Integrand size = 15, antiderivative size = 172 \[ \int \frac {x^4}{\left (a+b \sqrt {x}\right )^8} \, dx=\frac {2 a^9}{7 b^{10} \left (a+b \sqrt {x}\right )^7}-\frac {3 a^8}{b^{10} \left (a+b \sqrt {x}\right )^6}+\frac {72 a^7}{5 b^{10} \left (a+b \sqrt {x}\right )^5}-\frac {42 a^6}{b^{10} \left (a+b \sqrt {x}\right )^4}+\frac {84 a^5}{b^{10} \left (a+b \sqrt {x}\right )^3}-\frac {126 a^4}{b^{10} \left (a+b \sqrt {x}\right )^2}+\frac {168 a^3}{b^{10} \left (a+b \sqrt {x}\right )}-\frac {16 a \sqrt {x}}{b^9}+\frac {x}{b^8}+\frac {72 a^2 \log \left (a+b \sqrt {x}\right )}{b^{10}} \]
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Time = 0.11 (sec) , antiderivative size = 172, 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, 45} \[ \int \frac {x^4}{\left (a+b \sqrt {x}\right )^8} \, dx=\frac {2 a^9}{7 b^{10} \left (a+b \sqrt {x}\right )^7}-\frac {3 a^8}{b^{10} \left (a+b \sqrt {x}\right )^6}+\frac {72 a^7}{5 b^{10} \left (a+b \sqrt {x}\right )^5}-\frac {42 a^6}{b^{10} \left (a+b \sqrt {x}\right )^4}+\frac {84 a^5}{b^{10} \left (a+b \sqrt {x}\right )^3}-\frac {126 a^4}{b^{10} \left (a+b \sqrt {x}\right )^2}+\frac {168 a^3}{b^{10} \left (a+b \sqrt {x}\right )}+\frac {72 a^2 \log \left (a+b \sqrt {x}\right )}{b^{10}}-\frac {16 a \sqrt {x}}{b^9}+\frac {x}{b^8} \]
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Rule 45
Rule 272
Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \frac {x^9}{(a+b x)^8} \, dx,x,\sqrt {x}\right ) \\ & = 2 \text {Subst}\left (\int \left (-\frac {8 a}{b^9}+\frac {x}{b^8}-\frac {a^9}{b^9 (a+b x)^8}+\frac {9 a^8}{b^9 (a+b x)^7}-\frac {36 a^7}{b^9 (a+b x)^6}+\frac {84 a^6}{b^9 (a+b x)^5}-\frac {126 a^5}{b^9 (a+b x)^4}+\frac {126 a^4}{b^9 (a+b x)^3}-\frac {84 a^3}{b^9 (a+b x)^2}+\frac {36 a^2}{b^9 (a+b x)}\right ) \, dx,x,\sqrt {x}\right ) \\ & = \frac {2 a^9}{7 b^{10} \left (a+b \sqrt {x}\right )^7}-\frac {3 a^8}{b^{10} \left (a+b \sqrt {x}\right )^6}+\frac {72 a^7}{5 b^{10} \left (a+b \sqrt {x}\right )^5}-\frac {42 a^6}{b^{10} \left (a+b \sqrt {x}\right )^4}+\frac {84 a^5}{b^{10} \left (a+b \sqrt {x}\right )^3}-\frac {126 a^4}{b^{10} \left (a+b \sqrt {x}\right )^2}+\frac {168 a^3}{b^{10} \left (a+b \sqrt {x}\right )}-\frac {16 a \sqrt {x}}{b^9}+\frac {x}{b^8}+\frac {72 a^2 \log \left (a+b \sqrt {x}\right )}{b^{10}} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.83 \[ \int \frac {x^4}{\left (a+b \sqrt {x}\right )^8} \, dx=\frac {3349 a^9+20923 a^8 b \sqrt {x}+53949 a^7 b^2 x+72275 a^6 b^3 x^{3/2}+50225 a^5 b^4 x^2+12495 a^4 b^5 x^{5/2}-4655 a^3 b^6 x^3-3185 a^2 b^7 x^{7/2}-315 a b^8 x^4+35 b^9 x^{9/2}}{35 b^{10} \left (a+b \sqrt {x}\right )^7}+\frac {72 a^2 \log \left (a+b \sqrt {x}\right )}{b^{10}} \]
