Integrand size = 21, antiderivative size = 61 \[ \int (a+b x)^{-2+n} (c+d x)^{1-n} \, dx=-\frac {(a+b x)^{-1+n} (c+d x)^{2-n} \operatorname {Hypergeometric2F1}\left (1,1,n,-\frac {d (a+b x)}{b c-a d}\right )}{(b c-a d) (1-n)} \] Output:
-(b*x+a)^(-1+n)*(d*x+c)^(2-n)*hypergeom([1, 1],[n],-d*(b*x+a)/(-a*d+b*c))/ (-a*d+b*c)/(1-n)
Time = 0.07 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.23 \[ \int (a+b x)^{-2+n} (c+d x)^{1-n} \, dx=\frac {(a+b x)^{-1+n} (c+d x)^{1-n} \left (\frac {b (c+d x)}{b c-a d}\right )^{-1+n} \operatorname {Hypergeometric2F1}\left (-1+n,-1+n,n,\frac {d (a+b x)}{-b c+a d}\right )}{b (-1+n)} \] Input:
Integrate[(a + b*x)^(-2 + n)*(c + d*x)^(1 - n),x]
Output:
((a + b*x)^(-1 + n)*(c + d*x)^(1 - n)*((b*(c + d*x))/(b*c - a*d))^(-1 + n) *Hypergeometric2F1[-1 + n, -1 + n, n, (d*(a + b*x))/(-(b*c) + a*d)])/(b*(- 1 + n))
Time = 0.18 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.36, 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.
\(\displaystyle \int (a+b x)^{n-2} (c+d x)^{1-n} \, dx\) |
\(\Big \downarrow \) 80 |
\(\displaystyle \frac {(b c-a d) (c+d x)^{-n} \left (\frac {b (c+d x)}{b c-a d}\right )^n \int (a+b x)^{n-2} \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^{1-n}dx}{b}\) |
\(\Big \downarrow \) 79 |
\(\displaystyle -\frac {(b c-a d) (a+b x)^{n-1} (c+d x)^{-n} \left (\frac {b (c+d x)}{b c-a d}\right )^n \operatorname {Hypergeometric2F1}\left (n-1,n-1,n,-\frac {d (a+b x)}{b c-a d}\right )}{b^2 (1-n)}\) |
Input:
Int[(a + b*x)^(-2 + n)*(c + d*x)^(1 - n),x]
Output:
-(((b*c - a*d)*(a + b*x)^(-1 + n)*((b*(c + d*x))/(b*c - a*d))^n*Hypergeome tric2F1[-1 + n, -1 + n, n, -((d*(a + b*x))/(b*c - a*d))])/(b^2*(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])
\[\int \left (b x +a \right )^{n -2} \left (x d +c \right )^{-n +1}d x\]
Input:
int((b*x+a)^(n-2)*(d*x+c)^(-n+1),x)
Output:
int((b*x+a)^(n-2)*(d*x+c)^(-n+1),x)
\[ \int (a+b x)^{-2+n} (c+d x)^{1-n} \, dx=\int { {\left (b x + a\right )}^{n - 2} {\left (d x + c\right )}^{-n + 1} \,d x } \] Input:
integrate((b*x+a)^(-2+n)*(d*x+c)^(1-n),x, algorithm="fricas")
Output:
integral((b*x + a)^(n - 2)*(d*x + c)^(-n + 1), x)
\[ \int (a+b x)^{-2+n} (c+d x)^{1-n} \, dx=\int \left (a + b x\right )^{n - 2} \left (c + d x\right )^{1 - n}\, dx \] Input:
integrate((b*x+a)**(-2+n)*(d*x+c)**(1-n),x)
Output:
Integral((a + b*x)**(n - 2)*(c + d*x)**(1 - n), x)
\[ \int (a+b x)^{-2+n} (c+d x)^{1-n} \, dx=\int { {\left (b x + a\right )}^{n - 2} {\left (d x + c\right )}^{-n + 1} \,d x } \] Input:
integrate((b*x+a)^(-2+n)*(d*x+c)^(1-n),x, algorithm="maxima")
Output:
integrate((b*x + a)^(n - 2)*(d*x + c)^(-n + 1), x)
\[ \int (a+b x)^{-2+n} (c+d x)^{1-n} \, dx=\int { {\left (b x + a\right )}^{n - 2} {\left (d x + c\right )}^{-n + 1} \,d x } \] Input:
integrate((b*x+a)^(-2+n)*(d*x+c)^(1-n),x, algorithm="giac")
Output:
integrate((b*x + a)^(n - 2)*(d*x + c)^(-n + 1), x)
Timed out. \[ \int (a+b x)^{-2+n} (c+d x)^{1-n} \, dx=\int {\left (a+b\,x\right )}^{n-2}\,{\left (c+d\,x\right )}^{1-n} \,d x \] Input:
int((a + b*x)^(n - 2)*(c + d*x)^(1 - n),x)
Output:
int((a + b*x)^(n - 2)*(c + d*x)^(1 - n), x)
\[ \int (a+b x)^{-2+n} (c+d x)^{1-n} \, dx=\left (\int \frac {\left (b x +a \right )^{n}}{\left (d x +c \right )^{n} a^{2}+2 \left (d x +c \right )^{n} a b x +\left (d x +c \right )^{n} b^{2} x^{2}}d x \right ) c +\left (\int \frac {\left (b x +a \right )^{n} x}{\left (d x +c \right )^{n} a^{2}+2 \left (d x +c \right )^{n} a b x +\left (d x +c \right )^{n} b^{2} x^{2}}d x \right ) d \] Input:
int((b*x+a)^(-2+n)*(d*x+c)^(1-n),x)
Output:
int((a + b*x)**n/((c + d*x)**n*a**2 + 2*(c + d*x)**n*a*b*x + (c + d*x)**n* b**2*x**2),x)*c + int(((a + b*x)**n*x)/((c + d*x)**n*a**2 + 2*(c + d*x)**n *a*b*x + (c + d*x)**n*b**2*x**2),x)*d