Integrand size = 32, antiderivative size = 210 \[ \int \frac {(h+i x)^2 (a+b \log (c (e+f x)))^p}{d e+d f x} \, dx=\frac {(f h-e i)^2 (a+b \log (c (e+f x)))^{1+p}}{b d f^3 (1+p)}+\frac {2^{-1-p} e^{-\frac {2 a}{b}} i^2 \Gamma \left (1+p,-\frac {2 (a+b \log (c (e+f x)))}{b}\right ) (a+b \log (c (e+f x)))^p \left (-\frac {a+b \log (c (e+f x))}{b}\right )^{-p}}{c^2 d f^3}+\frac {2 e^{-\frac {a}{b}} i (f h-e i) \Gamma \left (1+p,-\frac {a+b \log (c (e+f x))}{b}\right ) (a+b \log (c (e+f x)))^p \left (-\frac {a+b \log (c (e+f x))}{b}\right )^{-p}}{c d f^3} \]
(-e*i+f*h)^2*(a+b*ln(c*(f*x+e)))^(p+1)/b/d/f^3/(p+1)+2^(-1-p)*i^2*GAMMA(p+ 1,-2*(a+b*ln(c*(f*x+e)))/b)*(a+b*ln(c*(f*x+e)))^p/c^2/d/exp(2*a/b)/f^3/((( -a-b*ln(c*(f*x+e)))/b)^p)+2*i*(-e*i+f*h)*GAMMA(p+1,(-a-b*ln(c*(f*x+e)))/b) *(a+b*ln(c*(f*x+e)))^p/c/d/exp(a/b)/f^3/(((-a-b*ln(c*(f*x+e)))/b)^p)
Time = 0.37 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.88 \[ \int \frac {(h+i x)^2 (a+b \log (c (e+f x)))^p}{d e+d f x} \, dx=\frac {2^{-1-p} e^{-\frac {2 a}{b}} (a+b \log (c (e+f x)))^p \left (-\frac {a+b \log (c (e+f x))}{b}\right )^{-p} \left (b i^2 \Gamma \left (2+p,-\frac {2 (a+b \log (c (e+f x)))}{b}\right )+2^{1+p} c e^{a/b} \left (2 b i (f h-e i) \Gamma \left (2+p,-\frac {a+b \log (c (e+f x))}{b}\right )-b c e^{a/b} f^2 (h+i x)^2 \left (-\frac {a+b \log (c (e+f x))}{b}\right )^{1+p}\right )\right )}{b c^2 d f^3 (1+p)} \]
(2^(-1 - p)*(a + b*Log[c*(e + f*x)])^p*(b*i^2*Gamma[2 + p, (-2*(a + b*Log[ c*(e + f*x)]))/b] + 2^(1 + p)*c*E^(a/b)*(2*b*i*(f*h - e*i)*Gamma[2 + p, -( (a + b*Log[c*(e + f*x)])/b)] - b*c*E^(a/b)*f^2*(h + i*x)^2*(-((a + b*Log[c *(e + f*x)])/b))^(1 + p))))/(b*c^2*d*E^((2*a)/b)*f^3*(1 + p)*(-((a + b*Log [c*(e + f*x)])/b))^p)
Time = 0.59 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.95, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2858, 27, 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.
