### 3.117 $$\int \frac{(d x)^m}{(b x+c x^2)^3} \, dx$$

Optimal. Leaf size=37 $-\frac{d^2 (d x)^{m-2} \, _2F_1\left (3,m-2;m-1;-\frac{c x}{b}\right )}{b^3 (2-m)}$

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0189416, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.118, Rules used = {647, 64} $-\frac{d^2 (d x)^{m-2} \, _2F_1\left (3,m-2;m-1;-\frac{c x}{b}\right )}{b^3 (2-m)}$

Antiderivative was successfully veriﬁed.

[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 647

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]

Rule 64

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(c^n*(b*x)^(m + 1)*Hypergeometric2F1[-n, m +
1, m + 2, -((d*x)/c)])/(b*(m + 1)), 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])))

Rubi steps

\begin{align*} \int \frac{(d x)^m}{\left (b x+c x^2\right )^3} \, dx &=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]  time = 0.0091514, size = 32, normalized size = 0.86 $\frac{(d x)^m \, _2F_1\left (3,m-2;m-1;-\frac{c x}{b}\right )}{b^3 (m-2) x^2}$

Antiderivative was successfully veriﬁed.

[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]  time = 0.409, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( dx \right ) ^{m}}{ \left ( c{x}^{2}+bx \right ) ^{3}}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (d x\right )^{m}}{{\left (c x^{2} + b x\right )}^{3}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\left (d x\right )^{m}}{c^{3} x^{6} + 3 \, b c^{2} x^{5} + 3 \, b^{2} c x^{4} + b^{3} x^{3}}, x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (d x\right )^{m}}{x^{3} \left (b + c x\right )^{3}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (d x\right )^{m}}{{\left (c x^{2} + b x\right )}^{3}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)