3.10.74 \(\int \frac {(a+b x)^n (c+d x)^{-n}}{x} \, dx\) [974]

Optimal. Leaf size=108 \[ -\frac {(a+b x)^n (c+d x)^{-n} \, _2F_1\left (1,n;1+n;\frac {c (a+b x)}{a (c+d x)}\right )}{n}+\frac {(a+b x)^n (c+d x)^{-n} \left (\frac {b (c+d x)}{b c-a d}\right )^n \, _2F_1\left (n,n;1+n;-\frac {d (a+b x)}{b c-a d}\right )}{n} \]

[Out]

-(b*x+a)^n*hypergeom([1, n],[1+n],c*(b*x+a)/a/(d*x+c))/n/((d*x+c)^n)+(b*x+a)^n*(b*(d*x+c)/(-a*d+b*c))^n*hyperg
eom([n, n],[1+n],-d*(b*x+a)/(-a*d+b*c))/n/((d*x+c)^n)

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Rubi [A]
time = 0.03, antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {132, 72, 71, 12, 133} \begin {gather*} \frac {(a+b x)^n (c+d x)^{-n} \left (\frac {b (c+d x)}{b c-a d}\right )^n \, _2F_1\left (n,n;n+1;-\frac {d (a+b x)}{b c-a d}\right )}{n}-\frac {(a+b x)^n (c+d x)^{-n} \, _2F_1\left (1,n;n+1;\frac {c (a+b x)}{a (c+d x)}\right )}{n} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^n/(x*(c + d*x)^n),x]

[Out]

-(((a + b*x)^n*Hypergeometric2F1[1, n, 1 + n, (c*(a + b*x))/(a*(c + d*x))])/(n*(c + d*x)^n)) + ((a + b*x)^n*((
b*(c + d*x))/(b*c - a*d))^n*Hypergeometric2F1[n, n, 1 + n, -((d*(a + b*x))/(b*c - a*d))])/(n*(c + d*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 132

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Dist[b*d^(m
+ n)*f^p, Int[(a + b*x)^(m - 1)/(c + d*x)^m, x], x] + Int[(a + b*x)^(m - 1)*((e + f*x)^p/(c + d*x)^m)*ExpandTo
Sum[(a + b*x)*(c + d*x)^(-p - 1) - (b*d^(-p - 1)*f^p)/(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x
] && EqQ[m + n + p + 1, 0] && ILtQ[p, 0] && (GtQ[m, 0] || SumSimplerQ[m, -1] ||  !(GtQ[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 (c+d x)^{-n}}{x} \, dx &=a \int \frac {(a+b x)^{-1+n} (c+d x)^{-n}}{x} \, dx+b \int (a+b x)^{-1+n} (c+d x)^{-n} \, dx\\ &=-\frac {(a+b x)^n (c+d x)^{-n} \, _2F_1\left (1,n;1+n;\frac {c (a+b x)}{a (c+d x)}\right )}{n}+\left (b (c+d x)^{-n} \left (\frac {b (c+d x)}{b c-a d}\right )^n\right ) \int (a+b x)^{-1+n} \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^{-n} \, dx\\ &=-\frac {(a+b x)^n (c+d x)^{-n} \, _2F_1\left (1,n;1+n;\frac {c (a+b x)}{a (c+d x)}\right )}{n}+\frac {(a+b x)^n (c+d x)^{-n} \left (\frac {b (c+d x)}{b c-a d}\right )^n \, _2F_1\left (n,n;1+n;-\frac {d (a+b x)}{b c-a d}\right )}{n}\\ \end {align*}

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Mathematica [A]
time = 0.06, size = 89, normalized size = 0.82 \begin {gather*} \frac {(a+b x)^n (c+d x)^{-n} \left (-\, _2F_1\left (1,n;1+n;\frac {c (a+b x)}{a (c+d x)}\right )+\left (\frac {b (c+d x)}{b c-a d}\right )^n \, _2F_1\left (n,n;1+n;\frac {d (a+b x)}{-b c+a d}\right )\right )}{n} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^n/(x*(c + d*x)^n),x]

[Out]

((a + b*x)^n*(-Hypergeometric2F1[1, n, 1 + n, (c*(a + b*x))/(a*(c + d*x))] + ((b*(c + d*x))/(b*c - a*d))^n*Hyp
ergeometric2F1[n, n, 1 + n, (d*(a + b*x))/(-(b*c) + a*d)]))/(n*(c + d*x)^n)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^n/x/((d*x+c)^n),x)

[Out]

int((b*x+a)^n/x/((d*x+c)^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)^n/x/((d*x+c)^n),x, algorithm="maxima")

[Out]

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

[Out]

integral((b*x + a)^n/((d*x + c)^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 (c + d x\right )^{- n}}{x}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**n/x/((d*x+c)**n),x)

[Out]

Integral((a + b*x)**n/(x*(c + d*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)^n/x/((d*x+c)^n),x, algorithm="giac")

[Out]

integrate((b*x + a)^n/((d*x + c)^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}{x\,{\left (c+d\,x\right )}^n} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^n/(x*(c + d*x)^n),x)

[Out]

int((a + b*x)^n/(x*(c + d*x)^n), x)

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