Optimal. Leaf size=129 \[ \frac {(d+e x)^{m+1} \left (a g^2-b f g+c f^2\right ) \, _2F_1\left (1,m+1;m+2;-\frac {g (d+e x)}{e f-d g}\right )}{g^2 (m+1) (e f-d g)}-\frac {(d+e x)^{m+1} (-b e g+c d g+c e f)}{e^2 g^2 (m+1)}+\frac {c (d+e x)^{m+2}}{e^2 g (m+2)} \]
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Rubi [A] time = 0.16, antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {951, 80, 68} \[ \frac {(d+e x)^{m+1} \left (a g^2-b f g+c f^2\right ) \, _2F_1\left (1,m+1;m+2;-\frac {g (d+e x)}{e f-d g}\right )}{g^2 (m+1) (e f-d g)}-\frac {(d+e x)^{m+1} (-b e g+c d g+c e f)}{e^2 g^2 (m+1)}+\frac {c (d+e x)^{m+2}}{e^2 g (m+2)} \]
Antiderivative was successfully verified.
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Rule 68
Rule 80
Rule 951
Rubi steps
\begin {align*} \int \frac {(d+e x)^m \left (a+b x+c x^2\right )}{f+g x} \, dx &=\frac {c (d+e x)^{2+m}}{e^2 g (2+m)}+\frac {\int \frac {(d+e x)^m (-e (c d f-a e g) (2+m)-e (c e f+c d g-b e g) (2+m) x)}{f+g x} \, dx}{e^2 g (2+m)}\\ &=-\frac {(c e f+c d g-b e g) (d+e x)^{1+m}}{e^2 g^2 (1+m)}+\frac {c (d+e x)^{2+m}}{e^2 g (2+m)}+\frac {\left (c f^2-b f g+a g^2\right ) \int \frac {(d+e x)^m}{f+g x} \, dx}{g^2}\\ &=-\frac {(c e f+c d g-b e g) (d+e x)^{1+m}}{e^2 g^2 (1+m)}+\frac {c (d+e x)^{2+m}}{e^2 g (2+m)}+\frac {\left (c f^2-b f g+a g^2\right ) (d+e x)^{1+m} \, _2F_1\left (1,1+m;2+m;-\frac {g (d+e x)}{e f-d g}\right )}{g^2 (e f-d g) (1+m)}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 111, normalized size = 0.86 \[ \frac {(d+e x)^{m+1} \left (\frac {\left (g (a g-b f)+c f^2\right ) \, _2F_1\left (1,m+1;m+2;\frac {g (d+e x)}{d g-e f}\right )}{(m+1) (e f-d g)}+\frac {b e g-c (d g+e f)}{e^2 (m+1)}+\frac {c g (d+e x)}{e^2 (m+2)}\right )}{g^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.75, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (c x^{2} + b x + a\right )} {\left (e x + d\right )}^{m}}{g x + f}, x\right ) \]
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 \[ \int \frac {{\left (c x^{2} + b x + a\right )} {\left (e x + d\right )}^{m}}{g x + f}\,{d x} \]
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 \[ \int \frac {\left (c \,x^{2}+b x +a \right ) \left (e x +d \right )^{m}}{g x +f}\, dx \]
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 \[ \int \frac {{\left (c x^{2} + b x + a\right )} {\left (e x + d\right )}^{m}}{g x + f}\,{d x} \]
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 \[ \int \frac {{\left (d+e\,x\right )}^m\,\left (c\,x^2+b\,x+a\right )}{f+g\,x} \,d x \]
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 \[ \int \frac {\left (d + e x\right )^{m} \left (a + b x + c x^{2}\right )}{f + g x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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