Integrand size = 15, antiderivative size = 95 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^2 x^3} \, dx=\frac {2 b^4}{a^5 \left (a+b \sqrt {x}\right )}-\frac {1}{2 a^2 x^2}+\frac {4 b}{3 a^3 x^{3/2}}-\frac {3 b^2}{a^4 x}+\frac {8 b^3}{a^5 \sqrt {x}}-\frac {10 b^4 \log \left (a+b \sqrt {x}\right )}{a^6}+\frac {5 b^4 \log (x)}{a^6} \]
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Time = 0.04 (sec) , antiderivative size = 95, 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} \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^2 x^3} \, dx=-\frac {10 b^4 \log \left (a+b \sqrt {x}\right )}{a^6}+\frac {5 b^4 \log (x)}{a^6}+\frac {2 b^4}{a^5 \left (a+b \sqrt {x}\right )}+\frac {8 b^3}{a^5 \sqrt {x}}-\frac {3 b^2}{a^4 x}+\frac {4 b}{3 a^3 x^{3/2}}-\frac {1}{2 a^2 x^2} \]
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Rule 46
Rule 272
Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \frac {1}{x^5 (a+b x)^2} \, dx,x,\sqrt {x}\right ) \\ & = 2 \text {Subst}\left (\int \left (\frac {1}{a^2 x^5}-\frac {2 b}{a^3 x^4}+\frac {3 b^2}{a^4 x^3}-\frac {4 b^3}{a^5 x^2}+\frac {5 b^4}{a^6 x}-\frac {b^5}{a^5 (a+b x)^2}-\frac {5 b^5}{a^6 (a+b x)}\right ) \, dx,x,\sqrt {x}\right ) \\ & = \frac {2 b^4}{a^5 \left (a+b \sqrt {x}\right )}-\frac {1}{2 a^2 x^2}+\frac {4 b}{3 a^3 x^{3/2}}-\frac {3 b^2}{a^4 x}+\frac {8 b^3}{a^5 \sqrt {x}}-\frac {10 b^4 \log \left (a+b \sqrt {x}\right )}{a^6}+\frac {5 b^4 \log (x)}{a^6} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.96 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^2 x^3} \, dx=\frac {\frac {a \left (-3 a^4+5 a^3 b \sqrt {x}-10 a^2 b^2 x+30 a b^3 x^{3/2}+60 b^4 x^2\right )}{\left (a+b \sqrt {x}\right ) x^2}-60 b^4 \log \left (a+b \sqrt {x}\right )+30 b^4 \log (x)}{6 a^6} \]
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Time = 6.03 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.88
method | result | size |
derivativedivides | \(-\frac {1}{2 a^{2} x^{2}}+\frac {4 b}{3 a^{3} x^{\frac {3}{2}}}-\frac {3 b^{2}}{a^{4} x}+\frac {5 b^{4} \ln \left (x \right )}{a^{6}}-\frac {10 b^{4} \ln \left (a +b \sqrt {x}\right )}{a^{6}}+\frac {8 b^{3}}{a^{5} \sqrt {x}}+\frac {2 b^{4}}{a^{5} \left (a +b \sqrt {x}\right )}\) | \(84\) |
default | \(-\frac {1}{2 a^{2} x^{2}}+\frac {4 b}{3 a^{3} x^{\frac {3}{2}}}-\frac {3 b^{2}}{a^{4} x}+\frac {5 b^{4} \ln \left (x \right )}{a^{6}}-\frac {10 b^{4} \ln \left (a +b \sqrt {x}\right )}{a^{6}}+\frac {8 b^{3}}{a^{5} \sqrt {x}}+\frac {2 b^{4}}{a^{5} \left (a +b \sqrt {x}\right )}\) | \(84\) |
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Time = 0.26 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.40 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^2 x^3} \, dx=-\frac {30 \, a^{2} b^{4} x^{2} - 15 \, a^{4} b^{2} x - 3 \, a^{6} + 60 \, {\left (b^{6} x^{3} - a^{2} b^{4} x^{2}\right )} \log \left (b \sqrt {x} + a\right ) - 60 \, {\left (b^{6} x^{3} - a^{2} b^{4} x^{2}\right )} \log \left (\sqrt {x}\right ) - 4 \, {\left (15 \, a b^{5} x^{2} - 10 \, a^{3} b^{3} x - 2 \, a^{5} b\right )} \sqrt {x}}{6 \, {\left (a^{6} b^{2} x^{3} - a^{8} x^{2}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 333 vs. \(2 (94) = 188\).
