\(\int (a+b x)^{-2+n} (c+d x)^{-n} \, dx\) [1861]

Optimal result

Integrand size = 19, antiderivative size = 39 \[ \int (a+b x)^{-2+n} (c+d x)^{-n} \, dx=-\frac {(a+b x)^{-1+n} (c+d x)^{1-n}}{(b c-a d) (1-n)} \]

[Out]

-(b*x+a)^(-1+n)*(d*x+c)^(1-n)/(-a*d+b*c)/(1-n)

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {37} \[ \int (a+b x)^{-2+n} (c+d x)^{-n} \, dx=-\frac {(a+b x)^{n-1} (c+d x)^{1-n}}{(1-n) (b c-a d)} \]

[In]

Int[(a + b*x)^(-2 + n)/(c + d*x)^n,x]

[Out]

-(((a + b*x)^(-1 + n)*(c + d*x)^(1 - n))/((b*c - a*d)*(1 - n)))

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n +
1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps \begin{align*} \text {integral}& = -\frac {(a+b x)^{-1+n} (c+d x)^{1-n}}{(b c-a d) (1-n)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.92 \[ \int (a+b x)^{-2+n} (c+d x)^{-n} \, dx=\frac {(a+b x)^{-1+n} (c+d x)^{1-n}}{(b c-a d) (-1+n)} \]

[In]

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

[Out]

((a + b*x)^(-1 + n)*(c + d*x)^(1 - n))/((b*c - a*d)*(-1 + n))

Maple [A] (verified)

Time = 0.49 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.15

method	result	size
gosper	\(-\frac {\left (b x +a \right )^{-1+n} \left (d x +c \right ) \left (d x +c \right )^{-n}}{a d n -b c n -a d +b c}\)	\(45\)
parallelrisch	\(-\frac {\left (x^{2} \left (b x +a \right )^{-2+n} b^{2} d^{2} n +x \left (b x +a \right )^{-2+n} a b \,d^{2} n +x \left (b x +a \right )^{-2+n} b^{2} c d n +\left (b x +a \right )^{-2+n} a b c d n \right ) \left (d x +c \right )^{-n}}{n \left (a d n -b c n -a d +b c \right ) b d}\)	\(110\)

[In]

int((b*x+a)^(-2+n)/((d*x+c)^n),x,method=_RETURNVERBOSE)

[Out]

-(b*x+a)^(-1+n)*(d*x+c)/((d*x+c)^n)/(a*d*n-b*c*n-a*d+b*c)

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.54 \[ \int (a+b x)^{-2+n} (c+d x)^{-n} \, dx=-\frac {{\left (b d x^{2} + a c + {\left (b c + a d\right )} x\right )} {\left (b x + a\right )}^{n - 2}}{{\left (b c - a d - {\left (b c - a d\right )} n\right )} {\left (d x + c\right )}^{n}} \]

[In]

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

[Out]

-(b*d*x^2 + a*c + (b*c + a*d)*x)*(b*x + a)^(n - 2)/((b*c - a*d - (b*c - a*d)*n)*(d*x + c)^n)

Sympy [F(-2)]

Exception generated. \[ \int (a+b x)^{-2+n} (c+d x)^{-n} \, dx=\text {Exception raised: HeuristicGCDFailed} \]

[In]

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

[Out]

Exception raised: HeuristicGCDFailed >> no luck

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.53 (sec) , antiderivative size = 102, normalized size of antiderivative = 2.62 \[ \int (a+b x)^{-2+n} (c+d x)^{-n} \, dx=-{\left (a+b\,x\right )}^{n-2}\,\left (\frac {a\,c}{\left (a\,d-b\,c\right )\,\left (n-1\right )\,{\left (c+d\,x\right )}^n}+\frac {x\,\left (a\,d+b\,c\right )}{\left (a\,d-b\,c\right )\,\left (n-1\right )\,{\left (c+d\,x\right )}^n}+\frac {b\,d\,x^2}{\left (a\,d-b\,c\right )\,\left (n-1\right )\,{\left (c+d\,x\right )}^n}\right ) \]

[In]

int((a + b*x)^(n - 2)/(c + d*x)^n,x)

[Out]

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