\(\int \frac {(d x)^m}{(b x+c x^2)^2} \, dx\) [116]

Optimal result

Integrand size = 17, antiderivative size = 33 \[ \int \frac {(d x)^m}{\left (b x+c x^2\right )^2} \, dx=-\frac {d (d x)^{-1+m} \operatorname {Hypergeometric2F1}\left (2,-1+m,m,-\frac {c x}{b}\right )}{b^2 (1-m)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 33, 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} \[ \int \frac {(d x)^m}{\left (b x+c x^2\right )^2} \, dx=-\frac {d (d x)^{m-1} \operatorname {Hypergeometric2F1}\left (2,m-1,m,-\frac {c x}{b}\right )}{b^2 (1-m)} \]

[In]

Int[(d*x)^m/(b*x + c*x^2)^2,x]

[Out]

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

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.91 \[ \int \frac {(d x)^m}{\left (b x+c x^2\right )^2} \, dx=\frac {(d x)^m \operatorname {Hypergeometric2F1}\left (2,-1+m,m,-\frac {c x}{b}\right )}{b^2 (-1+m) x} \]

[In]

Integrate[(d*x)^m/(b*x + c*x^2)^2,x]

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

integral((d*x)^m/(c^2*x^4 + 2*b*c*x^3 + b^2*x^2), x)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

integrate((d*x)^m/(c*x^2 + b*x)^2, x)

Giac [F]

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

[In]

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

[Out]

integrate((d*x)^m/(c*x^2 + b*x)^2, x)

Mupad [F(-1)]

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

[In]

int((d*x)^m/(b*x + c*x^2)^2,x)

[Out]

int((d*x)^m/(b*x + c*x^2)^2, x)