Integrand size = 35, antiderivative size = 225 \[ \int \frac {\sqrt {2-3 x} \sqrt {1+4 x}}{\sqrt {-5+2 x} (7+5 x)^3} \, dx=\frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{78 (7+5 x)^2}-\frac {361 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{481988 (7+5 x)}+\frac {361 \sqrt {11} \sqrt {-5+2 x} E\left (\arcsin \left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )}{1204970 \sqrt {5-2 x}}-\frac {6101 \sqrt {5-2 x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {1+4 x}\right ),\frac {1}{3}\right )}{231725 \sqrt {66} \sqrt {-5+2 x}}-\frac {6655867 \sqrt {5-2 x} \operatorname {EllipticPi}\left (\frac {55}{124},\arcsin \left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right ),-\frac {1}{2}\right )}{747081400 \sqrt {11} \sqrt {-5+2 x}} \]
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Time = 0.21 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.314, Rules used = {170, 1618, 1621, 174, 552, 551, 164, 115, 114, 122, 120} \[ \int \frac {\sqrt {2-3 x} \sqrt {1+4 x}}{\sqrt {-5+2 x} (7+5 x)^3} \, dx=-\frac {6101 \sqrt {5-2 x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right ),\frac {1}{3}\right )}{231725 \sqrt {66} \sqrt {2 x-5}}+\frac {361 \sqrt {11} \sqrt {2 x-5} E\left (\arcsin \left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )}{1204970 \sqrt {5-2 x}}-\frac {6655867 \sqrt {5-2 x} \operatorname {EllipticPi}\left (\frac {55}{124},\arcsin \left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right ),-\frac {1}{2}\right )}{747081400 \sqrt {11} \sqrt {2 x-5}}-\frac {361 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{481988 (5 x+7)}+\frac {\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{78 (5 x+7)^2} \]
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Rule 114
Rule 115
Rule 120
Rule 122
Rule 164
Rule 170
Rule 174
Rule 551
Rule 552
Rule 1618
Rule 1621
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{78 (7+5 x)^2}-\frac {1}{156} \int \frac {-37+100 x+24 x^2}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^2} \, dx \\ & = \frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{78 (7+5 x)^2}-\frac {361 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{481988 (7+5 x)}-\frac {\int \frac {-272145+485280 x+77976 x^2}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)} \, dx}{8675784} \\ & = \frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{78 (7+5 x)^2}-\frac {361 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{481988 (7+5 x)}-\frac {\int \frac {\frac {1880568}{25}+\frac {77976 x}{5}}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx}{8675784}+\frac {6655867 \int \frac {1}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)} \, dx}{72298200} \\ & = \frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{78 (7+5 x)^2}-\frac {361 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{481988 (7+5 x)}-\frac {1083 \int \frac {\sqrt {-5+2 x}}{\sqrt {2-3 x} \sqrt {1+4 x}} \, dx}{1204970}-\frac {6101 \int \frac {1}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx}{463450}-\frac {6655867 \text {Subst}\left (\int \frac {1}{\left (31-5 x^2\right ) \sqrt {\frac {11}{3}-\frac {4 x^2}{3}} \sqrt {-\frac {11}{3}-\frac {2 x^2}{3}}} \, dx,x,\sqrt {2-3 x}\right )}{36149100} \\ & = \frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{78 (7+5 x)^2}-\frac {361 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{481988 (7+5 x)}-\frac {\left (6101 \sqrt {5-2 x}\right ) \int \frac {1}{\sqrt {2-3 x} \sqrt {\frac {10}{11}-\frac {4 x}{11}} \sqrt {1+4 x}} \, dx}{231725 \sqrt {22} \sqrt {-5+2 x}}-\frac {\left (6655867 \sqrt {5-2 x}\right ) \text {Subst}\left (\int \frac {1}{\left (31-5 x^2\right ) \sqrt {\frac {11}{3}-\frac {4 x^2}{3}} \sqrt {1+\frac {2 x^2}{11}}} \, dx,x,\sqrt {2-3 x}\right )}{12049700 \sqrt {33} \sqrt {-5+2 x}}-\frac {\left (1083 \sqrt {-5+2 x}\right ) \int \frac {\sqrt {\frac {15}{11}-\frac {6 x}{11}}}{\sqrt {2-3 x} \sqrt {\frac {3}{11}+\frac {12 x}{11}}} \, dx}{1204970 \sqrt {5-2 x}} \\ & = \frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{78 (7+5 x)^2}-\frac {361 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{481988 (7+5 x)}+\frac {361 \sqrt {11} \sqrt {-5+2 x} E\left (\sin ^{-1}\left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )}{1204970 \sqrt {5-2 x}}-\frac {6101 \sqrt {5-2 x} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{11}} \sqrt {1+4 x}\right )|\frac {1}{3}\right )}{231725 \sqrt {66} \sqrt {-5+2 x}}-\frac {6655867 \sqrt {5-2 x} \Pi \left (\frac {55}{124};\sin ^{-1}\left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )}{747081400 \sqrt {11} \sqrt {-5+2 x}} \\ \end{align*}
Time = 5.29 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.60 \[ \int \frac {\sqrt {2-3 x} \sqrt {1+4 x}}{\sqrt {-5+2 x} (7+5 x)^3} \, dx=\frac {-\frac {17050 \sqrt {2-3 x} (-5+2 x) \sqrt {1+4 x} (-10957+5415 x)}{(7+5 x)^2}-3 \sqrt {55-22 x} \left (2462020 E\left (\arcsin \left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )-9834812 \operatorname {EllipticF}\left (\arcsin \left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right ),-\frac {1}{2}\right )+6655867 \operatorname {EllipticPi}\left (\frac {55}{124},\arcsin \left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right ),-\frac {1}{2}\right )\right )}{24653686200 \sqrt {-5+2 x}} \]
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Time = 1.64 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.21
method | result | size |
elliptic | \(\frac {\sqrt {-\left (-2+3 x \right ) \left (-5+2 x \right ) \left (1+4 x \right )}\, \left (\frac {\sqrt {-24 x^{3}+70 x^{2}-21 x -10}}{78 \left (7+5 x \right )^{2}}-\frac {361 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}{481988 \left (7+5 x \right )}-\frac {26119 \sqrt {11+44 x}\, \sqrt {22-33 x}\, \sqrt {110-44 x}\, F\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )}{364503425 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}-\frac {1083 \sqrt {11+44 x}\, \sqrt {22-33 x}\, \sqrt {110-44 x}\, \left (-\frac {11 E\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )}{12}+\frac {2 F\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )}{3}\right )}{72900685 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}+\frac {6655867 \sqrt {11+44 x}\, \sqrt {22-33 x}\, \sqrt {110-44 x}\, \Pi \left (\frac {\sqrt {11+44 x}}{11}, -\frac {55}{23}, \sqrt {3}\right )}{50301472650 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}\right )}{\sqrt {2-3 x}\, \sqrt {-5+2 x}\, \sqrt {1+4 x}}\) | \(273\) |
