Optimal. Leaf size=54 \[ -\frac {(d+e x)^m \, _2F_1\left (1,m;m+1;\frac {c d (d+e x)}{c d^2-a e^2}\right )}{m \left (c d^2-a e^2\right )} \]
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Rubi [A] time = 0.02, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {626, 68} \[ -\frac {(d+e x)^m \, _2F_1\left (1,m;m+1;\frac {c d (d+e x)}{c d^2-a e^2}\right )}{m \left (c d^2-a e^2\right )} \]
Antiderivative was successfully verified.
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Rule 68
Rule 626
Rubi steps
\begin {align*} \int \frac {(d+e x)^m}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx &=\int \frac {(d+e x)^{-1+m}}{a e+c d x} \, dx\\ &=-\frac {(d+e x)^m \, _2F_1\left (1,m;1+m;\frac {c d (d+e x)}{c d^2-a e^2}\right )}{\left (c d^2-a e^2\right ) m}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 54, normalized size = 1.00 \[ -\frac {(d+e x)^m \, _2F_1\left (1,m;m+1;\frac {c d (d+e x)}{c d^2-a e^2}\right )}{m \left (c d^2-a e^2\right )} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.67, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (e x + d\right )}^{m}}{c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}, x\right ) \]
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 \[ \int \frac {{\left (e x + d\right )}^{m}}{c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 2.07, size = 0, normalized size = 0.00 \[ \int \frac {\left (e x +d \right )^{m}}{c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x}\, dx \]
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 \[ \int \frac {{\left (e x + d\right )}^{m}}{c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (d+e\,x\right )}^m}{c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e} \,d x \]
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 \[ \int \frac {\left (d + e x\right )^{m}}{\left (d + e x\right ) \left (a e + c d x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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