\(\int (a+b x)^{2+n} (c+d x)^{-n} \, dx\) [1865]

Optimal result

Integrand size = 19, antiderivative size = 72 \[ \int (a+b x)^{2+n} (c+d x)^{-n} \, dx=\frac {(a+b x)^{3+n} (c+d x)^{-n} \left (\frac {b (c+d x)}{b c-a d}\right )^n \operatorname {Hypergeometric2F1}\left (n,3+n,4+n,-\frac {d (a+b x)}{b c-a d}\right )}{b (3+n)} \]

[Out]

(b*x+a)^(3+n)*(b*(d*x+c)/(-a*d+b*c))^n*hypergeom([n, 3+n],[4+n],-d*(b*x+a)/(-a*d+b*c))/b/(3+n)/((d*x+c)^n)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {72, 71} \[ \int (a+b x)^{2+n} (c+d x)^{-n} \, dx=\frac {(a+b x)^{n+3} (c+d x)^{-n} \left (\frac {b (c+d x)}{b c-a d}\right )^n \operatorname {Hypergeometric2F1}\left (n,n+3,n+4,-\frac {d (a+b x)}{b c-a d}\right )}{b (n+3)} \]

[In]

Int[(a + b*x)^(2 + n)/(c + d*x)^n,x]

[Out]

((a + b*x)^(3 + n)*((b*(c + d*x))/(b*c - a*d))^n*Hypergeometric2F1[n, 3 + n, 4 + n, -((d*(a + b*x))/(b*c - a*d
))])/(b*(3 + 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}& = \left ((c+d x)^{-n} \left (\frac {b (c+d x)}{b c-a d}\right )^n\right ) \int (a+b x)^{2+n} \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^{-n} \, dx \\ & = \frac {(a+b x)^{3+n} (c+d x)^{-n} \left (\frac {b (c+d x)}{b c-a d}\right )^n \, _2F_1\left (n,3+n;4+n;-\frac {d (a+b x)}{b c-a d}\right )}{b (3+n)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.28 \[ \int (a+b x)^{2+n} (c+d x)^{-n} \, dx=-\frac {(b c-a d)^2 (a+b x)^n \left (\frac {d (a+b x)}{-b c+a d}\right )^{-n} (c+d x)^{1-n} \operatorname {Hypergeometric2F1}\left (-2-n,1-n,2-n,\frac {b (c+d x)}{b c-a d}\right )}{d^3 (-1+n)} \]

[In]

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

[Out]

-(((b*c - a*d)^2*(a + b*x)^n*(c + d*x)^(1 - n)*Hypergeometric2F1[-2 - n, 1 - n, 2 - n, (b*(c + d*x))/(b*c - a*
d)])/(d^3*(-1 + n)*((d*(a + b*x))/(-(b*c) + a*d))^n))

Maple [F]

\[\int \left (b x +a \right )^{2+n} \left (d x +c \right )^{-n}d x\]

[In]

int((b*x+a)^(2+n)/((d*x+c)^n),x)

[Out]

int((b*x+a)^(2+n)/((d*x+c)^n),x)

Fricas [F]

\[ \int (a+b x)^{2+n} (c+d x)^{-n} \, dx=\int { \frac {{\left (b x + a\right )}^{n + 2}}{{\left (d x + c\right )}^{n}} \,d x } \]

[In]

integrate((b*x+a)^(2+n)/((d*x+c)^n),x, algorithm="fricas")

[Out]

integral((b*x + a)^(n + 2)/(d*x + c)^n, x)

Sympy [F(-2)]

Exception generated. \[ \int (a+b x)^{2+n} (c+d x)^{-n} \, dx=\text {Exception raised: HeuristicGCDFailed} \]

[In]

integrate((b*x+a)**(2+n)/((d*x+c)**n),x)

[Out]

Exception raised: HeuristicGCDFailed >> no luck

Maxima [F]

\[ \int (a+b x)^{2+n} (c+d x)^{-n} \, dx=\int { \frac {{\left (b x + a\right )}^{n + 2}}{{\left (d x + c\right )}^{n}} \,d x } \]

[In]

integrate((b*x+a)^(2+n)/((d*x+c)^n),x, algorithm="maxima")

[Out]

integrate((b*x + a)^(n + 2)/(d*x + c)^n, x)

Giac [F]

\[ \int (a+b x)^{2+n} (c+d x)^{-n} \, dx=\int { \frac {{\left (b x + a\right )}^{n + 2}}{{\left (d x + c\right )}^{n}} \,d x } \]

[In]

integrate((b*x+a)^(2+n)/((d*x+c)^n),x, algorithm="giac")

[Out]

integrate((b*x + a)^(n + 2)/(d*x + c)^n, x)

Mupad [F(-1)]

Timed out. \[ \int (a+b x)^{2+n} (c+d x)^{-n} \, dx=\int \frac {{\left (a+b\,x\right )}^{n+2}}{{\left (c+d\,x\right )}^n} \,d x \]

[In]

int((a + b*x)^(n + 2)/(c + d*x)^n,x)

[Out]

int((a + b*x)^(n + 2)/(c + d*x)^n, x)