Integrand size = 35, antiderivative size = 243 \[ \int \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^2 \, dx=-\frac {5256763 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{97200}-\frac {8141 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)}{2700}-\frac {61}{270} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^2+\frac {2}{45} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^3-\frac {17746949 \sqrt {11} \sqrt {-5+2 x} E\left (\arcsin \left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )}{29160 \sqrt {5-2 x}}+\frac {5592499 \sqrt {\frac {11}{6}} \sqrt {5-2 x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {1+4 x}\right ),\frac {1}{3}\right )}{3888 \sqrt {-5+2 x}} \]
[Out]
Time = 0.21 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.229, Rules used = {167, 1614, 1629, 164, 115, 114, 122, 120} \[ \int \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^2 \, dx=\frac {5592499 \sqrt {\frac {11}{6}} \sqrt {5-2 x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right ),\frac {1}{3}\right )}{3888 \sqrt {2 x-5}}-\frac {17746949 \sqrt {11} \sqrt {2 x-5} E\left (\arcsin \left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )}{29160 \sqrt {5-2 x}}+\frac {2}{45} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^3-\frac {61}{270} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^2-\frac {8141 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)}{2700}-\frac {5256763 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{97200} \]
[In]
[Out]
Rule 114
Rule 115
Rule 120
Rule 122
Rule 164
Rule 167
Rule 1614
Rule 1629
Rubi steps \begin{align*} \text {integral}& = \frac {2}{45} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^3+\frac {1}{45} \int \frac {(7+5 x)^2 \left (-3-1190 x+854 x^2\right )}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx \\ & = -\frac {61}{270} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^2+\frac {2}{45} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^3-\frac {\int \frac {(7+5 x) \left (299866+1013390 x-1367688 x^2\right )}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx}{7560} \\ & = -\frac {8141 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)}{2700}-\frac {61}{270} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^2+\frac {2}{45} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^3+\frac {\int \frac {-589706376-121654680 x+1766272368 x^2}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx}{907200} \\ & = -\frac {5256763 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{97200}-\frac {8141 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)}{2700}-\frac {61}{270} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^2+\frac {2}{45} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^3+\frac {\int \frac {-119325868200+357778491840 x}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx}{97977600} \\ & = -\frac {5256763 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{97200}-\frac {8141 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)}{2700}-\frac {61}{270} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^2+\frac {2}{45} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^3+\frac {17746949 \int \frac {\sqrt {-5+2 x}}{\sqrt {2-3 x} \sqrt {1+4 x}} \, dx}{9720}+\frac {61517489 \int \frac {1}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx}{7776} \\ & = -\frac {5256763 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{97200}-\frac {8141 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)}{2700}-\frac {61}{270} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^2+\frac {2}{45} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^3+\frac {\left (5592499 \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}{3888 \sqrt {-5+2 x}}+\frac {\left (17746949 \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}{9720 \sqrt {5-2 x}} \\ & = -\frac {5256763 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{97200}-\frac {8141 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)}{2700}-\frac {61}{270} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^2+\frac {2}{45} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^3-\frac {17746949 \sqrt {11} \sqrt {-5+2 x} E\left (\sin ^{-1}\left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )}{29160 \sqrt {5-2 x}}+\frac {5592499 \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 )}{3888 \sqrt {-5+2 x}} \\ \end{align*}
