3.9.26 \(\int \frac {(-a-b x)^{-n} (a+b x)^n}{x^3} \, dx\) [826]

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

[Out]

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

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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 = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {23, 30} \begin {gather*} -\frac {(-a-b x)^{-n} (a+b x)^n}{2 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^n/(x^3*(-a - b*x)^n),x]

[Out]

-1/2*(a + b*x)^n/(x^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 \frac {(-a-b x)^{-n} (a+b x)^n}{x^3} \, dx &=\left ((-a-b x)^{-n} (a+b x)^n\right ) \int \frac {1}{x^3} \, dx\\ &=-\frac {(-a-b x)^{-n} (a+b x)^n}{2 x^2}\\ \end {align*}

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

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^n/(x^3*(-a - b*x)^n),x]

[Out]

-1/2*(a + b*x)^n/(x^2*(-a - b*x)^n)

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Maple [A]
time = 0.10, size = 25, normalized size = 0.96

method	result	size
gosper	\(-\frac {\left (b x +a \right )^{n} \left (-b x -a \right )^{-n}}{2 x^{2}}\)	\(25\)
risch	\(-\frac {\left (b x +a \right )^{n} {\mathrm e}^{-n \left (i \pi \mathrm {csgn}\left (i \left (b x +a \right )\right )^{3}-i \pi \mathrm {csgn}\left (i \left (b x +a \right )\right )^{2}+i \pi +\ln \left (b x +a \right )\right )}}{2 x^{2}}\)	\(58\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.36, size = 8, normalized size = 0.31 \begin {gather*} -\frac {\left (-1\right )^{n}}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 1.49, size = 9, normalized size = 0.35 \begin {gather*} -\frac {\cos \left (\pi n\right )}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 3.59, size = 20, normalized size = 0.77 \begin {gather*} - \frac {\left (- a - b x\right )^{- n} \left (a + b x\right )^{n}}{2 x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(a + b*x)**n/(2*x**2*(-a - b*x)**n)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 1.04, size = 24, normalized size = 0.92 \begin {gather*} -\frac {{\left (a+b\,x\right )}^n}{2\,x^2\,{\left (-a-b\,x\right )}^n} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(a + b*x)^n/(2*x^2*(- a - b*x)^n)

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