Integrand size = 35, antiderivative size = 167 \[ \int \frac {\sqrt {2-3 x} (7+5 x)^2}{\sqrt {-5+2 x} \sqrt {1+4 x}} \, dx=\frac {68}{9} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}+\frac {1}{4} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)+\frac {44569 \sqrt {11} \sqrt {-5+2 x} E\left (\arcsin \left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )}{432 \sqrt {5-2 x}}-\frac {17533 \sqrt {\frac {11}{6}} \sqrt {5-2 x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {1+4 x}\right ),\frac {1}{3}\right )}{72 \sqrt {-5+2 x}} \]
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Time = 0.10 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {180, 1629, 164, 115, 114, 122, 120} \[ \int \frac {\sqrt {2-3 x} (7+5 x)^2}{\sqrt {-5+2 x} \sqrt {1+4 x}} \, dx=-\frac {17533 \sqrt {\frac {11}{6}} \sqrt {5-2 x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right ),\frac {1}{3}\right )}{72 \sqrt {2 x-5}}+\frac {44569 \sqrt {11} \sqrt {2 x-5} E\left (\arcsin \left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )}{432 \sqrt {5-2 x}}+\frac {1}{4} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)+\frac {68}{9} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} \]
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Rule 114
Rule 115
Rule 120
Rule 122
Rule 164
Rule 180
Rule 1629
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)-\frac {1}{40} \int \frac {-5155+3605 x+10880 x^2}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx \\ & = \frac {68}{9} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}+\frac {1}{4} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)-\frac {\int \frac {-899460+2674140 x}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx}{4320} \\ & = \frac {68}{9} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}+\frac {1}{4} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)-\frac {44569}{144} \int \frac {\sqrt {-5+2 x}}{\sqrt {2-3 x} \sqrt {1+4 x}} \, dx-\frac {192863}{144} \int \frac {1}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx \\ & = \frac {68}{9} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}+\frac {1}{4} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)-\frac {\left (17533 \sqrt {\frac {11}{2}} \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}{72 \sqrt {-5+2 x}}-\frac {\left (44569 \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}{144 \sqrt {5-2 x}} \\ & = \frac {68}{9} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}+\frac {1}{4} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)+\frac {44569 \sqrt {11} \sqrt {-5+2 x} E\left (\sin ^{-1}\left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )}{432 \sqrt {5-2 x}}-\frac {17533 \sqrt {\frac {11}{6}} \sqrt {5-2 x} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{11}} \sqrt {1+4 x}\right )|\frac {1}{3}\right )}{72 \sqrt {-5+2 x}} \\ \end{align*}
Time = 7.85 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.72 \[ \int \frac {\sqrt {2-3 x} (7+5 x)^2}{\sqrt {-5+2 x} \sqrt {1+4 x}} \, dx=\frac {120 \sqrt {2-3 x} \sqrt {1+4 x} \left (-335+89 x+18 x^2\right )+44569 \sqrt {66} \sqrt {5-2 x} E\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {1+4 x}\right )|\frac {1}{3}\right )-35066 \sqrt {66} \sqrt {5-2 x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {1+4 x}\right ),\frac {1}{3}\right )}{864 \sqrt {-5+2 x}} \]
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Time = 1.62 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.83
method | result | size |
