Integrand size = 15, antiderivative size = 111 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^3 x^3} \, dx=\frac {b^4}{a^5 \left (a+b \sqrt {x}\right )^2}+\frac {10 b^4}{a^6 \left (a+b \sqrt {x}\right )}-\frac {1}{2 a^3 x^2}+\frac {2 b}{a^4 x^{3/2}}-\frac {6 b^2}{a^5 x}+\frac {20 b^3}{a^6 \sqrt {x}}-\frac {30 b^4 \log \left (a+b \sqrt {x}\right )}{a^7}+\frac {15 b^4 \log (x)}{a^7} \]
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Time = 0.05 (sec) , antiderivative size = 111, 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} \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^3 x^3} \, dx=-\frac {30 b^4 \log \left (a+b \sqrt {x}\right )}{a^7}+\frac {15 b^4 \log (x)}{a^7}+\frac {10 b^4}{a^6 \left (a+b \sqrt {x}\right )}+\frac {20 b^3}{a^6 \sqrt {x}}+\frac {b^4}{a^5 \left (a+b \sqrt {x}\right )^2}-\frac {6 b^2}{a^5 x}+\frac {2 b}{a^4 x^{3/2}}-\frac {1}{2 a^3 x^2} \]
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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)^3} \, dx,x,\sqrt {x}\right ) \\ & = 2 \text {Subst}\left (\int \left (\frac {1}{a^3 x^5}-\frac {3 b}{a^4 x^4}+\frac {6 b^2}{a^5 x^3}-\frac {10 b^3}{a^6 x^2}+\frac {15 b^4}{a^7 x}-\frac {b^5}{a^5 (a+b x)^3}-\frac {5 b^5}{a^6 (a+b x)^2}-\frac {15 b^5}{a^7 (a+b x)}\right ) \, dx,x,\sqrt {x}\right ) \\ & = \frac {b^4}{a^5 \left (a+b \sqrt {x}\right )^2}+\frac {10 b^4}{a^6 \left (a+b \sqrt {x}\right )}-\frac {1}{2 a^3 x^2}+\frac {2 b}{a^4 x^{3/2}}-\frac {6 b^2}{a^5 x}+\frac {20 b^3}{a^6 \sqrt {x}}-\frac {30 b^4 \log \left (a+b \sqrt {x}\right )}{a^7}+\frac {15 b^4 \log (x)}{a^7} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.94 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^3 x^3} \, dx=\frac {\frac {a \left (-a^5+2 a^4 b \sqrt {x}-5 a^3 b^2 x+20 a^2 b^3 x^{3/2}+90 a b^4 x^2+60 b^5 x^{5/2}\right )}{\left (a+b \sqrt {x}\right )^2 x^2}-60 b^4 \log \left (a+b \sqrt {x}\right )+30 b^4 \log (x)}{2 a^7} \]
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Time = 5.84 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.90
method | result | size |
derivativedivides | \(-\frac {1}{2 a^{3} x^{2}}+\frac {2 b}{a^{4} x^{\frac {3}{2}}}-\frac {6 b^{2}}{a^{5} x}+\frac {15 b^{4} \ln \left (x \right )}{a^{7}}-\frac {30 b^{4} \ln \left (a +b \sqrt {x}\right )}{a^{7}}+\frac {20 b^{3}}{a^{6} \sqrt {x}}+\frac {b^{4}}{a^{5} \left (a +b \sqrt {x}\right )^{2}}+\frac {10 b^{4}}{a^{6} \left (a +b \sqrt {x}\right )}\) | \(100\) |
default | \(-\frac {1}{2 a^{3} x^{2}}+\frac {2 b}{a^{4} x^{\frac {3}{2}}}-\frac {6 b^{2}}{a^{5} x}+\frac {15 b^{4} \ln \left (x \right )}{a^{7}}-\frac {30 b^{4} \ln \left (a +b \sqrt {x}\right )}{a^{7}}+\frac {20 b^{3}}{a^{6} \sqrt {x}}+\frac {b^{4}}{a^{5} \left (a +b \sqrt {x}\right )^{2}}+\frac {10 b^{4}}{a^{6} \left (a +b \sqrt {x}\right )}\) | \(100\) |
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Time = 0.26 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.64 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^3 x^3} \, dx=-\frac {30 \, a^{2} b^{6} x^{3} - 45 \, a^{4} b^{4} x^{2} + 10 \, a^{6} b^{2} x + a^{8} + 60 \, {\left (b^{8} x^{4} - 2 \, a^{2} b^{6} x^{3} + a^{4} b^{4} x^{2}\right )} \log \left (b \sqrt {x} + a\right ) - 60 \, {\left (b^{8} x^{4} - 2 \, a^{2} b^{6} x^{3} + a^{4} b^{4} x^{2}\right )} \log \left (\sqrt {x}\right ) - 4 \, {\left (15 \, a b^{7} x^{3} - 25 \, a^{3} b^{5} x^{2} + 8 \, a^{5} b^{3} x + a^{7} b\right )} \sqrt {x}}{2 \, {\left (a^{7} b^{4} x^{4} - 2 \, a^{9} b^{2} x^{3} + a^{11} x^{2}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 612 vs. \(2 (109) = 218\).
