Optimal. Leaf size=129 \[ \frac{(d+e x)^{m+1} \left (a g^2-b f g+c f^2\right ) \, _2F_1\left (1,m+1;m+2;-\frac{g (d+e x)}{e f-d g}\right )}{g^2 (m+1) (e f-d g)}-\frac{(d+e x)^{m+1} (-b e g+c d g+c e f)}{e^2 g^2 (m+1)}+\frac{c (d+e x)^{m+2}}{e^2 g (m+2)} \]
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Rubi [A] time = 0.159453, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {951, 80, 68} \[ \frac{(d+e x)^{m+1} \left (a g^2-b f g+c f^2\right ) \, _2F_1\left (1,m+1;m+2;-\frac{g (d+e x)}{e f-d g}\right )}{g^2 (m+1) (e f-d g)}-\frac{(d+e x)^{m+1} (-b e g+c d g+c e f)}{e^2 g^2 (m+1)}+\frac{c (d+e x)^{m+2}}{e^2 g (m+2)} \]
Antiderivative was successfully verified.
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Rule 951
Rule 80
Rule 68
Rubi steps
\begin{align*} \int \frac{(d+e x)^m \left (a+b x+c x^2\right )}{f+g x} \, dx &=\frac{c (d+e x)^{2+m}}{e^2 g (2+m)}+\frac{\int \frac{(d+e x)^m (-e (c d f-a e g) (2+m)-e (c e f+c d g-b e g) (2+m) x)}{f+g x} \, dx}{e^2 g (2+m)}\\ &=-\frac{(c e f+c d g-b e g) (d+e x)^{1+m}}{e^2 g^2 (1+m)}+\frac{c (d+e x)^{2+m}}{e^2 g (2+m)}+\frac{\left (c f^2-b f g+a g^2\right ) \int \frac{(d+e x)^m}{f+g x} \, dx}{g^2}\\ &=-\frac{(c e f+c d g-b e g) (d+e x)^{1+m}}{e^2 g^2 (1+m)}+\frac{c (d+e x)^{2+m}}{e^2 g (2+m)}+\frac{\left (c f^2-b f g+a g^2\right ) (d+e x)^{1+m} \, _2F_1\left (1,1+m;2+m;-\frac{g (d+e x)}{e f-d g}\right )}{g^2 (e f-d g) (1+m)}\\ \end{align*}
Mathematica [A] time = 0.146815, size = 111, normalized size = 0.86 \[ \frac{(d+e x)^{m+1} \left (\frac{\left (g (a g-b f)+c f^2\right ) \, _2F_1\left (1,m+1;m+2;\frac{g (d+e x)}{d g-e f}\right )}{(m+1) (e f-d g)}+\frac{b e g-c (d g+e f)}{e^2 (m+1)}+\frac{c g (d+e x)}{e^2 (m+2)}\right )}{g^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.662, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( c{x}^{2}+bx+a \right ) \left ( ex+d \right ) ^{m}}{gx+f}}\, 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 (c x^{2} + b x + a\right )}{\left (e x + d\right )}^{m}}{g x + f}\,{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 (c x^{2} + b x + a\right )}{\left (e x + d\right )}^{m}}{g x + f}, 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 + e x\right )^{m} \left (a + b x + c x^{2}\right )}{f + g x}\, 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 (c x^{2} + b x + a\right )}{\left (e x + d\right )}^{m}}{g x + f}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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