\(\int (a+b x)^{1+n} (c+d x)^{-1-n} \, dx\) [1879]

Optimal result

Integrand size = 21, antiderivative size = 84 \[ \int (a+b x)^{1+n} (c+d x)^{-1-n} \, dx=\frac {(b c-a d) (a+b x)^n \left (-\frac {d (a+b x)}{b c-a d}\right )^{-n} (c+d x)^{-n} \operatorname {Hypergeometric2F1}\left (-1-n,-n,1-n,\frac {b (c+d x)}{b c-a d}\right )}{d^2 n} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {72, 71} \[ \int (a+b x)^{1+n} (c+d x)^{-1-n} \, dx=\frac {(b c-a d) (a+b x)^n (c+d x)^{-n} \left (-\frac {d (a+b x)}{b c-a d}\right )^{-n} \operatorname {Hypergeometric2F1}\left (-n-1,-n,1-n,\frac {b (c+d x)}{b c-a d}\right )}{d^2 n} \]

[In]

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

[Out]

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

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])

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

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.99 \[ \int (a+b x)^{1+n} (c+d x)^{-1-n} \, dx=\frac {(b c-a d) (a+b x)^n \left (\frac {d (a+b x)}{-b c+a d}\right )^{-n} (c+d x)^{-n} \operatorname {Hypergeometric2F1}\left (-1-n,-n,1-n,\frac {b (c+d x)}{b c-a d}\right )}{d^2 n} \]

[In]

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

[Out]

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

Maple [F]

\[\int \left (b x +a \right )^{1+n} \left (d x +c \right )^{-1-n}d x\]

[In]

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

[Out]

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

Fricas [F]

\[ \int (a+b x)^{1+n} (c+d x)^{-1-n} \, dx=\int { {\left (b x + a\right )}^{n + 1} {\left (d x + c\right )}^{-n - 1} \,d x } \]

[In]

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

[Out]

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

Sympy [F]

\[ \int (a+b x)^{1+n} (c+d x)^{-1-n} \, dx=\int \left (a + b x\right )^{n + 1} \left (c + d x\right )^{- n - 1}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int (a+b x)^{1+n} (c+d x)^{-1-n} \, dx=\int { {\left (b x + a\right )}^{n + 1} {\left (d x + c\right )}^{-n - 1} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int (a+b x)^{1+n} (c+d x)^{-1-n} \, dx=\int { {\left (b x + a\right )}^{n + 1} {\left (d x + c\right )}^{-n - 1} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int (a+b x)^{1+n} (c+d x)^{-1-n} \, dx=\int \frac {{\left (a+b\,x\right )}^{n+1}}{{\left (c+d\,x\right )}^{n+1}} \,d x \]

[In]

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

[Out]

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