Integrand size = 15, antiderivative size = 85 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^3 x^2} \, dx=\frac {b^2}{a^3 \left (a+b \sqrt {x}\right )^2}+\frac {6 b^2}{a^4 \left (a+b \sqrt {x}\right )}-\frac {1}{a^3 x}+\frac {6 b}{a^4 \sqrt {x}}-\frac {12 b^2 \log \left (a+b \sqrt {x}\right )}{a^5}+\frac {6 b^2 \log (x)}{a^5} \]
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Time = 0.05 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {272, 46} \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^3 x^2} \, dx=-\frac {12 b^2 \log \left (a+b \sqrt {x}\right )}{a^5}+\frac {6 b^2 \log (x)}{a^5}+\frac {6 b^2}{a^4 \left (a+b \sqrt {x}\right )}+\frac {6 b}{a^4 \sqrt {x}}+\frac {b^2}{a^3 \left (a+b \sqrt {x}\right )^2}-\frac {1}{a^3 x} \]
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Rule 46
Rule 272
Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \frac {1}{x^3 (a+b x)^3} \, dx,x,\sqrt {x}\right ) \\ & = 2 \text {Subst}\left (\int \left (\frac {1}{a^3 x^3}-\frac {3 b}{a^4 x^2}+\frac {6 b^2}{a^5 x}-\frac {b^3}{a^3 (a+b x)^3}-\frac {3 b^3}{a^4 (a+b x)^2}-\frac {6 b^3}{a^5 (a+b x)}\right ) \, dx,x,\sqrt {x}\right ) \\ & = \frac {b^2}{a^3 \left (a+b \sqrt {x}\right )^2}+\frac {6 b^2}{a^4 \left (a+b \sqrt {x}\right )}-\frac {1}{a^3 x}+\frac {6 b}{a^4 \sqrt {x}}-\frac {12 b^2 \log \left (a+b \sqrt {x}\right )}{a^5}+\frac {6 b^2 \log (x)}{a^5} \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.91 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^3 x^2} \, dx=\frac {\frac {a \left (-a^3+4 a^2 b \sqrt {x}+18 a b^2 x+12 b^3 x^{3/2}\right )}{\left (a+b \sqrt {x}\right )^2 x}-12 b^2 \log \left (a+b \sqrt {x}\right )+6 b^2 \log (x)}{a^5} \]
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Time = 10.29 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.92
method | result | size |
derivativedivides | \(-\frac {1}{a^{3} x}+\frac {6 b^{2} \ln \left (x \right )}{a^{5}}-\frac {12 b^{2} \ln \left (a +b \sqrt {x}\right )}{a^{5}}+\frac {6 b}{a^{4} \sqrt {x}}+\frac {b^{2}}{a^{3} \left (a +b \sqrt {x}\right )^{2}}+\frac {6 b^{2}}{a^{4} \left (a +b \sqrt {x}\right )}\) | \(78\) |
default | \(-\frac {1}{a^{3} x}+\frac {6 b^{2} \ln \left (x \right )}{a^{5}}-\frac {12 b^{2} \ln \left (a +b \sqrt {x}\right )}{a^{5}}+\frac {6 b}{a^{4} \sqrt {x}}+\frac {b^{2}}{a^{3} \left (a +b \sqrt {x}\right )^{2}}+\frac {6 b^{2}}{a^{4} \left (a +b \sqrt {x}\right )}\) | \(78\) |
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Leaf count of result is larger than twice the leaf count of optimal. 155 vs. \(2 (77) = 154\).
Time = 0.26 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.82 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^3 x^2} \, dx=-\frac {6 \, a^{2} b^{4} x^{2} - 9 \, a^{4} b^{2} x + a^{6} + 12 \, {\left (b^{6} x^{3} - 2 \, a^{2} b^{4} x^{2} + a^{4} b^{2} x\right )} \log \left (b \sqrt {x} + a\right ) - 12 \, {\left (b^{6} x^{3} - 2 \, a^{2} b^{4} x^{2} + a^{4} b^{2} x\right )} \log \left (\sqrt {x}\right ) - 2 \, {\left (6 \, a b^{5} x^{2} - 10 \, a^{3} b^{3} x + 3 \, a^{5} b\right )} \sqrt {x}}{a^{5} b^{4} x^{3} - 2 \, a^{7} b^{2} x^{2} + a^{9} x} \]
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Leaf count of result is larger than twice the leaf count of optimal. 481 vs. \(2 (82) = 164\).
