\(\int (\frac {d (a+b x)}{-b c+a d})^m (c+d x)^n \, dx\) [1890]

Optimal result

Integrand size = 27, antiderivative size = 45 \[ \int \left (\frac {d (a+b x)}{-b c+a d}\right )^m (c+d x)^n \, dx=\frac {(c+d x)^{1+n} \operatorname {Hypergeometric2F1}\left (-m,1+n,2+n,\frac {b (c+d x)}{b c-a d}\right )}{d (1+n)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {192, 71} \[ \int \left (\frac {d (a+b x)}{-b c+a d}\right )^m (c+d x)^n \, dx=\frac {(c+d x)^{n+1} \operatorname {Hypergeometric2F1}\left (-m,n+1,n+2,\frac {b (c+d x)}{b c-a d}\right )}{d (n+1)} \]

[In]

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

[Out]

((c + d*x)^(1 + n)*Hypergeometric2F1[-m, 1 + n, 2 + n, (b*(c + d*x))/(b*c - a*d)])/(d*(1 + 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 192

Int[(u_)^(m_.)*(v_)^(n_.), x_Symbol] :> Int[ExpandToSum[u, x]^m*ExpandToSum[v, x]^n, x] /; FreeQ[{m, n}, x] &&
LinearQ[{u, v}, x] &&  !LinearMatchQ[{u, v}, x]

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.96 \[ \int \left (\frac {d (a+b x)}{-b c+a d}\right )^m (c+d x)^n \, dx=\frac {(a+b x) \left (\frac {d (a+b x)}{-b c+a d}\right )^m (c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} \operatorname {Hypergeometric2F1}\left (1+m,-n,2+m,\frac {d (a+b x)}{-b c+a d}\right )}{b (1+m)} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F(-2)]

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

[In]

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

[Out]

Exception raised: HeuristicGCDFailed >> no luck

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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