Optimal. Leaf size=51 \[ \frac {e^2 x}{b^2}-\frac {(b d-a e)^2}{b^3 (a+b x)}+\frac {2 e (b d-a e) \log (a+b x)}{b^3} \]
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Rubi [A]
time = 0.03, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {27, 45}
\begin {gather*} -\frac {(b d-a e)^2}{b^3 (a+b x)}+\frac {2 e (b d-a e) \log (a+b x)}{b^3}+\frac {e^2 x}{b^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 45
Rubi steps
\begin {align*} \int \frac {(d+e x)^2}{a^2+2 a b x+b^2 x^2} \, dx &=\int \frac {(d+e x)^2}{(a+b x)^2} \, dx\\ &=\int \left (\frac {e^2}{b^2}+\frac {(b d-a e)^2}{b^2 (a+b x)^2}+\frac {2 e (b d-a e)}{b^2 (a+b x)}\right ) \, dx\\ &=\frac {e^2 x}{b^2}-\frac {(b d-a e)^2}{b^3 (a+b x)}+\frac {2 e (b d-a e) \log (a+b x)}{b^3}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 47, normalized size = 0.92 \begin {gather*} \frac {b e^2 x-\frac {(b d-a e)^2}{a+b x}+2 e (b d-a e) \log (a+b x)}{b^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.64, size = 63, normalized size = 1.24
method | result | size |
default | \(\frac {e^{2} x}{b^{2}}-\frac {a^{2} e^{2}-2 a b d e +b^{2} d^{2}}{b^{3} \left (b x +a \right )}-\frac {2 e \left (a e -b d \right ) \ln \left (b x +a \right )}{b^{3}}\) | \(63\) |
norman | \(\frac {\frac {e^{2} x^{2}}{b}-\frac {2 a^{2} e^{2}-2 a b d e +b^{2} d^{2}}{b^{3}}}{b x +a}-\frac {2 e \left (a e -b d \right ) \ln \left (b x +a \right )}{b^{3}}\) | \(68\) |
risch | \(\frac {e^{2} x}{b^{2}}-\frac {a^{2} e^{2}}{b^{3} \left (b x +a \right )}+\frac {2 a d e}{b^{2} \left (b x +a \right )}-\frac {d^{2}}{b \left (b x +a \right )}-\frac {2 e^{2} \ln \left (b x +a \right ) a}{b^{3}}+\frac {2 e \ln \left (b x +a \right ) d}{b^{2}}\) | \(86\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 66, normalized size = 1.29 \begin {gather*} -\frac {b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}}{b^{4} x + a b^{3}} + \frac {x e^{2}}{b^{2}} + \frac {2 \, {\left (b d e - a e^{2}\right )} \log \left (b x + a\right )}{b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.66, size = 86, normalized size = 1.69 \begin {gather*} -\frac {b^{2} d^{2} - 2 \, a b d e - {\left (b^{2} x^{2} + a b x - a^{2}\right )} e^{2} + 2 \, {\left ({\left (a b x + a^{2}\right )} e^{2} - {\left (b^{2} d x + a b d\right )} e\right )} \log \left (b x + a\right )}{b^{4} x + a b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.18, size = 60, normalized size = 1.18 \begin {gather*} \frac {- a^{2} e^{2} + 2 a b d e - b^{2} d^{2}}{a b^{3} + b^{4} x} + \frac {e^{2} x}{b^{2}} - \frac {2 e \left (a e - b d\right ) \log {\left (a + b x \right )}}{b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.90, size = 64, normalized size = 1.25 \begin {gather*} \frac {x e^{2}}{b^{2}} + \frac {2 \, {\left (b d e - a e^{2}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{3}} - \frac {b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}}{{\left (b x + a\right )} b^{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 {e^2\,x}{b^2}-\frac {a^2\,e^2-2\,a\,b\,d\,e+b^2\,d^2}{b\,\left (x\,b^3+a\,b^2\right )}-\frac {\ln \left (a+b\,x\right )\,\left (2\,a\,e^2-2\,b\,d\,e\right )}{b^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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