Integrand size = 26, antiderivative size = 113 \[ \int \frac {(2+3 x)^4}{\sqrt {1-2 x} (3+5 x)^{5/2}} \, dx=-\frac {2 \sqrt {1-2 x} (2+3 x)^3}{165 (3+5 x)^{3/2}}-\frac {602 \sqrt {1-2 x} (2+3 x)^2}{9075 \sqrt {3+5 x}}-\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} (12199+1020 x)}{242000}+\frac {8127 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{2000 \sqrt {10}} \]
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Time = 0.03 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {100, 155, 152, 56, 222} \[ \int \frac {(2+3 x)^4}{\sqrt {1-2 x} (3+5 x)^{5/2}} \, dx=\frac {8127 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{2000 \sqrt {10}}-\frac {2 \sqrt {1-2 x} (3 x+2)^3}{165 (5 x+3)^{3/2}}-\frac {602 \sqrt {1-2 x} (3 x+2)^2}{9075 \sqrt {5 x+3}}-\frac {7 \sqrt {1-2 x} \sqrt {5 x+3} (1020 x+12199)}{242000} \]
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Rule 56
Rule 100
Rule 152
Rule 155
Rule 222
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \sqrt {1-2 x} (2+3 x)^3}{165 (3+5 x)^{3/2}}-\frac {2}{165} \int \frac {\left (-112-\frac {273 x}{2}\right ) (2+3 x)^2}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx \\ & = -\frac {2 \sqrt {1-2 x} (2+3 x)^3}{165 (3+5 x)^{3/2}}-\frac {602 \sqrt {1-2 x} (2+3 x)^2}{9075 \sqrt {3+5 x}}-\frac {4 \int \frac {\left (-\frac {4809}{2}-\frac {1785 x}{4}\right ) (2+3 x)}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{9075} \\ & = -\frac {2 \sqrt {1-2 x} (2+3 x)^3}{165 (3+5 x)^{3/2}}-\frac {602 \sqrt {1-2 x} (2+3 x)^2}{9075 \sqrt {3+5 x}}-\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} (12199+1020 x)}{242000}+\frac {8127 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{4000} \\ & = -\frac {2 \sqrt {1-2 x} (2+3 x)^3}{165 (3+5 x)^{3/2}}-\frac {602 \sqrt {1-2 x} (2+3 x)^2}{9075 \sqrt {3+5 x}}-\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} (12199+1020 x)}{242000}+\frac {8127 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{2000 \sqrt {5}} \\ & = -\frac {2 \sqrt {1-2 x} (2+3 x)^3}{165 (3+5 x)^{3/2}}-\frac {602 \sqrt {1-2 x} (2+3 x)^2}{9075 \sqrt {3+5 x}}-\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} (12199+1020 x)}{242000}+\frac {8127 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{2000 \sqrt {10}} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.61 \[ \int \frac {(2+3 x)^4}{\sqrt {1-2 x} (3+5 x)^{5/2}} \, dx=-\frac {\sqrt {1-2 x} \left (2953931+10891910 x+11712195 x^2+2940300 x^3\right )}{726000 (3+5 x)^{3/2}}-\frac {8127 \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{2000 \sqrt {10}} \]
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Time = 5.27 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.15
method | result | size |
default | \(\frac {\left (73752525 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{2}-58806000 x^{3} \sqrt {-10 x^{2}-x +3}+88503030 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x -234243900 x^{2} \sqrt {-10 x^{2}-x +3}+26550909 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-217838200 x \sqrt {-10 x^{2}-x +3}-59078620 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}}{14520000 \sqrt {-10 x^{2}-x +3}\, \left (3+5 x \right )^{\frac {3}{2}}}\) | \(130\) |
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Time = 0.23 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.85 \[ \int \frac {(2+3 x)^4}{\sqrt {1-2 x} (3+5 x)^{5/2}} \, dx=-\frac {2950101 \, \sqrt {10} {\left (25 \, x^{2} + 30 \, x + 9\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 20 \, {\left (2940300 \, x^{3} + 11712195 \, x^{2} + 10891910 \, x + 2953931\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14520000 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \]
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\[ \int \frac {(2+3 x)^4}{\sqrt {1-2 x} (3+5 x)^{5/2}} \, dx=\int \frac {\left (3 x + 2\right )^{4}}{\sqrt {1 - 2 x} \left (5 x + 3\right )^{\frac {5}{2}}}\, dx \]
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Time = 0.28 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.81 \[ \int \frac {(2+3 x)^4}{\sqrt {1-2 x} (3+5 x)^{5/2}} \, dx=\frac {8127}{40000} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) - \frac {81}{500} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {4509}{10000} \, \sqrt {-10 \, x^{2} - x + 3} - \frac {2 \, \sqrt {-10 \, x^{2} - x + 3}}{20625 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} - \frac {32 \, \sqrt {-10 \, x^{2} - x + 3}}{9075 \, {\left (5 \, x + 3\right )}} \]
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Time = 0.37 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.52 \[ \int \frac {(2+3 x)^4}{\sqrt {1-2 x} (3+5 x)^{5/2}} \, dx=-\frac {27}{50000} \, {\left (12 \, \sqrt {5} {\left (5 \, x + 3\right )} + 131 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - \frac {\sqrt {10} {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}}{18150000 \, {\left (5 \, x + 3\right )}^{\frac {3}{2}}} + \frac {8127}{20000} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) - \frac {267 \, \sqrt {10} {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{1512500 \, \sqrt {5 \, x + 3}} + \frac {\sqrt {10} {\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (\frac {801 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} + 4\right )}}{1134375 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}} \]
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Timed out. \[ \int \frac {(2+3 x)^4}{\sqrt {1-2 x} (3+5 x)^{5/2}} \, dx=\int \frac {{\left (3\,x+2\right )}^4}{\sqrt {1-2\,x}\,{\left (5\,x+3\right )}^{5/2}} \,d x \]
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