Optimal. Leaf size=59 \[ -\frac {2 e (b d-a e)}{b^3 (a+b x)}-\frac {(b d-a e)^2}{2 b^3 (a+b x)^2}+\frac {e^2 \log (a+b x)}{b^3} \]
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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 = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {27, 43} \[ -\frac {2 e (b d-a e)}{b^3 (a+b x)}-\frac {(b d-a e)^2}{2 b^3 (a+b x)^2}+\frac {e^2 \log (a+b x)}{b^3} \]
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rubi steps
\begin {align*} \int \frac {(a+b x) (d+e x)^2}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx &=\int \frac {(d+e x)^2}{(a+b x)^3} \, dx\\ &=\int \left (\frac {(b d-a e)^2}{b^2 (a+b x)^3}+\frac {2 e (b d-a e)}{b^2 (a+b x)^2}+\frac {e^2}{b^2 (a+b x)}\right ) \, dx\\ &=-\frac {(b d-a e)^2}{2 b^3 (a+b x)^2}-\frac {2 e (b d-a e)}{b^3 (a+b x)}+\frac {e^2 \log (a+b x)}{b^3}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 49, normalized size = 0.83 \[ \frac {2 e^2 \log (a+b x)-\frac {(b d-a e) (3 a e+b (d+4 e x))}{(a+b x)^2}}{2 b^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.97, size = 99, normalized size = 1.68 \[ -\frac {b^{2} d^{2} + 2 \, a b d e - 3 \, a^{2} e^{2} + 4 \, {\left (b^{2} d e - a b e^{2}\right )} x - 2 \, {\left (b^{2} e^{2} x^{2} + 2 \, a b e^{2} x + a^{2} e^{2}\right )} \log \left (b x + a\right )}{2 \, {\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 67, normalized size = 1.14 \[ \frac {e^{2} \log \left ({\left | b x + a \right |}\right )}{b^{3}} - \frac {4 \, {\left (b d e - a e^{2}\right )} x + \frac {b^{2} d^{2} + 2 \, a b d e - 3 \, a^{2} e^{2}}{b}}{2 \, {\left (b x + a\right )}^{2} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 92, normalized size = 1.56 \[ -\frac {a^{2} e^{2}}{2 \left (b x +a \right )^{2} b^{3}}+\frac {a d e}{\left (b x +a \right )^{2} b^{2}}-\frac {d^{2}}{2 \left (b x +a \right )^{2} b}+\frac {2 a \,e^{2}}{\left (b x +a \right ) b^{3}}-\frac {2 d e}{\left (b x +a \right ) b^{2}}+\frac {e^{2} \ln \left (b x +a \right )}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.58, size = 79, normalized size = 1.34 \[ -\frac {b^{2} d^{2} + 2 \, a b d e - 3 \, a^{2} e^{2} + 4 \, {\left (b^{2} d e - a b e^{2}\right )} x}{2 \, {\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )}} + \frac {e^{2} \log \left (b x + a\right )}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 77, normalized size = 1.31 \[ \frac {e^2\,\ln \left (a+b\,x\right )}{b^3}-\frac {\frac {-3\,a^2\,e^2+2\,a\,b\,d\,e+b^2\,d^2}{2\,b^3}-\frac {2\,e\,x\,\left (a\,e-b\,d\right )}{b^2}}{a^2+2\,a\,b\,x+b^2\,x^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.48, size = 80, normalized size = 1.36 \[ \frac {3 a^{2} e^{2} - 2 a b d e - b^{2} d^{2} + x \left (4 a b e^{2} - 4 b^{2} d e\right )}{2 a^{2} b^{3} + 4 a b^{4} x + 2 b^{5} x^{2}} + \frac {e^{2} \log {\left (a + b x \right )}}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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