Optimal. Leaf size=85 \[ \frac {B (d+e x)^{1+m}}{b e (1+m)}-\frac {(A b-a B) (d+e x)^{1+m} \, _2F_1\left (1,1+m;2+m;\frac {b (d+e x)}{b d-a e}\right )}{b (b d-a e) (1+m)} \]
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Rubi [A]
time = 0.03, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {81, 70}
\begin {gather*} \frac {B (d+e x)^{m+1}}{b e (m+1)}-\frac {(A b-a B) (d+e x)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac {b (d+e x)}{b d-a e}\right )}{b (m+1) (b d-a e)} \end {gather*}
Antiderivative was successfully verified.
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Rule 70
Rule 81
Rubi steps
\begin {align*} \int \frac {(A+B x) (d+e x)^m}{a+b x} \, dx &=\frac {B (d+e x)^{1+m}}{b e (1+m)}+\frac {(A b e (1+m)-a B e (1+m)) \int \frac {(d+e x)^m}{a+b x} \, dx}{b e (1+m)}\\ &=\frac {B (d+e x)^{1+m}}{b e (1+m)}-\frac {(A b-a B) (d+e x)^{1+m} \, _2F_1\left (1,1+m;2+m;\frac {b (d+e x)}{b d-a e}\right )}{b (b d-a e) (1+m)}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 78, normalized size = 0.92 \begin {gather*} \frac {(d+e x)^{1+m} \left (B (b d-a e)+(-A b e+a B e) \, _2F_1\left (1,1+m;2+m;\frac {b (d+e x)}{b d-a e}\right )\right )}{b e (b d-a e) (1+m)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (B x +A \right ) \left (e x +d \right )^{m}}{b x +a}\, dx\]
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 \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 [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (A + B x\right ) \left (d + e x\right )^{m}}{a + b x}\, dx \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 = -0.01 \begin {gather*} \int \frac {\left (A+B\,x\right )\,{\left (d+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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