Optimal. Leaf size=37 \[ -\frac{d^2 (d x)^{m-2} \, _2F_1\left (3,m-2;m-1;-\frac{c x}{b}\right )}{b^3 (2-m)} \]
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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.
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Rule 647
Rule 64
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.
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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.
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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 )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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^{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.
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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.
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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 )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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