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

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

[Out]

(x^2*(a + b*x)^n)/(2*(-a - b*x)^n)

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Rubi [A]  time = 0.0030901, antiderivative size = 26, normalized size of antiderivative = 1., 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} \[ \frac{1}{2} x^2 (-a-b x)^{-n} (a+b x)^n \]

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 = 0.0027887, size = 26, normalized size = 1. \[ \frac{1}{2} x^2 (-a-b x)^{-n} (a+b x)^n \]

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)

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Maple [A]  time = 0.002, size = 25, normalized size = 1. \begin{align*}{\frac{{x}^{2} \left ( bx+a \right ) ^{n}}{2\, \left ( -bx-a \right ) ^{n}}} \end{align*}

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)

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Maxima [A]  time = 1.18332, size = 11, normalized size = 0.42 \begin{align*} \frac{1}{2} \, \left (-1\right )^{n} x^{2} \end{align*}

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

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Fricas [A]  time = 1.80333, size = 26, normalized size = 1. \begin{align*} \frac{1}{2} \, x^{2} \cos \left (\pi n\right ) \end{align*}

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)

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Sympy [A]  time = 20.1702, size = 19, normalized size = 0.73 \begin{align*} \frac{x^{2} \left (- a - b x\right )^{- n} \left (a + b x\right )^{n}}{2} \end{align*}

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

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Giac [A]  time = 1.40948, size = 7, normalized size = 0.27 \begin{align*} \frac{1}{2} \, x^{2} \end{align*}

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