Time = 1.37 (sec) , antiderivative size = 612, normalized size of antiderivative = 5.51 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^3 x^3} \, dx=\begin {cases} \frac {\tilde {\infty }}{x^{\frac {7}{2}}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {1}{2 a^{3} x^{2}} & \text {for}\: b = 0 \\- \frac {2}{7 b^{3} x^{\frac {7}{2}}} & \text {for}\: a = 0 \\- \frac {a^{6} \sqrt {x}}{2 a^{9} x^{\frac {5}{2}} + 4 a^{8} b x^{3} + 2 a^{7} b^{2} x^{\frac {7}{2}}} + \frac {2 a^{5} b x}{2 a^{9} x^{\frac {5}{2}} + 4 a^{8} b x^{3} + 2 a^{7} b^{2} x^{\frac {7}{2}}} - \frac {5 a^{4} b^{2} x^{\frac {3}{2}}}{2 a^{9} x^{\frac {5}{2}} + 4 a^{8} b x^{3} + 2 a^{7} b^{2} x^{\frac {7}{2}}} + \frac {20 a^{3} b^{3} x^{2}}{2 a^{9} x^{\frac {5}{2}} + 4 a^{8} b x^{3} + 2 a^{7} b^{2} x^{\frac {7}{2}}} + \frac {30 a^{2} b^{4} x^{\frac {5}{2}} \log {\left (x \right )}}{2 a^{9} x^{\frac {5}{2}} + 4 a^{8} b x^{3} + 2 a^{7} b^{2} x^{\frac {7}{2}}} - \frac {60 a^{2} b^{4} x^{\frac {5}{2}} \log {\left (\frac {a}{b} + \sqrt {x} \right )}}{2 a^{9} x^{\frac {5}{2}} + 4 a^{8} b x^{3} + 2 a^{7} b^{2} x^{\frac {7}{2}}} + \frac {90 a^{2} b^{4} x^{\frac {5}{2}}}{2 a^{9} x^{\frac {5}{2}} + 4 a^{8} b x^{3} + 2 a^{7} b^{2} x^{\frac {7}{2}}} + \frac {60 a b^{5} x^{3} \log {\left (x \right )}}{2 a^{9} x^{\frac {5}{2}} + 4 a^{8} b x^{3} + 2 a^{7} b^{2} x^{\frac {7}{2}}} - \frac {120 a b^{5} x^{3} \log {\left (\frac {a}{b} + \sqrt {x} \right )}}{2 a^{9} x^{\frac {5}{2}} + 4 a^{8} b x^{3} + 2 a^{7} b^{2} x^{\frac {7}{2}}} + \frac {60 a b^{5} x^{3}}{2 a^{9} x^{\frac {5}{2}} + 4 a^{8} b x^{3} + 2 a^{7} b^{2} x^{\frac {7}{2}}} + \frac {30 b^{6} x^{\frac {7}{2}} \log {\left (x \right )}}{2 a^{9} x^{\frac {5}{2}} + 4 a^{8} b x^{3} + 2 a^{7} b^{2} x^{\frac {7}{2}}} - \frac {60 b^{6} x^{\frac {7}{2}} \log {\left (\frac {a}{b} + \sqrt {x} \right )}}{2 a^{9} x^{\frac {5}{2}} + 4 a^{8} b x^{3} + 2 a^{7} b^{2} x^{\frac {7}{2}}} & \text {otherwise} \end {cases} \]
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Time = 0.19 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.99 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^3 x^3} \, dx=\frac {60 \, b^{5} x^{\frac {5}{2}} + 90 \, a b^{4} x^{2} + 20 \, a^{2} b^{3} x^{\frac {3}{2}} - 5 \, a^{3} b^{2} x + 2 \, a^{4} b \sqrt {x} - a^{5}}{2 \, {\left (a^{6} b^{2} x^{3} + 2 \, a^{7} b x^{\frac {5}{2}} + a^{8} x^{2}\right )}} - \frac {30 \, b^{4} \log \left (b \sqrt {x} + a\right )}{a^{7}} + \frac {15 \, b^{4} \log \left (x\right )}{a^{7}} \]
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Time = 0.28 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.91 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^3 x^3} \, dx=-\frac {30 \, b^{4} \log \left ({\left | b \sqrt {x} + a \right |}\right )}{a^{7}} + \frac {15 \, b^{4} \log \left ({\left | x \right |}\right )}{a^{7}} + \frac {60 \, a b^{5} x^{\frac {5}{2}} + 90 \, a^{2} b^{4} x^{2} + 20 \, a^{3} b^{3} x^{\frac {3}{2}} - 5 \, a^{4} b^{2} x + 2 \, a^{5} b \sqrt {x} - a^{6}}{2 \, {\left (b \sqrt {x} + a\right )}^{2} a^{7} x^{2}} \]
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Time = 0.11 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.92 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^3 x^3} \, dx=\frac {\frac {b\,\sqrt {x}}{a^2}-\frac {1}{2\,a}-\frac {5\,b^2\,x}{2\,a^3}+\frac {45\,b^4\,x^2}{a^5}+\frac {10\,b^3\,x^{3/2}}{a^4}+\frac {30\,b^5\,x^{5/2}}{a^6}}{a^2\,x^2+b^2\,x^3+2\,a\,b\,x^{5/2}}-\frac {60\,b^4\,\mathrm {atanh}\left (\frac {2\,b\,\sqrt {x}}{a}+1\right )}{a^7} \]
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