Time = 0.64 (sec) , antiderivative size = 333, normalized size of antiderivative = 3.51 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^2 x^3} \, dx=\begin {cases} \frac {\tilde {\infty }}{x^{3}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {1}{2 a^{2} x^{2}} & \text {for}\: b = 0 \\- \frac {1}{3 b^{2} x^{3}} & \text {for}\: a = 0 \\- \frac {3 a^{5} \sqrt {x}}{6 a^{7} x^{\frac {5}{2}} + 6 a^{6} b x^{3}} + \frac {5 a^{4} b x}{6 a^{7} x^{\frac {5}{2}} + 6 a^{6} b x^{3}} - \frac {10 a^{3} b^{2} x^{\frac {3}{2}}}{6 a^{7} x^{\frac {5}{2}} + 6 a^{6} b x^{3}} + \frac {30 a^{2} b^{3} x^{2}}{6 a^{7} x^{\frac {5}{2}} + 6 a^{6} b x^{3}} + \frac {30 a b^{4} x^{\frac {5}{2}} \log {\left (x \right )}}{6 a^{7} x^{\frac {5}{2}} + 6 a^{6} b x^{3}} - \frac {60 a b^{4} x^{\frac {5}{2}} \log {\left (\frac {a}{b} + \sqrt {x} \right )}}{6 a^{7} x^{\frac {5}{2}} + 6 a^{6} b x^{3}} + \frac {60 a b^{4} x^{\frac {5}{2}}}{6 a^{7} x^{\frac {5}{2}} + 6 a^{6} b x^{3}} + \frac {30 b^{5} x^{3} \log {\left (x \right )}}{6 a^{7} x^{\frac {5}{2}} + 6 a^{6} b x^{3}} - \frac {60 b^{5} x^{3} \log {\left (\frac {a}{b} + \sqrt {x} \right )}}{6 a^{7} x^{\frac {5}{2}} + 6 a^{6} b x^{3}} & \text {otherwise} \end {cases} \]
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Time = 0.21 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.93 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^2 x^3} \, dx=\frac {60 \, b^{4} x^{2} + 30 \, a b^{3} x^{\frac {3}{2}} - 10 \, a^{2} b^{2} x + 5 \, a^{3} b \sqrt {x} - 3 \, a^{4}}{6 \, {\left (a^{5} b x^{\frac {5}{2}} + a^{6} x^{2}\right )}} - \frac {10 \, b^{4} \log \left (b \sqrt {x} + a\right )}{a^{6}} + \frac {5 \, b^{4} \log \left (x\right )}{a^{6}} \]
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Time = 0.27 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.95 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^2 x^3} \, dx=-\frac {10 \, b^{4} \log \left ({\left | b \sqrt {x} + a \right |}\right )}{a^{6}} + \frac {5 \, b^{4} \log \left ({\left | x \right |}\right )}{a^{6}} + \frac {60 \, a b^{4} x^{2} + 30 \, a^{2} b^{3} x^{\frac {3}{2}} - 10 \, a^{3} b^{2} x + 5 \, a^{4} b \sqrt {x} - 3 \, a^{5}}{6 \, {\left (b \sqrt {x} + a\right )} a^{6} x^{2}} \]
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Time = 5.81 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.85 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^2 x^3} \, dx=\frac {\frac {5\,b\,\sqrt {x}}{6\,a^2}-\frac {1}{2\,a}-\frac {5\,b^2\,x}{3\,a^3}+\frac {10\,b^4\,x^2}{a^5}+\frac {5\,b^3\,x^{3/2}}{a^4}}{a\,x^2+b\,x^{5/2}}-\frac {20\,b^4\,\mathrm {atanh}\left (\frac {2\,b\,\sqrt {x}}{a}+1\right )}{a^6} \]
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