Optimal. Leaf size=56 \[ \frac {B x^{1+m}}{b (1+m)}+\frac {(A b-a B) x^{1+m} \, _2F_1\left (1,1+m;2+m;-\frac {b x}{a}\right )}{a b (1+m)} \]
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Rubi [A]
time = 0.02, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {81, 66}
\begin {gather*} \frac {x^{m+1} (A b-a B) \, _2F_1\left (1,m+1;m+2;-\frac {b x}{a}\right )}{a b (m+1)}+\frac {B x^{m+1}}{b (m+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 66
Rule 81
Rubi steps
\begin {align*} \int \frac {x^m (A+B x)}{a+b x} \, dx &=\frac {B x^{1+m}}{b (1+m)}+\frac {(A b (1+m)-a B (1+m)) \int \frac {x^m}{a+b x} \, dx}{b (1+m)}\\ &=\frac {B x^{1+m}}{b (1+m)}+\frac {(A b-a B) x^{1+m} \, _2F_1\left (1,1+m;2+m;-\frac {b x}{a}\right )}{a b (1+m)}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 45, normalized size = 0.80 \begin {gather*} \frac {x^{1+m} \left (a B+(A b-a B) \, _2F_1\left (1,1+m;2+m;-\frac {b x}{a}\right )\right )}{a b (1+m)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {x^{m} \left (B x +A \right )}{b x +a}\, dx\]
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 = 1.47, size = 136, normalized size = 2.43 \begin {gather*} \frac {A m x x^{m} \Phi \left (\frac {b x e^{i \pi }}{a}, 1, m + 1\right ) \Gamma \left (m + 1\right )}{a \Gamma \left (m + 2\right )} + \frac {A x x^{m} \Phi \left (\frac {b x e^{i \pi }}{a}, 1, m + 1\right ) \Gamma \left (m + 1\right )}{a \Gamma \left (m + 2\right )} + \frac {B 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 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 {x^m\,\left (A+B\,x\right )}{a+b\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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