Optimal. Leaf size=69 \[ \frac {e \sqrt {a^2+2 a b x+b^2 x^2}}{b^2}+\frac {(b d-a e) (a+b x) \log (a+b x)}{b^2 \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rubi [A]
time = 0.02, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {654, 622, 31}
\begin {gather*} \frac {(a+b x) (b d-a e) \log (a+b x)}{b^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {e \sqrt {a^2+2 a b x+b^2 x^2}}{b^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 622
Rule 654
Rubi steps
\begin {align*} \int \frac {d+e x}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx &=\frac {e \sqrt {a^2+2 a b x+b^2 x^2}}{b^2}+\frac {\left (2 b^2 d-2 a b e\right ) \int \frac {1}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx}{2 b^2}\\ &=\frac {e \sqrt {a^2+2 a b x+b^2 x^2}}{b^2}+\frac {\left (\left (2 b^2 d-2 a b e\right ) \left (a b+b^2 x\right )\right ) \int \frac {1}{a b+b^2 x} \, dx}{2 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {e \sqrt {a^2+2 a b x+b^2 x^2}}{b^2}+\frac {(b d-a e) (a+b x) \log (a+b x)}{b^2 \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 40, normalized size = 0.58 \begin {gather*} \frac {(a+b x) (b e x+(b d-a e) \log (a+b x))}{b^2 \sqrt {(a+b x)^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.54, size = 45, normalized size = 0.65
method | result | size |
default | \(-\frac {\left (b x +a \right ) \left (\ln \left (b x +a \right ) a e -\ln \left (b x +a \right ) b d -b e x \right )}{\sqrt {\left (b x +a \right )^{2}}\, b^{2}}\) | \(45\) |
risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, e x}{\left (b x +a \right ) b}-\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (a e -b d \right ) \ln \left (b x +a \right )}{\left (b x +a \right ) b^{2}}\) | \(59\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 54, normalized size = 0.78 \begin {gather*} \frac {d \log \left (x + \frac {a}{b}\right )}{b} - \frac {a e \log \left (x + \frac {a}{b}\right )}{b^{2}} + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} e}{b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.75, size = 26, normalized size = 0.38 \begin {gather*} \frac {b x e + {\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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Sympy [A]
time = 0.07, size = 20, normalized size = 0.29 \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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Giac [A]
time = 1.40, size = 46, normalized size = 0.67 \begin {gather*} \frac {x e \mathrm {sgn}\left (b x + a\right )}{b} + \frac {{\left (b d \mathrm {sgn}\left (b x + a\right ) - a e \mathrm {sgn}\left (b x + a\right )\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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Mupad [B]
time = 0.94, size = 79, normalized size = 1.14 \begin {gather*} \frac {e\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{b^2}+\frac {d\,\ln \left (a+b\,x+\sqrt {{\left (a+b\,x\right )}^2}\right )}{b}-\frac {a\,b\,e\,\ln \left (a\,b+\sqrt {{\left (a+b\,x\right )}^2}\,\sqrt {b^2}+b^2\,x\right )}{{\left (b^2\right )}^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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