default | \(\frac {\sqrt {2-3 x}\, \sqrt {-5+2 x}\, \sqrt {1+4 x}\, \left (16060 \sqrt {1+4 x}\, \sqrt {2-3 x}\, \sqrt {22}\, \sqrt {5-2 x}\, F\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )-44569 \sqrt {1+4 x}\, \sqrt {2-3 x}\, \sqrt {22}\, \sqrt {5-2 x}\, E\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )+25920 x^{4}+117360 x^{3}-540120 x^{2}+179640 x +80400\right )}{20736 x^{3}-60480 x^{2}+18144 x +8640}\) | \(139\) |
elliptic | \(\frac {\sqrt {-\left (-2+3 x \right ) \left (-5+2 x \right ) \left (1+4 x \right )}\, \left (\frac {5 x \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}{4}+\frac {335 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}{36}+\frac {4997 \sqrt {11+44 x}\, \sqrt {22-33 x}\, \sqrt {110-44 x}\, F\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )}{2904 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}-\frac {44569 \sqrt {11+44 x}\, \sqrt {22-33 x}\, \sqrt {110-44 x}\, \left (-\frac {11 E\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )}{12}+\frac {2 F\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )}{3}\right )}{8712 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}\right )}{\sqrt {2-3 x}\, \sqrt {-5+2 x}\, \sqrt {1+4 x}}\) | \(206\) |
risch | \(-\frac {5 \left (67+9 x \right ) \left (-2+3 x \right ) \sqrt {-5+2 x}\, \sqrt {1+4 x}\, \sqrt {\left (2-3 x \right ) \left (-5+2 x \right ) \left (1+4 x \right )}}{36 \sqrt {-\left (-2+3 x \right ) \left (-5+2 x \right ) \left (1+4 x \right )}\, \sqrt {2-3 x}}-\frac {\left (\frac {4997 \sqrt {22-33 x}\, \sqrt {-66 x +165}\, \sqrt {33+132 x}\, F\left (\frac {2 \sqrt {22-33 x}}{11}, \frac {i \sqrt {2}}{2}\right )}{8712 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}-\frac {44569 \sqrt {22-33 x}\, \sqrt {-66 x +165}\, \sqrt {33+132 x}\, \left (-\frac {11 E\left (\frac {2 \sqrt {22-33 x}}{11}, \frac {i \sqrt {2}}{2}\right )}{6}+\frac {5 F\left (\frac {2 \sqrt {22-33 x}}{11}, \frac {i \sqrt {2}}{2}\right )}{2}\right )}{26136 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}\right ) \sqrt {\left (2-3 x \right ) \left (-5+2 x \right ) \left (1+4 x \right )}}{\sqrt {2-3 x}\, \sqrt {-5+2 x}\, \sqrt {1+4 x}}\) | \(247\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.08 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.32 \[ \int \frac {\sqrt {2-3 x} (7+5 x)^2}{\sqrt {-5+2 x} \sqrt {1+4 x}} \, dx=\frac {5}{36} \, {\left (9 \, x + 67\right )} \sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2} + \frac {1020239}{15552} \, \sqrt {-6} {\rm weierstrassPInverse}\left (\frac {847}{108}, \frac {6655}{2916}, x - \frac {35}{36}\right ) - \frac {44569}{432} \, \sqrt {-6} {\rm weierstrassZeta}\left (\frac {847}{108}, \frac {6655}{2916}, {\rm weierstrassPInverse}\left (\frac {847}{108}, \frac {6655}{2916}, x - \frac {35}{36}\right )\right ) \]
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\[ \int \frac {\sqrt {2-3 x} (7+5 x)^2}{\sqrt {-5+2 x} \sqrt {1+4 x}} \, dx=\int \frac {\sqrt {2 - 3 x} \left (5 x + 7\right )^{2}}{\sqrt {2 x - 5} \sqrt {4 x + 1}}\, dx \]
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\[ \int \frac {\sqrt {2-3 x} (7+5 x)^2}{\sqrt {-5+2 x} \sqrt {1+4 x}} \, dx=\int { \frac {{\left (5 \, x + 7\right )}^{2} \sqrt {-3 \, x + 2}}{\sqrt {4 \, x + 1} \sqrt {2 \, x - 5}} \,d x } \]
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\[ \int \frac {\sqrt {2-3 x} (7+5 x)^2}{\sqrt {-5+2 x} \sqrt {1+4 x}} \, dx=\int { \frac {{\left (5 \, x + 7\right )}^{2} \sqrt {-3 \, x + 2}}{\sqrt {4 \, x + 1} \sqrt {2 \, x - 5}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {2-3 x} (7+5 x)^2}{\sqrt {-5+2 x} \sqrt {1+4 x}} \, dx=\int \frac {\sqrt {2-3\,x}\,{\left (5\,x+7\right )}^2}{\sqrt {4\,x+1}\,\sqrt {2\,x-5}} \,d x \]
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