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Time = 3.58 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.89
method | result | size |
derivativedivides | \(-\frac {2 \left (-\frac {b x}{2}+8 a \sqrt {x}\right )}{b^{9}}+\frac {72 a^{7}}{5 b^{10} \left (a +b \sqrt {x}\right )^{5}}+\frac {168 a^{3}}{b^{10} \left (a +b \sqrt {x}\right )}+\frac {84 a^{5}}{b^{10} \left (a +b \sqrt {x}\right )^{3}}-\frac {3 a^{8}}{b^{10} \left (a +b \sqrt {x}\right )^{6}}-\frac {42 a^{6}}{b^{10} \left (a +b \sqrt {x}\right )^{4}}-\frac {126 a^{4}}{b^{10} \left (a +b \sqrt {x}\right )^{2}}+\frac {2 a^{9}}{7 b^{10} \left (a +b \sqrt {x}\right )^{7}}+\frac {72 a^{2} \ln \left (a +b \sqrt {x}\right )}{b^{10}}\) | \(153\) |
default | \(-\frac {2 \left (-\frac {b x}{2}+8 a \sqrt {x}\right )}{b^{9}}+\frac {72 a^{7}}{5 b^{10} \left (a +b \sqrt {x}\right )^{5}}+\frac {168 a^{3}}{b^{10} \left (a +b \sqrt {x}\right )}+\frac {84 a^{5}}{b^{10} \left (a +b \sqrt {x}\right )^{3}}-\frac {3 a^{8}}{b^{10} \left (a +b \sqrt {x}\right )^{6}}-\frac {42 a^{6}}{b^{10} \left (a +b \sqrt {x}\right )^{4}}-\frac {126 a^{4}}{b^{10} \left (a +b \sqrt {x}\right )^{2}}+\frac {2 a^{9}}{7 b^{10} \left (a +b \sqrt {x}\right )^{7}}+\frac {72 a^{2} \ln \left (a +b \sqrt {x}\right )}{b^{10}}\) | \(153\) |
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Leaf count of result is larger than twice the leaf count of optimal. 348 vs. \(2 (150) = 300\).
Time = 0.33 (sec) , antiderivative size = 348, normalized size of antiderivative = 2.02 \[ \int \frac {x^4}{\left (a+b \sqrt {x}\right )^8} \, dx=\frac {35 \, b^{16} x^{8} - 245 \, a^{2} b^{14} x^{7} - 9555 \, a^{4} b^{12} x^{6} + 41405 \, a^{6} b^{10} x^{5} - 83720 \, a^{8} b^{8} x^{4} + 94745 \, a^{10} b^{6} x^{3} - 62139 \, a^{12} b^{4} x^{2} + 22183 \, a^{14} b^{2} x - 3349 \, a^{16} + 2520 \, {\left (a^{2} b^{14} x^{7} - 7 \, a^{4} b^{12} x^{6} + 21 \, a^{6} b^{10} x^{5} - 35 \, a^{8} b^{8} x^{4} + 35 \, a^{10} b^{6} x^{3} - 21 \, a^{12} b^{4} x^{2} + 7 \, a^{14} b^{2} x - a^{16}\right )} \log \left (b \sqrt {x} + a\right ) - 8 \, {\left (70 \, a b^{15} x^{7} - 1225 \, a^{3} b^{13} x^{6} + 4410 \, a^{5} b^{11} x^{5} - 8393 \, a^{7} b^{9} x^{4} + 9216 \, a^{9} b^{7} x^{3} - 5943 \, a^{11} b^{5} x^{2} + 2100 \, a^{13} b^{3} x - 315 \, a^{15} b\right )} \sqrt {x}}{35 \, {\left (b^{24} x^{7} - 7 \, a^{2} b^{22} x^{6} + 21 \, a^{4} b^{20} x^{5} - 35 \, a^{6} b^{18} x^{4} + 35 \, a^{8} b^{16} x^{3} - 21 \, a^{10} b^{14} x^{2} + 7 \, a^{12} b^{12} x - a^{14} b^{10}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1839 vs. \(2 (167) = 334\).
