Integrand size = 26, antiderivative size = 171 \[ \int \frac {(2+3 x)^6}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx=\frac {7591 \sqrt {1-2 x} (2+3 x)^3}{39930 (3+5 x)^{3/2}}-\frac {511 (2+3 x)^4}{242 \sqrt {1-2 x} (3+5 x)^{3/2}}+\frac {7 (2+3 x)^5}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac {261331 \sqrt {1-2 x} (2+3 x)^2}{2196150 \sqrt {3+5 x}}-\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} (190406711+78981180 x)}{117128000}+\frac {753543 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{8000 \sqrt {10}} \]
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Time = 0.05 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {100, 155, 152, 56, 222} \[ \int \frac {(2+3 x)^6}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx=\frac {753543 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{8000 \sqrt {10}}+\frac {7 (3 x+2)^5}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}-\frac {511 (3 x+2)^4}{242 \sqrt {1-2 x} (5 x+3)^{3/2}}+\frac {7591 \sqrt {1-2 x} (3 x+2)^3}{39930 (5 x+3)^{3/2}}+\frac {261331 \sqrt {1-2 x} (3 x+2)^2}{2196150 \sqrt {5 x+3}}-\frac {7 \sqrt {1-2 x} \sqrt {5 x+3} (78981180 x+190406711)}{117128000} \]
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Rule 56
Rule 100
Rule 152
Rule 155
Rule 222
Rubi steps \begin{align*} \text {integral}& = \frac {7 (2+3 x)^5}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac {1}{33} \int \frac {(2+3 x)^4 \left (204+\frac {717 x}{2}\right )}{(1-2 x)^{3/2} (3+5 x)^{5/2}} \, dx \\ & = -\frac {511 (2+3 x)^4}{242 \sqrt {1-2 x} (3+5 x)^{3/2}}+\frac {7 (2+3 x)^5}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac {1}{363} \int \frac {\left (-\frac {32415}{2}-\frac {115641 x}{4}\right ) (2+3 x)^3}{\sqrt {1-2 x} (3+5 x)^{5/2}} \, dx \\ & = \frac {7591 \sqrt {1-2 x} (2+3 x)^3}{39930 (3+5 x)^{3/2}}-\frac {511 (2+3 x)^4}{242 \sqrt {1-2 x} (3+5 x)^{3/2}}+\frac {7 (2+3 x)^5}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac {2 \int \frac {\left (-880026-\frac {11995011 x}{8}\right ) (2+3 x)^2}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx}{59895} \\ & = \frac {7591 \sqrt {1-2 x} (2+3 x)^3}{39930 (3+5 x)^{3/2}}-\frac {511 (2+3 x)^4}{242 \sqrt {1-2 x} (3+5 x)^{3/2}}+\frac {7 (2+3 x)^5}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac {261331 \sqrt {1-2 x} (2+3 x)^2}{2196150 \sqrt {3+5 x}}-\frac {4 \int \frac {\left (-\frac {127241163}{8}-\frac {414651195 x}{16}\right ) (2+3 x)}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{3294225} \\ & = \frac {7591 \sqrt {1-2 x} (2+3 x)^3}{39930 (3+5 x)^{3/2}}-\frac {511 (2+3 x)^4}{242 \sqrt {1-2 x} (3+5 x)^{3/2}}+\frac {7 (2+3 x)^5}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac {261331 \sqrt {1-2 x} (2+3 x)^2}{2196150 \sqrt {3+5 x}}-\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} (190406711+78981180 x)}{117128000}+\frac {753543 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{16000} \\ & = \frac {7591 \sqrt {1-2 x} (2+3 x)^3}{39930 (3+5 x)^{3/2}}-\frac {511 (2+3 x)^4}{242 \sqrt {1-2 x} (3+5 x)^{3/2}}+\frac {7 (2+3 x)^5}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac {261331 \sqrt {1-2 x} (2+3 x)^2}{2196150 \sqrt {3+5 x}}-\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} (190406711+78981180 x)}{117128000}+\frac {753543 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{8000 \sqrt {5}} \\ & = \frac {7591 \sqrt {1-2 x} (2+3 x)^3}{39930 (3+5 x)^{3/2}}-\frac {511 (2+3 x)^4}{242 \sqrt {1-2 x} (3+5 x)^{3/2}}+\frac {7 (2+3 x)^5}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac {261331 \sqrt {1-2 x} (2+3 x)^2}{2196150 \sqrt {3+5 x}}-\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} (190406711+78981180 x)}{117128000}+\frac {753543 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{8000 \sqrt {10}} \\ \end{align*}
Time = 0.25 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.46 \[ \int \frac {(2+3 x)^6}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx=-\frac {44437106459+19932058554 x-274128335769 x^2-252342435560 x^3+97980793020 x^4+12807946800 x^5}{351384000 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac {753543 \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{8000 \sqrt {10}} \]
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Time = 4.17 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.16
