Optimal. Leaf size=51 \[ \frac {e (d+e x)^{1+m} \, _2F_1\left (2,1+m;2+m;\frac {b (d+e x)}{b d-a e}\right )}{(b d-a e)^2 (1+m)} \]
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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 = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {27, 70}
\begin {gather*} \frac {e (d+e x)^{m+1} \, _2F_1\left (2,m+1;m+2;\frac {b (d+e x)}{b d-a e}\right )}{(m+1) (b d-a e)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 70
Rubi steps
\begin {align*} \int \frac {(d+e x)^m}{a^2+2 a b x+b^2 x^2} \, dx &=\int \frac {(d+e x)^m}{(a+b x)^2} \, dx\\ &=\frac {e (d+e x)^{1+m} \, _2F_1\left (2,1+m;2+m;\frac {b (d+e x)}{b d-a e}\right )}{(b d-a e)^2 (1+m)}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 52, normalized size = 1.02 \begin {gather*} \frac {e (d+e x)^{1+m} \, _2F_1\left (2,1+m;2+m;-\frac {b (d+e x)}{-b d+a e}\right )}{(-b d+a e)^2 (1+m)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.31, size = 0, normalized size = 0.00 \[\int \frac {\left (e x +d \right )^{m}}{b^{2} x^{2}+2 a b x +a^{2}}\, dx\]
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 \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 (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 = -0.02 \begin {gather*} \int \frac {{\left (d+e\,x\right )}^m}{a^2+2\,a\,b\,x+b^2\,x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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