Integrand size = 26, antiderivative size = 216 \[ \int (1-2 x)^{5/2} (2+3 x)^3 (3+5 x)^{5/2} \, dx=\frac {84729414253 \sqrt {1-2 x} \sqrt {3+5 x}}{6553600000}+\frac {7702674023 (1-2 x)^{3/2} \sqrt {3+5 x}}{1966080000}+\frac {700243093 (1-2 x)^{5/2} \sqrt {3+5 x}}{491520000}-\frac {63658463 (1-2 x)^{7/2} \sqrt {3+5 x}}{16384000}-\frac {5787133 (1-2 x)^{7/2} (3+5 x)^{3/2}}{3072000}-\frac {526103 (1-2 x)^{7/2} (3+5 x)^{5/2}}{768000}-\frac {1}{30} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{7/2}-\frac {(1-2 x)^{7/2} (3+5 x)^{7/2} (245011+170940 x)}{672000}+\frac {932023556783 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{6553600000 \sqrt {10}} \]
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Time = 0.05 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {102, 152, 52, 56, 222} \[ \int (1-2 x)^{5/2} (2+3 x)^3 (3+5 x)^{5/2} \, dx=\frac {932023556783 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{6553600000 \sqrt {10}}-\frac {1}{30} (3 x+2)^2 (5 x+3)^{7/2} (1-2 x)^{7/2}-\frac {526103 (5 x+3)^{5/2} (1-2 x)^{7/2}}{768000}-\frac {5787133 (5 x+3)^{3/2} (1-2 x)^{7/2}}{3072000}-\frac {(5 x+3)^{7/2} (170940 x+245011) (1-2 x)^{7/2}}{672000}-\frac {63658463 \sqrt {5 x+3} (1-2 x)^{7/2}}{16384000}+\frac {700243093 \sqrt {5 x+3} (1-2 x)^{5/2}}{491520000}+\frac {7702674023 \sqrt {5 x+3} (1-2 x)^{3/2}}{1966080000}+\frac {84729414253 \sqrt {5 x+3} \sqrt {1-2 x}}{6553600000} \]
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Rule 52
Rule 56
Rule 102
Rule 152
Rule 222
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{30} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{7/2}-\frac {1}{90} \int \left (-393-\frac {1221 x}{2}\right ) (1-2 x)^{5/2} (2+3 x) (3+5 x)^{5/2} \, dx \\ & = -\frac {1}{30} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{7/2}-\frac {(1-2 x)^{7/2} (3+5 x)^{7/2} (245011+170940 x)}{672000}+\frac {526103 \int (1-2 x)^{5/2} (3+5 x)^{5/2} \, dx}{64000} \\ & = -\frac {526103 (1-2 x)^{7/2} (3+5 x)^{5/2}}{768000}-\frac {1}{30} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{7/2}-\frac {(1-2 x)^{7/2} (3+5 x)^{7/2} (245011+170940 x)}{672000}+\frac {5787133 \int (1-2 x)^{5/2} (3+5 x)^{3/2} \, dx}{307200} \\ & = -\frac {5787133 (1-2 x)^{7/2} (3+5 x)^{3/2}}{3072000}-\frac {526103 (1-2 x)^{7/2} (3+5 x)^{5/2}}{768000}-\frac {1}{30} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{7/2}-\frac {(1-2 x)^{7/2} (3+5 x)^{7/2} (245011+170940 x)}{672000}+\frac {63658463 \int (1-2 x)^{5/2} \sqrt {3+5 x} \, dx}{2048000} \\ & = -\frac {63658463 (1-2 x)^{7/2} \sqrt {3+5 x}}{16384000}-\frac {5787133 (1-2 x)^{7/2} (3+5 x)^{3/2}}{3072000}-\frac {526103 (1-2 x)^{7/2} (3+5 x)^{5/2}}{768000}-\frac {1}{30} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{7/2}-\frac {(1-2 x)^{7/2} (3+5 x)^{7/2} (245011+170940 x)}{672000}+\frac {700243093 \int \frac {(1-2 x)^{5/2}}{\sqrt {3+5 x}} \, dx}{32768000} \\ & = \frac {700243093 (1-2 x)^{5/2} \sqrt {3+5 x}}{491520000}-\frac {63658463 (1-2 x)^{7/2} \sqrt {3+5 x}}{16384000}-\frac {5787133 (1-2 x)^{7/2} (3+5 x)^{3/2}}{3072000}-\frac {526103 (1-2 x)^{7/2} (3+5 x)^{5/2}}{768000}-\frac {1}{30} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{7/2}-\frac {(1-2 x)^{7/2} (3+5 