\(\int \frac {(d x)^m}{b x+c x^2} \, dx\) [115]

Optimal result

Integrand size = 17, antiderivative size = 25 \[ \int \frac {(d x)^m}{b x+c x^2} \, dx=\frac {(d x)^m \operatorname {Hypergeometric2F1}\left (1,m,1+m,-\frac {c x}{b}\right )}{b m} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 25, 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 = {661, 66} \[ \int \frac {(d x)^m}{b x+c x^2} \, dx=\frac {(d x)^m \operatorname {Hypergeometric2F1}\left (1,m,m+1,-\frac {c x}{b}\right )}{b m} \]

[In]

Int[(d*x)^m/(b*x + c*x^2),x]

[Out]

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

Rule 66

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[c^n*((b*x)^(m + 1)/(b*(m + 1)))*Hypergeometr
ic2F1[-n, m + 1, m + 2, (-d)*(x/c)], x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[m] && (IntegerQ[n] || (GtQ[
c, 0] &&  !(EqQ[n, -2^(-1)] && EqQ[c^2 - d^2, 0] && GtQ[-d/(b*c), 0])))

Rule 661

Int[((e_.)*(x_))^(m_.)*((b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[1/e^p, Int[(e*x)^(m + p)*(b + c*x)
^p, x], x] /; FreeQ[{b, c, e, m}, x] && IntegerQ[p]

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

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {(d x)^m}{b x+c x^2} \, dx=\frac {(d x)^m \operatorname {Hypergeometric2F1}\left (1,m,1+m,-\frac {c x}{b}\right )}{b m} \]

[In]

Integrate[(d*x)^m/(b*x + c*x^2),x]

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

integral((d*x)^m/(c*x^2 + b*x), x)

Sympy [F]

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

[In]

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

[Out]

Integral((d*x)**m/(x*(b + c*x)), x)

Maxima [F]

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

[In]

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

[Out]

integrate((d*x)^m/(c*x^2 + b*x), x)

Giac [F]

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

[In]

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

[Out]

integrate((d*x)^m/(c*x^2 + b*x), x)

Mupad [F(-1)]

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

[In]

int((d*x)^m/(b*x + c*x^2),x)

[Out]

int((d*x)^m/(b*x + c*x^2), x)