Integrand size = 35, antiderivative size = 205 \[ \int \frac {\sqrt {2-3 x} (7+5 x)^3}{\sqrt {-5+2 x} \sqrt {1+4 x}} \, dx=\frac {110743}{864} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}+\frac {121}{24} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)+\frac {5}{28} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^2+\frac {15629623 \sqrt {11} \sqrt {-5+2 x} E\left (\arcsin \left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )}{9072 \sqrt {5-2 x}}-\frac {25260049 \sqrt {\frac {11}{6}} \sqrt {5-2 x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {1+4 x}\right ),\frac {1}{3}\right )}{6048 \sqrt {-5+2 x}} \]
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Time = 0.14 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.229, Rules used = {180, 1614, 1629, 164, 115, 114, 122, 120} \[ \int \frac {\sqrt {2-3 x} (7+5 x)^3}{\sqrt {-5+2 x} \sqrt {1+4 x}} \, dx=-\frac {25260049 \sqrt {\frac {11}{6}} \sqrt {5-2 x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right ),\frac {1}{3}\right )}{6048 \sqrt {2 x-5}}+\frac {15629623 \sqrt {11} \sqrt {2 x-5} E\left (\arcsin \left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )}{9072 \sqrt {5-2 x}}+\frac {5}{28} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^2+\frac {121}{24} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)+\frac {110743}{864} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} \]
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Rule 114
Rule 115
Rule 120
Rule 122
Rule 164
Rule 180
Rule 1614
Rule 1629
Rubi steps \begin{align*} \text {integral}& = \frac {5}{28} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^2-\frac {1}{56} \int \frac {(7+5 x) \left (-7223+2667 x+16940 x^2\right )}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx \\ & = \frac {121}{24} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)+\frac {5}{28} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^2+\frac {\int \frac {10251500-9171580 x-31008040 x^2}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx}{6720} \\ & = \frac {110743}{864} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}+\frac {121}{24} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)+\frac {5}{28} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^2+\frac {\int \frac {2083915260-7502219040 x}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx}{725760} \\ & = \frac {110743}{864} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}+\frac {121}{24} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)+\frac {5}{28} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^2-\frac {15629623 \int \frac {\sqrt {-5+2 x}}{\sqrt {2-3 x} \sqrt {1+4 x}} \, dx}{3024}-\frac {277860539 \int \frac {1}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx}{12096} \\ & = \frac {110743}{864} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}+\frac {121}{24} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)+\frac {5}{28} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^2-\frac {\left (25260049 \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}{6048 \sqrt {-5+2 x}}-\frac {\left (15629623 \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}{3024 \sqrt {5-2 x}} \\ & = \frac {110743}{864} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}+\frac {121}{24} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)+\frac {5}{28} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^2+\frac {15629623 \sqrt {11} \sqrt {-5+2 x} E\left (\sin ^{-1}\left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )}{9072 \sqrt {5-2 x}}-\frac {25260049 \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 )}{6048 \sqrt {-5+2 x}} \\ \end{align*}
Time = 9.07 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.61 \[ \int \frac {\sqrt {2-3 x} (7+5 x)^3}{\sqrt {-5+2 x} \sqrt {1+4 x}} \, dx=\frac {30 \sqrt {2-3 x} \sqrt {1+4 x} \left (-1041565+188566 x+64224 x^2+10800 x^3\right )+31259246 \sqrt {66} \sqrt {5-2 x} E\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {1+4 x}\right )|\frac {1}{3}\right )-25260049 \sqrt {66} \sqrt {5-2 x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {1+4 x}\right ),\frac {1}{3}\right )}{36288 \sqrt {-5+2 x}} \]