Time = 4.89 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.53 \[ \int \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^2 \, dx=\frac {6 \sqrt {2-3 x} \sqrt {1+4 x} \left (6902575-2933650 x-1649952 x^2+147600 x^3+216000 x^4\right )-35493898 \sqrt {66} \sqrt {5-2 x} E\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {1+4 x}\right )|\frac {1}{3}\right )+27962495 \sqrt {66} \sqrt {5-2 x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {1+4 x}\right ),\frac {1}{3}\right )}{116640 \sqrt {-5+2 x}} \]
[In]
[Out]
Time = 1.63 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.61
method | result | size |
default | \(-\frac {\sqrt {2-3 x}\, \sqrt {-5+2 x}\, \sqrt {1+4 x}\, \left (-15552000 x^{6}-4147200 x^{5}+12899689 \sqrt {1+4 x}\, \sqrt {2-3 x}\, \sqrt {22}\, \sqrt {5-2 x}\, F\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )-35493898 \sqrt {1+4 x}\, \sqrt {2-3 x}\, \sqrt {22}\, \sqrt {5-2 x}\, E\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )+125816544 x^{4}+163495440 x^{3}-604794324 x^{2}+171873450 x +82830900\right )}{116640 \left (24 x^{3}-70 x^{2}+21 x +10\right )}\) | \(149\) |
elliptic | \(\frac {\sqrt {-\left (-2+3 x \right ) \left (-5+2 x \right ) \left (1+4 x \right )}\, \left (\frac {959 x \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}{540}-\frac {276103 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}{3888}-\frac {26089 \sqrt {11+44 x}\, \sqrt {22-33 x}\, \sqrt {110-44 x}\, F\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )}{2592 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}+\frac {146669 \sqrt {11+44 x}\, \sqrt {22-33 x}\, \sqrt {110-44 x}\, \left (-\frac {11 E\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )}{12}+\frac {2 F\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )}{3}\right )}{4860 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}+\frac {955 x^{2} \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}{54}+\frac {50 x^{3} \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}{9}\right )}{\sqrt {2-3 x}\, \sqrt {-5+2 x}\, \sqrt {1+4 x}}\) | \(250\) |
risch | \(-\frac {\left (108000 x^{3}+343800 x^{2}+34524 x -1380515\right ) \left (-2+3 x \right ) \sqrt {-5+2 x}\, \sqrt {1+4 x}\, \sqrt {\left (2-3 x \right ) \left (-5+2 x \right ) \left (1+4 x \right )}}{19440 \sqrt {-\left (-2+3 x \right ) \left (-5+2 x \right ) \left (1+4 x \right )}\, \sqrt {2-3 x}}-\frac {\left (-\frac {26089 \sqrt {22-33 x}\, \sqrt {-66 x +165}\, \sqrt {33+132 x}\, F\left (\frac {2 \sqrt {22-33 x}}{11}, \frac {i \sqrt {2}}{2}\right )}{7776 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}+\frac {146669 \sqrt {22-33 x}\, \sqrt {-66 x +165}\, \sqrt {33+132 x}\, \left (-\frac {11 E\left (\frac {2 \sqrt {22-33 x}}{11}, \frac {i \sqrt {2}}{2}\right )}{6}+\frac {5 F\left (\frac {2 \sqrt {22-33 x}}{11}, \frac {i \sqrt {2}}{2}\right )}{2}\right )}{14580 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}\right ) \sqrt {\left (2-3 x \right ) \left (-5+2 x \right ) \left (1+4 x \right )}}{\sqrt {2-3 x}\, \sqrt {-5+2 x}\, \sqrt {1+4 x}}\) | \(257\) |
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.08 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.26 \[ \int \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^2 \, dx=\frac {1}{19440} \, {\left (108000 \, x^{3} + 343800 \, x^{2} + 34524 \, x - 1380515\right )} \sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2} - \frac {163224523}{419904} \, \sqrt {-6} {\rm weierstrassPInverse}\left (\frac {847}{108}, \frac {6655}{2916}, x - \frac {35}{36}\right ) + \frac {17746949}{29160} \, \sqrt {-6} {\rm weierstrassZeta}\left (\frac {847}{108}, \frac {6655}{2916}, {\rm weierstrassPInverse}\left (\frac {847}{108}, \frac {6655}{2916}, x - \frac {35}{36}\right )\right ) \]
[In]
[Out]
\[ \int \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^2 \, dx=\int \sqrt {2 - 3 x} \sqrt {2 x - 5} \sqrt {4 x + 1} \left (5 x + 7\right )^{2}\, dx \]
[In]
[Out]
\[ \int \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^2 \, dx=\int { {\left (5 \, x + 7\right )}^{2} \sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2} \,d x } \]
[In]
[Out]
\[ \int \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^2 \, dx=\int { {\left (5 \, x + 7\right )}^{2} \sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2} \,d x } \]
[In]
[Out]
Timed out. \[ \int \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^2 \, dx=\int \sqrt {2-3\,x}\,\sqrt {4\,x+1}\,\sqrt {2\,x-5}\,{\left (5\,x+7\right )}^2 \,d x \]
[In]
[Out]