Integrand size = 28, antiderivative size = 191 \[ \int \frac {(2+3 x)^{9/2}}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx=-\frac {37 \sqrt {1-2 x} (2+3 x)^{5/2}}{605 \sqrt {3+5 x}}+\frac {7 (2+3 x)^{7/2}}{11 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {502941 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{151250}+\frac {10851 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{15125}+\frac {2911577 \sqrt {\frac {3}{11}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{34375}+\frac {175111 \sqrt {\frac {3}{11}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{68750} \]
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Time = 0.06 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {100, 155, 159, 164, 114, 120} \[ \int \frac {(2+3 x)^{9/2}}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx=\frac {175111 \sqrt {\frac {3}{11}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{68750}+\frac {2911577 \sqrt {\frac {3}{11}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{34375}+\frac {7 (3 x+2)^{7/2}}{11 \sqrt {1-2 x} \sqrt {5 x+3}}-\frac {37 \sqrt {1-2 x} (3 x+2)^{5/2}}{605 \sqrt {5 x+3}}+\frac {10851 \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{3/2}}{15125}+\frac {502941 \sqrt {1-2 x} \sqrt {5 x+3} \sqrt {3 x+2}}{151250} \]
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Rule 100
Rule 114
Rule 120
Rule 155
Rule 159
Rule 164
Rubi steps \begin{align*} \text {integral}& = \frac {7 (2+3 x)^{7/2}}{11 \sqrt {1-2 x} \sqrt {3+5 x}}-\frac {1}{11} \int \frac {(2+3 x)^{5/2} \left (\frac {367}{2}+312 x\right )}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx \\ & = -\frac {37 \sqrt {1-2 x} (2+3 x)^{5/2}}{605 \sqrt {3+5 x}}+\frac {7 (2+3 x)^{7/2}}{11 \sqrt {1-2 x} \sqrt {3+5 x}}-\frac {2}{605} \int \frac {(2+3 x)^{3/2} \left (\frac {13173}{4}+\frac {10851 x}{2}\right )}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx \\ & = -\frac {37 \sqrt {1-2 x} (2+3 x)^{5/2}}{605 \sqrt {3+5 x}}+\frac {7 (2+3 x)^{7/2}}{11 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {10851 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{15125}+\frac {2 \int \frac {\left (-\frac {929925}{4}-\frac {1508823 x}{4}\right ) \sqrt {2+3 x}}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{15125} \\ & = -\frac {37 \sqrt {1-2 x} (2+3 x)^{5/2}}{605 \sqrt {3+5 x}}+\frac {7 (2+3 x)^{7/2}}{11 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {502941 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{151250}+\frac {10851 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{15125}-\frac {2 \int \frac {\frac {66357261}{8}+\frac {26204193 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{226875} \\ & = -\frac {37 \sqrt {1-2 x} (2+3 x)^{5/2}}{605 \sqrt {3+5 x}}+\frac {7 (2+3 x)^{7/2}}{11 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {502941 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{151250}+\frac {10851 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{15125}-\frac {525333 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{137500}-\frac {8734731 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{378125} \\ & = -\frac {37 \sqrt {1-2 x} (2+3 x)^{5/2}}{605 \sqrt {3+5 x}}+\frac {7 (2+3 x)^{7/2}}{11 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {502941 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{151250}+\frac {10851 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{15125}+\frac {2911577 \sqrt {\frac {3}{11}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{34375}+\frac {175111 \sqrt {\frac {3}{11}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{68750} \\ \end{align*}
Result contains complex when optimal does not.
Time = 7.37 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.67 \[ \int \frac {(2+3 x)^{9/2}}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx=\frac {-5 \sqrt {2+3 x} \sqrt {3+5 x} \left (-2892883-3684629 x+2188890 x^2+490050 x^3\right )-5823154 i \sqrt {33-66 x} (3+5 x) E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )+5998265 i \sqrt {33-66 x} (3+5 x) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )}{756250 \sqrt {1-2 x} (3+5 x)} \]
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Time = 1.36 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.76
method | result | size |
default | \(-\frac {\sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (5823154 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-5655507 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-7350750 x^{4}-37733850 x^{3}+33380535 x^{2}+80239535 x +28928830\right )}{756250 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\) | \(145\) |
elliptic | \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (-\frac {2 \left (-20-30 x \right ) \left (\frac {900367}{1210000}+\frac {1500641 x}{1210000}\right )}{\sqrt {\left (-\frac {3}{10}+x^{2}+\frac {1}{10} x \right ) \left (-20-30 x \right )}}+\frac {81 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{250}+\frac {3537 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{2500}-\frac {7373029 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{5293750 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {5823154 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{2646875 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(234\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.07 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.48 \[ \int \frac {(2+3 x)^{9/2}}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx=\frac {450 \, {\left (490050 \, x^{3} + 2188890 \, x^{2} - 3684629 \, x - 2892883\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} + 197853763 \, \sqrt {-30} {\left (10 \, x^{2} + x - 3\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) - 524083860 \, \sqrt {-30} {\left (10 \, x^{2} + x - 3\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )}{68062500 \, {\left (10 \, x^{2} + x - 3\right )}} \]
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Timed out. \[ \int \frac {(2+3 x)^{9/2}}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {(2+3 x)^{9/2}}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx=\int { \frac {{\left (3 \, x + 2\right )}^{\frac {9}{2}}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {(2+3 x)^{9/2}}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx=\int { \frac {{\left (3 \, x + 2\right )}^{\frac {9}{2}}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(2+3 x)^{9/2}}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx=\int \frac {{\left (3\,x+2\right )}^{9/2}}{{\left (1-2\,x\right )}^{3/2}\,{\left (5\,x+3\right )}^{3/2}} \,d x \]
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