Integrand size = 28, antiderivative size = 129 \[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{3/2} \sqrt {2+3 x}} \, dx=\frac {335}{63} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {11 \sqrt {2+3 x} (3+5 x)^{3/2}}{7 \sqrt {1-2 x}}+\frac {4451}{126} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {67}{63} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right ) \]
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Time = 0.03 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {100, 159, 164, 114, 120} \[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{3/2} \sqrt {2+3 x}} \, dx=\frac {67}{63} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )+\frac {4451}{126} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {11 \sqrt {3 x+2} (5 x+3)^{3/2}}{7 \sqrt {1-2 x}}+\frac {335}{63} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3} \]
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Rule 100
Rule 114
Rule 120
Rule 159
Rule 164
Rubi steps \begin{align*} \text {integral}& = \frac {11 \sqrt {2+3 x} (3+5 x)^{3/2}}{7 \sqrt {1-2 x}}-\frac {1}{7} \int \frac {\sqrt {3+5 x} \left (\frac {435}{2}+335 x\right )}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx \\ & = \frac {335}{63} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {11 \sqrt {2+3 x} (3+5 x)^{3/2}}{7 \sqrt {1-2 x}}+\frac {1}{63} \int \frac {-7045-\frac {22255 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx \\ & = \frac {335}{63} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {11 \sqrt {2+3 x} (3+5 x)^{3/2}}{7 \sqrt {1-2 x}}-\frac {737}{126} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx-\frac {4451}{126} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx \\ & = \frac {335}{63} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {11 \sqrt {2+3 x} (3+5 x)^{3/2}}{7 \sqrt {1-2 x}}+\frac {4451}{126} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {67}{63} \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right ) \\ \end{align*}
Result contains complex when optimal does not.
Time = 5.39 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.78 \[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{3/2} \sqrt {2+3 x}} \, dx=\frac {6 (632-175 x) \sqrt {2+3 x} \sqrt {3+5 x}-4451 i \sqrt {33-66 x} E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )+4585 i \sqrt {33-66 x} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )}{378 \sqrt {1-2 x}} \]
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Time = 1.35 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.09
method | result | size |
default | \(\frac {\sqrt {3+5 x}\, \sqrt {1-2 x}\, \sqrt {2+3 x}\, \left (4323 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-4451 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )+15750 x^{3}-36930 x^{2}-65748 x -22752\right )}{11340 x^{3}+8694 x^{2}-2646 x -2268}\) | \(140\) |
elliptic | \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (\frac {25 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{18}-\frac {2818 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{1323 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {4451 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{1323 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {121 \left (-30 x^{2}-38 x -12\right )}{28 \sqrt {\left (x -\frac {1}{2}\right ) \left (-30 x^{2}-38 x -12\right )}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(214\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.07 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.57 \[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{3/2} \sqrt {2+3 x}} \, dx=\frac {540 \, {\left (175 \, x - 632\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} + 151247 \, \sqrt {-30} {\left (2 \, x - 1\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) - 400590 \, \sqrt {-30} {\left (2 \, x - 1\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )}{34020 \, {\left (2 \, x - 1\right )}} \]
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\[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{3/2} \sqrt {2+3 x}} \, dx=\int \frac {\left (5 x + 3\right )^{\frac {5}{2}}}{\left (1 - 2 x\right )^{\frac {3}{2}} \sqrt {3 x + 2}}\, dx \]
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\[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{3/2} \sqrt {2+3 x}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}}}{\sqrt {3 \, x + 2} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{3/2} \sqrt {2+3 x}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}}}{\sqrt {3 \, x + 2} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{3/2} \sqrt {2+3 x}} \, dx=\int \frac {{\left (5\,x+3\right )}^{5/2}}{{\left (1-2\,x\right )}^{3/2}\,\sqrt {3\,x+2}} \,d x \]
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