method | result | size |
default | \(\frac {\sqrt {1-2 x}\, \left (3309786918900 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{4}-256158936000 x^{5} \sqrt {-10 x^{2}-x +3}+661957383780 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{3}-1959615860400 x^{4} \sqrt {-10 x^{2}-x +3}-1952774282151 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{2}+5046848711200 x^{3} \sqrt {-10 x^{2}-x +3}-198587215134 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x +5482566715380 x^{2} \sqrt {-10 x^{2}-x +3}+297880822701 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-398641171080 x \sqrt {-10 x^{2}-x +3}-888742129180 \sqrt {-10 x^{2}-x +3}\right )}{7027680000 \left (-1+2 x \right )^{2} \sqrt {-10 x^{2}-x +3}\, \left (3+5 x \right )^{\frac {3}{2}}}\) | \(199\) |
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Time = 0.23 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.74 \[ \int \frac {(2+3 x)^6}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx=-\frac {33097869189 \, \sqrt {10} {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 20 \, {\left (12807946800 \, x^{5} + 97980793020 \, x^{4} - 252342435560 \, x^{3} - 274128335769 \, x^{2} + 19932058554 \, x + 44437106459\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{7027680000 \, {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )}} \]
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\[ \int \frac {(2+3 x)^6}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx=\int \frac {\left (3 x + 2\right )^{6}}{\left (1 - 2 x\right )^{\frac {5}{2}} \left (5 x + 3\right )^{\frac {5}{2}}}\, dx \]
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Time = 0.28 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.25 \[ \int \frac {(2+3 x)^6}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx=-\frac {729 \, x^{5}}{20 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} - \frac {111537 \, x^{4}}{400 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} + \frac {251181}{234256000} \, x {\left (\frac {7220 \, x}{\sqrt {-10 \, x^{2} - x + 3}} + \frac {439230 \, x^{2}}{{\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} + \frac {361}{\sqrt {-10 \, x^{2} - x + 3}} + \frac {21901 \, x}{{\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} - \frac {87483}{{\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}\right )} - \frac {753543}{160000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {90676341}{117128000} \, \sqrt {-10 \, x^{2} - x + 3} - \frac {170985889 \, x}{7027680 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {766611 \, x^{2}}{1000 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} + \frac {1005653687}{878460000 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {416356591 \, x}{3630000 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} - \frac {496819753}{3630000 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} \]
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Time = 0.35 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.20 \[ \int \frac {(2+3 x)^6}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx=-\frac {\sqrt {10} {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}}{2196150000 \, {\left (5 \, x + 3\right )}^{\frac {3}{2}}} + \frac {753543}{80000} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) - \frac {37 \, \sqrt {10} {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{16637500 \, \sqrt {5 \, x + 3}} - \frac {{\left (4 \, {\left (32019867 \, {\left (4 \, \sqrt {5} {\left (5 \, x + 3\right )} + 93 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} - 110347010662 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 1820310410259 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{219615000000 \, {\left (2 \, x - 1\right )}^{2}} + \frac {\sqrt {10} {\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (\frac {1221 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} + 4\right )}}{137259375 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}} \]
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Timed out. \[ \int \frac {(2+3 x)^6}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx=\int \frac {{\left (3\,x+2\right )}^6}{{\left (1-2\,x\right )}^{5/2}\,{\left (5\,x+3\right )}^{5/2}} \,d x \]
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