Optimal. Leaf size=39 \[ -\frac {e (d+e x)^{m-1} \, _2F_1\left (2,m-1;m;\frac {e x}{d}+1\right )}{c^2 d^2 (1-m)} \]
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Rubi [A] time = 0.02, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {626, 12, 65} \[ -\frac {e (d+e x)^{m-1} \, _2F_1\left (2,m-1;m;\frac {e x}{d}+1\right )}{c^2 d^2 (1-m)} \]
Antiderivative was successfully verified.
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Rule 12
Rule 65
Rule 626
Rubi steps
\begin {align*} \int \frac {(d+e x)^m}{\left (c d x+c e x^2\right )^2} \, dx &=\int \frac {(d+e x)^{-2+m}}{c^2 x^2} \, dx\\ &=\frac {\int \frac {(d+e x)^{-2+m}}{x^2} \, dx}{c^2}\\ &=-\frac {e (d+e x)^{-1+m} \, _2F_1\left (2,-1+m;m;1+\frac {e x}{d}\right )}{c^2 d^2 (1-m)}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 36, normalized size = 0.92 \[ \frac {e (d+e x)^{m-1} \, _2F_1\left (2,m-1;m;\frac {e x}{d}+1\right )}{c^2 d^2 (m-1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.78, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (e x + d\right )}^{m}}{c^{2} e^{2} x^{4} + 2 \, c^{2} d e x^{3} + c^{2} d^{2} x^{2}}, x\right ) \]
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 \[ \int \frac {{\left (e x + d\right )}^{m}}{{\left (c e x^{2} + c d x\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.70, size = 0, normalized size = 0.00 \[ \int \frac {\left (e x +d \right )^{m}}{\left (c e \,x^{2}+c d x \right )^{2}}\, dx \]
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 \[ \int \frac {{\left (e x + d\right )}^{m}}{{\left (c e x^{2} + c d x\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {{\left (d+e\,x\right )}^m}{{\left (c\,e\,x^2+c\,d\,x\right )}^2} \,d x \]
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 \[ \frac {\int \frac {\left (d + e x\right )^{m}}{d^{2} x^{2} + 2 d e x^{3} + e^{2} x^{4}}\, dx}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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