Optimal. Leaf size=55 \[ \frac {2 (e x)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac {b x}{a}\right )}{c e (m+1)}-\frac {(e x)^{m+1}}{c e (m+1)} \]
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Rubi [A] time = 0.02, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {80, 64} \[ \frac {2 (e x)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac {b x}{a}\right )}{c e (m+1)}-\frac {(e x)^{m+1}}{c e (m+1)} \]
Antiderivative was successfully verified.
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Rule 64
Rule 80
Rubi steps
\begin {align*} \int \frac {(e x)^m (a+b x)}{a c-b c x} \, dx &=-\frac {(e x)^{1+m}}{c e (1+m)}+(2 a) \int \frac {(e x)^m}{a c-b c x} \, dx\\ &=-\frac {(e x)^{1+m}}{c e (1+m)}+\frac {2 (e x)^{1+m} \, _2F_1\left (1,1+m;2+m;\frac {b x}{a}\right )}{c e (1+m)}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 33, normalized size = 0.60 \[ \frac {x (e x)^m \left (2 \, _2F_1\left (1,m+1;m+2;\frac {b x}{a}\right )-1\right )}{c (m+1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.72, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (b x + a\right )} \left (e x\right )^{m}}{b c x - a c}, x\right ) \]
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 \[ \int -\frac {{\left (b x + a\right )} \left (e x\right )^{m}}{b c x - a c}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.10, size = 0, normalized size = 0.00 \[ \int \frac {\left (b x +a \right ) \left (e x \right )^{m}}{-b c x +a c}\, dx \]
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 \[ -\int \frac {{\left (b x + a\right )} \left (e x\right )^{m}}{b c x - a c}\,{d x} \]
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 \[ \int \frac {{\left (e\,x\right )}^m\,\left (a+b\,x\right )}{a\,c-b\,c\,x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 5.69, size = 129, normalized size = 2.35 \[ \frac {e^{m} m x x^{m} \Phi \left (\frac {b x}{a}, 1, m + 1\right ) \Gamma \left (m + 1\right )}{c \Gamma \left (m + 2\right )} + \frac {e^{m} x x^{m} \Phi \left (\frac {b x}{a}, 1, m + 1\right ) \Gamma \left (m + 1\right )}{c \Gamma \left (m + 2\right )} + \frac {b e^{m} m x^{2} x^{m} \Phi \left (\frac {b x}{a}, 1, m + 2\right ) \Gamma \left (m + 2\right )}{a c \Gamma \left (m + 3\right )} + \frac {2 b e^{m} x^{2} x^{m} \Phi \left (\frac {b x}{a}, 1, m + 2\right ) \Gamma \left (m + 2\right )}{a c \Gamma \left (m + 3\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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