Integrand size = 33, antiderivative size = 193 \[ \int \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x) \, dx=-\frac {20911 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{3780}+\frac {136}{105} \sqrt {2-3 x} \sqrt {-5+2 x} (1+4 x)^{3/2}+\frac {5}{28} \sqrt {2-3 x} (-5+2 x)^{3/2} (1+4 x)^{3/2}-\frac {954811 \sqrt {11} \sqrt {-5+2 x} E\left (\arcsin \left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )}{22680 \sqrt {5-2 x}}+\frac {72479 \sqrt {\frac {11}{6}} \sqrt {5-2 x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {1+4 x}\right ),\frac {1}{3}\right )}{756 \sqrt {-5+2 x}} \]
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Time = 0.05 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {159, 164, 115, 114, 122, 120} \[ \int \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x) \, dx=\frac {72479 \sqrt {\frac {11}{6}} \sqrt {5-2 x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right ),\frac {1}{3}\right )}{756 \sqrt {2 x-5}}-\frac {954811 \sqrt {11} \sqrt {2 x-5} E\left (\arcsin \left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )}{22680 \sqrt {5-2 x}}+\frac {5}{28} \sqrt {2-3 x} (2 x-5)^{3/2} (4 x+1)^{3/2}+\frac {136}{105} \sqrt {2-3 x} \sqrt {2 x-5} (4 x+1)^{3/2}-\frac {20911 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{3780} \]
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Rule 114
Rule 115
Rule 120
Rule 122
Rule 159
Rule 164
Rubi steps \begin{align*} \text {integral}& = \frac {5}{28} \sqrt {2-3 x} (-5+2 x)^{3/2} (1+4 x)^{3/2}+\frac {1}{28} \int \frac {\left (\frac {1249}{2}-1088 x\right ) \sqrt {-5+2 x} \sqrt {1+4 x}}{\sqrt {2-3 x}} \, dx \\ & = \frac {136}{105} \sqrt {2-3 x} \sqrt {-5+2 x} (1+4 x)^{3/2}+\frac {5}{28} \sqrt {2-3 x} (-5+2 x)^{3/2} (1+4 x)^{3/2}-\frac {1}{840} \int \frac {(38731-41822 x) \sqrt {1+4 x}}{\sqrt {2-3 x} \sqrt {-5+2 x}} \, dx \\ & = -\frac {20911 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{3780}+\frac {136}{105} \sqrt {2-3 x} \sqrt {-5+2 x} (1+4 x)^{3/2}+\frac {5}{28} \sqrt {2-3 x} (-5+2 x)^{3/2} (1+4 x)^{3/2}+\frac {\int \frac {-787710+1909622 x}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx}{7560} \\ & = -\frac {20911 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{3780}+\frac {136}{105} \sqrt {2-3 x} \sqrt {-5+2 x} (1+4 x)^{3/2}+\frac {5}{28} \sqrt {2-3 x} (-5+2 x)^{3/2} (1+4 x)^{3/2}+\frac {954811 \int \frac {\sqrt {-5+2 x}}{\sqrt {2-3 x} \sqrt {1+4 x}} \, dx}{7560}+\frac {797269 \int \frac {1}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx}{1512} \\ & = -\frac {20911 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{3780}+\frac {136}{105} \sqrt {2-3 x} \sqrt {-5+2 x} (1+4 x)^{3/2}+\frac {5}{28} \sqrt {2-3 x} (-5+2 x)^{3/2} (1+4 x)^{3/2}+\frac {\left (72479 \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}{756 \sqrt {-5+2 x}}+\frac {\left (954811 \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}{7560 \sqrt {5-2 x}} \\ & = -\frac {20911 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{3780}+\frac {136}{105} \sqrt {2-3 x} \sqrt {-5+2 x} (1+4 x)^{3/2}+\frac {5}{28} \sqrt {2-3 x} (-5+2 x)^{3/2} (1+4 x)^{3/2}-\frac {954811 \sqrt {11} \sqrt {-5+2 x} E\left (\sin ^{-1}\left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )}{22680 \sqrt {5-2 x}}+\frac {72479 \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 )}{756 \sqrt {-5+2 x}} \\ \end{align*}
Time = 1.40 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.65 \[ \int \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x) \, dx=\frac {24 \sqrt {2-3 x} \sqrt {1+4 x} \left (48475-37975 x-6066 x^2+5400 x^3\right )-954811 \sqrt {66} \sqrt {5-2 x} E\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {1+4 x}\right )|\frac {1}{3}\right )+724790 \sqrt {66} \sqrt {5-2 x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {1+4 x}\right ),\frac {1}{3}\right )}{45360 \sqrt {-5+2 x}} \]
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Time = 1.61 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.75
