Optimal. Leaf size=59 \[ \frac {2 b (b d-a e)}{e^3 (d+e x)}-\frac {(b d-a e)^2}{2 e^3 (d+e x)^2}+\frac {b^2 \log (d+e x)}{e^3} \]
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Rubi [A] time = 0.04, antiderivative size = 59, 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} \[ \frac {2 b (b d-a e)}{e^3 (d+e x)}-\frac {(b d-a e)^2}{2 e^3 (d+e x)^2}+\frac {b^2 \log (d+e x)}{e^3} \]
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)^3} \, dx &=\int \frac {(a+b x)^2}{(d+e x)^3} \, dx\\ &=\int \left (\frac {(-b d+a e)^2}{e^2 (d+e x)^3}-\frac {2 b (b d-a e)}{e^2 (d+e x)^2}+\frac {b^2}{e^2 (d+e x)}\right ) \, dx\\ &=-\frac {(b d-a e)^2}{2 e^3 (d+e x)^2}+\frac {2 b (b d-a e)}{e^3 (d+e x)}+\frac {b^2 \log (d+e x)}{e^3}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 48, normalized size = 0.81 \[ \frac {\frac {(b d-a e) (a e+3 b d+4 b e x)}{(d+e x)^2}+2 b^2 \log (d+e x)}{2 e^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.77, size = 100, normalized size = 1.69 \[ \frac {3 \, b^{2} d^{2} - 2 \, a b d e - a^{2} e^{2} + 4 \, {\left (b^{2} d e - a b e^{2}\right )} x + 2 \, {\left (b^{2} e^{2} x^{2} + 2 \, b^{2} d e x + b^{2} d^{2}\right )} \log \left (e x + d\right )}{2 \, {\left (e^{5} x^{2} + 2 \, d e^{4} x + d^{2} e^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 69, normalized size = 1.17 \[ b^{2} e^{\left (-3\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {{\left (4 \, {\left (b^{2} d - a b e\right )} x + {\left (3 \, b^{2} d^{2} - 2 \, a b d e - a^{2} e^{2}\right )} e^{\left (-1\right )}\right )} e^{\left (-2\right )}}{2 \, {\left (x e + d\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 92, normalized size = 1.56 \[ -\frac {a^{2}}{2 \left (e x +d \right )^{2} e}+\frac {a b d}{\left (e x +d \right )^{2} e^{2}}-\frac {b^{2} d^{2}}{2 \left (e x +d \right )^{2} e^{3}}-\frac {2 a b}{\left (e x +d \right ) e^{2}}+\frac {2 b^{2} d}{\left (e x +d \right ) e^{3}}+\frac {b^{2} \ln \left (e x +d \right )}{e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.36, size = 80, normalized size = 1.36 \[ \frac {3 \, b^{2} d^{2} - 2 \, a b d e - a^{2} e^{2} + 4 \, {\left (b^{2} d e - a b e^{2}\right )} x}{2 \, {\left (e^{5} x^{2} + 2 \, d e^{4} x + d^{2} e^{3}\right )}} + \frac {b^{2} \log \left (e x + d\right )}{e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 77, normalized size = 1.31 \[ \frac {b^2\,\ln \left (d+e\,x\right )}{e^3}-\frac {\frac {a^2\,e^2+2\,a\,b\,d\,e-3\,b^2\,d^2}{2\,e^3}+\frac {2\,b\,x\,\left (a\,e-b\,d\right )}{e^2}}{d^2+2\,d\,e\,x+e^2\,x^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.46, size = 80, normalized size = 1.36 \[ \frac {b^{2} \log {\left (d + e x \right )}}{e^{3}} + \frac {- a^{2} e^{2} - 2 a b d e + 3 b^{2} d^{2} + x \left (- 4 a b e^{2} + 4 b^{2} d e\right )}{2 d^{2} e^{3} + 4 d e^{4} x + 2 e^{5} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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