risch | \(\frac {\left (-2+3 x \right ) \sqrt {-5+2 x}\, \sqrt {1+4 x}\, \left (-10957+5415 x \right ) \sqrt {\left (2-3 x \right ) \left (-5+2 x \right ) \left (1+4 x \right )}}{1445964 \left (7+5 x \right )^{2} \sqrt {-\left (-2+3 x \right ) \left (-5+2 x \right ) \left (1+4 x \right )}\, \sqrt {2-3 x}}-\frac {\left (-\frac {26119 \sqrt {22-33 x}\, \sqrt {-66 x +165}\, \sqrt {33+132 x}\, F\left (\frac {2 \sqrt {22-33 x}}{11}, \frac {i \sqrt {2}}{2}\right )}{1093510275 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}-\frac {361 \sqrt {22-33 x}\, \sqrt {-66 x +165}\, \sqrt {33+132 x}\, \left (-\frac {11 E\left (\frac {2 \sqrt {22-33 x}}{11}, \frac {i \sqrt {2}}{2}\right )}{6}+\frac {5 F\left (\frac {2 \sqrt {22-33 x}}{11}, \frac {i \sqrt {2}}{2}\right )}{2}\right )}{72900685 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}+\frac {6655867 \sqrt {22-33 x}\, \sqrt {-66 x +165}\, \sqrt {33+132 x}\, \Pi \left (\frac {2 \sqrt {22-33 x}}{11}, \frac {55}{124}, \frac {i \sqrt {2}}{2}\right )}{271190548200 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}\right ) \sqrt {\left (2-3 x \right ) \left (-5+2 x \right ) \left (1+4 x \right )}}{\sqrt {2-3 x}\, \sqrt {-5+2 x}\, \sqrt {1+4 x}}\) | \(311\) |
default | \(\frac {\sqrt {2-3 x}\, \sqrt {1+4 x}\, \sqrt {-5+2 x}\, \left (205130100 \sqrt {1+4 x}\, \sqrt {2-3 x}\, \sqrt {22}\, \sqrt {5-2 x}\, F\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right ) x^{2}-34249875 \sqrt {1+4 x}\, \sqrt {2-3 x}\, \sqrt {22}\, \sqrt {5-2 x}\, E\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right ) x^{2}-332793350 \sqrt {1+4 x}\, \sqrt {2-3 x}\, \sqrt {22}\, \sqrt {5-2 x}\, \Pi \left (\frac {\sqrt {11+44 x}}{11}, -\frac {55}{23}, \sqrt {3}\right ) x^{2}+574364280 \sqrt {1+4 x}\, \sqrt {2-3 x}\, \sqrt {22}\, \sqrt {5-2 x}\, F\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right ) x -95899650 \sqrt {1+4 x}\, \sqrt {2-3 x}\, \sqrt {22}\, \sqrt {5-2 x}\, E\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right ) x -931821380 \sqrt {1+4 x}\, \sqrt {2-3 x}\, \sqrt {22}\, \sqrt {5-2 x}\, \Pi \left (\frac {\sqrt {11+44 x}}{11}, -\frac {55}{23}, \sqrt {3}\right ) x +402054996 \sqrt {1+4 x}\, \sqrt {2-3 x}\, \sqrt {22}\, \sqrt {5-2 x}\, F\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )-67129755 \sqrt {1+4 x}\, \sqrt {2-3 x}\, \sqrt {22}\, \sqrt {5-2 x}\, E\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )-652274966 \sqrt {1+4 x}\, \sqrt {2-3 x}\, \sqrt {22}\, \sqrt {5-2 x}\, \Pi \left (\frac {\sqrt {11+44 x}}{11}, -\frac {55}{23}, \sqrt {3}\right )-821997000 x^{4}+4060763850 x^{3}-5570459125 x^{2}+1112864775 x +693030250\right )}{9145722300 \left (24 x^{3}-70 x^{2}+21 x +10\right ) \left (7+5 x \right )^{2}}\) | \(434\) |
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\[ \int \frac {\sqrt {2-3 x} \sqrt {1+4 x}}{\sqrt {-5+2 x} (7+5 x)^3} \, dx=\int { \frac {\sqrt {4 \, x + 1} \sqrt {-3 \, x + 2}}{{\left (5 \, x + 7\right )}^{3} \sqrt {2 \, x - 5}} \,d x } \]
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\[ \int \frac {\sqrt {2-3 x} \sqrt {1+4 x}}{\sqrt {-5+2 x} (7+5 x)^3} \, dx=\int \frac {\sqrt {2 - 3 x} \sqrt {4 x + 1}}{\sqrt {2 x - 5} \left (5 x + 7\right )^{3}}\, dx \]
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\[ \int \frac {\sqrt {2-3 x} \sqrt {1+4 x}}{\sqrt {-5+2 x} (7+5 x)^3} \, dx=\int { \frac {\sqrt {4 \, x + 1} \sqrt {-3 \, x + 2}}{{\left (5 \, x + 7\right )}^{3} \sqrt {2 \, x - 5}} \,d x } \]
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\[ \int \frac {\sqrt {2-3 x} \sqrt {1+4 x}}{\sqrt {-5+2 x} (7+5 x)^3} \, dx=\int { \frac {\sqrt {4 \, x + 1} \sqrt {-3 \, x + 2}}{{\left (5 \, x + 7\right )}^{3} \sqrt {2 \, x - 5}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {2-3 x} \sqrt {1+4 x}}{\sqrt {-5+2 x} (7+5 x)^3} \, dx=\int \frac {\sqrt {2-3\,x}\,\sqrt {4\,x+1}}{\sqrt {2\,x-5}\,{\left (5\,x+7\right )}^3} \,d x \]
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