3.1006 \(\int \frac{(a-b x)^{-n} (a+b x)^{1+n}}{x} \, dx\)

Optimal. Leaf size=142 \[ -\frac{a (a+b x)^n (a-b x)^{-n} \, _2F_1\left (1,n;n+1;\frac{a+b x}{a-b x}\right )}{n}+\frac{2^{-n-1} (2 n+1) \left (\frac{a-b x}{a}\right )^n (a+b x)^{n+1} (a-b x)^{-n} \, _2F_1\left (n,n+1;n+2;\frac{a+b x}{2 a}\right )}{n (n+1)}+\frac{(a+b x)^n (a-b x)^{1-n}}{2 n} \]

[Out]

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

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Rubi [A]  time = 0.0952356, antiderivative size = 173, normalized size of antiderivative = 1.22, number of steps used = 7, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {105, 70, 69, 131} \[ \frac{a (a+b x)^n (a-b x)^{-n} \, _2F_1\left (1,-n;1-n;\frac{a-b x}{a+b x}\right )}{n}-\frac{a 2^n (a+b x)^n \left (\frac{a+b x}{a}\right )^{-n} (a-b x)^{-n} \, _2F_1\left (-n,-n;1-n;\frac{a-b x}{2 a}\right )}{n}+\frac{2^{-n} \left (\frac{a-b x}{a}\right )^n (a+b x)^{n+1} (a-b x)^{-n} \, _2F_1\left (n,n+1;n+2;\frac{a+b x}{2 a}\right )}{n+1} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*(a + b*x)^n*Hypergeometric2F1[1, -n, 1 - n, (a - b*x)/(a + b*x)])/(n*(a - b*x)^n) - (2^n*a*(a + b*x)^n*Hype
rgeometric2F1[-n, -n, 1 - n, (a - b*x)/(2*a)])/(n*(a - b*x)^n*((a + b*x)/a)^n) + (((a - b*x)/a)^n*(a + b*x)^(1
+ n)*Hypergeometric2F1[n, 1 + n, 2 + n, (a + b*x)/(2*a)])/(2^n*(1 + n)*(a - b*x)^n)

Rule 105

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> Dist[b/f, Int[(a
+ b*x)^(m - 1)*(c + d*x)^n, x], x] - Dist[(b*e - a*f)/f, Int[((a + b*x)^(m - 1)*(c + d*x)^n)/(e + f*x), x], x]
/; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[Simplify[m + n + 1], 0] && (GtQ[m, 0] || ( !RationalQ[m] && (Su
mSimplerQ[m, -1] ||  !SumSimplerQ[n, -1])))

Rule 70

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), 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*Simp[(b*c)/(b*c - a*d) + (b*d*x)/(b*c -
a*d), x]^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] &&
(RationalQ[m] ||  !SimplerQ[n + 1, m + 1])

Rule 69

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*Hypergeometric2F1[
-n, m + 1, m + 2, -((d*(a + b*x))/(b*c - a*d))])/(b*(m + 1)*(b/(b*c - a*d))^n), x] /; FreeQ[{a, b, c, d, m, n}
, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] ||  !(Ra
tionalQ[n] && GtQ[-(d/(b*c - a*d)), 0]))

Rule 131

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)*Hypergeometric2F1[m + 1, -n, m + 2, -(((d*e - c*f)*(a + b*x))/((b*c - a*d)*(e + f*x))
)])/((m + 1)*(b*e - a*f)^(n + 1)*(e + f*x)^(m + 1)), x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[m + n + p
+ 2, 0] && ILtQ[n, 0]

Rubi steps

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

Mathematica [A]  time = 0.131531, size = 160, normalized size = 1.13 \[ \frac{2^{-n} (a-b x)^{-n} (a+b x)^n \left (\frac{b x}{a}+1\right )^{-n} \left (n (a+b x) \left (1-\frac{b^2 x^2}{a^2}\right )^n \, _2F_1\left (n,n+1;n+2;\frac{a+b x}{2 a}\right )+a (n+1) \left (\frac{2 b x}{a}+2\right )^n \, _2F_1\left (1,-n;1-n;\frac{a-b x}{a+b x}\right )-a 4^n (n+1) \, _2F_1\left (-n,-n;1-n;\frac{a-b x}{2 a}\right )\right )}{n (n+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.052, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( bx+a \right ) ^{1+n}}{x \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/((-b*x+a)^n),x)

[Out]

int((b*x+a)^(1+n)/x/((-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}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*x + a)^(n + 1)/((-b*x + a)^n*x), 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}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*x + a)^(n + 1)/((-b*x + a)^n*x), 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/((-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}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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