Optimal. Leaf size=48 \[ \frac {(e x)^{m+1} \, _2F_1\left (1,\frac {m+1}{2};\frac {m+3}{2};\frac {b^2 x^2}{a^2}\right )}{a^2 c e (m+1)} \]
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Rubi [A] time = 0.01, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {73, 364} \[ \frac {(e x)^{m+1} \, _2F_1\left (1,\frac {m+1}{2};\frac {m+3}{2};\frac {b^2 x^2}{a^2}\right )}{a^2 c e (m+1)} \]
Antiderivative was successfully verified.
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Rule 73
Rule 364
Rubi steps
\begin {align*} \int \frac {(e x)^m}{(a+b x) (a c-b c x)} \, dx &=\int \frac {(e x)^m}{a^2 c-b^2 c x^2} \, dx\\ &=\frac {(e x)^{1+m} \, _2F_1\left (1,\frac {1+m}{2};\frac {3+m}{2};\frac {b^2 x^2}{a^2}\right )}{a^2 c e (1+m)}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 46, normalized size = 0.96 \[ \frac {x (e x)^m \, _2F_1\left (1,\frac {m+1}{2};\frac {m+1}{2}+1;\frac {b^2 x^2}{a^2}\right )}{a^2 c (m+1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.72, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\left (e x\right )^{m}}{b^{2} c x^{2} - a^{2} c}, x\right ) \]
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 \[ \int -\frac {\left (e x\right )^{m}}{{\left (b c x - a c\right )} {\left (b x + a\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.11, size = 0, normalized size = 0.00 \[ \int \frac {\left (e x \right )^{m}}{\left (b x +a \right ) \left (-b c x +a c \right )}\, dx \]
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 \[ -\int \frac {\left (e x\right )^{m}}{{\left (b c x - a c\right )} {\left (b x + a\right )}}\,{d x} \]
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 \[ \int \frac {{\left (e\,x\right )}^m}{\left (a\,c-b\,c\,x\right )\,\left (a+b\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 2.22, size = 73, normalized size = 1.52 \[ - \frac {e^{m} m x^{m} \Phi \left (\frac {a}{b x}, 1, m e^{i \pi }\right ) \Gamma \left (- m\right )}{2 a b c \Gamma \left (1 - m\right )} + \frac {e^{m} m x^{m} \Phi \left (\frac {b x e^{i \pi }}{a}, 1, m\right ) \Gamma \left (- m\right )}{2 a b c \Gamma \left (1 - m\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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