Optimal. Leaf size=52 \[ \frac {2 c (e x)^{m+1} \, _2F_1\left (1,m+1;m+2;-\frac {b x}{a}\right )}{e (m+1)}-\frac {c (e x)^{m+1}}{e (m+1)} \]
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Rubi [A] time = 0.01, antiderivative size = 52, 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} \[ \frac {2 c (e x)^{m+1} \, _2F_1\left (1,m+1;m+2;-\frac {b x}{a}\right )}{e (m+1)}-\frac {c (e x)^{m+1}}{e (m+1)} \]
Antiderivative was successfully verified.
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Rule 64
Rule 80
Rubi steps
\begin {align*} \int \frac {(e x)^m (a c-b c x)}{a+b x} \, dx &=-\frac {c (e x)^{1+m}}{e (1+m)}+(2 a c) \int \frac {(e x)^m}{a+b x} \, dx\\ &=-\frac {c (e x)^{1+m}}{e (1+m)}+\frac {2 c (e x)^{1+m} \, _2F_1\left (1,1+m;2+m;-\frac {b x}{a}\right )}{e (1+m)}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 32, normalized size = 0.62 \[ \frac {c x (e x)^m \left (2 \, _2F_1\left (1,m+1;m+2;-\frac {b x}{a}\right )-1\right )}{m+1} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (b c x - a c\right )} \left (e x\right )^{m}}{b x + a}, x\right ) \]
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 \[ \int -\frac {{\left (b c x - a c\right )} \left (e x\right )^{m}}{b x + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.13, size = 0, normalized size = 0.00 \[ \int \frac {\left (-b c x +a c \right ) \left (e x \right )^{m}}{b x +a}\, dx \]
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 \[ -\int \frac {{\left (b c x - a c\right )} \left (e x\right )^{m}}{b x + a}\,{d x} \]
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 \[ \int \frac {\left (a\,c-b\,c\,x\right )\,{\left (e\,x\right )}^m}{a+b\,x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 4.46, size = 150, normalized size = 2.88 \[ \frac {c e^{m} m x x^{m} \Phi \left (\frac {b x e^{i \pi }}{a}, 1, m + 1\right ) \Gamma \left (m + 1\right )}{\Gamma \left (m + 2\right )} + \frac {c e^{m} x x^{m} \Phi \left (\frac {b x e^{i \pi }}{a}, 1, m + 1\right ) \Gamma \left (m + 1\right )}{\Gamma \left (m + 2\right )} - \frac {b c e^{m} m x^{2} x^{m} \Phi \left (\frac {b x e^{i \pi }}{a}, 1, m + 2\right ) \Gamma \left (m + 2\right )}{a \Gamma \left (m + 3\right )} - \frac {2 b c e^{m} x^{2} x^{m} \Phi \left (\frac {b x e^{i \pi }}{a}, 1, m + 2\right ) \Gamma \left (m + 2\right )}{a \Gamma \left (m + 3\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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