Optimal. Leaf size=101 \[ \frac {c d (a e+c d x) (d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m} \, _2F_1\left (2,1-m;2-m;-\frac {g (a e+c d x)}{c d f-a e g}\right )}{(c d f-a e g)^2 (1-m)} \]
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Rubi [A]
time = 0.09, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {905, 70}
\begin {gather*} \frac {c d (d+e x)^m (a e+c d x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{-m} \, _2F_1\left (2,1-m;2-m;-\frac {g (a e+c d x)}{c d f-a e g}\right )}{(1-m) (c d f-a e g)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 70
Rule 905
Rubi steps
\begin {align*} \int \frac {(d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m}}{(f+g x)^2} \, dx &=\left ((a e+c d x)^m (d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m}\right ) \int \frac {(a e+c d x)^{-m}}{(f+g x)^2} \, dx\\ &=\frac {c d (a e+c d x) (d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m} \, _2F_1\left (2,1-m;2-m;-\frac {g (a e+c d x)}{c d f-a e g}\right )}{(c d f-a e g)^2 (1-m)}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 84, normalized size = 0.83 \begin {gather*} -\frac {c d (d+e x)^{-1+m} ((a e+c d x) (d+e x))^{1-m} \, _2F_1\left (2,1-m;2-m;\frac {g (a e+c d x)}{-c d f+a e g}\right )}{(c d f-a e g)^2 (-1+m)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {\left (e x +d \right )^{m} \left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )^{-m}}{\left (g x +f \right )^{2}}\, dx\]
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(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 (f+g\,x\right )}^2\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^m} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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