Optimal. Leaf size=25 \[ \frac {(b d-a e) \log (a+b x)}{b^2}+\frac {e x}{b} \]
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Rubi [A] time = 0.02, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {27, 43} \begin {gather*} \frac {(b d-a e) \log (a+b x)}{b^2}+\frac {e x}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rubi steps
\begin {align*} \int \frac {(a+b x) (d+e x)}{a^2+2 a b x+b^2 x^2} \, dx &=\int \frac {d+e x}{a+b x} \, dx\\ &=\int \left (\frac {e}{b}+\frac {b d-a e}{b (a+b x)}\right ) \, dx\\ &=\frac {e x}{b}+\frac {(b d-a e) \log (a+b x)}{b^2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 25, normalized size = 1.00 \begin {gather*} \frac {(b d-a e) \log (a+b x)}{b^2}+\frac {e x}{b} \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+b x) (d+e x)}{a^2+2 a b x+b^2 x^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.38, size = 24, normalized size = 0.96 \begin {gather*} \frac {b e x + {\left (b d - a e\right )} \log \left (b x + a\right )}{b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 28, normalized size = 1.12 \begin {gather*} \frac {x e}{b} + \frac {{\left (b d - a e\right )} \log \left ({\left | b x + a \right |}\right )}{b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 32, normalized size = 1.28 \begin {gather*} -\frac {a e \ln \left (b x +a \right )}{b^{2}}+\frac {d \ln \left (b x +a \right )}{b}+\frac {e x}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.55, size = 25, normalized size = 1.00 \begin {gather*} \frac {e x}{b} + \frac {{\left (b d - a e\right )} \log \left (b x + a\right )}{b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 26, normalized size = 1.04 \begin {gather*} \frac {e\,x}{b}-\frac {\ln \left (a+b\,x\right )\,\left (a\,e-b\,d\right )}{b^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.19, size = 20, normalized size = 0.80 \begin {gather*} \frac {e x}{b} - \frac {\left (a e - b d\right ) \log {\left (a + b x \right )}}{b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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