\(\int (-a-b x)^{-n} (a+b x)^n \, dx\) [823]

Optimal result

Integrand size = 20, antiderivative size = 21 \[ \int (-a-b x)^{-n} (a+b x)^n \, dx=x (-a-b x)^{-n} (a+b x)^n \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {23, 8} \[ \int (-a-b x)^{-n} (a+b x)^n \, dx=x (-a-b x)^{-n} (a+b x)^n \]

[In]

Int[(a + b*x)^n/(-a - b*x)^n,x]

[Out]

(x*(a + b*x)^n)/(-a - b*x)^n

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 23

Int[(u_.)*((a_) + (b_.)*(v_))^(m_)*((c_) + (d_.)*(v_))^(n_), x_Symbol] :> Dist[(a + b*v)^m/(c + d*v)^m, Int[u*
(c + d*v)^(m + n), x], x] /; FreeQ[{a, b, c, d, m, n}, x] && EqQ[b*c - a*d, 0] &&  !(IntegerQ[m] || IntegerQ[n
] || GtQ[b/d, 0])

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

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.19 \[ \int (-a-b x)^{-n} (a+b x)^n \, dx=\frac {(-a-b x)^{-n} (a+b x)^{1+n}}{b} \]

[In]

Integrate[(a + b*x)^n/(-a - b*x)^n,x]

[Out]

(a + b*x)^(1 + n)/(b*(-a - b*x)^n)

Maple [A] (verified)

Time = 1.46 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.24

method	result	size
norman	\(x \,{\mathrm e}^{n \ln \left (b x +a \right )} {\mathrm e}^{-n \ln \left (-b x -a \right )}\)	\(26\)
risch	\(x \,{\mathrm e}^{-i n \pi \left (\operatorname {csgn}\left (i \left (b x +a \right )\right )^{3}-\operatorname {csgn}\left (i \left (b x +a \right )\right )^{2}+1\right )}\)	\(35\)
parallelrisch	\(-\frac {\left (-x \left (b x +a \right )^{n} b n +\left (b x +a \right )^{n} a n \right ) \left (-b x -a \right )^{-n}}{n b}\)	\(44\)

[In]

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

[Out]

x*exp(n*ln(b*x+a))/exp(n*ln(-b*x-a))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.22 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.33 \[ \int (-a-b x)^{-n} (a+b x)^n \, dx=x e^{\left (i \, \pi n\right )} \]

[In]

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

[Out]

x*e^(I*pi*n)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (15) = 30\).

Time = 1.51 (sec) , antiderivative size = 44, normalized size of antiderivative = 2.10 \[ \int (-a-b x)^{-n} (a+b x)^n \, dx=\begin {cases} - \frac {a \left (- a - b x\right )^{- n} \left (a + b x\right )^{n}}{b} + x \left (- a - b x\right )^{- n} \left (a + b x\right )^{n} & \text {for}\: b \neq 0 \\a^{n} x \left (- a\right )^{- n} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((-a*(a + b*x)**n/(b*(-a - b*x)**n) + x*(a + b*x)**n/(-a - b*x)**n, Ne(b, 0)), (a**n*x/(-a)**n, True)
)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 5, normalized size of antiderivative = 0.24 \[ \int (-a-b x)^{-n} (a+b x)^n \, dx=\left (-1\right )^{n} x \]

[In]

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

[Out]

(-1)^n*x

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 1, normalized size of antiderivative = 0.05 \[ \int (-a-b x)^{-n} (a+b x)^n \, dx=x \]

[In]

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

[Out]

x

Mupad [B] (verification not implemented)

Time = 1.33 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int (-a-b x)^{-n} (a+b x)^n \, dx=\frac {x\,{\left (a+b\,x\right )}^n}{{\left (-a-b\,x\right )}^n} \]

[In]

int((a + b*x)^n/(- a - b*x)^n,x)

[Out]

(x*(a + b*x)^n)/(- a - b*x)^n