Optimal. Leaf size=50 \[ \frac {(b d-a e)^2 \log (d+e x)}{e^3}-\frac {b x (b d-a e)}{e^2}+\frac {(a+b x)^2}{2 e} \]
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Rubi [A] time = 0.02, antiderivative size = 50, 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} \[ -\frac {b x (b d-a e)}{e^2}+\frac {(b d-a e)^2 \log (d+e x)}{e^3}+\frac {(a+b x)^2}{2 e} \]
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} \, dx &=\int \frac {(a+b x)^2}{d+e x} \, dx\\ &=\int \left (-\frac {b (b d-a e)}{e^2}+\frac {b (a+b x)}{e}+\frac {(-b d+a e)^2}{e^2 (d+e x)}\right ) \, dx\\ &=-\frac {b (b d-a e) x}{e^2}+\frac {(a+b x)^2}{2 e}+\frac {(b d-a e)^2 \log (d+e x)}{e^3}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 43, normalized size = 0.86 \[ \frac {b e x (4 a e-2 b d+b e x)+2 (b d-a e)^2 \log (d+e x)}{2 e^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.96, size = 62, normalized size = 1.24 \[ \frac {b^{2} e^{2} x^{2} - 2 \, {\left (b^{2} d e - 2 \, a b e^{2}\right )} x + 2 \, {\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\right )} \log \left (e x + d\right )}{2 \, e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 61, normalized size = 1.22 \[ {\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\right )} e^{\left (-3\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {1}{2} \, {\left (b^{2} x^{2} e - 2 \, b^{2} d x + 4 \, a b x e\right )} e^{\left (-2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 74, normalized size = 1.48 \[ \frac {b^{2} x^{2}}{2 e}+\frac {a^{2} \ln \left (e x +d \right )}{e}-\frac {2 a b d \ln \left (e x +d \right )}{e^{2}}+\frac {2 a b x}{e}+\frac {b^{2} d^{2} \ln \left (e x +d \right )}{e^{3}}-\frac {b^{2} d x}{e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.38, size = 60, normalized size = 1.20 \[ \frac {b^{2} e x^{2} - 2 \, {\left (b^{2} d - 2 \, a b e\right )} x}{2 \, e^{2}} + \frac {{\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\right )} \log \left (e x + d\right )}{e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.52, size = 62, normalized size = 1.24 \[ \frac {\ln \left (d+e\,x\right )\,\left (a^2\,e^2-2\,a\,b\,d\,e+b^2\,d^2\right )}{e^3}-x\,\left (\frac {b^2\,d}{e^2}-\frac {2\,a\,b}{e}\right )+\frac {b^2\,x^2}{2\,e} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.22, size = 44, normalized size = 0.88 \[ \frac {b^{2} x^{2}}{2 e} + x \left (\frac {2 a b}{e} - \frac {b^{2} d}{e^{2}}\right ) + \frac {\left (a e - b d\right )^{2} \log {\left (d + e x \right )}}{e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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