Integrand size = 15, antiderivative size = 203 \[ \int \frac {x^5}{\left (a+b \sqrt {x}\right )^8} \, dx=\frac {2 a^{11}}{7 b^{12} \left (a+b \sqrt {x}\right )^7}-\frac {11 a^{10}}{3 b^{12} \left (a+b \sqrt {x}\right )^6}+\frac {22 a^9}{b^{12} \left (a+b \sqrt {x}\right )^5}-\frac {165 a^8}{2 b^{12} \left (a+b \sqrt {x}\right )^4}+\frac {220 a^7}{b^{12} \left (a+b \sqrt {x}\right )^3}-\frac {462 a^6}{b^{12} \left (a+b \sqrt {x}\right )^2}+\frac {924 a^5}{b^{12} \left (a+b \sqrt {x}\right )}-\frac {240 a^3 \sqrt {x}}{b^{11}}+\frac {36 a^2 x}{b^{10}}-\frac {16 a x^{3/2}}{3 b^9}+\frac {x^2}{2 b^8}+\frac {660 a^4 \log \left (a+b \sqrt {x}\right )}{b^{12}} \]
[Out]
Time = 0.16 (sec) , antiderivative size = 203, 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} \[ \int \frac {x^5}{\left (a+b \sqrt {x}\right )^8} \, dx=\frac {2 a^{11}}{7 b^{12} \left (a+b \sqrt {x}\right )^7}-\frac {11 a^{10}}{3 b^{12} \left (a+b \sqrt {x}\right )^6}+\frac {22 a^9}{b^{12} \left (a+b \sqrt {x}\right )^5}-\frac {165 a^8}{2 b^{12} \left (a+b \sqrt {x}\right )^4}+\frac {220 a^7}{b^{12} \left (a+b \sqrt {x}\right )^3}-\frac {462 a^6}{b^{12} \left (a+b \sqrt {x}\right )^2}+\frac {924 a^5}{b^{12} \left (a+b \sqrt {x}\right )}+\frac {660 a^4 \log \left (a+b \sqrt {x}\right )}{b^{12}}-\frac {240 a^3 \sqrt {x}}{b^{11}}+\frac {36 a^2 x}{b^{10}}-\frac {16 a x^{3/2}}{3 b^9}+\frac {x^2}{2 b^8} \]
[In]
[Out]
Rule 45
Rule 272
Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \frac {x^{11}}{(a+b x)^8} \, dx,x,\sqrt {x}\right ) \\ & = 2 \text {Subst}\left (\int \left (-\frac {120 a^3}{b^{11}}+\frac {36 a^2 x}{b^{10}}-\frac {8 a x^2}{b^9}+\frac {x^3}{b^8}-\frac {a^{11}}{b^{11} (a+b x)^8}+\frac {11 a^{10}}{b^{11} (a+b x)^7}-\frac {55 a^9}{b^{11} (a+b x)^6}+\frac {165 a^8}{b^{11} (a+b x)^5}-\frac {330 a^7}{b^{11} (a+b x)^4}+\frac {462 a^6}{b^{11} (a+b x)^3}-\frac {462 a^5}{b^{11} (a+b x)^2}+\frac {330 a^4}{b^{11} (a+b x)}\right ) \, dx,x,\sqrt {x}\right ) \\ & = \frac {2 a^{11}}{7 b^{12} \left (a+b \sqrt {x}\right )^7}-\frac {11 a^{10}}{3 b^{12} \left (a+b \sqrt {x}\right )^6}+\frac {22 a^9}{b^{12} \left (a+b \sqrt {x}\right )^5}-\frac {165 a^8}{2 b^{12} \left (a+b \sqrt {x}\right )^4}+\frac {220 a^7}{b^{12} \left (a+b \sqrt {x}\right )^3}-\frac {462 a^6}{b^{12} \left (a+b \sqrt {x}\right )^2}+\frac {924 a^5}{b^{12} \left (a+b \sqrt {x}\right )}-\frac {240 a^3 \sqrt {x}}{b^{11}}+\frac {36 a^2 x}{b^{10}}-\frac {16 a x^{3/2}}{3 b^9}+\frac {x^2}{2 b^8}+\frac {660 a^4 \log \left (a+b \sqrt {x}\right )}{b^{12}} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.82 \[ \int \frac {x^5}{\left (a+b \sqrt {x}\right )^8} \, dx=\frac {25961 a^{11}+154007 a^{10} b \sqrt {x}+365001 a^9 b^2 x+414295 a^8 b^3 x^{3/2}+171745 a^7 b^4 x^2-90993 a^6 b^5 x^{5/2}-127351 a^5 b^6 x^3-45913 a^4 b^7 x^{7/2}-3465 a^3 b^8 x^4+385 a^2 b^9 x^{9/2}-77 a b^{10} x^5+21 b^{11} x^{11/2}}{42 b^{12} \left (a+b \sqrt {x}\right )^7}+\frac {660 a^4 \log \left (a+b \sqrt {x}\right )}{b^{12}} \]
