Integrand size = 28, antiderivative size = 160 \[ \int \frac {(2+3 x)^{7/2}}{(1-2 x)^{3/2} \sqrt {3+5 x}} \, dx=\frac {14517 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{2750}+\frac {312}{275} \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}+\frac {7 (2+3 x)^{5/2} \sqrt {3+5 x}}{11 \sqrt {1-2 x}}+\frac {168123 \sqrt {\frac {3}{11}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1250}+\frac {5057 \sqrt {\frac {3}{11}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{1250} \]
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Time = 0.04 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {100, 159, 164, 114, 120} \[ \int \frac {(2+3 x)^{7/2}}{(1-2 x)^{3/2} \sqrt {3+5 x}} \, dx=\frac {5057 \sqrt {\frac {3}{11}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{1250}+\frac {168123 \sqrt {\frac {3}{11}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1250}+\frac {7 \sqrt {5 x+3} (3 x+2)^{5/2}}{11 \sqrt {1-2 x}}+\frac {312}{275} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{3/2}+\frac {14517 \sqrt {1-2 x} \sqrt {5 x+3} \sqrt {3 x+2}}{2750} \]
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Rule 100
Rule 114
Rule 120
Rule 159
Rule 164
Rubi steps \begin{align*} \text {integral}& = \frac {7 (2+3 x)^{5/2} \sqrt {3+5 x}}{11 \sqrt {1-2 x}}-\frac {1}{11} \int \frac {(2+3 x)^{3/2} \left (\frac {381}{2}+312 x\right )}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx \\ & = \frac {312}{275} \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}+\frac {7 (2+3 x)^{5/2} \sqrt {3+5 x}}{11 \sqrt {1-2 x}}+\frac {1}{275} \int \frac {\left (-13425-\frac {43551 x}{2}\right ) \sqrt {2+3 x}}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx \\ & = \frac {14517 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{2750}+\frac {312}{275} \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}+\frac {7 (2+3 x)^{5/2} \sqrt {3+5 x}}{11 \sqrt {1-2 x}}-\frac {\int \frac {\frac {1915857}{4}+\frac {1513107 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{4125} \\ & = \frac {14517 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{2750}+\frac {312}{275} \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}+\frac {7 (2+3 x)^{5/2} \sqrt {3+5 x}}{11 \sqrt {1-2 x}}-\frac {15171 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{2500}-\frac {504369 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{13750} \\ & = \frac {14517 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{2750}+\frac {312}{275} \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}+\frac {7 (2+3 x)^{5/2} \sqrt {3+5 x}}{11 \sqrt {1-2 x}}+\frac {168123 \sqrt {\frac {3}{11}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1250}+\frac {5057 \sqrt {\frac {3}{11}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1250} \\ \end{align*}
Result contains complex when optimal does not.
Time = 6.40 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.66 \[ \int \frac {(2+3 x)^{7/2}}{(1-2 x)^{3/2} \sqrt {3+5 x}} \, dx=\frac {-5 \sqrt {2+3 x} \sqrt {3+5 x} \left (-27757+11154 x+2970 x^2\right )-168123 i \sqrt {33-66 x} E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )+173180 i \sqrt {33-66 x} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )}{13750 \sqrt {1-2 x}} \]
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Time = 1.37 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.91
method | result | size |
default | \(\frac {\sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (163284 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-168123 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )+222750 x^{4}+1118700 x^{3}-933045 x^{2}-2302295 x -832710\right )}{412500 x^{3}+316250 x^{2}-96250 x -82500}\) | \(145\) |
elliptic | \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (\frac {27 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{50}+\frac {1149 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{500}-\frac {212873 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{96250 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {168123 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{48125 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {343 \left (-30 x^{2}-38 x -12\right )}{88 \sqrt {\left (x -\frac {1}{2}\right ) \left (-30 x^{2}-38 x -12\right )}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(234\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.07 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.49 \[ \int \frac {(2+3 x)^{7/2}}{(1-2 x)^{3/2} \sqrt {3+5 x}} \, dx=\frac {75 \, {\left (2970 \, x^{2} + 11154 \, x - 27757\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} + 952076 \, \sqrt {-30} {\left (2 \, x - 1\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) - 2521845 \, \sqrt {-30} {\left (2 \, x - 1\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )}{206250 \, {\left (2 \, x - 1\right )}} \]
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Timed out. \[ \int \frac {(2+3 x)^{7/2}}{(1-2 x)^{3/2} \sqrt {3+5 x}} \, dx=\text {Timed out} \]
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\[ \int \frac {(2+3 x)^{7/2}}{(1-2 x)^{3/2} \sqrt {3+5 x}} \, dx=\int { \frac {{\left (3 \, x + 2\right )}^{\frac {7}{2}}}{\sqrt {5 \, x + 3} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {(2+3 x)^{7/2}}{(1-2 x)^{3/2} \sqrt {3+5 x}} \, dx=\int { \frac {{\left (3 \, x + 2\right )}^{\frac {7}{2}}}{\sqrt {5 \, x + 3} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(2+3 x)^{7/2}}{(1-2 x)^{3/2} \sqrt {3+5 x}} \, dx=\int \frac {{\left (3\,x+2\right )}^{7/2}}{{\left (1-2\,x\right )}^{3/2}\,\sqrt {5\,x+3}} \,d x \]
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