Integrand size = 18, antiderivative size = 115 \[ \int x (a+b x)^n (c+d x)^{-n} \, dx=\frac {(a+b x)^{1+n} (c+d x)^{1-n}}{2 b d}-\frac {(a d (1-n)+b c (1+n)) (a+b x)^{1+n} (c+d x)^{1-n} \operatorname {Hypergeometric2F1}\left (1,2,2+n,-\frac {d (a+b x)}{b c-a d}\right )}{2 b d (b c-a d) (1+n)} \] Output:
1/2*(b*x+a)^(1+n)*(d*x+c)^(1-n)/b/d-1/2*(a*d*(1-n)+b*c*(1+n))*(b*x+a)^(1+n )*(d*x+c)^(1-n)*hypergeom([1, 2],[2+n],-d*(b*x+a)/(-a*d+b*c))/b/d/(-a*d+b* c)/(1+n)
Time = 0.07 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.88 \[ \int x (a+b x)^n (c+d x)^{-n} \, dx=\frac {(a+b x)^{1+n} (c+d x)^{-n} \left (b (c+d x)-\frac {(-a d (-1+n)+b c (1+n)) \left (\frac {b (c+d x)}{b c-a d}\right )^n \operatorname {Hypergeometric2F1}\left (n,1+n,2+n,\frac {d (a+b x)}{-b c+a d}\right )}{1+n}\right )}{2 b^2 d} \] Input:
Integrate[(x*(a + b*x)^n)/(c + d*x)^n,x]
Output:
((a + b*x)^(1 + n)*(b*(c + d*x) - ((-(a*d*(-1 + n)) + b*c*(1 + n))*((b*(c + d*x))/(b*c - a*d))^n*Hypergeometric2F1[n, 1 + n, 2 + n, (d*(a + b*x))/(- (b*c) + a*d)])/(1 + n)))/(2*b^2*d*(c + d*x)^n)
Time = 0.28 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.09, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, 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.
\(\displaystyle \int x (a+b x)^n (c+d x)^{-n} \, dx\) |
\(\Big \downarrow \) 90 |
\(\displaystyle \frac {(a+b x)^{n+1} (c+d x)^{1-n}}{2 b d}-\frac {1}{2} \left (\frac {a (1-n)}{b}+\frac {c (n+1)}{d}\right ) \int (a+b x)^n (c+d x)^{-n}dx\) |
\(\Big \downarrow \) 80 |
\(\displaystyle \frac {(a+b x)^{n+1} (c+d x)^{1-n}}{2 b d}-\frac {1}{2} (c+d x)^{-n} \left (\frac {a (1-n)}{b}+\frac {c (n+1)}{d}\right ) \left (\frac {b (c+d x)}{b c-a d}\right )^n \int (a+b x)^n \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^{-n}dx\) |
\(\Big \downarrow \) 79 |
\(\displaystyle \frac {(a+b x)^{n+1} (c+d x)^{1-n}}{2 b d}-\frac {(a+b x)^{n+1} (c+d x)^{-n} \left (\frac {a (1-n)}{b}+\frac {c (n+1)}{d}\right ) \left (\frac {b (c+d x)}{b c-a d}\right )^n \operatorname {Hypergeometric2F1}\left (n,n+1,n+2,-\frac {d (a+b x)}{b c-a d}\right )}{2 b (n+1)}\) |
Input:
Int[(x*(a + b*x)^n)/(c + d*x)^n,x]
Output:
((a + b*x)^(1 + n)*(c + d*x)^(1 - n))/(2*b*d) - (((a*(1 - n))/b + (c*(1 + n))/d)*(a + b*x)^(1 + n)*((b*(c + d*x))/(b*c - a*d))^n*Hypergeometric2F1[n , 1 + n, 2 + n, -((d*(a + b*x))/(b*c - a*d))])/(2*b*(1 + n)*(c + d*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]
\[\int x \left (b x +a \right )^{n} \left (x d +c \right )^{-n}d x\]
Input:
int(x*(b*x+a)^n/((d*x+c)^n),x)
Output:
int(x*(b*x+a)^n/((d*x+c)^n),x)
\[ \int x (a+b x)^n (c+d x)^{-n} \, dx=\int { \frac {{\left (b x + a\right )}^{n} x}{{\left (d x + c\right )}^{n}} \,d x } \] Input:
integrate(x*(b*x+a)^n/((d*x+c)^n),x, algorithm="fricas")
Output:
integral((b*x + a)^n*x/(d*x + c)^n, x)
Exception generated. \[ \int x (a+b x)^n (c+d x)^{-n} \, dx=\text {Exception raised: HeuristicGCDFailed} \] Input:
integrate(x*(b*x+a)**n/((d*x+c)**n),x)
Output:
Exception raised: HeuristicGCDFailed >> no luck
\[ \int x (a+b x)^n (c+d x)^{-n} \, dx=\int { \frac {{\left (b x + a\right )}^{n} x}{{\left (d x + c\right )}^{n}} \,d x } \] Input:
integrate(x*(b*x+a)^n/((d*x+c)^n),x, algorithm="maxima")
Output:
integrate((b*x + a)^n*x/(d*x + c)^n, x)
\[ \int x (a+b x)^n (c+d x)^{-n} \, dx=\int { \frac {{\left (b x + a\right )}^{n} x}{{\left (d x + c\right )}^{n}} \,d x } \] Input:
integrate(x*(b*x+a)^n/((d*x+c)^n),x, algorithm="giac")
Output:
integrate((b*x + a)^n*x/(d*x + c)^n, x)
Timed out. \[ \int x (a+b x)^n (c+d x)^{-n} \, dx=\int \frac {x\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^n} \,d x \] Input:
int((x*(a + b*x)^n)/(c + d*x)^n,x)
Output:
int((x*(a + b*x)^n)/(c + d*x)^n, x)
\[ \int x (a+b x)^n (c+d x)^{-n} \, dx=\int \frac {\left (b x +a \right )^{n} x}{\left (d x +c \right )^{n}}d x \] Input:
int(x*(b*x+a)^n/((d*x+c)^n),x)
Output:
int(((a + b*x)**n*x)/(c + d*x)**n,x)