Optimal. Leaf size=184 \[ \frac {-a-b x}{3 a x^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {b (a+b x)}{2 a^2 x^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {b^2 (a+b x)}{a^3 x \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {b^3 (a+b x) \log (x)}{a^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {b^3 (a+b x) \log (a+b x)}{a^4 \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rubi [A]
time = 0.04, antiderivative size = 181, 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)}{2 a^2 x^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {a+b x}{3 a x^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {b^3 \log (x) (a+b x)}{a^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {b^3 (a+b x) \log (a+b x)}{a^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {b^2 (a+b x)}{a^3 x \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^4 \sqrt {a^2+2 a b x+b^2 x^2}} \, dx &=\frac {\left (a b+b^2 x\right ) \int \frac {1}{x^4 \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^4}-\frac {1}{a^2 x^3}+\frac {b}{a^3 x^2}-\frac {b^2}{a^4 x}+\frac {b^3}{a^4 (a+b x)}\right ) \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {a+b x}{3 a x^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {b (a+b x)}{2 a^2 x^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {b^2 (a+b x)}{a^3 x \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {b^3 (a+b x) \log (x)}{a^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {b^3 (a+b x) \log (a+b x)}{a^4 \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 72, normalized size = 0.39 \begin {gather*} -\frac {(a+b x) \left (a \left (2 a^2-3 a b x+6 b^2 x^2\right )+6 b^3 x^3 \log (x)-6 b^3 x^3 \log (a+b x)\right )}{6 a^4 x^3 \sqrt {(a+b x)^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.41, size = 69, normalized size = 0.38
method | result | size |
default | \(-\frac {\left (b x +a \right ) \left (6 b^{3} \ln \left (x \right ) x^{3}-6 b^{3} \ln \left (b x +a \right ) x^{3}+6 a \,b^{2} x^{2}-3 a^{2} b x +2 a^{3}\right )}{6 \sqrt {\left (b x +a \right )^{2}}\, a^{4} x^{3}}\) | \(69\) |
risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (-\frac {1}{3 a}+\frac {b x}{2 a^{2}}-\frac {b^{2} x^{2}}{a^{3}}\right )}{\left (b x +a \right ) x^{3}}-\frac {\sqrt {\left (b x +a \right )^{2}}\, b^{3} \ln \left (x \right )}{\left (b x +a \right ) a^{4}}+\frac {\sqrt {\left (b x +a \right )^{2}}\, b^{3} \ln \left (-b x -a \right )}{\left (b x +a \right ) a^{4}}\) | \(104\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 123, normalized size = 0.67 \begin {gather*} \frac {\left (-1\right )^{2 \, a b x + 2 \, a^{2}} b^{3} \log \left (\frac {2 \, a b x}{{\left | x \right |}} + \frac {2 \, a^{2}}{{\left | x \right |}}\right )}{a^{4}} - \frac {11 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b^{2}}{6 \, a^{4} x} + \frac {5 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b}{6 \, a^{3} x^{2}} - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}}}{3 \, a^{2} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.26, size = 54, normalized size = 0.29 \begin {gather*} \frac {6 \, b^{3} x^{3} \log \left (b x + a\right ) - 6 \, b^{3} x^{3} \log \left (x\right ) - 6 \, a b^{2} x^{2} + 3 \, a^{2} b x - 2 \, a^{3}}{6 \, a^{4} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.09, size = 44, normalized size = 0.24 \begin {gather*} \frac {- 2 a^{2} + 3 a b x - 6 b^{2} x^{2}}{6 a^{3} x^{3}} + \frac {b^{3} \left (- \log {\left (x \right )} + \log {\left (\frac {a}{b} + x \right )}\right )}{a^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.22, size = 65, normalized size = 0.35 \begin {gather*} \frac {1}{6} \, {\left (\frac {6 \, b^{3} \log \left ({\left | b x + a \right |}\right )}{a^{4}} - \frac {6 \, b^{3} \log \left ({\left | x \right |}\right )}{a^{4}} - \frac {6 \, a b^{2} x^{2} - 3 \, a^{2} b x + 2 \, a^{3}}{a^{4} x^{3}}\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^4\,\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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