Optimal. Leaf size=112 \[ -\frac {(A b-a B) (d+e x)^{1+m}}{b (b d-a e) (a+b x)}+\frac {(a B e (1+m)-b (B d+A e m)) (d+e x)^{1+m} \, _2F_1\left (1,1+m;2+m;\frac {b (d+e x)}{b d-a e}\right )}{b (b d-a e)^2 (1+m)} \]
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Rubi [A]
time = 0.04, antiderivative size = 112, 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 = {79, 70}
\begin {gather*} \frac {(d+e x)^{m+1} (a B e (m+1)-b (A e m+B d)) \, _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)^2}-\frac {(A b-a B) (d+e x)^{m+1}}{b (a+b x) (b d-a e)} \end {gather*}
Antiderivative was successfully verified.
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Rule 70
Rule 79
Rubi steps
\begin {align*} \int \frac {(A+B x) (d+e x)^m}{(a+b x)^2} \, dx &=-\frac {(A b-a B) (d+e x)^{1+m}}{b (b d-a e) (a+b x)}-\frac {(a B e (1+m)-b (B d+A e m)) \int \frac {(d+e x)^m}{a+b x} \, dx}{b (b d-a e)}\\ &=-\frac {(A b-a B) (d+e x)^{1+m}}{b (b d-a e) (a+b x)}+\frac {(a B e (1+m)-b (B d+A e m)) (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)^2 (1+m)}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 99, normalized size = 0.88 \begin {gather*} \frac {(d+e x)^{1+m} \left (-\frac {(A b-a B) (b d-a e)}{a+b x}+\frac {(a B e (1+m)-b (B d+A e m)) \, _2F_1\left (1,1+m;2+m;\frac {b (d+e x)}{b d-a e}\right )}{1+m}\right )}{b (b d-a e)^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (B x +A \right ) \left (e x +d \right )^{m}}{\left (b x +a \right )^{2}}\, dx\]
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}}{\left (a + b x\right )^{2}}\, 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}{{\left (a+b\,x\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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