Optimal. Leaf size=67 \[ -\frac{2 (d (b+2 c x))^{m+1} \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right )}{d (m+1) \left (b^2-4 a c\right )} \]
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Rubi [A] time = 0.0437272, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {694, 364} \[ -\frac{2 (d (b+2 c x))^{m+1} \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right )}{d (m+1) \left (b^2-4 a c\right )} \]
Antiderivative was successfully verified.
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Rule 694
Rule 364
Rubi steps
\begin{align*} \int \frac{(b d+2 c d x)^m}{a+b x+c x^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^m}{a-\frac{b^2}{4 c}+\frac{x^2}{4 c d^2}} \, dx,x,b d+2 c d x\right )}{2 c d}\\ &=-\frac{2 (d (b+2 c x))^{1+m} \, _2F_1\left (1,\frac{1+m}{2};\frac{3+m}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right )}{\left (b^2-4 a c\right ) d (1+m)}\\ \end{align*}
Mathematica [A] time = 0.0378771, size = 68, normalized size = 1.01 \[ -\frac{2 (b+2 c x) (d (b+2 c x))^m \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right )}{(m+1) \left (b^2-4 a c\right )} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.204, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( 2\,cdx+bd \right ) ^{m}}{c{x}^{2}+bx+a}}\, 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 (2 \, c d x + b d\right )}^{m}}{c x^{2} + b x + a}\,{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 (2 \, c d x + b d\right )}^{m}}{c x^{2} + b x + a}, 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 \left (b + 2 c x\right )\right )^{m}}{a + b x + c x^{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 (2 \, c d x + b d\right )}^{m}}{c x^{2} + b x + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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