\(\int \frac {(c+d x)^n}{a+b x} \, dx\) [1856]

Optimal result

Integrand size = 15, antiderivative size = 51 \[ \int \frac {(c+d x)^n}{a+b x} \, dx=-\frac {(c+d x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {b (c+d x)}{b c-a d}\right )}{(b c-a d) (1+n)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {70} \[ \int \frac {(c+d x)^n}{a+b x} \, dx=-\frac {(c+d x)^{n+1} \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {b (c+d x)}{b c-a d}\right )}{(n+1) (b c-a d)} \]

[In]

Int[(c + d*x)^n/(a + b*x),x]

[Out]

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

Rule 70

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(b*c - a*d)^n*((a + b*x)^(m + 1)/(b^(
n + 1)*(m + 1)))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m
}, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] && IntegerQ[n]

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

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.00 \[ \int \frac {(c+d x)^n}{a+b x} \, dx=-\frac {(c+d x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {b (c+d x)}{b c-a d}\right )}{(b c-a d) (1+n)} \]

[In]

Integrate[(c + d*x)^n/(a + b*x),x]

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

integral((d*x + c)^n/(b*x + a), x)

Sympy [F]

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

[In]

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

[Out]

Integral((c + d*x)**n/(a + b*x), x)

Maxima [F]

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

[In]

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

[Out]

integrate((d*x + c)^n/(b*x + a), x)

Giac [F]

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

[In]

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

[Out]

integrate((d*x + c)^n/(b*x + a), x)

Mupad [F(-1)]

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

[In]

int((c + d*x)^n/(a + b*x),x)

[Out]

int((c + d*x)^n/(a + b*x), x)