m e^{-\frac {a (m+2 r+1)}{b n}} \left (c x^n\right )^{-\frac {m+2 r+1}{n}} \left (a+b \log \left (c x^n\right )\right )^p \left (-\frac {(m+2 r+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p} \Gamma \left (p+1,-\frac {(m+2 r+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{m+2 r+1}+\frac {e^3 x^{3 r+1} (f x)^m e^{-\frac {a (m+3 r+1)}{b n}} \left (c x^n\right )^{-\frac {m+3 r+1}{n}} \left (a+b \log \left (c x^n\right )\right )^p \left (-\frac {(m+3 r+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p} \Gamma \left (p+1,-\frac {(m+3 r+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{m+3 r+1}\) |
(d^3*(f*x)^(1 + m)*Gamma[1 + p, -(((1 + m)*(a + b*Log[c*x^n]))/(b*n))]*(a + b*Log[c*x^n])^p)/(E^((a*(1 + m))/(b*n))*f*(1 + m)*(c*x^n)^((1 + m)/n)*(- (((1 + m)*(a + b*Log[c*x^n]))/(b*n)))^p) + (3*d^2*e*x^(1 + r)*(f*x)^m*Gamm a[1 + p, -(((1 + m + r)*(a + b*Log[c*x^n]))/(b*n))]*(a + b*Log[c*x^n])^p)/ (E^((a*(1 + m + r))/(b*n))*(1 + m + r)*(c*x^n)^((1 + m + r)/n)*(-(((1 + m + r)*(a + b*Log[c*x^n]))/(b*n)))^p) + (3*d*e^2*x^(1 + 2*r)*(f*x)^m*Gamma[1 + p, -(((1 + m + 2*r)*(a + b*Log[c*x^n]))/(b*n))]*(a + b*Log[c*x^n])^p)/( E^((a*(1 + m + 2*r))/(b*n))*(1 + m + 2*r)*(c*x^n)^((1 + m + 2*r)/n)*(-(((1 + m + 2*r)*(a + b*Log[c*x^n]))/(b*n)))^p) + (e^3*x^(1 + 3*r)*(f*x)^m*Gamm a[1 + p, -(((1 + m + 3*r)*(a + b*Log[c*x^n]))/(b*n))]*(a + b*Log[c*x^n])^p )/(E^((a*(1 + m + 3*r))/(b*n))*(1 + m + 3*r)*(c*x^n)^((1 + m + 3*r)/n)*(-( ((1 + m + 3*r)*(a + b*Log[c*x^n]))/(b*n)))^p)
3.5.48.3.1 Defintions of rubi rules used
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[ c*x^n])^p, (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b , c, d, e, f, m, n, p, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IGtQ[p, 0 ] && IntegerQ[m] && IntegerQ[r]))
\[\int \left (f x \right )^{m} \left (d +e \,x^{r}\right )^{3} {\left (a +b \ln \left (c \,x^{n}\right )\right )}^{p}d x\]
\[ \int (f x)^m \left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )^p \, dx=\int { {\left (e x^{r} + d\right )}^{3} \left (f x\right )^{m} {\left (b \log \left (c x^{n}\right ) + a\right )}^{p} \,d x } \]
integral((e^3*x^(3*r) + 3*d*e^2*x^(2*r) + 3*d^2*e*x^r + d^3)*(f*x)^m*(b*lo g(c*x^n) + a)^p, x)
Timed out. \[ \int (f x)^m \left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )^p \, dx=\text {Timed out} \]
Exception generated. \[ \int (f x)^m \left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )^p \, dx=\text {Exception raised: RuntimeError} \]
Exception raised: RuntimeError >> ECL says: In function CAR, the value of the first argument is 0which is not of the expected type LIST
\[ \int (f x)^m \left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )^p \, dx=\int { {\left (e x^{r} + d\right )}^{3} \left (f x\right )^{m} {\left (b \log \left (c x^{n}\right ) + a\right )}^{p} \,d x } \]
Timed out. \[ \int (f x)^m \left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )^p \, dx=\int {\left (f\,x\right )}^m\,{\left (d+e\,x^r\right )}^3\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^p \,d x \]