3.8.97 \(\int x (-a-b x)^{-n} (a+b x)^n \, dx\)

Optimal. Leaf size=26 \[ \frac {1}{2} x^2 (-a-b x)^{-n} (a+b x)^n \]

________________________________________________________________________________________

Rubi [A]  time = 0.00, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {23, 30} \begin {gather*} \frac {1}{2} x^2 (-a-b x)^{-n} (a+b x)^n \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x*(a + b*x)^n)/(-a - b*x)^n,x]

[Out]

(x^2*(a + b*x)^n)/(2*(-a - b*x)^n)

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])

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.00, size = 26, normalized size = 1.00 \begin {gather*} \frac {1}{2} x^2 (-a-b x)^{-n} (a+b x)^n \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x^2*(a + b*x)^n)/(2*(-a - b*x)^n)

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.05, size = 34, normalized size = 1.31 \begin {gather*} \frac {(a-b x) (-a-b x)^{1-n} (a+b x)^n}{2 b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(x*(a + b*x)^n)/(-a - b*x)^n,x]

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 1.56, size = 9, normalized size = 0.35 \begin {gather*} \frac {1}{2} \, x^{2} \cos \left (\pi n\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^2*cos(pi*n)

________________________________________________________________________________________

giac [A]  time = 1.20, size = 5, normalized size = 0.19 \begin {gather*} \frac {1}{2} \, x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^2

________________________________________________________________________________________

maple [A]  time = 0.00, size = 25, normalized size = 0.96 \begin {gather*} \frac {x^{2} \left (-b x -a \right )^{-n} \left (b x +a \right )^{n}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.78, size = 8, normalized size = 0.31 \begin {gather*} \frac {1}{2} \, \left (-1\right )^{n} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(-1)^n*x^2

________________________________________________________________________________________

mupad [B]  time = 0.97, size = 24, normalized size = 0.92 \begin {gather*} \frac {x^2\,{\left (a+b\,x\right )}^n}{2\,{\left (-a-b\,x\right )}^n} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x*(a + b*x)^n)/(- a - b*x)^n,x)

[Out]

(x^2*(a + b*x)^n)/(2*(- a - b*x)^n)

________________________________________________________________________________________

sympy [A]  time = 8.36, size = 19, normalized size = 0.73 \begin {gather*} \frac {x^{2} \left (- a - b x\right )^{- n} \left (a + b x\right )^{n}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**2*(-a - b*x)**(-n)*(a + b*x)**n/2

________________________________________________________________________________________