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 = 0.08, antiderivative size = 88, normalized size of antiderivative = 1.00, 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 68
Rule 947
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*}
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Mathematica [A] time = 0.09, 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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fricas [F] time = 1.01, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (c e x^{2} + 2 \, c d x + a\right )} {\left (g x + f\right )}^{n}}{e^{2} x^{2} + 2 \, d e x + d^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (c e x^{2} + 2 \, c d x + a\right )} {\left (g x + f\right )}^{n}}{{\left (e x + d\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.06, size = 0, normalized size = 0.00 \[ \int \frac {\left (c e \,x^{2}+2 c d x +a \right ) \left (g x +f \right )^{n}}{\left (e x +d \right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (c e x^{2} + 2 \, c d x + a\right )} {\left (g x + f\right )}^{n}}{{\left (e x + d\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (f+g\,x\right )}^n\,\left (c\,e\,x^2+2\,c\,d\,x+a\right )}{{\left (d+e\,x\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: HeuristicGCDFailed} \]
Verification of antiderivative is not currently implemented for this CAS.
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