Integrand size = 18, antiderivative size = 89 \[ \int (d+e x)^m \log \left (c (a+b x)^p\right ) \, dx=\frac {b p (d+e x)^{2+m} \operatorname {Hypergeometric2F1}\left (1,2+m,3+m,\frac {b (d+e x)}{b d-a e}\right )}{e (b d-a e) (1+m) (2+m)}+\frac {(d+e x)^{1+m} \log \left (c (a+b x)^p\right )}{e (1+m)} \] Output:
b*p*(e*x+d)^(2+m)*hypergeom([1, 2+m],[3+m],b*(e*x+d)/(-a*e+b*d))/e/(-a*e+b *d)/(1+m)/(2+m)+(e*x+d)^(1+m)*ln(c*(b*x+a)^p)/e/(1+m)
Time = 0.06 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.87 \[ \int (d+e x)^m \log \left (c (a+b x)^p\right ) \, dx=\frac {(d+e x)^{1+m} \left (\frac {b p (d+e x) \operatorname {Hypergeometric2F1}\left (1,2+m,3+m,\frac {b (d+e x)}{b d-a e}\right )}{(b d-a e) (2+m)}+\log \left (c (a+b x)^p\right )\right )}{e (1+m)} \] Input:
Integrate[(d + e*x)^m*Log[c*(a + b*x)^p],x]
Output:
((d + e*x)^(1 + m)*((b*p*(d + e*x)*Hypergeometric2F1[1, 2 + m, 3 + m, (b*( d + e*x))/(b*d - a*e)])/((b*d - a*e)*(2 + m)) + Log[c*(a + b*x)^p]))/(e*(1 + m))
Time = 0.38 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2842, 78}
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.
| \(\displaystyle \int (d+e x)^m \log \left (c (a+b x)^p\right ) \, dx\) |
| \(\Big \downarrow \) 2842 |
| \(\displaystyle \frac {(d+e x)^{m+1} \log \left (c (a+b x)^p\right )}{e (m+1)}-\frac {b p \int \frac {(d+e x)^{m+1}}{a+b x}dx}{e (m+1)}\) |
| \(\Big \downarrow \) 78 |
| \(\displaystyle \frac {(d+e x)^{m+1} \log \left (c (a+b x)^p\right )}{e (m+1)}+\frac {b p (d+e x)^{m+2} \operatorname {Hypergeometric2F1}\left (1,m+2,m+3,\frac {b (d+e x)}{b d-a e}\right )}{e (m+1) (m+2) (b d-a e)}\) |
Input:
Int[(d + e*x)^m*Log[c*(a + b*x)^p],x]
Output:
(b*p*(d + e*x)^(2 + m)*Hypergeometric2F1[1, 2 + m, 3 + m, (b*(d + e*x))/(b *d - a*e)])/(e*(b*d - a*e)*(1 + m)*(2 + m)) + ((d + e*x)^(1 + m)*Log[c*(a + b*x)^p])/(e*(1 + m))
Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(b *c - a*d)^n*((a + b*x)^(m + 1)/(b^(n + 1)*(m + 1)))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m}, x] && !IntegerQ[m] && IntegerQ[n]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*((f_.) + (g_.)*(x_ ))^(q_.), x_Symbol] :> Simp[(f + g*x)^(q + 1)*((a + b*Log[c*(d + e*x)^n])/( g*(q + 1))), x] - Simp[b*e*(n/(g*(q + 1))) Int[(f + g*x)^(q + 1)/(d + e*x ), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*f - d*g, 0] && NeQ[q, -1]
\[\int \left (e x +d \right )^{m} \ln \left (c \left (b x +a \right )^{p}\right )d x\]
Input:
int((e*x+d)^m*ln(c*(b*x+a)^p),x)
Output:
int((e*x+d)^m*ln(c*(b*x+a)^p),x)
\[ \int (d+e x)^m \log \left (c (a+b x)^p\right ) \, dx=\int { {\left (e x + d\right )}^{m} \log \left ({\left (b x + a\right )}^{p} c\right ) \,d x } \] Input:
integrate((e*x+d)^m*log(c*(b*x+a)^p),x, algorithm="fricas")
Output:
integral((e*x + d)^m*log((b*x + a)^p*c), x)
Exception generated. \[ \int (d+e x)^m \log \left (c (a+b x)^p\right ) \, dx=\text {Exception raised: HeuristicGCDFailed} \] Input:
integrate((e*x+d)**m*ln(c*(b*x+a)**p),x)
Output:
Exception raised: HeuristicGCDFailed >> no luck
\[ \int (d+e x)^m \log \left (c (a+b x)^p\right ) \, dx=\int { {\left (e x + d\right )}^{m} \log \left ({\left (b x + a\right )}^{p} c\right ) \,d x } \] Input:
integrate((e*x+d)^m*log(c*(b*x+a)^p),x, algorithm="maxima")
Output:
(e*x + d)*(e*x + d)^m*log((b*x + a)^p)/(e*(m + 1)) + integrate((a*e*(m + 1 )*log(c) - b*d*p + (e*(m + 1)*log(c) - e*p)*b*x)*(e*x + d)^m/(b*e*(m + 1)* x + a*e*(m + 1)), x)
\[ \int (d+e x)^m \log \left (c (a+b x)^p\right ) \, dx=\int { {\left (e x + d\right )}^{m} \log \left ({\left (b x + a\right )}^{p} c\right ) \,d x } \] Input:
integrate((e*x+d)^m*log(c*(b*x+a)^p),x, algorithm="giac")
Output:
integrate((e*x + d)^m*log((b*x + a)^p*c), x)
Timed out. \[ \int (d+e x)^m \log \left (c (a+b x)^p\right ) \, dx=\int \ln \left (c\,{\left (a+b\,x\right )}^p\right )\,{\left (d+e\,x\right )}^m \,d x \] Input:
int(log(c*(a + b*x)^p)*(d + e*x)^m,x)
Output:
int(log(c*(a + b*x)^p)*(d + e*x)^m, x)
\[ \int (d+e x)^m \log \left (c (a+b x)^p\right ) \, dx=\frac {\left (e x +d \right )^{m} \mathrm {log}\left (\left (b x +a \right )^{p} c \right ) a d e \,m^{2}+\left (e x +d \right )^{m} \mathrm {log}\left (\left (b x +a \right )^{p} c \right ) a d e m +\left (e x +d \right )^{m} \mathrm {log}\left (\left (b x +a \right )^{p} c \right ) a \,e^{2} m^{2} x +\left (e x +d \right )^{m} \mathrm {log}\left (\left (b x +a \right )^{p} c \right ) a \,e^{2} m x +\left (e x +d \right )^{m} a d e p -\left (e x +d \right )^{m} a \,e^{2} m p x -\left (e x +d \right )^{m} b \,d^{2} m p -\left (e x +d \right )^{m} b \,d^{2} p +\left (\int \frac {\left (e x +d \right )^{m} x}{b e m \,x^{2}+a e m x +b d m x +b e \,x^{2}+a d m +a e x +b d x +a d}d x \right ) a^{2} e^{3} m^{3} p +2 \left (\int \frac {\left (e x +d \right )^{m} x}{b e m \,x^{2}+a e m x +b d m x +b e \,x^{2}+a d m +a e x +b d x +a d}d x \right ) a^{2} e^{3} m^{2} p +\left (\int \frac {\left (e x +d \right )^{m} x}{b e m \,x^{2}+a e m x +b d m x +b