[In]
[Out]
Time = 3.66 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.86
method | result | size |
derivativedivides | \(-\frac {2 \left (-\frac {b^{3} x^{2}}{4}+\frac {8 a \,b^{2} x^{\frac {3}{2}}}{3}-18 a^{2} b x +120 a^{3} \sqrt {x}\right )}{b^{11}}+\frac {660 a^{4} \ln \left (a +b \sqrt {x}\right )}{b^{12}}+\frac {22 a^{9}}{b^{12} \left (a +b \sqrt {x}\right )^{5}}-\frac {11 a^{10}}{3 b^{12} \left (a +b \sqrt {x}\right )^{6}}+\frac {220 a^{7}}{b^{12} \left (a +b \sqrt {x}\right )^{3}}-\frac {165 a^{8}}{2 b^{12} \left (a +b \sqrt {x}\right )^{4}}-\frac {462 a^{6}}{b^{12} \left (a +b \sqrt {x}\right )^{2}}+\frac {924 a^{5}}{b^{12} \left (a +b \sqrt {x}\right )}+\frac {2 a^{11}}{7 b^{12} \left (a +b \sqrt {x}\right )^{7}}\) | \(175\) |
default | \(-\frac {2 \left (-\frac {b^{3} x^{2}}{4}+\frac {8 a \,b^{2} x^{\frac {3}{2}}}{3}-18 a^{2} b x +120 a^{3} \sqrt {x}\right )}{b^{11}}+\frac {660 a^{4} \ln \left (a +b \sqrt {x}\right )}{b^{12}}+\frac {22 a^{9}}{b^{12} \left (a +b \sqrt {x}\right )^{5}}-\frac {11 a^{10}}{3 b^{12} \left (a +b \sqrt {x}\right )^{6}}+\frac {220 a^{7}}{b^{12} \left (a +b \sqrt {x}\right )^{3}}-\frac {165 a^{8}}{2 b^{12} \left (a +b \sqrt {x}\right )^{4}}-\frac {462 a^{6}}{b^{12} \left (a +b \sqrt {x}\right )^{2}}+\frac {924 a^{5}}{b^{12} \left (a +b \sqrt {x}\right )}+\frac {2 a^{11}}{7 b^{12} \left (a +b \sqrt {x}\right )^{7}}\) | \(175\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 370 vs. \(2 (173) = 346\).
Time = 0.37 (sec) , antiderivative size = 370, normalized size of antiderivative = 1.82 \[ \int \frac {x^5}{\left (a+b \sqrt {x}\right )^8} \, dx=\frac {21 \, b^{18} x^{9} + 1365 \, a^{2} b^{16} x^{8} - 10143 \, a^{4} b^{14} x^{7} - 27195 \, a^{6} b^{12} x^{6} + 227094 \, a^{8} b^{10} x^{5} - 540190 \, a^{10} b^{8} x^{4} + 661465 \, a^{12} b^{6} x^{3} - 455091 \, a^{14} b^{4} x^{2} + 167867 \, a^{16} b^{2} x - 25961 \, a^{18} + 27720 \, {\left (a^{4} b^{14} x^{7} - 7 \, a^{6} b^{12} x^{6} + 21 \, a^{8} b^{10} x^{5} - 35 \, a^{10} b^{8} x^{4} + 35 \, a^{12} b^{6} x^{3} - 21 \, a^{14} b^{4} x^{2} + 7 \, a^{16} b^{2} x - a^{18}\right )} \log \left (b \sqrt {x} + a\right ) - 8 \, {\left (28 \, a b^{17} x^{8} + 1064 \, a^{3} b^{15} x^{7} - 13083 \, a^{5} b^{13} x^{6} + 48580 \, a^{7} b^{11} x^{5} - 92323 \, a^{9} b^{9} x^{4} + 101376 \, a^{11} b^{7} x^{3} - 65373 \, a^{13} b^{5} x^{2} + 23100 \, a^{15} b^{3} x - 3465 \, a^{17} b\right )} \sqrt {x}}{42 \, {\left (b^{26} x^{7} - 7 \, a^{2} b^{24} x^{6} + 21 \, a^{4} b^{22} x^{5} - 35 \, a^{6} b^{20} x^{4} + 35 \, a^{8} b^{18} x^{3} - 21 \, a^{10} b^{16} x^{2} + 7 \, a^{12} b^{14} x - a^{14} b^{12}\right )}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 2048 vs. \(2 (197) = 394\).
