\(\int (a+b x)^{-2+n} (c+d x)^{1-n} \, dx\) [1878]

Optimal result

Integrand size = 21, antiderivative size = 83 \[ \int (a+b x)^{-2+n} (c+d x)^{1-n} \, dx=-\frac {(b c-a d) (a+b x)^{-1+n} (c+d x)^{-n} \left (\frac {b (c+d x)}{b c-a d}\right )^n \operatorname {Hypergeometric2F1}\left (-1+n,-1+n,n,-\frac {d (a+b x)}{b c-a d}\right )}{b^2 (1-n)} \]

[Out]

-(-a*d+b*c)*(b*x+a)^(-1+n)*(b*(d*x+c)/(-a*d+b*c))^n*hypergeom([-1+n, -1+n],[n],-d*(b*x+a)/(-a*d+b*c))/b^2/(1-n
)/((d*x+c)^n)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {72, 71} \[ \int (a+b x)^{-2+n} (c+d x)^{1-n} \, dx=-\frac {(b c-a d) (a+b x)^{n-1} (c+d x)^{-n} \left (\frac {b (c+d x)}{b c-a d}\right )^n \operatorname {Hypergeometric2F1}\left (n-1,n-1,n,-\frac {d (a+b x)}{b c-a d}\right )}{b^2 (1-n)} \]

[In]

Int[(a + b*x)^(-2 + n)*(c + d*x)^(1 - n),x]

[Out]

-(((b*c - a*d)*(a + b*x)^(-1 + n)*((b*(c + d*x))/(b*c - a*d))^n*Hypergeometric2F1[-1 + n, -1 + n, n, -((d*(a +
b*x))/(b*c - a*d))])/(b^2*(1 - n)*(c + d*x)^n))

Rule 71

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] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] ||  !(Ra
tionalQ[n] && GtQ[-d/(b*c - a*d), 0]))

Rule 72

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Dist[(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] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] &&
(RationalQ[m] ||  !SimplerQ[n + 1, m + 1])

Rubi steps \begin{align*} \text {integral}& = \frac {\left ((b c-a d) (c+d x)^{-n} \left (\frac {b (c+d x)}{b c-a d}\right )^n\right ) \int (a+b x)^{-2+n} \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^{1-n} \, dx}{b} \\ & = -\frac {(b c-a d) (a+b x)^{-1+n} (c+d x)^{-n} \left (\frac {b (c+d x)}{b c-a d}\right )^n \, _2F_1\left (-1+n,-1+n;n;-\frac {d (a+b x)}{b c-a d}\right )}{b^2 (1-n)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.90 \[ \int (a+b x)^{-2+n} (c+d x)^{1-n} \, dx=\frac {(a+b x)^{-1+n} (c+d x)^{1-n} \left (\frac {b (c+d x)}{b c-a d}\right )^{-1+n} \operatorname {Hypergeometric2F1}\left (-1+n,-1+n,n,\frac {d (a+b x)}{-b c+a d}\right )}{b (-1+n)} \]

[In]

Integrate[(a + b*x)^(-2 + n)*(c + d*x)^(1 - n),x]

[Out]

((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))

Maple [F]

\[\int \left (b x +a \right )^{-2+n} \left (d x +c \right )^{1-n}d x\]

[In]

int((b*x+a)^(-2+n)*(d*x+c)^(1-n),x)

[Out]

int((b*x+a)^(-2+n)*(d*x+c)^(1-n),x)

Fricas [F]

\[ \int (a+b x)^{-2+n} (c+d x)^{1-n} \, dx=\int { {\left (b x + a\right )}^{n - 2} {\left (d x + c\right )}^{-n + 1} \,d x } \]

[In]

integrate((b*x+a)^(-2+n)*(d*x+c)^(1-n),x, algorithm="fricas")

[Out]

integral((b*x + a)^(n - 2)*(d*x + c)^(-n + 1), x)

Sympy [F]

\[ \int (a+b x)^{-2+n} (c+d x)^{1-n} \, dx=\int \left (a + b x\right )^{n - 2} \left (c + d x\right )^{1 - n}\, dx \]

[In]

integrate((b*x+a)**(-2+n)*(d*x+c)**(1-n),x)

[Out]

Integral((a + b*x)**(n - 2)*(c + d*x)**(1 - n), x)

Maxima [F]

\[ \int (a+b x)^{-2+n} (c+d x)^{1-n} \, dx=\int { {\left (b x + a\right )}^{n - 2} {\left (d x + c\right )}^{-n + 1} \,d x } \]

[In]

integrate((b*x+a)^(-2+n)*(d*x+c)^(1-n),x, algorithm="maxima")

[Out]

integrate((b*x + a)^(n - 2)*(d*x + c)^(-n + 1), x)

Giac [F]

\[ \int (a+b x)^{-2+n} (c+d x)^{1-n} \, dx=\int { {\left (b x + a\right )}^{n - 2} {\left (d x + c\right )}^{-n + 1} \,d x } \]

[In]

integrate((b*x+a)^(-2+n)*(d*x+c)^(1-n),x, algorithm="giac")

[Out]

integrate((b*x + a)^(n - 2)*(d*x + c)^(-n + 1), x)

Mupad [F(-1)]

Timed out. \[ \int (a+b x)^{-2+n} (c+d x)^{1-n} \, dx=\int {\left (a+b\,x\right )}^{n-2}\,{\left (c+d\,x\right )}^{1-n} \,d x \]

[In]

int((a + b*x)^(n - 2)*(c + d*x)^(1 - n),x)

[Out]

int((a + b*x)^(n - 2)*(c + d*x)^(1 - n), x)