e \,x^{2}+a d m +a e x +b d x +a d}d x \right ) a^{2} e^{3} m p -2 \left (\int \frac {\left (e x +d \right )^{m} x}{b e m \,x^{2}+a e m x +b d m x +b e \,x^{2}+a d m +a e x +b d x +a d}d x \right ) a b d \,e^{2} m^{3} p -4 \left (\int \frac {\left (e x +d \right )^{m} x}{b e m \,x^{2}+a e m x +b d m x +b e \,x^{2}+a d m +a e x +b d x +a d}d x \right ) a b d \,e^{2} m^{2} p -2 \left (\int \frac {\left (e x +d \right )^{m} x}{b e m \,x^{2}+a e m x +b d m x +b e \,x^{2}+a d m +a e x +b d x +a d}d x \right ) a b d \,e^{2} m p +\left (\int \frac {\left (e x +d \right )^{m} x}{b e m \,x^{2}+a e m x +b d m x +b e \,x^{2}+a d m +a e x +b d x +a d}d x \right ) b^{2} d^{2} e \,m^{3} p +2 \left (\int \frac {\left (e x +d \right )^{m} x}{b e m \,x^{2}+a e m x +b d m x +b e \,x^{2}+a d m +a e x +b d x +a d}d x \right ) b^{2} d^{2} e \,m^{2} p +\left (\int \frac {\left (e x +d \right )^{m} x}{b e m \,x^{2}+a e m x +b d m x +b e \,x^{2}+a d m +a e x +b d x +a d}d x \right ) b^{2} d^{2} e m p}{a \,e^{2} m \left (m^{2}+2 m +1\right )} \] Input:
int((e*x+d)^m*log(c*(b*x+a)^p),x)
Output:
((d + e*x)**m*log((a + b*x)**p*c)*a*d*e*m**2 + (d + e*x)**m*log((a + b*x)* *p*c)*a*d*e*m + (d + e*x)**m*log((a + b*x)**p*c)*a*e**2*m**2*x + (d + e*x) **m*log((a + b*x)**p*c)*a*e**2*m*x + (d + e*x)**m*a*d*e*p - (d + e*x)**m*a *e**2*m*p*x - (d + e*x)**m*b*d**2*m*p - (d + e*x)**m*b*d**2*p + int(((d + e*x)**m*x)/(a*d*m + a*d + a*e*m*x + a*e*x + b*d*m*x + b*d*x + b*e*m*x**2 + b*e*x**2),x)*a**2*e**3*m**3*p + 2*int(((d + e*x)**m*x)/(a*d*m + a*d + a*e *m*x + a*e*x + b*d*m*x + b*d*x + b*e*m*x**2 + b*e*x**2),x)*a**2*e**3*m**2* p + int(((d + e*x)**m*x)/(a*d*m + a*d + a*e*m*x + a*e*x + b*d*m*x + b*d*x + b*e*m*x**2 + b*e*x**2),x)*a**2*e**3*m*p - 2*int(((d + e*x)**m*x)/(a*d*m + a*d + a*e*m*x + a*e*x + b*d*m*x + b*d*x + b*e*m*x**2 + b*e*x**2),x)*a*b* d*e**2*m**3*p - 4*int(((d + e*x)**m*x)/(a*d*m + a*d + a*e*m*x + a*e*x + b* d*m*x + b*d*x + b*e*m*x**2 + b*e*x**2),x)*a*b*d*e**2*m**2*p - 2*int(((d + e*x)**m*x)/(a*d*m + a*d + a*e*m*x + a*e*x + b*d*m*x + b*d*x + b*e*m*x**2 + b*e*x**2),x)*a*b*d*e**2*m*p + int(((d + e*x)**m*x)/(a*d*m + a*d + a*e*m*x + a*e*x + b*d*m*x + b*d*x + b*e*m*x**2 + b*e*x**2),x)*b**2*d**2*e*m**3*p + 2*int(((d + e*x)**m*x)/(a*d*m + a*d + a*e*m*x + a*e*x + b*d*m*x + b*d*x + b*e*m*x**2 + b*e*x**2),x)*b**2*d**2*e*m**2*p + int(((d + e*x)**m*x)/(a*d *m + a*d + a*e*m*x + a*e*x + b*d*m*x + b*d*x + b*e*m*x**2 + b*e*x**2),x)*b **2*d**2*e*m*p)/(a*e**2*m*(m**2 + 2*m + 1))