3.1007 \(\int \frac{(a-b x)^{-n} (a+b x)^{1+n}}{x^2} \, dx\)

Optimal. Leaf size=140 \[ \frac{b (2 n+1) (a+b x)^n (a-b x)^{-n} \, _2F_1\left (1,-n;1-n;\frac{a-b x}{a+b x}\right )}{n}-\frac{b 2^n (a+b x)^n \left (\frac{a+b x}{a}\right )^{-n} (a-b x)^{-n} \, _2F_1\left (-n,-n;1-n;\frac{a-b x}{2 a}\right )}{n}-\frac{(a+b x)^{n+1} (a-b x)^{-n}}{x} \]

[Out]

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

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Rubi [C]  time = 0.0374907, antiderivative size = 76, normalized size of antiderivative = 0.54, 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 = {137, 136} \[ \frac{b 2^{-n} (a-b x)^{-n} \left (\frac{a-b x}{a}\right )^n (a+b x)^{n+2} F_1\left (n+2;n,2;n+3;\frac{a+b x}{2 a},\frac{a+b x}{a}\right )}{a^2 (n+2)} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(b*((a - b*x)/a)^n*(a + b*x)^(2 + n)*AppellF1[2 + n, n, 2, 3 + n, (a + b*x)/(2*a), (a + b*x)/a])/(2^n*a^2*(2 +
 n)*(a - b*x)^n)

Rule 137

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Dist[(c + d*x)^
FracPart[n]/((b/(b*c - a*d))^IntPart[n]*((b*(c + d*x))/(b*c - a*d))^FracPart[n]), Int[(a + b*x)^m*((b*c)/(b*c
- a*d) + (b*d*x)/(b*c - a*d))^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] &&  !IntegerQ[m] &&
 !IntegerQ[n] && IntegerQ[p] &&  !GtQ[b/(b*c - a*d), 0] &&  !SimplerQ[c + d*x, a + b*x]

Rule 136

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Simp[((b*e - a*
f)^p*(a + b*x)^(m + 1)*AppellF1[m + 1, -n, -p, m + 2, -((d*(a + b*x))/(b*c - a*d)), -((f*(a + b*x))/(b*e - a*f
))])/(b^(p + 1)*(m + 1)*(b/(b*c - a*d))^n), x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] &&  !IntegerQ[m] &&  !Int
egerQ[n] && IntegerQ[p] && GtQ[b/(b*c - a*d), 0] &&  !(GtQ[d/(d*a - c*b), 0] && SimplerQ[c + d*x, a + b*x])

Rubi steps

\begin{align*} \int \frac{(a-b x)^{-n} (a+b x)^{1+n}}{x^2} \, dx &=\left (2^{-n} (a-b x)^{-n} \left (\frac{a-b x}{a}\right )^n\right ) \int \frac{(a+b x)^{1+n} \left (\frac{1}{2}-\frac{b x}{2 a}\right )^{-n}}{x^2} \, dx\\ &=\frac{2^{-n} b (a-b x)^{-n} \left (\frac{a-b x}{a}\right )^n (a+b x)^{2+n} F_1\left (2+n;n,2;3+n;\frac{a+b x}{2 a},\frac{a+b x}{a}\right )}{a^2 (2+n)}\\ \end{align*}

Mathematica [C]  time = 0.24904, size = 146, normalized size = 1.04 \[ \frac{(a-b x)^{-n} (a+b x)^n \left (\frac{b 2^n (a-b x) \left (\frac{b x}{a}+1\right )^{-n} F_1\left (1-n;-n,1;2-n;\frac{a-b x}{2 a},1-\frac{b x}{a}\right )}{n-1}-\frac{a^2 \left (1-\frac{a}{b x}\right )^n \left (\frac{a}{b x}+1\right )^{-n} F_1\left (1;n,-n;2;\frac{a}{b x},-\frac{a}{b x}\right )}{x}\right )}{a} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((a + b*x)^n*(-((a^2*(1 - a/(b*x))^n*AppellF1[1, n, -n, 2, a/(b*x), -(a/(b*x))])/((1 + a/(b*x))^n*x)) + (2^n*b
*(a - b*x)*AppellF1[1 - n, -n, 1, 2 - n, (a - b*x)/(2*a), 1 - (b*x)/a])/((-1 + n)*(1 + (b*x)/a)^n)))/(a*(a - b
*x)^n)

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Maple [F]  time = 0.056, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( bx+a \right ) ^{1+n}}{{x}^{2} \left ( -bx+a \right ) ^{n}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x + a\right )}^{n + 1}}{{\left (-b x + a\right )}^{n} x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(1+n)/x^2/((-b*x+a)^n),x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b x + a\right )}^{n + 1}}{{\left (-b x + a\right )}^{n} x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(1+n)/x^2/((-b*x+a)^n),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x + a\right )}^{n + 1}}{{\left (-b x + a\right )}^{n} x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(1+n)/x^2/((-b*x+a)^n),x, algorithm="giac")

[Out]

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