x)^{7/2} (245011+170940 x)}{672000}+\frac {7702674023 \int \frac {(1-2 x)^{3/2}}{\sqrt {3+5 x}} \, dx}{196608000} \\ & = \frac {7702674023 (1-2 x)^{3/2} \sqrt {3+5 x}}{1966080000}+\frac {700243093 (1-2 x)^{5/2} \sqrt {3+5 x}}{491520000}-\frac {63658463 (1-2 x)^{7/2} \sqrt {3+5 x}}{16384000}-\frac {5787133 (1-2 x)^{7/2} (3+5 x)^{3/2}}{3072000}-\frac {526103 (1-2 x)^{7/2} (3+5 x)^{5/2}}{768000}-\frac {1}{30} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{7/2}-\frac {(1-2 x)^{7/2} (3+5 x)^{7/2} (245011+170940 x)}{672000}+\frac {84729414253 \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx}{1310720000} \\ & = \frac {84729414253 \sqrt {1-2 x} \sqrt {3+5 x}}{6553600000}+\frac {7702674023 (1-2 x)^{3/2} \sqrt {3+5 x}}{1966080000}+\frac {700243093 (1-2 x)^{5/2} \sqrt {3+5 x}}{491520000}-\frac {63658463 (1-2 x)^{7/2} \sqrt {3+5 x}}{16384000}-\frac {5787133 (1-2 x)^{7/2} (3+5 x)^{3/2}}{3072000}-\frac {526103 (1-2 x)^{7/2} (3+5 x)^{5/2}}{768000}-\frac {1}{30} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{7/2}-\frac {(1-2 x)^{7/2} (3+5 x)^{7/2} (245011+170940 x)}{672000}+\frac {932023556783 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{13107200000} \\ & = \frac {84729414253 \sqrt {1-2 x} \sqrt {3+5 x}}{6553600000}+\frac {7702674023 (1-2 x)^{3/2} \sqrt {3+5 x}}{1966080000}+\frac {700243093 (1-2 x)^{5/2} \sqrt {3+5 x}}{491520000}-\frac {63658463 (1-2 x)^{7/2} \sqrt {3+5 x}}{16384000}-\frac {5787133 (1-2 x)^{7/2} (3+5 x)^{3/2}}{3072000}-\frac {526103 (1-2 x)^{7/2} (3+5 x)^{5/2}}{768000}-\frac {1}{30} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{7/2}-\frac {(1-2 x)^{7/2} (3+5 x)^{7/2} (245011+170940 x)}{672000}+\frac {932023556783 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{6553600000 \sqrt {5}} \\ & = \frac {84729414253 \sqrt {1-2 x} \sqrt {3+5 x}}{6553600000}+\frac {7702674023 (1-2 x)^{3/2} \sqrt {3+5 x}}{1966080000}+\frac {700243093 (1-2 x)^{5/2} \sqrt {3+5 x}}{491520000}-\frac {63658463 (1-2 x)^{7/2} \sqrt {3+5 x}}{16384000}-\frac {5787133 (1-2 x)^{7/2} (3+5 x)^{3/2}}{3072000}-\frac {526103 (1-2 x)^{7/2} (3+5 x)^{5/2}}{768000}-\frac {1}{30} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{7/2}-\frac {(1-2 x)^{7/2} (3+5 x)^{7/2} (245011+170940 x)}{672000}+\frac {932023556783 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{6553600000 \sqrt {10}} \\ \end{align*}
Time = 0.33 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.48 \[ \int (1-2 x)^{5/2} (2+3 x)^3 (3+5 x)^{5/2} \, dx=\frac {10 \sqrt {1-2 x} \left (-4490138164311+11708962285875 x+81324587821180 x^2+65091129546400 x^3-209312726736000 x^4-440233726720000 x^5-52760857600000 x^6+605463552000000 x^7+636088320000000 x^8+206438400000000 x^9\right )-19572494692443 \sqrt {30+50 x} \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{1376256000000 \sqrt {3+5 x}} \]
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Time = 1.22 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.59
method | result | size |
