Integrand size = 26, antiderivative size = 281 \[ \int (g+h x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^n \, dx=\frac {2^{-1-n} e^{-\frac {2 a}{b p q}} h (e+f x)^2 \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {2}{p q}} \Gamma \left (1+n,-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{b p q}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^n \left (-\frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{b p q}\right )^{-n}}{f^2}+\frac {e^{-\frac {a}{b p q}} (f g-e h) (e+f x) \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {1}{p q}} \Gamma \left (1+n,-\frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{b p q}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^n \left (-\frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{b p q}\right )^{-n}}{f^2} \]
2^(-1-n)*h*(f*x+e)^2*GAMMA(1+n,-2*(a+b*ln(c*(d*(f*x+e)^p)^q))/b/p/q)*(a+b* ln(c*(d*(f*x+e)^p)^q))^n/exp(2*a/b/p/q)/f^2/((c*(d*(f*x+e)^p)^q)^(2/p/q))/ (((-a-b*ln(c*(d*(f*x+e)^p)^q))/b/p/q)^n)+(-e*h+f*g)*(f*x+e)*GAMMA(1+n,(-a- b*ln(c*(d*(f*x+e)^p)^q))/b/p/q)*(a+b*ln(c*(d*(f*x+e)^p)^q))^n/exp(a/b/p/q) /f^2/((c*(d*(f*x+e)^p)^q)^(1/p/q))/(((-a-b*ln(c*(d*(f*x+e)^p)^q))/b/p/q)^n )
Time = 0.18 (sec) , antiderivative size = 227, normalized size of antiderivative = 0.81 \[ \int (g+h x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^n \, dx=\frac {2^{-1-n} e^{-\frac {2 a}{b p q}} (e+f x) \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {2}{p q}} \left (h (e+f x) \Gamma \left (1+n,-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{b p q}\right )+2^{1+n} e^{\frac {a}{b p q}} (f g-e h) \left (c \left (d (e+f x)^p\right )^q\right )^{\frac {1}{p q}} \Gamma \left (1+n,-\frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{b p q}\right )\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^n \left (-\frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{b p q}\right )^{-n}}{f^2} \]
(2^(-1 - n)*(e + f*x)*(h*(e + f*x)*Gamma[1 + n, (-2*(a + b*Log[c*(d*(e + f *x)^p)^q]))/(b*p*q)] + 2^(1 + n)*E^(a/(b*p*q))*(f*g - e*h)*(c*(d*(e + f*x) ^p)^q)^(1/(p*q))*Gamma[1 + n, -((a + b*Log[c*(d*(e + f*x)^p)^q])/(b*p*q))] )*(a + b*Log[c*(d*(e + f*x)^p)^q])^n)/(E^((2*a)/(b*p*q))*f^2*(c*(d*(e + f* x)^p)^q)^(2/(p*q))*(-((a + b*Log[c*(d*(e + f*x)^p)^q])/(b*p*q)))^n)
Time = 0.93 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {2895, 2848, 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.
\(\displaystyle \int (g+h x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^n \, dx\) |
\(\Big \downarrow \) 2895 |
\(\displaystyle \int (g+h x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^ndx\) |
\(\Big \downarrow \) 2848 |
\(\displaystyle \int \left (\frac {(f g-e h) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^n}{f}+\frac {h (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^n}{f}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {(e+f x) e^{-\frac {a}{b p q}} (f g-e h) \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {1}{p q}} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^n \left (-\frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{b p q}\right )^{-n} \Gamma \left (n+1,-\frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{b p q}\right )}{f^2}+\frac {h 2^{-n-1} (e+f x)^2 e^{-\frac {2 a}{b p q}} \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {2}{p q}} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^n \left (-\frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{b p q}\right )^{-n} \Gamma \left (n+1,-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{b p q}\right )}{f^2}\) |
(2^(-1 - n)*h*(e + f*x)^2*Gamma[1 + n, (-2*(a + b*Log[c*(d*(e + f*x)^p)^q] ))/(b*p*q)]*(a + b*Log[c*(d*(e + f*x)^p)^q])^n)/(E^((2*a)/(b*p*q))*f^2*(c* (d*(e + f*x)^p)^q)^(2/(p*q))*(-((a + b*Log[c*(d*(e + f*x)^p)^q])/(b*p*q))) ^n) + ((f*g - e*h)*(e + f*x)*Gamma[1 + n, -((a + b*Log[c*(d*(e + f*x)^p)^q ])/(b*p*q))]*(a + b*Log[c*(d*(e + f*x)^p)^q])^n)/(E^(a/(b*p*q))*f^2*(c*(d* (e + f*x)^p)^q)^(1/(p*q))*(-((a + b*Log[c*(d*(e + f*x)^p)^q])/(b*p*q)))^n)
3.6.15.3.1 Defintions of rubi rules used
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)*((f_.) + (g_. )*(x_))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(f + g*x)^q*(a + b*Log[c*(d + e*x)^n])^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && NeQ[e*f - d*g, 0] && IGtQ[q, 0]
Int[((a_.) + Log[(c_.)*((d_.)*((e_.) + (f_.)*(x_))^(m_.))^(n_)]*(b_.))^(p_. )*(u_.), x_Symbol] :> Subst[Int[u*(a + b*Log[c*d^n*(e + f*x)^(m*n)])^p, x], c*d^n*(e + f*x)^(m*n), c*(d*(e + f*x)^m)^n] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && !IntegerQ[n] && !(EqQ[d, 1] && EqQ[m, 1]) && IntegralFreeQ[ IntHide[u*(a + b*Log[c*d^n*(e + f*x)^(m*n)])^p, x]]
\[\int \left (h x +g \right ) {\left (a +b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )\right )}^{n}d x\]
\[ \int (g+h x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^n \, dx=\int { {\left (h x + g\right )} {\left (b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}^{n} \,d x } \]
\[ \int (g+h x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^n \, dx=\int \left (a + b \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}\right )^{n} \left (g + h x\right )\, dx \]
Exception generated. \[ \int (g+h x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^n \, dx=\text {Exception raised: RuntimeError} \]
Exception raised: RuntimeError >> ECL says: In function CAR, the value of the first argument is 0which is not of the expected type LIST
\[ \int (g+h x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^n \, dx=\int { {\left (h x + g\right )} {\left (b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}^{n} \,d x } \]
Timed out. \[ \int (g+h x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^n \, dx=\int \left (g+h\,x\right )\,{\left (a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )\right )}^n \,d x \]