Optimal. Leaf size=63 \[ \frac {(b x)^{1+m} (c+d x)^n \left (1+\frac {d x}{c}\right )^{-n} F_1\left (1+m;-n,2;2+m;-\frac {d x}{c},-\frac {f x}{e}\right )}{b e^2 (1+m)} \]
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Rubi [A]
time = 0.02, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {140, 138}
\begin {gather*} \frac {(b x)^{m+1} (c+d x)^n \left (\frac {d x}{c}+1\right )^{-n} F_1\left (m+1;-n,2;m+2;-\frac {d x}{c},-\frac {f x}{e}\right )}{b e^2 (m+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 138
Rule 140
Rubi steps
\begin {align*} \int \frac {(b x)^m (c+d x)^n}{(e+f x)^2} \, dx &=\left ((c+d x)^n \left (1+\frac {d x}{c}\right )^{-n}\right ) \int \frac {(b x)^m \left (1+\frac {d x}{c}\right )^n}{(e+f x)^2} \, dx\\ &=\frac {(b x)^{1+m} (c+d x)^n \left (1+\frac {d x}{c}\right )^{-n} F_1\left (1+m;-n,2;2+m;-\frac {d x}{c},-\frac {f x}{e}\right )}{b e^2 (1+m)}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 60, normalized size = 0.95 \begin {gather*} \frac {x (b x)^m (c+d x)^n \left (\frac {c+d x}{c}\right )^{-n} F_1\left (1+m;-n,2;2+m;-\frac {d x}{c},-\frac {f x}{e}\right )}{e^2 (1+m)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (b x \right )^{m} \left (d x +c \right )^{n}}{\left (f x +e \right )^{2}}\, dx\]
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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (b x\right )^{m} \left (c + d x\right )^{n}}{\left (e + f x\right )^{2}}\, dx \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 = -0.02 \begin {gather*} \int \frac {{\left (b\,x\right )}^m\,{\left (c+d\,x\right )}^n}{{\left (e+f\,x\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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