Integrand size = 28, antiderivative size = 129 \[ \int \frac {(3+5 x)^{5/2}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx=-\frac {62}{27} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {1}{3} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {4141}{270} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {62}{135} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right ) \]
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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 = {104, 159, 164, 114, 120} \[ \int \frac {(3+5 x)^{5/2}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx=-\frac {62}{135} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {4141}{270} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {1}{3} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}-\frac {62}{27} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3} \]
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Rule 104
Rule 114
Rule 120
Rule 159
Rule 164
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{3} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {1}{15} \int \frac {\left (-\frac {405}{2}-310 x\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx \\ & = -\frac {62}{27} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {1}{3} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}+\frac {1}{135} \int \frac {\frac {13105}{2}+\frac {20705 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx \\ & = -\frac {62}{27} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {1}{3} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}+\frac {341}{135} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx+\frac {4141}{270} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx \\ & = -\frac {62}{27} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {1}{3} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {4141}{270} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {62}{135} \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 = 3.29 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.74 \[ \int \frac {(3+5 x)^{5/2}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx=\frac {1}{810} \left (4141 i \sqrt {33} E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-5 \left (6 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x} (89+45 x)+853 i \sqrt {33} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )\right ) \]
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Time = 1.35 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.12
method | result | size |
default | \(-\frac {\sqrt {3+5 x}\, \sqrt {1-2 x}\, \sqrt {2+3 x}\, \left (28149 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-28987 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )+283500 x^{4}+778050 x^{3}+363720 x^{2}-187530 x -112140\right )}{5670 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\) | \(145\) |
elliptic | \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (-\frac {5 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{3}-\frac {89 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{27}+\frac {2621 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2835 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {4141 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{2835 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(206\) |
risch | \(\frac {\left (89+45 x \right ) \left (-1+2 x \right ) \sqrt {3+5 x}\, \sqrt {2+3 x}\, \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{27 \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \sqrt {1-2 x}}+\frac {\left (\frac {2621 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {-110 x +55}\, F\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{2970 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {4141 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {-110 x +55}\, \left (\frac {E\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{15}-\frac {2 F\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{3}\right )}{2970 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right ) \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(246\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.07 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.42 \[ \int \frac {(3+5 x)^{5/2}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx=-\frac {1}{27} \, {\left (45 \, x + 89\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - \frac {140647}{72900} \, \sqrt {-30} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + \frac {4141}{810} \, \sqrt {-30} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right ) \]
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\[ \int \frac {(3+5 x)^{5/2}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx=\int \frac {\left (5 x + 3\right )^{\frac {5}{2}}}{\sqrt {1 - 2 x} \sqrt {3 x + 2}}\, dx \]
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\[ \int \frac {(3+5 x)^{5/2}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}}}{\sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}} \,d x } \]
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\[ \int \frac {(3+5 x)^{5/2}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}}}{\sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}} \,d x } \]
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Timed out. \[ \int \frac {(3+5 x)^{5/2}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx=\int \frac {{\left (5\,x+3\right )}^{5/2}}{\sqrt {1-2\,x}\,\sqrt {3\,x+2}} \,d x \]
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