Optimal. Leaf size=165 \[ -\frac {b}{2 a^2 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 b \log (x) (a+b x)}{a^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3 b (a+b x) \log (a+b x)}{a^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 b}{a^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {a+b x}{a^3 x \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rubi [A] time = 0.07, antiderivative size = 165, normalized size of antiderivative = 1.00, 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 = {646, 44} \[ -\frac {b}{2 a^2 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 b}{a^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {a+b x}{a^3 x \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 b \log (x) (a+b x)}{a^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3 b (a+b x) \log (a+b x)}{a^4 \sqrt {a^2+2 a b x+b^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 44
Rule 646
Rubi steps
\begin {align*} \int \frac {1}{x^2 \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx &=\frac {\left (b^2 \left (a b+b^2 x\right )\right ) \int \frac {1}{x^2 \left (a b+b^2 x\right )^3} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {\left (b^2 \left (a b+b^2 x\right )\right ) \int \left (\frac {1}{a^3 b^3 x^2}-\frac {3}{a^4 b^2 x}+\frac {1}{a^2 b (a+b x)^3}+\frac {2}{a^3 b (a+b x)^2}+\frac {3}{a^4 b (a+b x)}\right ) \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {2 b}{a^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {b}{2 a^2 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {a+b x}{a^3 x \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 b (a+b x) \log (x)}{a^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3 b (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 = 0.03, size = 81, normalized size = 0.49 \[ \frac {-a \left (2 a^2+9 a b x+6 b^2 x^2\right )-6 b x \log (x) (a+b x)^2+6 b x (a+b x)^2 \log (a+b x)}{2 a^4 x (a+b x) \sqrt {(a+b x)^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.89, size = 109, normalized size = 0.66 \[ -\frac {6 \, a b^{2} x^{2} + 9 \, a^{2} b x + 2 \, a^{3} - 6 \, {\left (b^{3} x^{3} + 2 \, a b^{2} x^{2} + a^{2} b x\right )} \log \left (b x + a\right ) + 6 \, {\left (b^{3} x^{3} + 2 \, a b^{2} x^{2} + a^{2} b x\right )} \log \relax (x)}{2 \, {\left (a^{4} b^{2} x^{3} + 2 \, a^{5} b x^{2} + a^{6} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 117, normalized size = 0.71 \[ \frac {\left (-6 b^{3} x^{3} \ln \relax (x )+6 b^{3} x^{3} \ln \left (b x +a \right )-12 a \,b^{2} x^{2} \ln \relax (x )+12 a \,b^{2} x^{2} \ln \left (b x +a \right )-6 a^{2} b x \ln \relax (x )+6 a^{2} b x \ln \left (b x +a \right )-6 a \,b^{2} x^{2}-9 a^{2} b x -2 a^{3}\right ) \left (b x +a \right )}{2 \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}} a^{4} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.36, size = 107, normalized size = 0.65 \[ \frac {3 \, \left (-1\right )^{2 \, a b x + 2 \, a^{2}} b \log \left (\frac {2 \, a b x}{{\left | x \right |}} + \frac {2 \, a^{2}}{{\left | x \right |}}\right )}{a^{4}} - \frac {3 \, b}{\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{3}} - \frac {1}{\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{2} x} - \frac {1}{2 \, a^{2} b {\left (x + \frac {a}{b}\right )}^{2}} \]
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 \[ \int \frac {1}{x^2\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{2} \left (\left (a + b x\right )^{2}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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