3.11.6 \(\int \frac {(a-b x)^{-n} (a+b x)^{1+n}}{x} \, dx\) [1006]

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

[Out]

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

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Rubi [A]
time = 0.07, antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {131, 80, 72, 71, 12, 133} \begin {gather*} -\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} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[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)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 71

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c
- a*d))^n))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*((a + b*x)/(b*c - a*d))], 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 72

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 80

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

Rule 131

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

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} \, 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*}

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Mathematica [A]
time = 0.13, size = 160, normalized size = 1.13 \begin {gather*} \frac {2^{-n} (a-b x)^{-n} (a+b x)^n \left (1+\frac {b x}{a}\right )^{-n} \left (a (1+n) \left (2+\frac {2 b x}{a}\right )^n \, _2F_1\left (1,-n;1-n;\frac {a-b x}{a+b x}\right )-4^n a (1+n) \, _2F_1\left (-n,-n;1-n;\frac {a-b x}{2 a}\right )+n (a+b x) \left (1-\frac {b^2 x^2}{a^2}\right )^n \, _2F_1\left (n,1+n;2+n;\frac {a+b x}{2 a}\right )\right )}{n (1+n)} \end {gather*}

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

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 = 0.00, 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/((-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 = 0.00, 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/((-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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a - b x\right )^{- n} \left (a + b x\right )^{n + 1}}{x}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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/((-b*x+a)^n),x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (a+b\,x\right )}^{n+1}}{x\,{\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*(a - b*x)^n),x)

[Out]

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

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