\(\int \frac {(a+b x)^m}{(c+d x)^2} \, dx\) [1850]

Optimal result

Integrand size = 15, antiderivative size = 52 \[ \int \frac {(a+b x)^m}{(c+d x)^2} \, dx=\frac {b (a+b x)^{1+m} \operatorname {Hypergeometric2F1}\left (2,1+m,2+m,-\frac {d (a+b x)}{b c-a d}\right )}{(b c-a d)^2 (1+m)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 52, 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} \[ \int \frac {(a+b x)^m}{(c+d x)^2} \, dx=\frac {b (a+b x)^{m+1} \operatorname {Hypergeometric2F1}\left (2,m+1,m+2,-\frac {d (a+b x)}{b c-a d}\right )}{(m+1) (b c-a d)^2} \]

[In]

Int[(a + b*x)^m/(c + d*x)^2,x]

[Out]

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

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 {b (a+b x)^{1+m} \, _2F_1\left (2,1+m;2+m;-\frac {d (a+b x)}{b c-a d}\right )}{(b c-a d)^2 (1+m)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x)^m}{(c+d x)^2} \, dx=\frac {b (a+b x)^{1+m} \operatorname {Hypergeometric2F1}\left (2,1+m,2+m,-\frac {d (a+b x)}{b c-a d}\right )}{(b c-a d)^2 (1+m)} \]

[In]

Integrate[(a + b*x)^m/(c + d*x)^2,x]

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

Integral((a + b*x)**m/(c + d*x)**2, x)

Maxima [F]

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

[In]

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

[Out]

integrate((b*x + a)^m/(d*x + c)^2, x)

Giac [F]

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

[In]

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

[Out]

integrate((b*x + a)^m/(d*x + c)^2, x)

Mupad [F(-1)]

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

[In]

int((a + b*x)^m/(c + d*x)^2,x)

[Out]

int((a + b*x)^m/(c + d*x)^2, x)