\(\int \frac {(a+b x)^n (c+d x)^{-n}}{x^2} \, dx\) [975]

Optimal result

Integrand size = 20, antiderivative size = 62 \[ \int \frac {(a+b x)^n (c+d x)^{-n}}{x^2} \, dx=\frac {(b c-a d) (a+b x)^{1+n} (c+d x)^{-1-n} \operatorname {Hypergeometric2F1}\left (2,1+n,2+n,\frac {c (a+b x)}{a (c+d x)}\right )}{a^2 (1+n)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {133} \[ \int \frac {(a+b x)^n (c+d x)^{-n}}{x^2} \, dx=\frac {(b c-a d) (a+b x)^{n+1} (c+d x)^{-n-1} \operatorname {Hypergeometric2F1}\left (2,n+1,n+2,\frac {c (a+b x)}{a (c+d x)}\right )}{a^2 (n+1)} \]

[In]

Int[(a + b*x)^n/(x^2*(c + d*x)^n),x]

[Out]

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

Rule 133

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Simp[(b*c - a
*d)^n*((a + b*x)^(m + 1)/((m + 1)*(b*e - a*f)^(n + 1)*(e + f*x)^(m + 1)))*Hypergeometric2F1[m + 1, -n, m + 2,
(-(d*e - c*f))*((a + b*x)/((b*c - a*d)*(e + f*x)))], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[m + n + p
+ 2, 0] && ILtQ[n, 0] && (SumSimplerQ[m, 1] ||  !SumSimplerQ[p, 1]) &&  !ILtQ[m, 0]

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

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x)^n (c+d x)^{-n}}{x^2} \, dx=\frac {(b c-a d) (a+b x)^{1+n} (c+d x)^{-1-n} \operatorname {Hypergeometric2F1}\left (2,1+n,2+n,\frac {c (a+b x)}{a (c+d x)}\right )}{a^2 (1+n)} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int((a + b*x)^n/(x^2*(c + d*x)^n),x)

[Out]

int((a + b*x)^n/(x^2*(c + d*x)^n), x)