Optimal. Leaf size=68 \[ \frac{8 c (d (b+2 c x))^{m+1} \, _2F_1\left (2,\frac{m+1}{2};\frac{m+3}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right )}{d (m+1) \left (b^2-4 a c\right )^2} \]
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Rubi [A] time = 0.0466202, antiderivative size = 68, 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} \[ \frac{8 c (d (b+2 c x))^{m+1} \, _2F_1\left (2,\frac{m+1}{2};\frac{m+3}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right )}{d (m+1) \left (b^2-4 a c\right )^2} \]
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}{\left (a+b x+c x^2\right )^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^m}{\left (a-\frac{b^2}{4 c}+\frac{x^2}{4 c d^2}\right )^2} \, dx,x,b d+2 c d x\right )}{2 c d}\\ &=\frac{8 c (d (b+2 c x))^{1+m} \, _2F_1\left (2,\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 )^2 d (1+m)}\\ \end{align*}
Mathematica [A] time = 0.0370594, size = 69, normalized size = 1.01 \[ \frac{8 c (b+2 c x) (d (b+2 c x))^m \, _2F_1\left (2,\frac{m+1}{2};\frac{m+3}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right )}{(m+1) \left (b^2-4 a c\right )^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.211, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( 2\,cdx+bd \right ) ^{m}}{ \left ( c{x}^{2}+bx+a \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 (2 \, c d x + b d\right )}^{m}}{{\left (c x^{2} + b x + a\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 (2 \, c d x + b d\right )}^{m}}{c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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}}{{\left (c x^{2} + b x + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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