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Time = 1.60 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.70
method | result | size |
default | \(\frac {\sqrt {2-3 x}\, \sqrt {-5+2 x}\, \sqrt {1+4 x}\, \left (13261655 \sqrt {1+4 x}\, \sqrt {2-3 x}\, \sqrt {22}\, \sqrt {5-2 x}\, F\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )-31259246 \sqrt {1+4 x}\, \sqrt {2-3 x}\, \sqrt {22}\, \sqrt {5-2 x}\, E\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )+3888000 x^{5}+21500640 x^{4}+57602160 x^{3}-407101740 x^{2}+144920790 x +62493900\right )}{870912 x^{3}-2540160 x^{2}+762048 x +362880}\) | \(144\) |
elliptic | \(\frac {\sqrt {-\left (-2+3 x \right ) \left (-5+2 x \right ) \left (1+4 x \right )}\, \left (\frac {905 x \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}{24}+\frac {148795 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}{864}+\frac {1653901 \sqrt {11+44 x}\, \sqrt {22-33 x}\, \sqrt {110-44 x}\, F\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )}{69696 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}-\frac {15629623 \sqrt {11+44 x}\, \sqrt {22-33 x}\, \sqrt {110-44 x}\, \left (-\frac {11 E\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )}{12}+\frac {2 F\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )}{3}\right )}{182952 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}+\frac {125 x^{2} \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}{28}\right )}{\sqrt {2-3 x}\, \sqrt {-5+2 x}\, \sqrt {1+4 x}}\) | \(228\) |
risch | \(-\frac {5 \left (5400 x^{2}+45612 x +208313\right ) \left (-2+3 x \right ) \sqrt {-5+2 x}\, \sqrt {1+4 x}\, \sqrt {\left (2-3 x \right ) \left (-5+2 x \right ) \left (1+4 x \right )}}{6048 \sqrt {-\left (-2+3 x \right ) \left (-5+2 x \right ) \left (1+4 x \right )}\, \sqrt {2-3 x}}-\frac {\left (\frac {1653901 \sqrt {22-33 x}\, \sqrt {-66 x +165}\, \sqrt {33+132 x}\, F\left (\frac {2 \sqrt {22-33 x}}{11}, \frac {i \sqrt {2}}{2}\right )}{209088 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}-\frac {15629623 \sqrt {22-33 x}\, \sqrt {-66 x +165}\, \sqrt {33+132 x}\, \left (-\frac {11 E\left (\frac {2 \sqrt {22-33 x}}{11}, \frac {i \sqrt {2}}{2}\right )}{6}+\frac {5 F\left (\frac {2 \sqrt {22-33 x}}{11}, \frac {i \sqrt {2}}{2}\right )}{2}\right )}{548856 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}\right ) \sqrt {\left (2-3 x \right ) \left (-5+2 x \right ) \left (1+4 x \right )}}{\sqrt {2-3 x}\, \sqrt {-5+2 x}\, \sqrt {1+4 x}}\) | \(252\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.08 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.29 \[ \int \frac {\sqrt {2-3 x} (7+5 x)^3}{\sqrt {-5+2 x} \sqrt {1+4 x}} \, dx=\frac {5}{6048} \, {\left (5400 \, x^{2} + 45612 \, x + 208313\right )} \sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2} + \frac {111640903}{93312} \, \sqrt {-6} {\rm weierstrassPInverse}\left (\frac {847}{108}, \frac {6655}{2916}, x - \frac {35}{36}\right ) - \frac {15629623}{9072} \, \sqrt {-6} {\rm weierstrassZeta}\left (\frac {847}{108}, \frac {6655}{2916}, {\rm weierstrassPInverse}\left (\frac {847}{108}, \frac {6655}{2916}, x - \frac {35}{36}\right )\right ) \]
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\[ \int \frac {\sqrt {2-3 x} (7+5 x)^3}{\sqrt {-5+2 x} \sqrt {1+4 x}} \, dx=\int \frac {\sqrt {2 - 3 x} \left (5 x + 7\right )^{3}}{\sqrt {2 x - 5} \sqrt {4 x + 1}}\, dx \]
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\[ \int \frac {\sqrt {2-3 x} (7+5 x)^3}{\sqrt {-5+2 x} \sqrt {1+4 x}} \, dx=\int { \frac {{\left (5 \, x + 7\right )}^{3} \sqrt {-3 \, x + 2}}{\sqrt {4 \, x + 1} \sqrt {2 \, x - 5}} \,d x } \]
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\[ \int \frac {\sqrt {2-3 x} (7+5 x)^3}{\sqrt {-5+2 x} \sqrt {1+4 x}} \, dx=\int { \frac {{\left (5 \, x + 7\right )}^{3} \sqrt {-3 \, x + 2}}{\sqrt {4 \, x + 1} \sqrt {2 \, x - 5}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {2-3 x} (7+5 x)^3}{\sqrt {-5+2 x} \sqrt {1+4 x}} \, dx=\int \frac {\sqrt {2-3\,x}\,{\left (5\,x+7\right )}^3}{\sqrt {4\,x+1}\,\sqrt {2\,x-5}} \,d x \]
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