Integrand size = 22, antiderivative size = 125 \[ \int (a+b x) (c+d x)^n (e+f x)^{-n} \, dx=\frac {b (c+d x)^{1+n} (e+f x)^{1-n}}{2 d f}+\frac {(2 a d f-b c f (1-n)-b d e (1+n)) (c+d x)^{1+n} (e+f x)^{1-n} \operatorname {Hypergeometric2F1}\left (1,2,2+n,-\frac {f (c+d x)}{d e-c f}\right )}{2 d f (d e-c f) (1+n)} \] Output:
1/2*b*(d*x+c)^(1+n)*(f*x+e)^(1-n)/d/f+1/2*(2*a*d*f-b*c*f*(1-n)-b*d*e*(1+n) )*(d*x+c)^(1+n)*(f*x+e)^(1-n)*hypergeom([1, 2],[2+n],-f*(d*x+c)/(-c*f+d*e) )/d/f/(-c*f+d*e)/(1+n)
Time = 0.10 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.87 \[ \int (a+b x) (c+d x)^n (e+f x)^{-n} \, dx=\frac {(c+d x)^{1+n} (e+f x)^{-n} \left (b d (e+f x)-\frac {(-2 a d f-b c f (-1+n)+b d e (1+n)) \left (\frac {d (e+f x)}{d e-c f}\right )^n \operatorname {Hypergeometric2F1}\left (n,1+n,2+n,\frac {f (c+d x)}{-d e+c f}\right )}{1+n}\right )}{2 d^2 f} \] Input:
Integrate[((a + b*x)*(c + d*x)^n)/(e + f*x)^n,x]
Output:
((c + d*x)^(1 + n)*(b*d*(e + f*x) - ((-2*a*d*f - b*c*f*(-1 + n) + b*d*e*(1 + n))*((d*(e + f*x))/(d*e - c*f))^n*Hypergeometric2F1[n, 1 + n, 2 + n, (f *(c + d*x))/(-(d*e) + c*f)])/(1 + n)))/(2*d^2*f*(e + f*x)^n)
Time = 0.25 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.07, 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.
\(\displaystyle \int (a+b x) (c+d x)^n (e+f x)^{-n} \, dx\) |
\(\Big \downarrow \) 90 |
\(\displaystyle \frac {(2 a d f-b c f (1-n)-b d e (n+1)) \int (c+d x)^n (e+f x)^{-n}dx}{2 d f}+\frac {b (c+d x)^{n+1} (e+f x)^{1-n}}{2 d f}\) |
\(\Big \downarrow \) 80 |
\(\displaystyle \frac {(e+f x)^{-n} \left (\frac {d (e+f x)}{d e-c f}\right )^n (2 a d f-b c f (1-n)-b d e (n+1)) \int (c+d x)^n \left (\frac {d e}{d e-c f}+\frac {d f x}{d e-c f}\right )^{-n}dx}{2 d f}+\frac {b (c+d x)^{n+1} (e+f x)^{1-n}}{2 d f}\) |
\(\Big \downarrow \) 79 |
\(\displaystyle \frac {(c+d x)^{n+1} (e+f x)^{-n} \left (\frac {d (e+f x)}{d e-c f}\right )^n (2 a d f-b c f (1-n)-b d e (n+1)) \operatorname {Hypergeometric2F1}\left (n,n+1,n+2,-\frac {f (c+d x)}{d e-c f}\right )}{2 d^2 f (n+1)}+\frac {b (c+d x)^{n+1} (e+f x)^{1-n}}{2 d f}\) |
Input:
Int[((a + b*x)*(c + d*x)^n)/(e + f*x)^n,x]
Output:
(b*(c + d*x)^(1 + n)*(e + f*x)^(1 - n))/(2*d*f) + ((2*a*d*f - b*c*f*(1 - n ) - b*d*e*(1 + n))*(c + d*x)^(1 + n)*((d*(e + f*x))/(d*e - c*f))^n*Hyperge ometric2F1[n, 1 + n, 2 + n, -((f*(c + d*x))/(d*e - c*f))])/(2*d^2*f*(1 + n )*(e + f*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]
\[\int \left (b x +a \right ) \left (x d +c \right )^{n} \left (f x +e \right )^{-n}d x\]
Input:
int((b*x+a)*(d*x+c)^n/((f*x+e)^n),x)
Output:
int((b*x+a)*(d*x+c)^n/((f*x+e)^n),x)
\[ \int (a+b x) (c+d x)^n (e+f x)^{-n} \, dx=\int { \frac {{\left (b x + a\right )} {\left (d x + c\right )}^{n}}{{\left (f x + e\right )}^{n}} \,d x } \] Input:
integrate((b*x+a)*(d*x+c)^n/((f*x+e)^n),x, algorithm="fricas")
Output:
integral((b*x + a)*(d*x + c)^n/(f*x + e)^n, x)
Exception generated. \[ \int (a+b x) (c+d x)^n (e+f x)^{-n} \, dx=\text {Exception raised: HeuristicGCDFailed} \] Input:
integrate((b*x+a)*(d*x+c)**n/((f*x+e)**n),x)
Output:
Exception raised: HeuristicGCDFailed >> no luck
\[ \int (a+b x) (c+d x)^n (e+f x)^{-n} \, dx=\int { \frac {{\left (b x + a\right )} {\left (d x + c\right )}^{n}}{{\left (f x + e\right )}^{n}} \,d x } \] Input:
integrate((b*x+a)*(d*x+c)^n/((f*x+e)^n),x, algorithm="maxima")
Output:
integrate((b*x + a)*(d*x + c)^n/(f*x + e)^n, x)
\[ \int (a+b x) (c+d x)^n (e+f x)^{-n} \, dx=\int { \frac {{\left (b x + a\right )} {\left (d x + c\right )}^{n}}{{\left (f x + e\right )}^{n}} \,d x } \] Input:
integrate((b*x+a)*(d*x+c)^n/((f*x+e)^n),x, algorithm="giac")
Output:
integrate((b*x + a)*(d*x + c)^n/(f*x + e)^n, x)
Timed out. \[ \int (a+b x) (c+d x)^n (e+f x)^{-n} \, dx=\int \frac {\left (a+b\,x\right )\,{\left (c+d\,x\right )}^n}{{\left (e+f\,x\right )}^n} \,d x \] Input:
int(((a + b*x)*(c + d*x)^n)/(e + f*x)^n,x)
Output:
int(((a + b*x)*(c + d*x)^n)/(e + f*x)^n, x)
\[ \int (a+b x) (c+d x)^n (e+f x)^{-n} \, dx=\left (\int \frac {\left (d x +c \right )^{n}}{\left (f x +e \right )^{n}}d x \right ) a +\left (\int \frac {\left (d x +c \right )^{n} x}{\left (f x +e \right )^{n}}d x \right ) b \] Input:
int((b*x+a)*(d*x+c)^n/((f*x+e)^n),x)
Output:
int((c + d*x)**n/(e + f*x)**n,x)*a + int(((c + d*x)**n*x)/(e + f*x)**n,x)* b