Optimal. Leaf size=88 \[ \frac {c (f+g x)^{1+n}}{e g (1+n)}-\frac {\left (c d^2-a e\right ) g (f+g x)^{1+n} \, _2F_1\left (2,1+n;2+n;\frac {e (f+g x)}{e f-d g}\right )}{e (e f-d g)^2 (1+n)} \]
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Rubi [A]
time = 0.06, 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 = {961, 70}
\begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 70
Rule 961
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 [F]
time = 0.18, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(f+g x)^n \left (a+2 c d x+c e x^2\right )}{(d+e x)^2} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {\left (g x +f \right )^{n} \left (c e \,x^{2}+2 c d x +a \right )}{\left (e x +d \right )^{2}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \text {Exception raised: HeuristicGCDFailed} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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