Integrand size = 30, antiderivative size = 115 \[ \int \frac {(h+i x) (a+b \log (c (e+f x)))^p}{d e+d f x} \, dx=\frac {(f h-e i) (a+b \log (c (e+f x)))^{1+p}}{b d f^2 (1+p)}+\frac {e^{-\frac {a}{b}} i \Gamma \left (1+p,-\frac {a+b \log (c (e+f x))}{b}\right ) (a+b \log (c (e+f x)))^p \left (-\frac {a+b \log (c (e+f x))}{b}\right )^{-p}}{c d f^2} \] Output:
(-e*i+f*h)*(a+b*ln(c*(f*x+e)))^(p+1)/b/d/f^2/(p+1)+i*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^2/((-(a+b*ln(c*(f*x+e))) /b)^p)
Time = 0.41 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.99 \[ \int \frac {(h+i x) (a+b \log (c (e+f x)))^p}{d e+d f x} \, dx=(a+b \log (c (e+f x)))^p \left (\frac {(h+i x) (a+b \log (c (e+f x)))}{b d f+b d f p}+\frac {e^{-\frac {a}{b}} i \Gamma \left (2+p,-\frac {a+b \log (c (e+f x))}{b}\right ) \left (-\frac {a+b \log (c (e+f x))}{b}\right )^{-p}}{c d f^2+c d f^2 p}\right ) \] Input:
Integrate[((h + i*x)*(a + b*Log[c*(e + f*x)])^p)/(d*e + d*f*x),x]
Output:
(a + b*Log[c*(e + f*x)])^p*(((h + i*x)*(a + b*Log[c*(e + f*x)]))/(b*d*f + b*d*f*p) + (i*Gamma[2 + p, -((a + b*Log[c*(e + f*x)])/b)])/(E^(a/b)*(c*d*f ^2 + c*d*f^2*p)*(-((a + b*Log[c*(e + f*x)])/b))^p))
Time = 0.76 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.96, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, 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.
| \(\displaystyle \int \frac {(h+i x) (a+b \log (c (e+f x)))^p}{d e+d f x} \, dx\) |
| \(\Big \downarrow \) 2858 |
| \(\displaystyle \frac {\int \frac {\left (f \left (h-\frac {e i}{f}\right )+i (e+f x)\right ) (a+b \log (c (e+f x)))^p}{d f (e+f x)}d(e+f x)}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(f h-e i+i (e+f x)) (a+b \log (c (e+f x)))^p}{e+f x}d(e+f x)}{d f^2}\) |
| \(\Big \downarrow \) 2795 |
| \(\displaystyle \frac {\int \left (i (a+b \log (c (e+f x)))^p+\frac {(f h-e i) (a+b \log (c (e+f x)))^p}{e+f x}\right )d(e+f x)}{d f^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {(f h-e i) (a+b \log (c (e+f x)))^{p+1}}{b (p+1)}+\frac {i e^{-\frac {a}{b}} (a+b \log (c (e+f x)))^p \left (-\frac {a+b \log (c (e+f x))}{b}\right )^{-p} \Gamma \left (p+1,-\frac {a+b \log (c (e+f x))}{b}\right )}{c}}{d f^2}\) |
Input:
Int[((h + i*x)*(a + b*Log[c*(e + f*x)])^p)/(d*e + d*f*x),x]
Output:
(((f*h - e*i)*(a + b*Log[c*(e + f*x)])^(1 + p))/(b*(1 + p)) + (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^2)
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]
\[\int \frac {\left (i x +h \right ) \left (a +b \ln \left (c \left (f x +e \right )\right )\right )^{p}}{d f x +d e}d x\]
Input:
int((i*x+h)*(a+b*ln(c*(f*x+e)))^p/(d*f*x+d*e),x)
Output:
int((i*x+h)*(a+b*ln(c*(f*x+e)))^p/(d*f*x+d*e),x)
Time = 0.10 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.03 \[ \int \frac {(h+i x) (a+b \log (c (e+f x)))^p}{d e+d f x} \, dx=\frac {{\left (b i p + b i\right )} e^{\left (-\frac {b p \log \left (-\frac {1}{b}\right ) + a}{b}\right )} \Gamma \left (p + 1, -\frac {b \log \left (c f x + c e\right ) + a}{b}\right ) + {\left (a c f h - a c e i + {\left (b c f h - b c e i\right )} \log \left (c f x + c e\right )\right )} {\left (b \log \left (c f x + c e\right ) + a\right )}^{p}}{b c d f^{2} p + b c d f^{2}} \] Input:
integrate((i*x+h)*(a+b*log(c*(f*x+e)))^p/(d*f*x+d*e),x, algorithm="fricas" )
Output:
