Optimal. Leaf size=61 \[ -\frac {e (d+e x)^{m-1} \, _2F_1\left (2,m-1;m;\frac {c d (d+e x)}{c d^2-a e^2}\right )}{(1-m) \left (c d^2-a e^2\right )^2} \]
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Rubi [A] time = 0.02, antiderivative size = 61, 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} \[ -\frac {e (d+e x)^{m-1} \, _2F_1\left (2,m-1;m;\frac {c d (d+e x)}{c d^2-a e^2}\right )}{(1-m) \left (c d^2-a e^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 68
Rule 626
Rubi steps
\begin {align*} \int \frac {(d+e x)^m}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx &=\int \frac {(d+e x)^{-2+m}}{(a e+c d x)^2} \, dx\\ &=-\frac {e (d+e x)^{-1+m} \, _2F_1\left (2,-1+m;m;\frac {c d (d+e x)}{c d^2-a e^2}\right )}{\left (c d^2-a e^2\right )^2 (1-m)}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 59, normalized size = 0.97 \[ \frac {e (d+e x)^{m-1} \, _2F_1\left (2,m-1;m;-\frac {c d (d+e x)}{a e^2-c d^2}\right )}{(m-1) \left (a e^2-c d^2\right )^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.73, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (e x + d\right )}^{m}}{c^{2} d^{2} e^{2} x^{4} + a^{2} d^{2} e^{2} + 2 \, {\left (c^{2} d^{3} e + a c d e^{3}\right )} x^{3} + {\left (c^{2} d^{4} + 4 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} x^{2} + 2 \, {\left (a c d^{3} e + a^{2} d e^{3}\right )} x}, x\right ) \]
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 \[ \int \frac {{\left (e x + d\right )}^{m}}{{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.70, size = 0, normalized size = 0.00 \[ \int \frac {\left (e x +d \right )^{m}}{\left (c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x \right )^{2}}\, dx \]
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 \[ \int \frac {{\left (e x + d\right )}^{m}}{{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (d+e\,x\right )}^m}{{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^2} \,d x \]
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 \[ \int \frac {\left (d + e x\right )^{m}}{\left (d + e x\right )^{2} \left (a e + c d x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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