| \(\displaystyle \int \frac {(h+i x)^2 (a+b \log (c (e+f x)))^p}{d e+d f x} \, dx\) |
| \(\Big \downarrow \) 2858 |
| \(\displaystyle \frac {\int \frac {\left (f \left (h-\frac {e i}{f}\right )+i (e+f x)\right )^2 (a+b \log (c (e+f x)))^p}{d f^2 (e+f x)}d(e+f x)}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(f h-e i+i (e+f x))^2 (a+b \log (c (e+f x)))^p}{e+f x}d(e+f x)}{d f^3}\) |
| \(\Big \downarrow \) 2795 |
| \(\displaystyle \frac {\int \left (2 i (f h-e i) (a+b \log (c (e+f x)))^p+i^2 (e+f x) (a+b \log (c (e+f x)))^p+\frac {(f h-e i)^2 (a+b \log (c (e+f x)))^p}{e+f x}\right )d(e+f x)}{d f^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {i^2 2^{-p-1} e^{-\frac {2 a}{b}} (a+b \log (c (e+f x)))^p \left (-\frac {a+b \log (c (e+f x))}{b}\right )^{-p} \Gamma \left (p+1,-\frac {2 (a+b \log (c (e+f x)))}{b}\right )}{c^2}+\frac {(f h-e i)^2 (a+b \log (c (e+f x)))^{p+1}}{b (p+1)}+\frac {2 i e^{-\frac {a}{b}} (f h-e i) (a+b \log (c (e+f x)))^p \left (-\frac {a+b \log (c (e+f x))}{b}\right )^{-p} \Gamma \left (p+1,-\frac {a+b \log (c (e+f x))}{b}\right )}{c}}{d f^3}\) |
(((f*h - e*i)^2*(a + b*Log[c*(e + f*x)])^(1 + p))/(b*(1 + p)) + (2^(-1 - p )*i^2*Gamma[1 + p, (-2*(a + b*Log[c*(e + f*x)]))/b]*(a + b*Log[c*(e + f*x) ])^p)/(c^2*E^((2*a)/b)*(-((a + b*Log[c*(e + f*x)])/b))^p) + (2*i*(f*h - e* i)*Gamma[1 + p, -((a + b*Log[c*(e + f*x)])/b)]*(a + b*Log[c*(e + f*x)])^p) /(c*E^(a/b)*(-((a + b*Log[c*(e + f*x)])/b))^p))/(d*f^3)
3.3.11.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_ .)*(x_))^(q_.)*((h_.) + (i_.)*(x_))^(r_.), x_Symbol] :> Simp[1/e Subst[In t[(g*(x/e))^q*((e*h - d*i)/e + i*(x/e))^r*(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - d*g, 0] && (IGtQ[p, 0] || IGtQ[r, 0]) && IntegerQ[2*r]
\[\int \frac {\left (i x +h \right )^{2} \left (a +b \ln \left (c \left (f x +e \right )\right )\right )^{p}}{d f x +d e}d x\]
\[ \int \frac {(h+i x)^2 (a+b \log (c (e+f x)))^p}{d e+d f x} \, dx=\int { \frac {{\left (i x + h\right )}^{2} {\left (b \log \left ({\left (f x + e\right )} c\right ) + a\right )}^{p}}{d f x + d e} \,d x } \]
Timed out. \[ \int \frac {(h+i x)^2 (a+b \log (c (e+f x)))^p}{d e+d f x} \, dx=\text {Timed out} \]
\[ \int \frac {(h+i x)^2 (a+b \log (c (e+f x)))^p}{d e+d f x} \, dx=\int { \frac {{\left (i x + h\right )}^{2} {\left (b \log \left ({\left (f x + e\right )} c\right ) + a\right )}^{p}}{d f x + d e} \,d x } \]
(b*c*log(c*f*x + c*e) + a*c)*(b*log(c*f*x + c*e) + a)^p*h^2/(b*c*d*f*(p + 1)) + integrate((i^2*x^2 + 2*h*i*x)*(b*log(f*x + e) + b*log(c) + a)^p/(d*f *x + d*e), x)
\[ \int \frac {(h+i x)^2 (a+b \log (c (e+f x)))^p}{d e+d f x} \, dx=\int { \frac {{\left (i x + h\right )}^{2} {\left (b \log \left ({\left (f x + e\right )} c\right ) + a\right )}^{p}}{d f x + d e} \,d x } \]
Timed out. \[ \int \frac {(h+i x)^2 (a+b \log (c (e+f x)))^p}{d e+d f x} \, dx=\int \frac {{\left (h+i\,x\right )}^2\,{\left (a+b\,\ln \left (c\,\left (e+f\,x\right )\right )\right )}^p}{d\,e+d\,f\,x} \,d x \]