Optimal. Leaf size=64 \[ \frac {e^2 (d+e x)^{m-2} \, _2F_1\left (3,m-2;m-1;\frac {c d (d+e x)}{c d^2-a e^2}\right )}{(2-m) \left (c d^2-a e^2\right )^3} \]
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Rubi [A] time = 0.03, antiderivative size = 64, 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} \[ \frac {e^2 (d+e x)^{m-2} \, _2F_1\left (3,m-2;m-1;\frac {c d (d+e x)}{c d^2-a e^2}\right )}{(2-m) \left (c d^2-a e^2\right )^3} \]
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 )^3} \, dx &=\int \frac {(d+e x)^{-3+m}}{(a e+c d x)^3} \, dx\\ &=\frac {e^2 (d+e x)^{-2+m} \, _2F_1\left (3,-2+m;-1+m;\frac {c d (d+e x)}{c d^2-a e^2}\right )}{\left (c d^2-a e^2\right )^3 (2-m)}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 63, normalized size = 0.98 \[ \frac {e^2 (d+e x)^{m-2} \, _2F_1\left (3,m-2;m-1;-\frac {c d (d+e x)}{a e^2-c d^2}\right )}{(m-2) \left (a e^2-c d^2\right )^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.84, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (e x + d\right )}^{m}}{c^{3} d^{3} e^{3} x^{6} + a^{3} d^{3} e^{3} + 3 \, {\left (c^{3} d^{4} e^{2} + a c^{2} d^{2} e^{4}\right )} x^{5} + 3 \, {\left (c^{3} d^{5} e + 3 \, a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x^{4} + {\left (c^{3} d^{6} + 9 \, a c^{2} d^{4} e^{2} + 9 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )} x^{3} + 3 \, {\left (a c^{2} d^{5} e + 3 \, a^{2} c d^{3} e^{3} + a^{3} d e^{5}\right )} x^{2} + 3 \, {\left (a^{2} c d^{4} e^{2} + a^{3} d^{2} e^{4}\right )} x}, x\right ) \]
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 \[ \int \frac {{\left (e x + d\right )}^{m}}{{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.85, size = 0, normalized size = 0.00 \[ \int \frac {\left (e x +d \right )^{m}}{\left (c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x \right )^{3}}\, dx \]
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 \[ \int \frac {{\left (e x + d\right )}^{m}}{{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{3}}\,{d x} \]
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 \[ \int \frac {{\left (d+e\,x\right )}^m}{{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^3} \,d x \]
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 \[ \int \frac {\left (d + e x\right )^{m}}{\left (d + e x\right )^{3} \left (a e + c d x\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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