Optimal. Leaf size=88 \[ \frac{c (f+g x)^{n+1}}{e g (n+1)}-\frac{g \left (c d^2-a e\right ) (f+g x)^{n+1} \, _2F_1\left (2,n+1;n+2;\frac{e (f+g x)}{e f-d g}\right )}{e (n+1) (e f-d g)^2} \]
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Rubi [A] time = 0.0830026, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {947, 68} \[ \frac{c (f+g x)^{n+1}}{e g (n+1)}-\frac{g \left (c d^2-a e\right ) (f+g x)^{n+1} \, _2F_1\left (2,n+1;n+2;\frac{e (f+g x)}{e f-d g}\right )}{e (n+1) (e f-d g)^2} \]
Antiderivative was successfully verified.
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Rule 947
Rule 68
Rubi steps
\begin{align*} \int \frac{(f+g x)^n \left (a+2 c d x+c e x^2\right )}{(d+e x)^2} \, dx &=\int \left (\frac{c (f+g x)^n}{e}+\frac{\left (-c d^2+a e\right ) (f+g x)^n}{e (d+e x)^2}\right ) \, dx\\ &=\frac{c (f+g x)^{1+n}}{e g (1+n)}+\frac{\left (-c d^2+a e\right ) \int \frac{(f+g x)^n}{(d+e x)^2} \, dx}{e}\\ &=\frac{c (f+g x)^{1+n}}{e g (1+n)}-\frac{\left (c d^2-a e\right ) g (f+g x)^{1+n} \, _2F_1\left (2,1+n;2+n;\frac{e (f+g x)}{e f-d g}\right )}{e (e f-d g)^2 (1+n)}\\ \end{align*}
Mathematica [A] time = 0.0926034, size = 83, normalized size = 0.94 \[ \frac{(f+g x)^{n+1} \left (g^2 \left (a e-c d^2\right ) \, _2F_1\left (2,n+1;n+2;\frac{e (f+g x)}{e f-d g}\right )+c (e f-d g)^2\right )}{e g (n+1) (e f-d g)^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.692, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( gx+f \right ) ^{n} \left ( ce{x}^{2}+2\,cdx+a \right ) }{ \left ( ex+d \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 (c e x^{2} + 2 \, c d x + a\right )}{\left (g x + f\right )}^{n}}{{\left (e x + d\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 (c e x^{2} + 2 \, c d x + a\right )}{\left (g x + f\right )}^{n}}{e^{2} x^{2} + 2 \, d e x + d^{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 (f + g x\right )^{n} \left (a + 2 c d x + c e x^{2}\right )}{\left (d + e 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 (c e x^{2} + 2 \, c d x + a\right )}{\left (g x + f\right )}^{n}}{{\left (e x + d\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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