\(\int \frac {(a-b x)^{-n} (a+b x)^{1+n}}{x^3} \, dx\) [1008]

Optimal result

Integrand size = 23, antiderivative size = 62 \[ \int \frac {(a-b x)^{-n} (a+b x)^{1+n}}{x^3} \, dx=-\frac {4 b^2 (a-b x)^{1-n} (a+b x)^{-1+n} \operatorname {Hypergeometric2F1}\left (3,1-n,2-n,\frac {a-b x}{a+b x}\right )}{a (1-n)} \]

[Out]

-4*b^2*(-b*x+a)^(1-n)*(b*x+a)^(-1+n)*hypergeom([3, 1-n],[2-n],(-b*x+a)/(b*x+a))/a/(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, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {133} \[ \int \frac {(a-b x)^{-n} (a+b x)^{1+n}}{x^3} \, dx=-\frac {4 b^2 (a-b x)^{1-n} (a+b x)^{n-1} \operatorname {Hypergeometric2F1}\left (3,1-n,2-n,\frac {a-b x}{a+b x}\right )}{a (1-n)} \]

[In]

Int[(a + b*x)^(1 + n)/(x^3*(a - b*x)^n),x]

[Out]

(-4*b^2*(a - b*x)^(1 - n)*(a + b*x)^(-1 + n)*Hypergeometric2F1[3, 1 - n, 2 - n, (a - b*x)/(a + b*x)])/(a*(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 {4 b^2 (a-b x)^{1-n} (a+b x)^{-1+n} \, _2F_1\left (3,1-n;2-n;\frac {a-b x}{a+b x}\right )}{a (1-n)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00 \[ \int \frac {(a-b x)^{-n} (a+b x)^{1+n}}{x^3} \, dx=-\frac {4 b^2 (a-b x)^{1-n} (a+b x)^{-1+n} \operatorname {Hypergeometric2F1}\left (3,1-n,2-n,\frac {a-b x}{a+b x}\right )}{a (1-n)} \]

[In]

Integrate[(a + b*x)^(1 + n)/(x^3*(a - b*x)^n),x]

[Out]

(-4*b^2*(a - b*x)^(1 - n)*(a + b*x)^(-1 + n)*Hypergeometric2F1[3, 1 - n, 2 - n, (a - b*x)/(a + b*x)])/(a*(1 -
n))

Maple [F]

\[\int \frac {\left (b x +a \right )^{1+n} \left (-b x +a \right )^{-n}}{x^{3}}d x\]

[In]

int((b*x+a)^(1+n)/x^3/((-b*x+a)^n),x)

[Out]

int((b*x+a)^(1+n)/x^3/((-b*x+a)^n),x)

Fricas [F]

\[ \int \frac {(a-b x)^{-n} (a+b x)^{1+n}}{x^3} \, dx=\int { \frac {{\left (b x + a\right )}^{n + 1}}{{\left (-b x + a\right )}^{n} x^{3}} \,d x } \]

[In]

integrate((b*x+a)^(1+n)/x^3/((-b*x+a)^n),x, algorithm="fricas")

[Out]

integral((b*x + a)^(n + 1)/((-b*x + a)^n*x^3), x)

Sympy [F]

\[ \int \frac {(a-b x)^{-n} (a+b x)^{1+n}}{x^3} \, dx=\int \frac {\left (a - b x\right )^{- n} \left (a + b x\right )^{n + 1}}{x^{3}}\, dx \]

[In]

integrate((b*x+a)**(1+n)/x**3/((-b*x+a)**n),x)

[Out]

Integral((a + b*x)**(n + 1)/(x**3*(a - b*x)**n), x)

Maxima [F]

\[ \int \frac {(a-b x)^{-n} (a+b x)^{1+n}}{x^3} \, dx=\int { \frac {{\left (b x + a\right )}^{n + 1}}{{\left (-b x + a\right )}^{n} x^{3}} \,d x } \]

[In]

integrate((b*x+a)^(1+n)/x^3/((-b*x+a)^n),x, algorithm="maxima")

[Out]

integrate((b*x + a)^(n + 1)/((-b*x + a)^n*x^3), x)

Giac [F]

\[ \int \frac {(a-b x)^{-n} (a+b x)^{1+n}}{x^3} \, dx=\int { \frac {{\left (b x + a\right )}^{n + 1}}{{\left (-b x + a\right )}^{n} x^{3}} \,d x } \]

[In]

integrate((b*x+a)^(1+n)/x^3/((-b*x+a)^n),x, algorithm="giac")

[Out]

integrate((b*x + a)^(n + 1)/((-b*x + a)^n*x^3), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(a-b x)^{-n} (a+b x)^{1+n}}{x^3} \, dx=\int \frac {{\left (a+b\,x\right )}^{n+1}}{x^3\,{\left (a-b\,x\right )}^n} \,d x \]

[In]

int((a + b*x)^(n + 1)/(x^3*(a - b*x)^n),x)

[Out]

int((a + b*x)^(n + 1)/(x^3*(a - b*x)^n), x)