### 3.115 $$\int \frac{(d x)^m}{b x+c x^2} \, dx$$

Optimal. Leaf size=25 $\frac{(d x)^m \, _2F_1\left (1,m;m+1;-\frac{c x}{b}\right )}{b m}$

[Out]

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

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Rubi [A]  time = 0.0130205, antiderivative size = 25, 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} $\frac{(d x)^m \, _2F_1\left (1,m;m+1;-\frac{c x}{b}\right )}{b m}$

Antiderivative was successfully veriﬁed.

[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 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}{b x+c x^2} \, dx &=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]  time = 0.006013, size = 25, normalized size = 1. $\frac{(d x)^m \, _2F_1\left (1,m;m+1;-\frac{c x}{b}\right )}{b m}$

Antiderivative was successfully veriﬁed.

[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)

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Maple [F]  time = 0.374, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( dx \right ) ^{m}}{c{x}^{2}+bx}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((d*x)^m/(c*x^2 + b*x), 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 x^{2} + b x}, x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((d*x)^m/(c*x^2 + b*x), x)

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((d*x)**m/(x*(b + c*x)), x)

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((d*x)^m/(c*x^2 + b*x), x)