Integrand size = 22, antiderivative size = 129 \[ \int (a+b x)^{-n} (c+d x) (e+f x)^n \, dx=\frac {d (a+b x)^{1-n} (e+f x)^{1+n}}{2 b f}+\frac {(2 b c f-b d e (1-n)-a d f (1+n)) (a+b x)^{1-n} (e+f x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,2,2-n,-\frac {f (a+b x)}{b e-a f}\right )}{2 b f (b e-a f) (1-n)} \] Output:
1/2*d*(b*x+a)^(1-n)*(f*x+e)^(1+n)/b/f+1/2*(2*b*c*f-b*d*e*(1-n)-a*d*f*(1+n) )*(b*x+a)^(1-n)*(f*x+e)^(1+n)*hypergeom([1, 2],[2-n],-f*(b*x+a)/(-a*f+b*e) )/b/f/(-a*f+b*e)/(1-n)
Time = 0.11 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.84 \[ \int (a+b x)^{-n} (c+d x) (e+f x)^n \, dx=\frac {(a+b x)^{-n} (e+f x)^{1+n} \left (d f (a+b x)+\frac {(2 b c f+b d e (-1+n)-a d f (1+n)) \left (\frac {f (a+b x)}{-b e+a f}\right )^n \operatorname {Hypergeometric2F1}\left (n,1+n,2+n,\frac {b (e+f x)}{b e-a f}\right )}{1+n}\right )}{2 b f^2} \] Input:
Integrate[((c + d*x)*(e + f*x)^n)/(a + b*x)^n,x]
Output:
((e + f*x)^(1 + n)*(d*f*(a + b*x) + ((2*b*c*f + b*d*e*(-1 + n) - a*d*f*(1 + n))*((f*(a + b*x))/(-(b*e) + a*f))^n*Hypergeometric2F1[n, 1 + n, 2 + n, (b*(e + f*x))/(b*e - a*f)])/(1 + n)))/(2*b*f^2*(a + b*x)^n)
Time = 0.24 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.04, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {90, 80, 79}
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.
\(\displaystyle \int (c+d x) (a+b x)^{-n} (e+f x)^n \, dx\) |
\(\Big \downarrow \) 90 |
\(\displaystyle \frac {(-a d f (n+1)+2 b c f-b d e (1-n)) \int (a+b x)^{-n} (e+f x)^ndx}{2 b f}+\frac {d (a+b x)^{1-n} (e+f x)^{n+1}}{2 b f}\) |
\(\Big \downarrow \) 80 |
\(\displaystyle \frac {(a+b x)^{-n} \left (-\frac {f (a+b x)}{b e-a f}\right )^n (-a d f (n+1)+2 b c f-b d e (1-n)) \int (e+f x)^n \left (-\frac {b x f}{b e-a f}-\frac {a f}{b e-a f}\right )^{-n}dx}{2 b f}+\frac {d (a+b x)^{1-n} (e+f x)^{n+1}}{2 b f}\) |
\(\Big \downarrow \) 79 |
\(\displaystyle \frac {(a+b x)^{-n} (e+f x)^{n+1} \left (-\frac {f (a+b x)}{b e-a f}\right )^n (-a d f (n+1)+2 b c f-b d e (1-n)) \operatorname {Hypergeometric2F1}\left (n,n+1,n+2,\frac {b (e+f x)}{b e-a f}\right )}{2 b f^2 (n+1)}+\frac {d (a+b x)^{1-n} (e+f x)^{n+1}}{2 b f}\) |
Input:
Int[((c + d*x)*(e + f*x)^n)/(a + b*x)^n,x]
Output:
(d*(a + b*x)^(1 - n)*(e + f*x)^(1 + n))/(2*b*f) + ((2*b*c*f - b*d*e*(1 - n ) - a*d*f*(1 + n))*(-((f*(a + b*x))/(b*e - a*f)))^n*(e + f*x)^(1 + n)*Hype rgeometric2F1[n, 1 + n, 2 + n, (b*(e + f*x))/(b*e - a*f)])/(2*b*f^2*(1 + n )*(a + b*x)^n)
Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(( a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c - a*d))^n))*Hypergeometric2F1[-n, m + 1 , m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m, n}, x] && !IntegerQ[m] && !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] || !(RationalQ[n] && GtQ[-d/(b*c - a*d), 0]))
Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(c + d*x)^FracPart[n]/((b/(b*c - a*d))^IntPart[n]*(b*((c + d*x)/(b*c - a*d))) ^FracPart[n]) Int[(a + b*x)^m*Simp[b*(c/(b*c - a*d)) + b*d*(x/(b*c - a*d) ), x]^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && !IntegerQ[m] && !Integ erQ[n] && (RationalQ[m] || !SimplerQ[n + 1, m + 1])
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p + 2)) Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0]
\[\int \left (x d +c \right ) \left (f x +e \right )^{n} \left (b x +a \right )^{-n}d x\]
Input:
int((d*x+c)*(f*x+e)^n/((b*x+a)^n),x)
Output:
int((d*x+c)*(f*x+e)^n/((b*x+a)^n),x)
\[ \int (a+b x)^{-n} (c+d x) (e+f x)^n \, dx=\int { \frac {{\left (d x + c\right )} {\left (f x + e\right )}^{n}}{{\left (b x + a\right )}^{n}} \,d x } \] Input:
integrate((d*x+c)*(f*x+e)^n/((b*x+a)^n),x, algorithm="fricas")
Output:
integral((d*x + c)*(f*x + e)^n/(b*x + a)^n, x)
Exception generated. \[ \int (a+b x)^{-n} (c+d x) (e+f x)^n \, dx=\text {Exception raised: HeuristicGCDFailed} \] Input:
integrate((d*x+c)*(f*x+e)**n/((b*x+a)**n),x)
Output:
Exception raised: HeuristicGCDFailed >> no luck
\[ \int (a+b x)^{-n} (c+d x) (e+f x)^n \, dx=\int { \frac {{\left (d x + c\right )} {\left (f x + e\right )}^{n}}{{\left (b x + a\right )}^{n}} \,d x } \] Input:
integrate((d*x+c)*(f*x+e)^n/((b*x+a)^n),x, algorithm="maxima")
Output:
integrate((d*x + c)*(f*x + e)^n/(b*x + a)^n, x)
\[ \int (a+b x)^{-n} (c+d x) (e+f x)^n \, dx=\int { \frac {{\left (d x + c\right )} {\left (f x + e\right )}^{n}}{{\left (b x + a\right )}^{n}} \,d x } \] Input:
integrate((d*x+c)*(f*x+e)^n/((b*x+a)^n),x, algorithm="giac")
Output:
integrate((d*x + c)*(f*x + e)^n/(b*x + a)^n, x)
Timed out. \[ \int (a+b x)^{-n} (c+d x) (e+f x)^n \, dx=\int \frac {{\left (e+f\,x\right )}^n\,\left (c+d\,x\right )}{{\left (a+b\,x\right )}^n} \,d x \] Input:
int(((e + f*x)^n*(c + d*x))/(a + b*x)^n,x)
Output:
int(((e + f*x)^n*(c + d*x))/(a + b*x)^n, x)
\[ \int (a+b x)^{-n} (c+d x) (e+f x)^n \, dx=\left (\int \frac {\left (f x +e \right )^{n}}{\left (b x +a \right )^{n}}d x \right ) c +\left (\int \frac {\left (f x +e \right )^{n} x}{\left (b x +a \right )^{n}}d x \right ) d \] Input:
int((d*x+c)*(f*x+e)^n/((b*x+a)^n),x)
Output:
int((e + f*x)**n/(a + b*x)**n,x)*c + int(((e + f*x)**n*x)/(a + b*x)**n,x)* d