Optimal. Leaf size=114 \[ \frac {\left (c d^2-a e\right ) (f+g x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac {e (f+g x)}{e f-d g}\right )}{e (n+1) (e f-d g)}-\frac {c (e f-d g) (f+g x)^{n+1}}{e g^2 (n+1)}+\frac {c (f+g x)^{n+2}}{g^2 (n+2)} \]
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Rubi [A] time = 0.15, antiderivative size = 114, normalized size of antiderivative = 1.00, 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 = {951, 80, 68} \[ \frac {\left (c d^2-a e\right ) (f+g x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac {e (f+g x)}{e f-d g}\right )}{e (n+1) (e f-d g)}-\frac {c (e f-d g) (f+g x)^{n+1}}{e g^2 (n+1)}+\frac {c (f+g x)^{n+2}}{g^2 (n+2)} \]
Antiderivative was successfully verified.
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Rule 68
Rule 80
Rule 951
Rubi steps
\begin {align*} \int \frac {(f+g x)^n \left (a+2 c d x+c e x^2\right )}{d+e x} \, dx &=\frac {c (f+g x)^{2+n}}{g^2 (2+n)}+\frac {\int \frac {(f+g x)^n (-e g (c d f-a g) (2+n)-c e g (e f-d g) (2+n) x)}{d+e x} \, dx}{e g^2 (2+n)}\\ &=-\frac {c (e f-d g) (f+g x)^{1+n}}{e g^2 (1+n)}+\frac {c (f+g x)^{2+n}}{g^2 (2+n)}-\frac {\left (c d^2-a e\right ) \int \frac {(f+g x)^n}{d+e x} \, dx}{e}\\ &=-\frac {c (e f-d g) (f+g x)^{1+n}}{e g^2 (1+n)}+\frac {c (f+g x)^{2+n}}{g^2 (2+n)}+\frac {\left (c d^2-a e\right ) (f+g x)^{1+n} \, _2F_1\left (1,1+n;2+n;\frac {e (f+g x)}{e f-d g}\right )}{e (e f-d g) (1+n)}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 93, normalized size = 0.82 \[ \frac {(f+g x)^{n+1} \left (\frac {\left (c d^2-a e\right ) \, _2F_1\left (1,n+1;n+2;\frac {e (f+g x)}{e f-d g}\right )}{e f-d g}+\frac {c (d g (n+2)-e f+e g (n+1) x)}{g^2 (n+2)}\right )}{e (n+1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.25, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (c e x^{2} + 2 \, c d x + a\right )} {\left (g x + f\right )}^{n}}{e x + d}, x\right ) \]
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 \[ \int \frac {{\left (c e x^{2} + 2 \, c d x + a\right )} {\left (g x + f\right )}^{n}}{e x + d}\,{d x} \]
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 \[ \int \frac {\left (c e \,x^{2}+2 c d x +a \right ) \left (g x +f \right )^{n}}{e x +d}\, dx \]
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 \[ \int \frac {{\left (c e x^{2} + 2 \, c d x + a\right )} {\left (g x + f\right )}^{n}}{e x + d}\,{d x} \]
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 \[ \int \frac {{\left (f+g\,x\right )}^n\,\left (c\,e\,x^2+2\,c\,d\,x+a\right )}{d+e\,x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (f + g x\right )^{n} \left (a + 2 c d x + c e x^{2}\right )}{d + e x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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