\(\int (a+b x)^{-n} (c+d x)^n \, dx\) [1866]

Optimal result

Integrand size = 17, antiderivative size = 72 \[ \int (a+b x)^{-n} (c+d x)^n \, dx=\frac {(a+b x)^{-n} \left (-\frac {d (a+b x)}{b c-a d}\right )^n (c+d x)^{1+n} \operatorname {Hypergeometric2F1}\left (n,1+n,2+n,\frac {b (c+d x)}{b c-a d}\right )}{d (1+n)} \]

[Out]

(-d*(b*x+a)/(-a*d+b*c))^n*(d*x+c)^(1+n)*hypergeom([n, 1+n],[2+n],b*(d*x+c)/(-a*d+b*c))/d/(1+n)/((b*x+a)^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, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {72, 71} \[ \int (a+b x)^{-n} (c+d x)^n \, dx=\frac {(a+b x)^{-n} (c+d x)^{n+1} \left (-\frac {d (a+b x)}{b c-a d}\right )^n \operatorname {Hypergeometric2F1}\left (n,n+1,n+2,\frac {b (c+d x)}{b c-a d}\right )}{d (n+1)} \]

[In]

Int[(c + d*x)^n/(a + b*x)^n,x]

[Out]

((-((d*(a + b*x))/(b*c - a*d)))^n*(c + d*x)^(1 + n)*Hypergeometric2F1[n, 1 + n, 2 + n, (b*(c + d*x))/(b*c - a*
d)])/(d*(1 + n)*(a + b*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 ((a+b x)^{-n} \left (\frac {d (a+b x)}{-b c+a d}\right )^n\right ) \int (c+d x)^n \left (-\frac {a d}{b c-a d}-\frac {b d x}{b c-a d}\right )^{-n} \, dx \\ & = \frac {(a+b x)^{-n} \left (-\frac {d (a+b x)}{b c-a d}\right )^n (c+d x)^{1+n} \, _2F_1\left (n,1+n;2+n;\frac {b (c+d x)}{b c-a d}\right )}{d (1+n)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.99 \[ \int (a+b x)^{-n} (c+d x)^n \, dx=\frac {(a+b x)^{-n} \left (\frac {d (a+b x)}{-b c+a d}\right )^n (c+d x)^{1+n} \operatorname {Hypergeometric2F1}\left (n,1+n,2+n,\frac {b (c+d x)}{b c-a d}\right )}{d (1+n)} \]

[In]

Integrate[(c + d*x)^n/(a + b*x)^n,x]

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

integral((d*x + c)^n/(b*x + a)^n, x)

Sympy [F(-2)]

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

[In]

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

[Out]

Exception raised: HeuristicGCDFailed >> no luck

Maxima [F]

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

[In]

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

[Out]

integrate((d*x + c)^n/(b*x + a)^n, x)

Giac [F]

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

[In]

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

[Out]

integrate((d*x + c)^n/(b*x + a)^n, x)

Mupad [F(-1)]

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

[In]

int((c + d*x)^n/(a + b*x)^n,x)

[Out]

int((c + d*x)^n/(a + b*x)^n, x)