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

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Rubi [A]  time = 0.02, antiderivative size = 37, normalized size of antiderivative = 1.00, 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 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])))

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]

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*}

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Mathematica [A]  time = 0.01, 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)

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fricas [F]  time = 1.01, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\left (d x\right )^{m}}{c^{3} x^{6} + 3 \, b c^{2} x^{5} + 3 \, b^{2} c x^{4} + b^{3} x^{3}}, x\right ) \]

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)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (d x\right )^{m}}{{\left (c x^{2} + b x\right )}^{3}}\,{d x} \]

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)

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maple [F]  time = 0.51, size = 0, normalized size = 0.00 \[ \int \frac {\left (d x \right )^{m}}{\left (c \,x^{2}+b x \right )^{3}}\, dx \]

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)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (d x\right )^{m}}{{\left (c x^{2} + b x\right )}^{3}}\,{d x} \]

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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {{\left (d\,x\right )}^m}{{\left (c\,x^2+b\,x\right )}^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^m/(b*x + c*x^2)^3,x)

[Out]

int((d*x)^m/(b*x + c*x^2)^3, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (d x\right )^{m}}{x^{3} \left (b + c x\right )^{3}}\, dx \]

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)

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