Optimal. Leaf size=51 \[ -\frac {(d+e x)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac {b (d+e x)}{b d-a e}\right )}{(m+1) (b d-a e)} \]
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Rubi [A] time = 0.01, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {27, 68} \[ -\frac {(d+e x)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac {b (d+e x)}{b d-a e}\right )}{(m+1) (b d-a e)} \]
Antiderivative was successfully verified.
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Rule 27
Rule 68
Rubi steps
\begin {align*} \int \frac {(a+b x) (d+e x)^m}{a^2+2 a b x+b^2 x^2} \, dx &=\int \frac {(d+e x)^m}{a+b x} \, dx\\ &=-\frac {(d+e x)^{1+m} \, _2F_1\left (1,1+m;2+m;\frac {b (d+e x)}{b d-a e}\right )}{(b d-a e) (1+m)}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 51, normalized size = 1.00 \[ -\frac {(d+e x)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac {b (d+e x)}{b d-a e}\right )}{(m+1) (b d-a e)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.81, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (e x + d\right )}^{m}}{b x + a}, x\right ) \]
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 \[ \int \frac {{\left (b x + a\right )} {\left (e x + d\right )}^{m}}{b^{2} x^{2} + 2 \, a b x + a^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.25, size = 0, normalized size = 0.00 \[ \int \frac {\left (b x +a \right ) \left (e x +d \right )^{m}}{b^{2} x^{2}+2 a b x +a^{2}}\, dx \]
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 \[ \int \frac {{\left (b x + a\right )} {\left (e x + d\right )}^{m}}{b^{2} x^{2} + 2 \, a b x + a^{2}}\,{d x} \]
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 \[ \int \frac {\left (a+b\,x\right )\,{\left (d+e\,x\right )}^m}{a^2+2\,a\,b\,x+b^2\,x^2} \,d x \]
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 \[ \int \frac {\left (d + e x\right )^{m}}{a + b x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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