Optimal. Leaf size=32 \[ -\frac {(d+e x)^m \, _2F_1\left (1,m;m+1;\frac {e x}{d}+1\right )}{c d m} \]
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Rubi [A] time = 0.02, antiderivative size = 32, 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} \[ -\frac {(d+e x)^m \, _2F_1\left (1,m;m+1;\frac {e x}{d}+1\right )}{c d m} \]
Antiderivative was successfully verified.
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Rule 12
Rule 65
Rule 626
Rubi steps
\begin {align*} \int \frac {(d+e x)^m}{c d x+c e x^2} \, dx &=\int \frac {(d+e x)^{-1+m}}{c x} \, dx\\ &=\frac {\int \frac {(d+e x)^{-1+m}}{x} \, dx}{c}\\ &=-\frac {(d+e x)^m \, _2F_1\left (1,m;1+m;1+\frac {e x}{d}\right )}{c d m}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 32, normalized size = 1.00 \[ -\frac {(d+e x)^m \, _2F_1\left (1,m;m+1;\frac {e x}{d}+1\right )}{c d m} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.88, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (e x + d\right )}^{m}}{c e x^{2} + c d x}, x\right ) \]
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 \[ \int \frac {{\left (e x + d\right )}^{m}}{c e x^{2} + c d x}\,{d x} \]
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 \[ \int \frac {\left (e x +d \right )^{m}}{c e \,x^{2}+c d x}\, dx \]
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 \[ \int \frac {{\left (e x + d\right )}^{m}}{c e x^{2} + c d x}\,{d x} \]
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 \[ \int \frac {{\left (d+e\,x\right )}^m}{c\,e\,x^2+c\,d\,x} \,d x \]
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 \[ \frac {\int \frac {\left (d + e x\right )^{m}}{d x + e x^{2}}\, dx}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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