((b*i*p + b*i)*e^(-(b*p*log(-1/b) + a)/b)*gamma(p + 1, -(b*log(c*f*x + c*e ) + a)/b) + (a*c*f*h - a*c*e*i + (b*c*f*h - b*c*e*i)*log(c*f*x + c*e))*(b* log(c*f*x + c*e) + a)^p)/(b*c*d*f^2*p + b*c*d*f^2)
\[ \int \frac {(h+i x) (a+b \log (c (e+f x)))^p}{d e+d f x} \, dx=\frac {\int \frac {h \left (a + b \log {\left (c e + c f x \right )}\right )^{p}}{e + f x}\, dx + \int \frac {i x \left (a + b \log {\left (c e + c f x \right )}\right )^{p}}{e + f x}\, dx}{d} \] Input:
integrate((i*x+h)*(a+b*ln(c*(f*x+e)))**p/(d*f*x+d*e),x)
Output:
(Integral(h*(a + b*log(c*e + c*f*x))**p/(e + f*x), x) + Integral(i*x*(a + b*log(c*e + c*f*x))**p/(e + f*x), x))/d
\[ \int \frac {(h+i x) (a+b \log (c (e+f x)))^p}{d e+d f x} \, dx=\int { \frac {{\left (i x + h\right )} {\left (b \log \left ({\left (f x + e\right )} c\right ) + a\right )}^{p}}{d f x + d e} \,d x } \] Input:
integrate((i*x+h)*(a+b*log(c*(f*x+e)))^p/(d*f*x+d*e),x, algorithm="maxima" )
Output:
i*integrate((b*log(f*x + e) + b*log(c) + a)^p*x/(d*f*x + d*e), x) + (b*c*l og(c*f*x + c*e) + a*c)*(b*log(c*f*x + c*e) + a)^p*h/(b*c*d*f*(p + 1))
\[ \int \frac {(h+i x) (a+b \log (c (e+f x)))^p}{d e+d f x} \, dx=\int { \frac {{\left (i x + h\right )} {\left (b \log \left ({\left (f x + e\right )} c\right ) + a\right )}^{p}}{d f x + d e} \,d x } \] Input:
integrate((i*x+h)*(a+b*log(c*(f*x+e)))^p/(d*f*x+d*e),x, algorithm="giac")
Output:
integrate((i*x + h)*(b*log((f*x + e)*c) + a)^p/(d*f*x + d*e), x)
Timed out. \[ \int \frac {(h+i x) (a+b \log (c (e+f x)))^p}{d e+d f x} \, dx=\int \frac {\left (h+i\,x\right )\,{\left (a+b\,\ln \left (c\,\left (e+f\,x\right )\right )\right )}^p}{d\,e+d\,f\,x} \,d x \] Input:
int(((h + i*x)*(a + b*log(c*(e + f*x)))^p)/(d*e + d*f*x),x)
Output:
int(((h + i*x)*(a + b*log(c*(e + f*x)))^p)/(d*e + d*f*x), x)
\[ \int \frac {(h+i x) (a+b \log (c (e+f x)))^p}{d e+d f x} \, dx =\text {Too large to display} \] Input:
int((i*x+h)*(a+b*log(c*(f*x+e)))^p/(d*f*x+d*e),x)
Output:
( - (log(c*e + c*f*x)*b + a)**p*log(c*e + c*f*x)*a*b*e*i + (log(c*e + c*f* x)*b + a)**p*log(c*e + c*f*x)*a*b*f*h + (log(c*e + c*f*x)*b + a)**p*log(c* e + c*f*x)*b**2*f*h*p - (log(c*e + c*f*x)*b + a)**p*a**2*e*i + (log(c*e + c*f*x)*b + a)**p*a**2*f*h + (log(c*e + c*f*x)*b + a)**p*a*b*f*h*p + (log(c *e + c*f*x)*b + a)**p*a*b*f*i*p*x + (log(c*e + c*f*x)*b + a)**p*a*b*f*i*x + int(((log(c*e + c*f*x)*b + a)**p*log(c*e + c*f*x)*x)/(log(c*e + c*f*x)*a *b*e + log(c*e + c*f*x)*a*b*f*x + log(c*e + c*f*x)*b**2*e*p + log(c*e + c* f*x)*b**2*f*p*x + a**2*e + a**2*f*x + a*b*e*p + a*b*f*p*x),x)*a*b**3*f**2* i*p**2 + int(((log(c*e + c*f*x)*b + a)**p*log(c*e + c*f*x)*x)/(log(c*e + c *f*x)*a*b*e + log(c*e + c*f*x)*a*b*f*x + log(c*e + c*f*x)*b**2*e*p + log(c *e + c*f*x)*b**2*f*p*x + a**2*e + a**2*f*x + a*b*e*p + a*b*f*p*x),x)*a*b** 3*f**2*i*p + int(((log(c*e + c*f*x)*b + a)**p*log(c*e + c*f*x)*x)/(log(c*e + c*f*x)*a*b*e + log(c*e + c*f*x)*a*b*f*x + log(c*e + c*f*x)*b**2*e*p + l og(c*e + c*f*x)*b**2*f*p*x + a**2*e + a**2*f*x + a*b*e*p + a*b*f*p*x),x)*b **4*f**2*i*p**3 + int(((log(c*e + c*f*x)*b + a)**p*log(c*e + c*f*x)*x)/(lo g(c*e + c*f*x)*a*b*e + log(c*e + c*f*x)*a*b*f*x + log(c*e + c*f*x)*b**2*e* p + log(c*e + c*f*x)*b**2*f*p*x + a**2*e + a**2*f*x + a*b*e*p + a*b*f*p*x) ,x)*b**4*f**2*i*p**2)/(b*d*f**2*(a*p + a + b*p**2 + b*p))