\(\int \frac {(d x)^m}{(b x+c x^2)^3} \, dx\) [117]

Optimal result

Integrand size = 17, antiderivative size = 37 \[ \int \frac {(d x)^m}{\left (b x+c x^2\right )^3} \, dx=-\frac {d^2 (d x)^{-2+m} \operatorname {Hypergeometric2F1}\left (3,-2+m,-1+m,-\frac {c x}{b}\right )}{b^3 (2-m)} \]

[Out]

-d^2*(d*x)^(-2+m)*hypergeom([3, -2+m],[-1+m],-c*x/b)/b^3/(2-m)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 37, 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} \[ \int \frac {(d x)^m}{\left (b x+c x^2\right )^3} \, dx=-\frac {d^2 (d x)^{m-2} \operatorname {Hypergeometric2F1}\left (3,m-2,m-1,-\frac {c x}{b}\right )}{b^3 (2-m)} \]

[In]

Int[(d*x)^m/(b*x + c*x^2)^3,x]

[Out]

-((d^2*(d*x)^(-2 + m)*Hypergeometric2F1[3, -2 + m, -1 + m, -((c*x)/b)])/(b^3*(2 - 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^3 \int \frac {(d x)^{-3+m}}{(b+c x)^3} \, dx \\ & = -\frac {d^2 (d x)^{-2+m} \, _2F_1\left (3,-2+m;-1+m;-\frac {c x}{b}\right )}{b^3 (2-m)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.86 \[ \int \frac {(d x)^m}{\left (b x+c x^2\right )^3} \, dx=\frac {(d x)^m \operatorname {Hypergeometric2F1}\left (3,-2+m,-1+m,-\frac {c x}{b}\right )}{b^3 (-2+m) x^2} \]

[In]

Integrate[(d*x)^m/(b*x + c*x^2)^3,x]

[Out]

((d*x)^m*Hypergeometric2F1[3, -2 + m, -1 + m, -((c*x)/b)])/(b^3*(-2 + m)*x^2)

Maple [F]

\[\int \frac {\left (d x \right )^{m}}{\left (c \,x^{2}+b x \right )^{3}}d x\]

[In]

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

[Out]

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

Fricas [F]

\[ \int \frac {(d x)^m}{\left (b x+c x^2\right )^3} \, dx=\int { \frac {\left (d x\right )^{m}}{{\left (c x^{2} + b x\right )}^{3}} \,d x } \]

[In]

integrate((d*x)^m/(c*x^2+b*x)^3,x, algorithm="fricas")

[Out]

integral((d*x)^m/(c^3*x^6 + 3*b*c^2*x^5 + 3*b^2*c*x^4 + b^3*x^3), x)

Sympy [F]

\[ \int \frac {(d x)^m}{\left (b x+c x^2\right )^3} \, dx=\int \frac {\left (d x\right )^{m}}{x^{3} \left (b + c x\right )^{3}}\, dx \]

[In]

integrate((d*x)**m/(c*x**2+b*x)**3,x)

[Out]

Integral((d*x)**m/(x**3*(b + c*x)**3), x)

Maxima [F]

\[ \int \frac {(d x)^m}{\left (b x+c x^2\right )^3} \, dx=\int { \frac {\left (d x\right )^{m}}{{\left (c x^{2} + b x\right )}^{3}} \,d x } \]

[In]

integrate((d*x)^m/(c*x^2+b*x)^3,x, algorithm="maxima")

[Out]

integrate((d*x)^m/(c*x^2 + b*x)^3, x)

Giac [F]

\[ \int \frac {(d x)^m}{\left (b x+c x^2\right )^3} \, dx=\int { \frac {\left (d x\right )^{m}}{{\left (c x^{2} + b x\right )}^{3}} \,d x } \]

[In]

integrate((d*x)^m/(c*x^2+b*x)^3,x, algorithm="giac")

[Out]

integrate((d*x)^m/(c*x^2 + b*x)^3, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(d x)^m}{\left (b x+c x^2\right )^3} \, dx=\int \frac {{\left (d\,x\right )}^m}{{\left (c\,x^2+b\,x\right )}^3} \,d x \]

[In]

int((d*x)^m/(b*x + c*x^2)^3,x)

[Out]

int((d*x)^m/(b*x + c*x^2)^3, x)