risch | \(-\frac {\left (41287680000000 x^{8}+102445056000000 x^{7}+59625676800000 x^{6}-46327577600000 x^{5}-60250198784000 x^{4}-5712426076800 x^{3}+16445681555360 x^{2}+6397508631020 x -1496712721437\right ) \left (-1+2 x \right ) \sqrt {3+5 x}\, \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{137625600000 \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {932023556783 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{131072000000 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(128\) |
default | \(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (825753600000000 \sqrt {-10 x^{2}-x +3}\, x^{8}+2048901120000000 \sqrt {-10 x^{2}-x +3}\, x^{7}+1192513536000000 \sqrt {-10 x^{2}-x +3}\, x^{6}-926551552000000 x^{5} \sqrt {-10 x^{2}-x +3}-1205003975680000 x^{4} \sqrt {-10 x^{2}-x +3}-114248521536000 x^{3} \sqrt {-10 x^{2}-x +3}+328913631107200 x^{2} \sqrt {-10 x^{2}-x +3}+19572494692443 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+127950172620400 x \sqrt {-10 x^{2}-x +3}-29934254428740 \sqrt {-10 x^{2}-x +3}\right )}{2752512000000 \sqrt {-10 x^{2}-x +3}}\) | \(189\) |
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Time = 0.23 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.45 \[ \int (1-2 x)^{5/2} (2+3 x)^3 (3+5 x)^{5/2} \, dx=\frac {1}{137625600000} \, {\left (41287680000000 \, x^{8} + 102445056000000 \, x^{7} + 59625676800000 \, x^{6} - 46327577600000 \, x^{5} - 60250198784000 \, x^{4} - 5712426076800 \, x^{3} + 16445681555360 \, x^{2} + 6397508631020 \, x - 1496712721437\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {932023556783}{131072000000} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \]
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\[ \int (1-2 x)^{5/2} (2+3 x)^3 (3+5 x)^{5/2} \, dx=\int \left (1 - 2 x\right )^{\frac {5}{2}} \left (3 x + 2\right )^{3} \left (5 x + 3\right )^{\frac {5}{2}}\, dx \]
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Time = 0.29 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.67 \[ \int (1-2 x)^{5/2} (2+3 x)^3 (3+5 x)^{5/2} \, dx=-\frac {3}{10} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {7}{2}} x^{2} - \frac {1047}{1600} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {7}{2}} x - \frac {111537}{224000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {7}{2}} + \frac {526103}{384000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} x + \frac {526103}{7680000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} + \frac {63658463}{12288000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + \frac {63658463}{245760000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {7702674023}{327680000} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {932023556783}{131072000000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {7702674023}{6553600000} \, \sqrt {-10 \, x^{2} - x + 3} \]
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Leaf count of result is larger than twice the leaf count of optimal. 653 vs. \(2 (159) = 318\).
Time = 0.43 (sec) , antiderivative size = 653, normalized size of antiderivative = 3.02 \[ \int (1-2 x)^{5/2} (2+3 x)^3 (3+5 x)^{5/2} \, dx=\text {Too large to display} \]
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Timed out. \[ \int (1-2 x)^{5/2} (2+3 x)^3 (3+5 x)^{5/2} \, dx=\int {\left (1-2\,x\right )}^{5/2}\,{\left (3\,x+2\right )}^3\,{\left (5\,x+3\right )}^{5/2} \,d x \]
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