Optimal. Leaf size=33 \[ -\frac{d (d x)^{m-1} \, _2F_1\left (2,m-1;m;-\frac{c x}{b}\right )}{b^2 (1-m)} \]
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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.
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Rule 647
Rule 64
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.
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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.
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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.
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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.
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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.
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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.
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