Optimal. Leaf size=51 \[ -\frac {(b d-a e)^2}{e^3 (d+e x)}-\frac {2 b (b d-a e) \log (d+e x)}{e^3}+\frac {b^2 x}{e^2} \]
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Rubi [A] time = 0.05, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {27, 43} \begin {gather*} -\frac {(b d-a e)^2}{e^3 (d+e x)}-\frac {2 b (b d-a e) \log (d+e x)}{e^3}+\frac {b^2 x}{e^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rubi steps
\begin {align*} \int \frac {a^2+2 a b x+b^2 x^2}{(d+e x)^2} \, dx &=\int \frac {(a+b x)^2}{(d+e x)^2} \, dx\\ &=\int \left (\frac {b^2}{e^2}+\frac {(-b d+a e)^2}{e^2 (d+e x)^2}-\frac {2 b (b d-a e)}{e^2 (d+e x)}\right ) \, dx\\ &=\frac {b^2 x}{e^2}-\frac {(b d-a e)^2}{e^3 (d+e x)}-\frac {2 b (b d-a e) \log (d+e x)}{e^3}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 47, normalized size = 0.92 \begin {gather*} \frac {-\frac {(b d-a e)^2}{d+e x}+2 b (a e-b d) \log (d+e x)+b^2 e x}{e^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a^2+2 a b x+b^2 x^2}{(d+e x)^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.39, size = 92, normalized size = 1.80 \begin {gather*} \frac {b^{2} e^{2} x^{2} + b^{2} d e x - b^{2} d^{2} + 2 \, a b d e - a^{2} e^{2} - 2 \, {\left (b^{2} d^{2} - a b d e + {\left (b^{2} d e - a b e^{2}\right )} x\right )} \log \left (e x + d\right )}{e^{4} x + d e^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.16, size = 111, normalized size = 2.18 \begin {gather*} -2 \, {\left (e^{\left (-1\right )} \log \left (\frac {{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) - \frac {d e^{\left (-1\right )}}{x e + d}\right )} a b e^{\left (-1\right )} + {\left (2 \, d e^{\left (-3\right )} \log \left (\frac {{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) + {\left (x e + d\right )} e^{\left (-3\right )} - \frac {d^{2} e^{\left (-3\right )}}{x e + d}\right )} b^{2} - \frac {a^{2} e^{\left (-1\right )}}{x e + d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 86, normalized size = 1.69 \begin {gather*} -\frac {a^{2}}{\left (e x +d \right ) e}+\frac {2 a b d}{\left (e x +d \right ) e^{2}}+\frac {2 a b \ln \left (e x +d \right )}{e^{2}}-\frac {b^{2} d^{2}}{\left (e x +d \right ) e^{3}}-\frac {2 b^{2} d \ln \left (e x +d \right )}{e^{3}}+\frac {b^{2} x}{e^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.35, size = 67, normalized size = 1.31 \begin {gather*} \frac {b^{2} x}{e^{2}} - \frac {b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}}{e^{4} x + d e^{3}} - \frac {2 \, {\left (b^{2} d - a b e\right )} \log \left (e x + d\right )}{e^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 71, normalized size = 1.39 \begin {gather*} \frac {b^2\,x}{e^2}-\frac {a^2\,e^2-2\,a\,b\,d\,e+b^2\,d^2}{e\,\left (x\,e^3+d\,e^2\right )}-\frac {\ln \left (d+e\,x\right )\,\left (2\,b^2\,d-2\,a\,b\,e\right )}{e^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.34, size = 60, normalized size = 1.18 \begin {gather*} \frac {b^{2} x}{e^{2}} + \frac {2 b \left (a e - b d\right ) \log {\left (d + e x \right )}}{e^{3}} + \frac {- a^{2} e^{2} + 2 a b d e - b^{2} d^{2}}{d e^{3} + e^{4} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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