Optimal. Leaf size=35 \[ \frac {(a+b x) \log (d+e x)}{e \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rubi [A] time = 0.03, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {770, 21, 31} \begin {gather*} \frac {(a+b x) \log (d+e x)}{e \sqrt {a^2+2 a b x+b^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rule 31
Rule 770
Rubi steps
\begin {align*} \int \frac {a+b x}{(d+e x) \sqrt {a^2+2 a b x+b^2 x^2}} \, dx &=\frac {\left (a b+b^2 x\right ) \int \frac {a+b x}{\left (a b+b^2 x\right ) (d+e x)} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {\left (a b+b^2 x\right ) \int \frac {1}{d+e x} \, dx}{b \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {(a+b x) \log (d+e x)}{e \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 26, normalized size = 0.74 \begin {gather*} \frac {(a+b x) \log (d+e x)}{e \sqrt {(a+b x)^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.54, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a+b x}{(d+e x) \sqrt {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 = 0.44, size = 10, normalized size = 0.29 \begin {gather*} \frac {\log \left (e x + d\right )}{e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 17, normalized size = 0.49 \begin {gather*} e^{\left (-1\right )} \log \left ({\left | x e + d \right |}\right ) \mathrm {sgn}\left (b x + a\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 25, normalized size = 0.71 \begin {gather*} \frac {\left (b x +a \right ) \ln \left (e x +d \right )}{\sqrt {\left (b x +a \right )^{2}}\, e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {a+b\,x}{\sqrt {{\left (a+b\,x\right )}^2}\,\left (d+e\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.10, size = 7, normalized size = 0.20 \begin {gather*} \frac {\log {\left (d + e x \right )}}{e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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