3.1867 \(\int (a+b x)^{-1-n} (c+d x)^n \, dx\)

Optimal. Leaf size=75 \[ -\frac{(a+b x)^{-n} (c+d x)^n \left (\frac{b (c+d x)}{b c-a d}\right )^{-n} \, _2F_1\left (-n,-n;1-n;-\frac{d (a+b x)}{b c-a d}\right )}{b n} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0238464, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {70, 69} \[ -\frac{(a+b x)^{-n} (c+d x)^n \left (\frac{b (c+d x)}{b c-a d}\right )^{-n} \, _2F_1\left (-n,-n;1-n;-\frac{d (a+b x)}{b c-a d}\right )}{b n} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^(-1 - n)*(c + d*x)^n,x]

[Out]

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

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]))

Rubi steps

\begin{align*} \int (a+b x)^{-1-n} (c+d x)^n \, dx &=\left ((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 \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 )}{b n}\\ \end{align*}

Mathematica [A]  time = 0.0198404, size = 74, normalized size = 0.99 \[ -\frac{(a+b x)^{-n} (c+d x)^n \left (\frac{b (c+d x)}{b c-a d}\right )^{-n} \, _2F_1\left (-n,-n;1-n;\frac{d (a+b x)}{a d-b c}\right )}{b n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [F]  time = 0.069, size = 0, normalized size = 0. \begin{align*} \int \left ( bx+a \right ) ^{-1-n} \left ( dx+c \right ) ^{n}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*x + a)^(-n - 1)*(d*x + c)^n, x)

________________________________________________________________________________________

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)*(d*x+c)**n,x)

[Out]

Timed out

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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