\(\int x^{-3-n} (a+b x)^n \, dx\) [754]

Optimal result

Integrand size = 15, antiderivative size = 58 \[ \int x^{-3-n} (a+b x)^n \, dx=-\frac {x^{-2-n} (a+b x)^{1+n}}{a (2+n)}+\frac {b x^{-1-n} (a+b x)^{1+n}}{a^2 (1+n) (2+n)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 58, 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 = {47, 37} \[ \int x^{-3-n} (a+b x)^n \, dx=\frac {b x^{-n-1} (a+b x)^{n+1}}{a^2 (n+1) (n+2)}-\frac {x^{-n-2} (a+b x)^{n+1}}{a (n+2)} \]

[In]

Int[x^(-3 - n)*(a + b*x)^n,x]

[Out]

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

Rule 47

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] - Dist[d*(Simplify[m + n + 2]/((b*c - a*d)*(m + 1))), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.69 \[ \int x^{-3-n} (a+b x)^n \, dx=-\frac {x^{-2-n} (a+a n-b x) (a+b x)^{1+n}}{a^2 (1+n) (2+n)} \]

[In]

Integrate[x^(-3 - n)*(a + b*x)^n,x]

[Out]

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

Maple [A] (verified)

Time = 0.18 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.71

method	result	size
gosper	\(-\frac {x^{-2-n} \left (b x +a \right )^{1+n} \left (a n -b x +a \right )}{a^{2} \left (1+n \right ) \left (2+n \right )}\)	\(41\)

[In]

int(x^(-3-n)*(b*x+a)^n,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.10 \[ \int x^{-3-n} (a+b x)^n \, dx=-\frac {{\left (a b n x^{2} - b^{2} x^{3} + {\left (a^{2} n + a^{2}\right )} x\right )} {\left (b x + a\right )}^{n} x^{-n - 3}}{a^{2} n^{2} + 3 \, a^{2} n + 2 \, a^{2}} \]

[In]

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

[Out]

-(a*b*n*x^2 - b^2*x^3 + (a^2*n + a^2)*x)*(b*x + a)^n*x^(-n - 3)/(a^2*n^2 + 3*a^2*n + 2*a^2)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 134 vs. \(2 (46) = 92\).

Time = 1.96 (sec) , antiderivative size = 134, normalized size of antiderivative = 2.31 \[ \int x^{-3-n} (a+b x)^n \, dx=- \frac {a a^{n} n x^{- n - 2} \left (1 + \frac {b x}{a}\right )^{n + 2} \Gamma \left (- n - 2\right )}{a \Gamma \left (- n\right ) + b x \Gamma \left (- n\right )} - \frac {a a^{n} x^{- n - 2} \left (1 + \frac {b x}{a}\right )^{n + 2} \Gamma \left (- n - 2\right )}{a \Gamma \left (- n\right ) + b x \Gamma \left (- n\right )} + \frac {a^{n} b x x^{- n - 2} \left (1 + \frac {b x}{a}\right )^{n + 2} \Gamma \left (- n - 2\right )}{a \Gamma \left (- n\right ) + b x \Gamma \left (- n\right )} \]

[In]

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

[Out]

-a*a**n*n*x**(-n - 2)*(1 + b*x/a)**(n + 2)*gamma(-n - 2)/(a*gamma(-n) + b*x*gamma(-n)) - a*a**n*x**(-n - 2)*(1
+ b*x/a)**(n + 2)*gamma(-n - 2)/(a*gamma(-n) + b*x*gamma(-n)) + a**n*b*x*x**(-n - 2)*(1 + b*x/a)**(n + 2)*gam
ma(-n - 2)/(a*gamma(-n) + b*x*gamma(-n))

Maxima [F]

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

[In]

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

[Out]

integrate((b*x + a)^n*x^(-n - 3), x)

Giac [F]

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

[In]

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

[Out]

integrate((b*x + a)^n*x^(-n - 3), x)

Mupad [B] (verification not implemented)

Time = 0.62 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.48 \[ \int x^{-3-n} (a+b x)^n \, dx=-{\left (a+b\,x\right )}^n\,\left (\frac {x\,\left (n+1\right )}{x^{n+3}\,\left (n^2+3\,n+2\right )}-\frac {b^2\,x^3}{a^2\,x^{n+3}\,\left (n^2+3\,n+2\right )}+\frac {b\,n\,x^2}{a\,x^{n+3}\,\left (n^2+3\,n+2\right )}\right ) \]

[In]

int((a + b*x)^n/x^(n + 3),x)

[Out]

-(a + b*x)^n*((x*(n + 1))/(x^(n + 3)*(3*n + n^2 + 2)) - (b^2*x^3)/(a^2*x^(n + 3)*(3*n + n^2 + 2)) + (b*n*x^2)/
(a*x^(n + 3)*(3*n + n^2 + 2)))