\(\int x^{1+n} (a+b x)^{-n} \, dx\) [744]

Optimal result

Integrand size = 15, antiderivative size = 45 \[ \int x^{1+n} (a+b x)^{-n} \, dx=\frac {x^{2+n} (a+b x)^{-n} \left (1+\frac {b x}{a}\right )^n \operatorname {Hypergeometric2F1}\left (n,2+n,3+n,-\frac {b x}{a}\right )}{2+n} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {68, 66} \[ \int x^{1+n} (a+b x)^{-n} \, dx=\frac {x^{n+2} (a+b x)^{-n} \left (\frac {b x}{a}+1\right )^n \operatorname {Hypergeometric2F1}\left (n,n+2,n+3,-\frac {b x}{a}\right )}{n+2} \]

[In]

Int[x^(1 + n)/(a + b*x)^n,x]

[Out]

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

Rule 66

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[c^n*((b*x)^(m + 1)/(b*(m + 1)))*Hypergeometr
ic2F1[-n, m + 1, m + 2, (-d)*(x/c)], x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[m] && (IntegerQ[n] || (GtQ[
c, 0] &&  !(EqQ[n, -2^(-1)] && EqQ[c^2 - d^2, 0] && GtQ[-d/(b*c), 0])))

Rule 68

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Dist[c^IntPart[n]*((c + d*x)^FracPart[n]/(1 + d*(
x/c))^FracPart[n]), Int[(b*x)^m*(1 + d*(x/c))^n, x], x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[m] &&  !Int
egerQ[n] &&  !GtQ[c, 0] &&  !GtQ[-d/(b*c), 0] && ((RationalQ[m] &&  !(EqQ[n, -2^(-1)] && EqQ[c^2 - d^2, 0])) |
|  !RationalQ[n])

Rubi steps \begin{align*} \text {integral}& = \left ((a+b x)^{-n} \left (1+\frac {b x}{a}\right )^n\right ) \int x^{1+n} \left (1+\frac {b x}{a}\right )^{-n} \, dx \\ & = \frac {x^{2+n} (a+b x)^{-n} \left (1+\frac {b x}{a}\right )^n \, _2F_1\left (n,2+n;3+n;-\frac {b x}{a}\right )}{2+n} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00 \[ \int x^{1+n} (a+b x)^{-n} \, dx=\frac {x^{2+n} (a+b x)^{-n} \left (1+\frac {b x}{a}\right )^n \operatorname {Hypergeometric2F1}\left (n,2+n,3+n,-\frac {b x}{a}\right )}{2+n} \]

[In]

Integrate[x^(1 + n)/(a + b*x)^n,x]

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

integral(x^(n + 1)/(b*x + a)^n, x)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 21.33 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.71 \[ \int x^{1+n} (a+b x)^{-n} \, dx=\frac {a^{- n} x^{n + 2} \Gamma \left (n + 2\right ) {{}_{2}F_{1}\left (\begin {matrix} n, n + 2 \\ n + 3 \end {matrix}\middle | {\frac {b x e^{i \pi }}{a}} \right )}}{\Gamma \left (n + 3\right )} \]

[In]

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

[Out]

x**(n + 2)*gamma(n + 2)*hyper((n, n + 2), (n + 3,), b*x*exp_polar(I*pi)/a)/(a**n*gamma(n + 3))

Maxima [F]

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

[In]

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

[Out]

integrate(x^(n + 1)/(b*x + a)^n, x)

Giac [F]

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

[In]

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

[Out]

integrate(x^(n + 1)/(b*x + a)^n, x)

Mupad [F(-1)]

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

[In]

int(x^(n + 1)/(a + b*x)^n,x)

[Out]

int(x^(n + 1)/(a + b*x)^n, x)