Optimal. Leaf size=52 \[ -\frac {c (e x)^{1+m}}{e (1+m)}+\frac {2 c (e x)^{1+m} \, _2F_1\left (1,1+m;2+m;-\frac {b x}{a}\right )}{e (1+m)} \]
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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 = {81, 66}
\begin {gather*} \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)} \end {gather*}
Antiderivative was successfully verified.
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Rule 66
Rule 81
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 = 0.04, size = 32, normalized size = 0.62 \begin {gather*} \frac {c x (e x)^m \left (-1+2 \, _2F_1\left (1,1+m;2+m;-\frac {b x}{a}\right )\right )}{1+m} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (e x \right )^{m} \left (-b c x +a c \right )}{b x +a}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 1.51, size = 150, normalized size = 2.88 \begin {gather*} \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 )} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\left (a\,c-b\,c\,x\right )\,{\left (e\,x\right )}^m}{a+b\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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