### 3.116 $$\int \frac{(d x)^m}{(b x+c x^2)^2} \, dx$$

Optimal. Leaf size=33 $-\frac{d (d x)^{m-1} \, _2F_1\left (2,m-1;m;-\frac{c x}{b}\right )}{b^2 (1-m)}$

[Out]

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

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Rubi [A]  time = 0.0172126, antiderivative size = 33, 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} $-\frac{d (d x)^{m-1} \, _2F_1\left (2,m-1;m;-\frac{c x}{b}\right )}{b^2 (1-m)}$

Antiderivative was successfully veriﬁed.

[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 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 )^2} \, dx &=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]  time = 0.0088843, size = 30, normalized size = 0.91 $\frac{(d x)^m \, _2F_1\left (2,m-1;m;-\frac{c x}{b}\right )}{b^2 (m-1) x}$

Antiderivative was successfully veriﬁed.

[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)

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Maple [F]  time = 0.39, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( dx \right ) ^{m}}{ \left ( c{x}^{2}+bx \right ) ^{2}}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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 )}^{2}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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 )}^{2}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)