3.11.8 \(\int \frac {(a-b x)^{-n} (a+b x)^{1+n}}{x^3} \, dx\) [1008]

Optimal. Leaf size=62 \[ -\frac {4 b^2 (a-b x)^{1-n} (a+b x)^{-1+n} \, _2F_1\left (3,1-n;2-n;\frac {a-b x}{a+b x}\right )}{a (1-n)} \]

[Out]

-4*b^2*(-b*x+a)^(1-n)*(b*x+a)^(-1+n)*hypergeom([3, 1-n],[2-n],(-b*x+a)/(b*x+a))/a/(1-n)

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Rubi [A]
time = 0.01, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {133} \begin {gather*} -\frac {4 b^2 (a-b x)^{1-n} (a+b x)^{n-1} \, _2F_1\left (3,1-n;2-n;\frac {a-b x}{a+b x}\right )}{a (1-n)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^(1 + n)/(x^3*(a - b*x)^n),x]

[Out]

(-4*b^2*(a - b*x)^(1 - n)*(a + b*x)^(-1 + n)*Hypergeometric2F1[3, 1 - n, 2 - n, (a - b*x)/(a + b*x)])/(a*(1 -
n))

Rule 133

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Simp[(b*c - a
*d)^n*((a + b*x)^(m + 1)/((m + 1)*(b*e - a*f)^(n + 1)*(e + f*x)^(m + 1)))*Hypergeometric2F1[m + 1, -n, m + 2,
(-(d*e - c*f))*((a + b*x)/((b*c - a*d)*(e + f*x)))], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[m + n + p
+ 2, 0] && ILtQ[n, 0] && (SumSimplerQ[m, 1] ||  !SumSimplerQ[p, 1]) &&  !ILtQ[m, 0]

Rubi steps

\begin {align*} \int \frac {(a-b x)^{-n} (a+b x)^{1+n}}{x^3} \, dx &=-\frac {4 b^2 (a-b x)^{1-n} (a+b x)^{-1+n} \, _2F_1\left (3,1-n;2-n;\frac {a-b x}{a+b x}\right )}{a (1-n)}\\ \end {align*}

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Mathematica [A]
time = 0.06, size = 62, normalized size = 1.00 \begin {gather*} -\frac {4 b^2 (a-b x)^{1-n} (a+b x)^{-1+n} \, _2F_1\left (3,1-n;2-n;\frac {a-b x}{a+b x}\right )}{a (1-n)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^(1 + n)/(x^3*(a - b*x)^n),x]

[Out]

(-4*b^2*(a - b*x)^(1 - n)*(a + b*x)^(-1 + n)*Hypergeometric2F1[3, 1 - n, 2 - n, (a - b*x)/(a + b*x)])/(a*(1 -
n))

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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (b x +a \right )^{1+n} \left (-b x +a \right )^{-n}}{x^{3}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(1+n)/x^3/((-b*x+a)^n),x, algorithm="maxima")

[Out]

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

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Fricas [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)^(1+n)/x^3/((-b*x+a)^n),x, algorithm="fricas")

[Out]

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

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**(1+n)/x**3/((-b*x+a)**n),x)

[Out]

Timed out

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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)^(1+n)/x^3/((-b*x+a)^n),x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {{\left (a+b\,x\right )}^{n+1}}{x^3\,{\left (a-b\,x\right )}^n} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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