Optimal. Leaf size=86 \[ \frac {(a+b x) \log (a+b x)}{(b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(a+b x) \log (d+e x)}{(b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rubi [A]
time = 0.02, antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {660, 36, 31}
\begin {gather*} \frac {(a+b x) \log (a+b x)}{\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac {(a+b x) \log (d+e x)}{\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 36
Rule 660
Rubi steps
\begin {align*} \int \frac {1}{(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 {1}{\left (a b+b^2 x\right ) (d+e x)} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {\left (b \left (a b+b^2 x\right )\right ) \int \frac {1}{a b+b^2 x} \, dx}{(b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (e \left (a b+b^2 x\right )\right ) \int \frac {1}{d+e x} \, dx}{b (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {(a+b x) \log (a+b x)}{(b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(a+b x) \log (d+e x)}{(b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 42, normalized size = 0.49 \begin {gather*} \frac {(a+b x) (\log (a+b x)-\log (d+e x))}{(b d-a e) \sqrt {(a+b x)^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.54, size = 42, normalized size = 0.49
method | result | size |
default | \(-\frac {\left (b x +a \right ) \left (\ln \left (b x +a \right )-\ln \left (e x +d \right )\right )}{\sqrt {\left (b x +a \right )^{2}}\, \left (a e -b d \right )}\) | \(42\) |
risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \ln \left (-e x -d \right )}{\left (b x +a \right ) \left (a e -b d \right )}-\frac {\sqrt {\left (b x +a \right )^{2}}\, \ln \left (b x +a \right )}{\left (b x +a \right ) \left (a e -b d \right )}\) | \(72\) |
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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Fricas [A]
time = 3.78, size = 28, normalized size = 0.33 \begin {gather*} \frac {\log \left (b x + a\right ) - \log \left (x e + d\right )}{b d - a e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 128 vs.
\(2 (56) = 112\).
time = 0.17, size = 128, normalized size = 1.49 \begin {gather*} \frac {\log {\left (x + \frac {- \frac {a^{2} e^{2}}{a e - b d} + \frac {2 a b d e}{a e - b d} + a e - \frac {b^{2} d^{2}}{a e - b d} + b d}{2 b e} \right )}}{a e - b d} - \frac {\log {\left (x + \frac {\frac {a^{2} e^{2}}{a e - b d} - \frac {2 a b d e}{a e - b d} + a e + \frac {b^{2} d^{2}}{a e - b d} + b d}{2 b e} \right )}}{a e - b d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.62, size = 56, normalized size = 0.65 \begin {gather*} {\left (\frac {b \log \left ({\left | b x + a \right |}\right )}{b^{2} d - a b e} - \frac {e \log \left ({\left | x e + d \right |}\right )}{b d e - a e^{2}}\right )} \mathrm {sgn}\left (b x + a\right ) \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 = -0.01 \begin {gather*} \int \frac {1}{\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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