Optimal. Leaf size=216 \[ -\frac {1}{30} (3 x+2)^2 (5 x+3)^{7/2} (1-2 x)^{7/2}-\frac {526103 (5 x+3)^{5/2} (1-2 x)^{7/2}}{768000}-\frac {5787133 (5 x+3)^{3/2} (1-2 x)^{7/2}}{3072000}-\frac {(5 x+3)^{7/2} (170940 x+245011) (1-2 x)^{7/2}}{672000}-\frac {63658463 \sqrt {5 x+3} (1-2 x)^{7/2}}{16384000}+\frac {700243093 \sqrt {5 x+3} (1-2 x)^{5/2}}{491520000}+\frac {7702674023 \sqrt {5 x+3} (1-2 x)^{3/2}}{1966080000}+\frac {84729414253 \sqrt {5 x+3} \sqrt {1-2 x}}{6553600000}+\frac {932023556783 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{6553600000 \sqrt {10}} \]
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Rubi [A] time = 0.08, antiderivative size = 216, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {100, 147, 50, 54, 216} \[ -\frac {1}{30} (3 x+2)^2 (5 x+3)^{7/2} (1-2 x)^{7/2}-\frac {526103 (5 x+3)^{5/2} (1-2 x)^{7/2}}{768000}-\frac {5787133 (5 x+3)^{3/2} (1-2 x)^{7/2}}{3072000}-\frac {(5 x+3)^{7/2} (170940 x+245011) (1-2 x)^{7/2}}{672000}-\frac {63658463 \sqrt {5 x+3} (1-2 x)^{7/2}}{16384000}+\frac {700243093 \sqrt {5 x+3} (1-2 x)^{5/2}}{491520000}+\frac {7702674023 \sqrt {5 x+3} (1-2 x)^{3/2}}{1966080000}+\frac {84729414253 \sqrt {5 x+3} \sqrt {1-2 x}}{6553600000}+\frac {932023556783 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{6553600000 \sqrt {10}} \]
Antiderivative was successfully verified.
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Rule 50
Rule 54
Rule 100
Rule 147
Rule 216
Rubi steps
\begin {align*} \int (1-2 x)^{5/2} (2+3 x)^3 (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}{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 \operatorname {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*}
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Mathematica [A] time = 0.24, size = 99, normalized size = 0.46 \[ \frac {19572494692443 \sqrt {20 x-10} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )-10 \sqrt {5 x+3} \left (82575360000000 x^9+163602432000000 x^8+16806297600000 x^7-152280832000000 x^6-74172819968000 x^5+48825346630400 x^4+38603789187520 x^3-3650664293320 x^2-9390934073894 x+1496712721437\right )}{1376256000000 \sqrt {1-2 x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.89, size = 97, normalized size = 0.45 \[ \frac {1}{137625600000} \, {\left (41287680000000 \, x^{8} + 102445056000000 \, x^{7} + 59625676800000 \, x^{6} - 46327577600000 \, x^{5} - 60250198784000 \, x^{4} - 5712426076800 \, x^{3} + 16445681555360 \, x^{2} + 6397508631020 \, x - 1496712721437\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {932023556783}{131072000000} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 2.07, size = 653, normalized size = 3.02 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 189, normalized size = 0.88 \[ \frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (825753600000000 \sqrt {-10 x^{2}-x +3}\, x^{8}+2048901120000000 \sqrt {-10 x^{2}-x +3}\, x^{7}+1192513536000000 \sqrt {-10 x^{2}-x +3}\, x^{6}-926551552000000 \sqrt {-10 x^{2}-x +3}\, x^{5}-1205003975680000 \sqrt {-10 x^{2}-x +3}\, x^{4}-114248521536000 \sqrt {-10 x^{2}-x +3}\, x^{3}+328913631107200 \sqrt {-10 x^{2}-x +3}\, x^{2}+127950172620400 \sqrt {-10 x^{2}-x +3}\, x +19572494692443 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-29934254428740 \sqrt {-10 x^{2}-x +3}\right )}{2752512000000 \sqrt {-10 x^{2}-x +3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.12, size = 145, normalized size = 0.67 \[ -\frac {3}{10} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {7}{2}} x^{2} - \frac {1047}{1600} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {7}{2}} x - \frac {111537}{224000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {7}{2}} + \frac {526103}{384000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} x + \frac {526103}{7680000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} + \frac {63658463}{12288000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + \frac {63658463}{245760000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {7702674023}{327680000} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {932023556783}{131072000000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {7702674023}{6553600000} \, \sqrt {-10 \, x^{2} - x + 3} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (1-2\,x\right )}^{5/2}\,{\left (3\,x+2\right )}^3\,{\left (5\,x+3\right )}^{5/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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