Time = 2.39 (sec) , antiderivative size = 2048, normalized size of antiderivative = 10.09 \[ \int \frac {x^5}{\left (a+b \sqrt {x}\right )^8} \, dx=\text {Too large to display} \]
[In]
[Out]
none
Time = 0.19 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.97 \[ \int \frac {x^5}{\left (a+b \sqrt {x}\right )^8} \, dx=\frac {660 \, a^{4} \log \left (b \sqrt {x} + a\right )}{b^{12}} + \frac {{\left (b \sqrt {x} + a\right )}^{4}}{2 \, b^{12}} - \frac {22 \, {\left (b \sqrt {x} + a\right )}^{3} a}{3 \, b^{12}} + \frac {55 \, {\left (b \sqrt {x} + a\right )}^{2} a^{2}}{b^{12}} - \frac {330 \, {\left (b \sqrt {x} + a\right )} a^{3}}{b^{12}} + \frac {924 \, a^{5}}{{\left (b \sqrt {x} + a\right )} b^{12}} - \frac {462 \, a^{6}}{{\left (b \sqrt {x} + a\right )}^{2} b^{12}} + \frac {220 \, a^{7}}{{\left (b \sqrt {x} + a\right )}^{3} b^{12}} - \frac {165 \, a^{8}}{2 \, {\left (b \sqrt {x} + a\right )}^{4} b^{12}} + \frac {22 \, a^{9}}{{\left (b \sqrt {x} + a\right )}^{5} b^{12}} - \frac {11 \, a^{10}}{3 \, {\left (b \sqrt {x} + a\right )}^{6} b^{12}} + \frac {2 \, a^{11}}{7 \, {\left (b \sqrt {x} + a\right )}^{7} b^{12}} \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.70 \[ \int \frac {x^5}{\left (a+b \sqrt {x}\right )^8} \, dx=\frac {660 \, a^{4} \log \left ({\left | b \sqrt {x} + a \right |}\right )}{b^{12}} + \frac {38808 \, a^{5} b^{6} x^{3} + 213444 \, a^{6} b^{5} x^{\frac {5}{2}} + 494340 \, a^{7} b^{4} x^{2} + 615615 \, a^{8} b^{3} x^{\frac {3}{2}} + 434049 \, a^{9} b^{2} x + 164087 \, a^{10} b \sqrt {x} + 25961 \, a^{11}}{42 \, {\left (b \sqrt {x} + a\right )}^{7} b^{12}} + \frac {3 \, b^{24} x^{2} - 32 \, a b^{23} x^{\frac {3}{2}} + 216 \, a^{2} b^{22} x - 1440 \, a^{3} b^{21} \sqrt {x}}{6 \, b^{32}} \]
[In]
[Out]
Time = 5.82 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.00 \[ \int \frac {x^5}{\left (a+b \sqrt {x}\right )^8} \, dx=\frac {\frac {25961\,a^{11}}{42\,b}+\frac {23441\,a^{10}\,\sqrt {x}}{6}+11770\,a^7\,b^3\,x^2+924\,a^5\,b^5\,x^3+\frac {29315\,a^8\,b^2\,x^{3/2}}{2}+5082\,a^6\,b^4\,x^{5/2}+\frac {20669\,a^9\,b\,x}{2}}{a^7\,b^{11}+b^{18}\,x^{7/2}+21\,a^5\,b^{13}\,x+7\,a\,b^{17}\,x^3+35\,a^3\,b^{15}\,x^2+7\,a^6\,b^{12}\,\sqrt {x}+35\,a^4\,b^{14}\,x^{3/2}+21\,a^2\,b^{16}\,x^{5/2}}+\frac {x^2}{2\,b^8}+\frac {36\,a^2\,x}{b^{10}}-\frac {16\,a\,x^{3/2}}{3\,b^9}+\frac {660\,a^4\,\ln \left (a+b\,\sqrt {x}\right )}{b^{12}}-\frac {240\,a^3\,\sqrt {x}}{b^{11}} \]
[In]
[Out]