method | result | size |
default | \(-\frac {\sqrt {2-3 x}\, \sqrt {-5+2 x}\, \sqrt {1+4 x}\, \left (-1555200 x^{5}+264748 \sqrt {1+4 x}\, \sqrt {2-3 x}\, \sqrt {22}\, \sqrt {5-2 x}\, F\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )-954811 \sqrt {1+4 x}\, \sqrt {2-3 x}\, \sqrt {22}\, \sqrt {5-2 x}\, E\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )+2395008 x^{4}+10468080 x^{3}-18808968 x^{2}+3994200 x +2326800\right )}{45360 \left (24 x^{3}-70 x^{2}+21 x +10\right )}\) | \(144\) |
elliptic | \(\frac {\sqrt {-\left (-2+3 x \right ) \left (-5+2 x \right ) \left (1+4 x \right )}\, \left (\frac {59 x \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}{30}-\frac {277 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}{54}-\frac {31 \sqrt {11+44 x}\, \sqrt {22-33 x}\, \sqrt {110-44 x}\, F\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )}{36 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}+\frac {7891 \sqrt {11+44 x}\, \sqrt {22-33 x}\, \sqrt {110-44 x}\, \left (-\frac {11 E\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )}{12}+\frac {2 F\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )}{3}\right )}{3780 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}+\frac {10 x^{2} \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}{7}\right )}{\sqrt {2-3 x}\, \sqrt {-5+2 x}\, \sqrt {1+4 x}}\) | \(228\) |
risch | \(-\frac {\left (2700 x^{2}+3717 x -9695\right ) \left (-2+3 x \right ) \sqrt {-5+2 x}\, \sqrt {1+4 x}\, \sqrt {\left (2-3 x \right ) \left (-5+2 x \right ) \left (1+4 x \right )}}{1890 \sqrt {-\left (-2+3 x \right ) \left (-5+2 x \right ) \left (1+4 x \right )}\, \sqrt {2-3 x}}-\frac {\left (-\frac {31 \sqrt {22-33 x}\, \sqrt {-66 x +165}\, \sqrt {33+132 x}\, F\left (\frac {2 \sqrt {22-33 x}}{11}, \frac {i \sqrt {2}}{2}\right )}{108 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}+\frac {7891 \sqrt {22-33 x}\, \sqrt {-66 x +165}\, \sqrt {33+132 x}\, \left (-\frac {11 E\left (\frac {2 \sqrt {22-33 x}}{11}, \frac {i \sqrt {2}}{2}\right )}{6}+\frac {5 F\left (\frac {2 \sqrt {22-33 x}}{11}, \frac {i \sqrt {2}}{2}\right )}{2}\right )}{11340 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}\right ) \sqrt {\left (2-3 x \right ) \left (-5+2 x \right ) \left (1+4 x \right )}}{\sqrt {2-3 x}\, \sqrt {-5+2 x}\, \sqrt {1+4 x}}\) | \(252\) |
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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 = 0.31 \[ \int \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x) \, dx=\frac {1}{1890} \, {\left (2700 \, x^{2} + 3717 \, x - 9695\right )} \sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2} - \frac {549703}{23328} \, \sqrt {-6} {\rm weierstrassPInverse}\left (\frac {847}{108}, \frac {6655}{2916}, x - \frac {35}{36}\right ) + \frac {954811}{22680} \, \sqrt {-6} {\rm weierstrassZeta}\left (\frac {847}{108}, \frac {6655}{2916}, {\rm weierstrassPInverse}\left (\frac {847}{108}, \frac {6655}{2916}, x - \frac {35}{36}\right )\right ) \]
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\[ \int \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x) \, dx=\int \sqrt {2 - 3 x} \sqrt {2 x - 5} \sqrt {4 x + 1} \cdot \left (5 x + 7\right )\, dx \]
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\[ \int \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x) \, dx=\int { {\left (5 \, x + 7\right )} \sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2} \,d x } \]
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\[ \int \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x) \, dx=\int { {\left (5 \, x + 7\right )} \sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2} \,d x } \]
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Timed out. \[ \int \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x) \, dx=\int \sqrt {2-3\,x}\,\sqrt {4\,x+1}\,\sqrt {2\,x-5}\,\left (5\,x+7\right ) \,d x \]
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