Integrand size = 21, antiderivative size = 57 \[ \int (a+b x)^{1+n} (c+d x)^{-1-n} \, dx=\frac {(a+b x)^{2+n} (c+d x)^{-n} \operatorname {Hypergeometric2F1}\left (1,2,1-n,\frac {b (c+d x)}{b c-a d}\right )}{(b c-a d) n} \] Output:
(b*x+a)^(2+n)*hypergeom([1, 2],[1-n],b*(d*x+c)/(-a*d+b*c))/(-a*d+b*c)/n/(( d*x+c)^n)
Time = 0.06 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.46 \[ \int (a+b x)^{1+n} (c+d x)^{-1-n} \, dx=\frac {(b c-a d) (a+b x)^n \left (\frac {d (a+b x)}{-b c+a d}\right )^{-n} (c+d x)^{-n} \operatorname {Hypergeometric2F1}\left (-1-n,-n,1-n,\frac {b (c+d x)}{b c-a d}\right )}{d^2 n} \] Input:
Integrate[(a + b*x)^(1 + n)*(c + d*x)^(-1 - n),x]
Output:
((b*c - a*d)*(a + b*x)^n*Hypergeometric2F1[-1 - n, -n, 1 - n, (b*(c + d*x) )/(b*c - a*d)])/(d^2*n*((d*(a + b*x))/(-(b*c) + a*d))^n*(c + d*x)^n)
Time = 0.19 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.47, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {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.
\(\displaystyle \int (a+b x)^{n+1} (c+d x)^{-n-1} \, dx\) |
\(\Big \downarrow \) 80 |
\(\displaystyle -\frac {(b c-a d) (a+b x)^n \left (-\frac {d (a+b x)}{b c-a d}\right )^{-n} \int (c+d x)^{-n-1} \left (-\frac {b x d}{b c-a d}-\frac {a d}{b c-a d}\right )^{n+1}dx}{d}\) |
\(\Big \downarrow \) 79 |
\(\displaystyle \frac {(b c-a d) (a+b x)^n (c+d x)^{-n} \left (-\frac {d (a+b x)}{b c-a d}\right )^{-n} \operatorname {Hypergeometric2F1}\left (-n-1,-n,1-n,\frac {b (c+d x)}{b c-a d}\right )}{d^2 n}\) |
Input:
Int[(a + b*x)^(1 + n)*(c + d*x)^(-1 - n),x]
Output:
((b*c - a*d)*(a + b*x)^n*Hypergeometric2F1[-1 - n, -n, 1 - n, (b*(c + d*x) )/(b*c - a*d)])/(d^2*n*(-((d*(a + b*x))/(b*c - a*d)))^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])
\[\int \left (b x +a \right )^{1+n} \left (x d +c \right )^{-1-n}d x\]
Input:
int((b*x+a)^(1+n)*(d*x+c)^(-1-n),x)
Output:
int((b*x+a)^(1+n)*(d*x+c)^(-1-n),x)
\[ \int (a+b x)^{1+n} (c+d x)^{-1-n} \, dx=\int { {\left (b x + a\right )}^{n + 1} {\left (d x + c\right )}^{-n - 1} \,d x } \] Input:
integrate((b*x+a)^(1+n)*(d*x+c)^(-1-n),x, algorithm="fricas")
Output:
integral((b*x + a)^(n + 1)*(d*x + c)^(-n - 1), x)
\[ \int (a+b x)^{1+n} (c+d x)^{-1-n} \, dx=\int \left (a + b x\right )^{n + 1} \left (c + d x\right )^{- n - 1}\, dx \] Input:
integrate((b*x+a)**(1+n)*(d*x+c)**(-1-n),x)
Output:
Integral((a + b*x)**(n + 1)*(c + d*x)**(-n - 1), x)
\[ \int (a+b x)^{1+n} (c+d x)^{-1-n} \, dx=\int { {\left (b x + a\right )}^{n + 1} {\left (d x + c\right )}^{-n - 1} \,d x } \] Input:
integrate((b*x+a)^(1+n)*(d*x+c)^(-1-n),x, algorithm="maxima")
Output:
integrate((b*x + a)^(n + 1)*(d*x + c)^(-n - 1), x)
\[ \int (a+b x)^{1+n} (c+d x)^{-1-n} \, dx=\int { {\left (b x + a\right )}^{n + 1} {\left (d x + c\right )}^{-n - 1} \,d x } \] Input:
integrate((b*x+a)^(1+n)*(d*x+c)^(-1-n),x, algorithm="giac")
Output:
integrate((b*x + a)^(n + 1)*(d*x + c)^(-n - 1), x)
Timed out. \[ \int (a+b x)^{1+n} (c+d x)^{-1-n} \, dx=\int \frac {{\left (a+b\,x\right )}^{n+1}}{{\left (c+d\,x\right )}^{n+1}} \,d x \] Input:
int((a + b*x)^(n + 1)/(c + d*x)^(n + 1),x)
Output:
int((a + b*x)^(n + 1)/(c + d*x)^(n + 1), x)
\[ \int (a+b x)^{1+n} (c+d x)^{-1-n} \, dx=\left (\int \frac {\left (b x +a \right )^{n}}{\left (d x +c \right )^{n} c +\left (d x +c \right )^{n} d x}d x \right ) a +\left (\int \frac {\left (b x +a \right )^{n} x}{\left (d x +c \right )^{n} c +\left (d x +c \right )^{n} d x}d x \right ) b \] Input:
int((b*x+a)^(1+n)*(d*x+c)^(-1-n),x)
Output:
int((a + b*x)**n/((c + d*x)**n*c + (c + d*x)**n*d*x),x)*a + int(((a + b*x) **n*x)/((c + d*x)**n*c + (c + d*x)**n*d*x),x)*b