Optimal. Leaf size=197 \[ \frac{g (f+g x)^{n+1} \left (a e g^2 \left (n^2-3 n+2\right )+c \left (d^2 g^2 \left (-n^2+3 n+4\right )-12 d e f g+6 e^2 f^2\right )\right ) \, _2F_1\left (2,n+1;n+2;\frac{e (f+g x)}{e f-d g}\right )}{6 e (n+1) (e f-d g)^4}-\frac{g (2-n) \left (c d^2-a e\right ) (f+g x)^{n+1}}{6 e (d+e x)^2 (e f-d g)^2}-\frac{\left (a-\frac{c d^2}{e}\right ) (f+g x)^{n+1}}{3 (d+e x)^3 (e f-d g)} \]
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Rubi [A] time = 0.233455, antiderivative size = 197, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107, Rules used = {949, 78, 68} \[ \frac{g (f+g x)^{n+1} \left (a e g^2 \left (n^2-3 n+2\right )+c \left (d^2 g^2 \left (-n^2+3 n+4\right )-12 d e f g+6 e^2 f^2\right )\right ) \, _2F_1\left (2,n+1;n+2;\frac{e (f+g x)}{e f-d g}\right )}{6 e (n+1) (e f-d g)^4}-\frac{g (2-n) \left (c d^2-a e\right ) (f+g x)^{n+1}}{6 e (d+e x)^2 (e f-d g)^2}-\frac{\left (a-\frac{c d^2}{e}\right ) (f+g x)^{n+1}}{3 (d+e x)^3 (e f-d g)} \]
Antiderivative was successfully verified.
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Rule 949
Rule 78
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)^4} \, dx &=-\frac{\left (a-\frac{c d^2}{e}\right ) (f+g x)^{1+n}}{3 (e f-d g) (d+e x)^3}-\frac{\int \frac{(f+g x)^n \left (a g (2-n)-\frac{c d (3 e f-d g (1+n))}{e}-3 c (e f-d g) x\right )}{(d+e x)^3} \, dx}{3 (e f-d g)}\\ &=-\frac{\left (a-\frac{c d^2}{e}\right ) (f+g x)^{1+n}}{3 (e f-d g) (d+e x)^3}-\frac{\left (c d^2-a e\right ) g (2-n) (f+g x)^{1+n}}{6 e (e f-d g)^2 (d+e x)^2}+\frac{\left (a e g^2 \left (2-3 n+n^2\right )+c \left (6 e^2 f^2-12 d e f g+d^2 g^2 \left (4+3 n-n^2\right )\right )\right ) \int \frac{(f+g x)^n}{(d+e x)^2} \, dx}{6 e (e f-d g)^2}\\ &=-\frac{\left (a-\frac{c d^2}{e}\right ) (f+g x)^{1+n}}{3 (e f-d g) (d+e x)^3}-\frac{\left (c d^2-a e\right ) g (2-n) (f+g x)^{1+n}}{6 e (e f-d g)^2 (d+e x)^2}+\frac{g \left (a e g^2 \left (2-3 n+n^2\right )+c \left (6 e^2 f^2-12 d e f g+d^2 g^2 \left (4+3 n-n^2\right )\right )\right ) (f+g x)^{1+n} \, _2F_1\left (2,1+n;2+n;\frac{e (f+g x)}{e f-d g}\right )}{6 e (e f-d g)^4 (1+n)}\\ \end{align*}
Mathematica [A] time = 0.111262, size = 106, normalized size = 0.54 \[ \frac{g (f+g x)^{n+1} \left (g^2 \left (a e-c d^2\right ) \, _2F_1\left (4,n+1;n+2;\frac{e (f+g x)}{e f-d g}\right )+c (e f-d g)^2 \, _2F_1\left (2,n+1;n+2;\frac{e (f+g x)}{e f-d g}\right )\right )}{e (n+1) (e f-d g)^4} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.767, 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 ) ^{4}}}\, 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 )}^{4}}\,{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^{4} x^{4} + 4 \, d e^{3} x^{3} + 6 \, d^{2} e^{2} x^{2} + 4 \, d^{3} e x + d^{4}}, 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 )^{4}}\, 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 )}^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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