Optimal. Leaf size=157 \[ \frac {c (d+e x)^{1+m}}{e g^2 (1+m)}+\frac {\left (a+\frac {f (c f-b g)}{g^2}\right ) (d+e x)^{1+m}}{(e f-d g) (f+g x)}+\frac {(c f (2 d g-e f (2+m))-g (a e g m+b (d g-e f (1+m)))) (d+e x)^{1+m} \, _2F_1\left (1,1+m;2+m;-\frac {g (d+e x)}{e f-d g}\right )}{g^2 (e f-d g)^2 (1+m)} \]
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Rubi [A]
time = 0.13, antiderivative size = 157, 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 = {963, 81, 70}
\begin {gather*} -\frac {(d+e x)^{m+1} \, _2F_1\left (1,m+1;m+2;-\frac {g (d+e x)}{e f-d g}\right ) (g (a e g m+b d g-b e f (m+1))-c f (2 d g-e f (m+2)))}{g^2 (m+1) (e f-d g)^2}+\frac {(d+e x)^{m+1} \left (a+\frac {f (c f-b g)}{g^2}\right )}{(f+g x) (e f-d g)}+\frac {c (d+e x)^{m+1}}{e g^2 (m+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 70
Rule 81
Rule 963
Rubi steps
\begin {align*} \int \frac {(d+e x)^m \left (a+b x+c x^2\right )}{(f+g x)^2} \, dx &=\frac {\left (a+\frac {f (c f-b g)}{g^2}\right ) (d+e x)^{1+m}}{(e f-d g) (f+g x)}+\frac {\int \frac {(d+e x)^m \left (\frac {c d f g-a e g^2 m-c e f^2 (1+m)-b g (d g-e f (1+m))}{g^2}-c \left (d-\frac {e f}{g}\right ) x\right )}{f+g x} \, dx}{e f-d g}\\ &=\frac {c (d+e x)^{1+m}}{e g^2 (1+m)}+\frac {\left (a+\frac {f (c f-b g)}{g^2}\right ) (d+e x)^{1+m}}{(e f-d g) (f+g x)}-\frac {(g (b d g+a e g m-b e f (1+m))-c f (2 d g-e f (2+m))) \int \frac {(d+e x)^m}{f+g x} \, dx}{g^2 (e f-d g)}\\ &=\frac {c (d+e x)^{1+m}}{e g^2 (1+m)}+\frac {\left (a+\frac {f (c f-b g)}{g^2}\right ) (d+e x)^{1+m}}{(e f-d g) (f+g x)}-\frac {(g (b d g+a e g m-b e f (1+m))-c f (2 d g-e f (2+m))) (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)^2 (1+m)}\\ \end {align*}
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Mathematica [F]
time = 0.21, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(d+e x)^m \left (a+b x+c x^2\right )}{(f+g 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 \[\int \frac {\left (e x +d \right )^{m} \left (c \,x^{2}+b x +a \right )}{\left (g x +f \right )^{2}}\, dx\]
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 (d+e\,x\right )}^m\,\left (c\,x^2+b\,x+a\right )}{{\left (f+g\,x\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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