Optimal. Leaf size=145 \[ \frac {-a-b x}{2 a x^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {b (a+b x)}{a^2 x \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {b^2 (a+b x) \log (x)}{a^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {b^2 (a+b x) \log (a+b x)}{a^3 \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rubi [A]
time = 0.03, antiderivative size = 142, normalized size of antiderivative = 0.98, number of steps
used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {660, 46}
\begin {gather*} \frac {b (a+b x)}{a^2 x \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {a+b x}{2 a x^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {b^2 \log (x) (a+b x)}{a^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {b^2 (a+b x) \log (a+b x)}{a^3 \sqrt {a^2+2 a b x+b^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 46
Rule 660
Rubi steps
\begin {align*} \int \frac {1}{x^3 \sqrt {a^2+2 a b x+b^2 x^2}} \, dx &=\frac {\left (a b+b^2 x\right ) \int \frac {1}{x^3 \left (a b+b^2 x\right )} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {\left (a b+b^2 x\right ) \int \left (\frac {1}{a b x^3}-\frac {1}{a^2 x^2}+\frac {b}{a^3 x}-\frac {b^2}{a^3 (a+b x)}\right ) \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {a+b x}{2 a x^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {b (a+b x)}{a^2 x \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {b^2 (a+b x) \log (x)}{a^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {b^2 (a+b x) \log (a+b x)}{a^3 \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 59, normalized size = 0.41 \begin {gather*} -\frac {(a+b x) \left (a (a-2 b x)-2 b^2 x^2 \log (x)+2 b^2 x^2 \log (a+b x)\right )}{2 a^3 x^2 \sqrt {(a+b x)^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.42, size = 58, normalized size = 0.40
method | result | size |
default | \(\frac {\left (b x +a \right ) \left (2 b^{2} \ln \left (x \right ) x^{2}-2 b^{2} \ln \left (b x +a \right ) x^{2}+2 a b x -a^{2}\right )}{2 \sqrt {\left (b x +a \right )^{2}}\, a^{3} x^{2}}\) | \(58\) |
risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (\frac {b x}{a^{2}}-\frac {1}{2 a}\right )}{\left (b x +a \right ) x^{2}}-\frac {\sqrt {\left (b x +a \right )^{2}}\, b^{2} \ln \left (b x +a \right )}{\left (b x +a \right ) a^{3}}+\frac {\sqrt {\left (b x +a \right )^{2}}\, b^{2} \ln \left (-x \right )}{\left (b x +a \right ) a^{3}}\) | \(91\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 95, normalized size = 0.66 \begin {gather*} -\frac {\left (-1\right )^{2 \, a b x + 2 \, a^{2}} b^{2} \log \left (\frac {2 \, a b x}{{\left | x \right |}} + \frac {2 \, a^{2}}{{\left | x \right |}}\right )}{a^{3}} + \frac {3 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b}{2 \, a^{3} x} - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}}}{2 \, a^{2} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.78, size = 41, normalized size = 0.28 \begin {gather*} -\frac {2 \, b^{2} x^{2} \log \left (b x + a\right ) - 2 \, b^{2} x^{2} \log \left (x\right ) - 2 \, a b x + a^{2}}{2 \, a^{3} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.08, size = 31, normalized size = 0.21 \begin {gather*} \frac {- a + 2 b x}{2 a^{2} x^{2}} + \frac {b^{2} \left (\log {\left (x \right )} - \log {\left (\frac {a}{b} + x \right )}\right )}{a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.55, size = 54, normalized size = 0.37 \begin {gather*} -\frac {1}{2} \, {\left (\frac {2 \, b^{2} \log \left ({\left | b x + a \right |}\right )}{a^{3}} - \frac {2 \, b^{2} \log \left ({\left | x \right |}\right )}{a^{3}} - \frac {2 \, a b x - a^{2}}{a^{3} x^{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}{x^3\,\sqrt {{\left (a+b\,x\right )}^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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