Integrand size = 35, antiderivative size = 129 \[ \int \frac {(7+5 x)^2}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx=-\frac {25}{36} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}-\frac {2135 \sqrt {11} \sqrt {-5+2 x} E\left (\arcsin \left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )}{108 \sqrt {5-2 x}}+\frac {24353 \sqrt {5-2 x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {1+4 x}\right ),\frac {1}{3}\right )}{36 \sqrt {66} \sqrt {-5+2 x}} \]
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Time = 0.04 (sec) , antiderivative size = 129, 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 = {173, 24, 164, 115, 114, 122, 120} \[ \int \frac {(7+5 x)^2}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx=\frac {24353 \sqrt {5-2 x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right ),\frac {1}{3}\right )}{36 \sqrt {66} \sqrt {2 x-5}}-\frac {2135 \sqrt {11} \sqrt {2 x-5} E\left (\arcsin \left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )}{108 \sqrt {5-2 x}}-\frac {25}{36} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} \]
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Rule 24
Rule 114
Rule 115
Rule 120
Rule 122
Rule 164
Rule 173
Rubi steps \begin{align*} \text {integral}& = -\frac {25}{36} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}+\frac {1}{72} \int \frac {21021+74795 x+42700 x^2}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)} \, dx \\ & = -\frac {25}{36} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}+\frac {\int \frac {75075+213500 x}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx}{1800} \\ & = -\frac {25}{36} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}+\frac {2135}{36} \int \frac {\sqrt {-5+2 x}}{\sqrt {2-3 x} \sqrt {1+4 x}} \, dx+\frac {24353}{72} \int \frac {1}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx \\ & = -\frac {25}{36} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}+\frac {\left (24353 \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}{36 \sqrt {22} \sqrt {-5+2 x}}+\frac {\left (2135 \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}{36 \sqrt {5-2 x}} \\ & = -\frac {25}{36} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}-\frac {2135 \sqrt {11} \sqrt {-5+2 x} E\left (\sin ^{-1}\left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )}{108 \sqrt {5-2 x}}+\frac {24353 \sqrt {5-2 x} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{11}} \sqrt {1+4 x}\right )|\frac {1}{3}\right )}{36 \sqrt {66} \sqrt {-5+2 x}} \\ \end{align*}
Time = 16.62 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.89 \[ \int \frac {(7+5 x)^2}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx=\frac {1650 \sqrt {2-3 x} (5-2 x) \sqrt {1+4 x}-23485 \sqrt {66} \sqrt {5-2 x} E\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {1+4 x}\right )|\frac {1}{3}\right )+24353 \sqrt {66} \sqrt {5-2 x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {1+4 x}\right ),\frac {1}{3}\right )}{2376 \sqrt {-5+2 x}} \]
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Time = 1.64 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.04
method | result | size |
default | \(-\frac {\sqrt {2-3 x}\, \sqrt {-5+2 x}\, \sqrt {1+4 x}\, \left (26089 \sqrt {1+4 x}\, \sqrt {2-3 x}\, \sqrt {22}\, \sqrt {5-2 x}\, F\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )-23485 \sqrt {1+4 x}\, \sqrt {2-3 x}\, \sqrt {22}\, \sqrt {5-2 x}\, E\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )+39600 x^{3}-115500 x^{2}+34650 x +16500\right )}{2376 \left (24 x^{3}-70 x^{2}+21 x +10\right )}\) | \(134\) |
elliptic | \(\frac {\sqrt {-\left (-2+3 x \right ) \left (-5+2 x \right ) \left (1+4 x \right )}\, \left (-\frac {25 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}{36}+\frac {91 \sqrt {11+44 x}\, \sqrt {22-33 x}\, \sqrt {110-44 x}\, F\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )}{264 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}+\frac {2135 \sqrt {11+44 x}\, \sqrt {22-33 x}\, \sqrt {110-44 x}\, \left (-\frac {11 E\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )}{12}+\frac {2 F\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )}{3}\right )}{2178 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}\right )}{\sqrt {2-3 x}\, \sqrt {-5+2 x}\, \sqrt {1+4 x}}\) | \(186\) |
risch | \(\frac {25 \left (-2+3 x \right ) \sqrt {-5+2 x}\, \sqrt {1+4 x}\, \sqrt {\left (2-3 x \right ) \left (-5+2 x \right ) \left (1+4 x \right )}}{36 \sqrt {-\left (-2+3 x \right ) \left (-5+2 x \right ) \left (1+4 x \right )}\, \sqrt {2-3 x}}+\frac {\left (-\frac {91 \sqrt {22-33 x}\, \sqrt {-66 x +165}\, \sqrt {33+132 x}\, F\left (\frac {2 \sqrt {22-33 x}}{11}, \frac {i \sqrt {2}}{2}\right )}{792 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}-\frac {2135 \sqrt {22-33 x}\, \sqrt {-66 x +165}\, \sqrt {33+132 x}\, \left (-\frac {11 E\left (\frac {2 \sqrt {22-33 x}}{11}, \frac {i \sqrt {2}}{2}\right )}{6}+\frac {5 F\left (\frac {2 \sqrt {22-33 x}}{11}, \frac {i \sqrt {2}}{2}\right )}{2}\right )}{6534 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}\right ) \sqrt {\left (2-3 x \right ) \left (-5+2 x \right ) \left (1+4 x \right )}}{\sqrt {2-3 x}\, \sqrt {-5+2 x}\, \sqrt {1+4 x}}\) | \(241\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.08 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.38 \[ \int \frac {(7+5 x)^2}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx=-\frac {25}{36} \, \sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2} - \frac {12719}{486} \, \sqrt {-6} {\rm weierstrassPInverse}\left (\frac {847}{108}, \frac {6655}{2916}, x - \frac {35}{36}\right ) + \frac {2135}{108} \, \sqrt {-6} {\rm weierstrassZeta}\left (\frac {847}{108}, \frac {6655}{2916}, {\rm weierstrassPInverse}\left (\frac {847}{108}, \frac {6655}{2916}, x - \frac {35}{36}\right )\right ) \]
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\[ \int \frac {(7+5 x)^2}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx=\int \frac {\left (5 x + 7\right )^{2}}{\sqrt {2 - 3 x} \sqrt {2 x - 5} \sqrt {4 x + 1}}\, dx \]
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\[ \int \frac {(7+5 x)^2}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx=\int { \frac {{\left (5 \, x + 7\right )}^{2}}{\sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2}} \,d x } \]
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\[ \int \frac {(7+5 x)^2}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx=\int { \frac {{\left (5 \, x + 7\right )}^{2}}{\sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2}} \,d x } \]
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Timed out. \[ \int \frac {(7+5 x)^2}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx=\int \frac {{\left (5\,x+7\right )}^2}{\sqrt {2-3\,x}\,\sqrt {4\,x+1}\,\sqrt {2\,x-5}} \,d x \]
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