Optimal. Leaf size=63 \[ \frac {(b x)^{m+1} (c+d x)^n \left (\frac {d x}{c}+1\right )^{-n} F_1\left (m+1;-n,2;m+2;-\frac {d x}{c},-\frac {f x}{e}\right )}{b e^2 (m+1)} \]
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Rubi [A] time = 0.03, 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 = {135, 133} \[ \frac {(b x)^{m+1} (c+d x)^n \left (\frac {d x}{c}+1\right )^{-n} F_1\left (m+1;-n,2;m+2;-\frac {d x}{c},-\frac {f x}{e}\right )}{b e^2 (m+1)} \]
Antiderivative was successfully verified.
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Rule 133
Rule 135
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 = 0.06, size = 60, normalized size = 0.95 \[ \frac {x (b x)^m (c+d x)^n \left (\frac {c+d x}{c}\right )^{-n} F_1\left (m+1;-n,2;m+2;-\frac {d x}{c},-\frac {f x}{e}\right )}{e^2 (m+1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.96, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\left (b x\right )^{m} {\left (d x + c\right )}^{n}}{f^{2} x^{2} + 2 \, e f x + e^{2}}, x\right ) \]
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 \[ \int \frac {\left (b x\right )^{m} {\left (d x + c\right )}^{n}}{{\left (f x + e\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.14, 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 \[ \int \frac {\left (b x\right )^{m} {\left (d x + c\right )}^{n}}{{\left (f x + e\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (b\,x\right )}^m\,{\left (c+d\,x\right )}^n}{{\left (e+f\,x\right )}^2} \,d x \]
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 \[ \int \frac {\left (b x\right )^{m} \left (c + d x\right )^{n}}{\left (e + f x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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