Time = 0.84 (sec) , antiderivative size = 481, normalized size of antiderivative = 5.66 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^3 x^2} \, dx=\begin {cases} \frac {\tilde {\infty }}{x^{\frac {5}{2}}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {1}{a^{3} x} & \text {for}\: b = 0 \\- \frac {2}{5 b^{3} x^{\frac {5}{2}}} & \text {for}\: a = 0 \\- \frac {a^{4} \sqrt {x}}{a^{7} x^{\frac {3}{2}} + 2 a^{6} b x^{2} + a^{5} b^{2} x^{\frac {5}{2}}} + \frac {4 a^{3} b x}{a^{7} x^{\frac {3}{2}} + 2 a^{6} b x^{2} + a^{5} b^{2} x^{\frac {5}{2}}} + \frac {6 a^{2} b^{2} x^{\frac {3}{2}} \log {\left (x \right )}}{a^{7} x^{\frac {3}{2}} + 2 a^{6} b x^{2} + a^{5} b^{2} x^{\frac {5}{2}}} - \frac {12 a^{2} b^{2} x^{\frac {3}{2}} \log {\left (\frac {a}{b} + \sqrt {x} \right )}}{a^{7} x^{\frac {3}{2}} + 2 a^{6} b x^{2} + a^{5} b^{2} x^{\frac {5}{2}}} + \frac {18 a^{2} b^{2} x^{\frac {3}{2}}}{a^{7} x^{\frac {3}{2}} + 2 a^{6} b x^{2} + a^{5} b^{2} x^{\frac {5}{2}}} + \frac {12 a b^{3} x^{2} \log {\left (x \right )}}{a^{7} x^{\frac {3}{2}} + 2 a^{6} b x^{2} + a^{5} b^{2} x^{\frac {5}{2}}} - \frac {24 a b^{3} x^{2} \log {\left (\frac {a}{b} + \sqrt {x} \right )}}{a^{7} x^{\frac {3}{2}} + 2 a^{6} b x^{2} + a^{5} b^{2} x^{\frac {5}{2}}} + \frac {12 a b^{3} x^{2}}{a^{7} x^{\frac {3}{2}} + 2 a^{6} b x^{2} + a^{5} b^{2} x^{\frac {5}{2}}} + \frac {6 b^{4} x^{\frac {5}{2}} \log {\left (x \right )}}{a^{7} x^{\frac {3}{2}} + 2 a^{6} b x^{2} + a^{5} b^{2} x^{\frac {5}{2}}} - \frac {12 b^{4} x^{\frac {5}{2}} \log {\left (\frac {a}{b} + \sqrt {x} \right )}}{a^{7} x^{\frac {3}{2}} + 2 a^{6} b x^{2} + a^{5} b^{2} x^{\frac {5}{2}}} & \text {otherwise} \end {cases} \]
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Time = 0.19 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^3 x^2} \, dx=\frac {12 \, b^{3} x^{\frac {3}{2}} + 18 \, a b^{2} x + 4 \, a^{2} b \sqrt {x} - a^{3}}{a^{4} b^{2} x^{2} + 2 \, a^{5} b x^{\frac {3}{2}} + a^{6} x} - \frac {12 \, b^{2} \log \left (b \sqrt {x} + a\right )}{a^{5}} + \frac {6 \, b^{2} \log \left (x\right )}{a^{5}} \]
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Time = 0.28 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.87 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^3 x^2} \, dx=-\frac {12 \, b^{2} \log \left ({\left | b \sqrt {x} + a \right |}\right )}{a^{5}} + \frac {6 \, b^{2} \log \left ({\left | x \right |}\right )}{a^{5}} + \frac {12 \, b^{3} x^{\frac {3}{2}} + 18 \, a b^{2} x + 4 \, a^{2} b \sqrt {x} - a^{3}}{{\left (b x + a \sqrt {x}\right )}^{2} a^{4}} \]
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Time = 5.85 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.93 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^3 x^2} \, dx=\frac {\frac {4\,b\,\sqrt {x}}{a^2}-\frac {1}{a}+\frac {18\,b^2\,x}{a^3}+\frac {12\,b^3\,x^{3/2}}{a^4}}{a^2\,x+b^2\,x^2+2\,a\,b\,x^{3/2}}-\frac {24\,b^2\,\mathrm {atanh}\left (\frac {2\,b\,\sqrt {x}}{a}+1\right )}{a^5} \]
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