Optimal. Leaf size=54 \[ -\frac {(d+e x)^m \, _2F_1\left (1,m;1+m;\frac {c d (d+e x)}{c d^2-a e^2}\right )}{\left (c d^2-a e^2\right ) m} \]
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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 = {640, 70}
\begin {gather*} -\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 )} \end {gather*}
Antiderivative was successfully verified.
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Rule 70
Rule 640
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 = 0.06, size = 54, normalized size = 1.00 \begin {gather*} -\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 {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.38, size = 0, normalized size = 0.00 \[\int \frac {\left (e x +d \right )^{m}}{a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}}\, dx\]
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 (d + e x\right )^{m}}{\left (d + e x\right ) \left (a e + c d x\right )}\, 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 (d+e\,x\right )}^m}{c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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