Time = 1.62 (sec) , antiderivative size = 1839, normalized size of antiderivative = 10.69 \[ \int \frac {x^4}{\left (a+b \sqrt {x}\right )^8} \, dx=\text {Too large to display} \]
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Time = 0.20 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.94 \[ \int \frac {x^4}{\left (a+b \sqrt {x}\right )^8} \, dx=\frac {72 \, a^{2} \log \left (b \sqrt {x} + a\right )}{b^{10}} + \frac {{\left (b \sqrt {x} + a\right )}^{2}}{b^{10}} - \frac {18 \, {\left (b \sqrt {x} + a\right )} a}{b^{10}} + \frac {168 \, a^{3}}{{\left (b \sqrt {x} + a\right )} b^{10}} - \frac {126 \, a^{4}}{{\left (b \sqrt {x} + a\right )}^{2} b^{10}} + \frac {84 \, a^{5}}{{\left (b \sqrt {x} + a\right )}^{3} b^{10}} - \frac {42 \, a^{6}}{{\left (b \sqrt {x} + a\right )}^{4} b^{10}} + \frac {72 \, a^{7}}{5 \, {\left (b \sqrt {x} + a\right )}^{5} b^{10}} - \frac {3 \, a^{8}}{{\left (b \sqrt {x} + a\right )}^{6} b^{10}} + \frac {2 \, a^{9}}{7 \, {\left (b \sqrt {x} + a\right )}^{7} b^{10}} \]
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Time = 0.30 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.69 \[ \int \frac {x^4}{\left (a+b \sqrt {x}\right )^8} \, dx=\frac {72 \, a^{2} \log \left ({\left | b \sqrt {x} + a \right |}\right )}{b^{10}} + \frac {b^{8} x - 16 \, a b^{7} \sqrt {x}}{b^{16}} + \frac {5880 \, a^{3} b^{6} x^{3} + 30870 \, a^{4} b^{5} x^{\frac {5}{2}} + 69090 \, a^{5} b^{4} x^{2} + 83790 \, a^{6} b^{3} x^{\frac {3}{2}} + 57834 \, a^{7} b^{2} x + 21483 \, a^{8} b \sqrt {x} + 3349 \, a^{9}}{35 \, {\left (b \sqrt {x} + a\right )}^{7} b^{10}} \]
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Time = 0.10 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.04 \[ \int \frac {x^4}{\left (a+b \sqrt {x}\right )^8} \, dx=\frac {x}{b^8}+\frac {\frac {3349\,a^9}{35\,b}+\frac {3069\,a^8\,\sqrt {x}}{5}+1974\,a^5\,b^3\,x^2+168\,a^3\,b^5\,x^3+2394\,a^6\,b^2\,x^{3/2}+882\,a^4\,b^4\,x^{5/2}+\frac {8262\,a^7\,b\,x}{5}}{a^7\,b^9+b^{16}\,x^{7/2}+21\,a^5\,b^{11}\,x+7\,a\,b^{15}\,x^3+35\,a^3\,b^{13}\,x^2+7\,a^6\,b^{10}\,\sqrt {x}+35\,a^4\,b^{12}\,x^{3/2}+21\,a^2\,b^{14}\,x^{5/2}}-\frac {16\,a\,\sqrt {x}}{b^9}+\frac {72\,a^2\,\ln \left (a+b\,\sqrt {x}\right )}{b^{10}} \]
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