Optimal. Leaf size=165 \[ -\frac {13}{80} (1-2 x)^{3/2} (5 x+3)^{7/2}-\frac {1}{20} (1-2 x)^{3/2} (3 x+2) (5 x+3)^{7/2}-\frac {1069 (1-2 x)^{3/2} (5 x+3)^{5/2}}{1280}-\frac {11759 (1-2 x)^{3/2} (5 x+3)^{3/2}}{3072}-\frac {129349 (1-2 x)^{3/2} \sqrt {5 x+3}}{8192}+\frac {1422839 \sqrt {1-2 x} \sqrt {5 x+3}}{81920}+\frac {15651229 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{81920 \sqrt {10}} \]
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Rubi [A] time = 0.05, antiderivative size = 165, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {90, 80, 50, 54, 216} \begin {gather*} -\frac {13}{80} (1-2 x)^{3/2} (5 x+3)^{7/2}-\frac {1}{20} (1-2 x)^{3/2} (3 x+2) (5 x+3)^{7/2}-\frac {1069 (1-2 x)^{3/2} (5 x+3)^{5/2}}{1280}-\frac {11759 (1-2 x)^{3/2} (5 x+3)^{3/2}}{3072}-\frac {129349 (1-2 x)^{3/2} \sqrt {5 x+3}}{8192}+\frac {1422839 \sqrt {1-2 x} \sqrt {5 x+3}}{81920}+\frac {15651229 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{81920 \sqrt {10}} \end {gather*}
Antiderivative was successfully verified.
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Rule 50
Rule 54
Rule 80
Rule 90
Rule 216
Rubi steps
\begin {align*} \int \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{5/2} \, dx &=-\frac {1}{20} (1-2 x)^{3/2} (2+3 x) (3+5 x)^{7/2}-\frac {1}{60} \int \left (-318-\frac {975 x}{2}\right ) \sqrt {1-2 x} (3+5 x)^{5/2} \, dx\\ &=-\frac {13}{80} (1-2 x)^{3/2} (3+5 x)^{7/2}-\frac {1}{20} (1-2 x)^{3/2} (2+3 x) (3+5 x)^{7/2}+\frac {1069}{160} \int \sqrt {1-2 x} (3+5 x)^{5/2} \, dx\\ &=-\frac {1069 (1-2 x)^{3/2} (3+5 x)^{5/2}}{1280}-\frac {13}{80} (1-2 x)^{3/2} (3+5 x)^{7/2}-\frac {1}{20} (1-2 x)^{3/2} (2+3 x) (3+5 x)^{7/2}+\frac {11759}{512} \int \sqrt {1-2 x} (3+5 x)^{3/2} \, dx\\ &=-\frac {11759 (1-2 x)^{3/2} (3+5 x)^{3/2}}{3072}-\frac {1069 (1-2 x)^{3/2} (3+5 x)^{5/2}}{1280}-\frac {13}{80} (1-2 x)^{3/2} (3+5 x)^{7/2}-\frac {1}{20} (1-2 x)^{3/2} (2+3 x) (3+5 x)^{7/2}+\frac {129349 \int \sqrt {1-2 x} \sqrt {3+5 x} \, dx}{2048}\\ &=-\frac {129349 (1-2 x)^{3/2} \sqrt {3+5 x}}{8192}-\frac {11759 (1-2 x)^{3/2} (3+5 x)^{3/2}}{3072}-\frac {1069 (1-2 x)^{3/2} (3+5 x)^{5/2}}{1280}-\frac {13}{80} (1-2 x)^{3/2} (3+5 x)^{7/2}-\frac {1}{20} (1-2 x)^{3/2} (2+3 x) (3+5 x)^{7/2}+\frac {1422839 \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx}{16384}\\ &=\frac {1422839 \sqrt {1-2 x} \sqrt {3+5 x}}{81920}-\frac {129349 (1-2 x)^{3/2} \sqrt {3+5 x}}{8192}-\frac {11759 (1-2 x)^{3/2} (3+5 x)^{3/2}}{3072}-\frac {1069 (1-2 x)^{3/2} (3+5 x)^{5/2}}{1280}-\frac {13}{80} (1-2 x)^{3/2} (3+5 x)^{7/2}-\frac {1}{20} (1-2 x)^{3/2} (2+3 x) (3+5 x)^{7/2}+\frac {15651229 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{163840}\\ &=\frac {1422839 \sqrt {1-2 x} \sqrt {3+5 x}}{81920}-\frac {129349 (1-2 x)^{3/2} \sqrt {3+5 x}}{8192}-\frac {11759 (1-2 x)^{3/2} (3+5 x)^{3/2}}{3072}-\frac {1069 (1-2 x)^{3/2} (3+5 x)^{5/2}}{1280}-\frac {13}{80} (1-2 x)^{3/2} (3+5 x)^{7/2}-\frac {1}{20} (1-2 x)^{3/2} (2+3 x) (3+5 x)^{7/2}+\frac {15651229 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{81920 \sqrt {5}}\\ &=\frac {1422839 \sqrt {1-2 x} \sqrt {3+5 x}}{81920}-\frac {129349 (1-2 x)^{3/2} \sqrt {3+5 x}}{8192}-\frac {11759 (1-2 x)^{3/2} (3+5 x)^{3/2}}{3072}-\frac {1069 (1-2 x)^{3/2} (3+5 x)^{5/2}}{1280}-\frac {13}{80} (1-2 x)^{3/2} (3+5 x)^{7/2}-\frac {1}{20} (1-2 x)^{3/2} (2+3 x) (3+5 x)^{7/2}+\frac {15651229 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{81920 \sqrt {10}}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 84, normalized size = 0.51 \begin {gather*} \frac {46953687 \sqrt {20 x-10} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )-10 \sqrt {5 x+3} \left (18432000 x^6+47001600 x^5+37666560 x^4+128192 x^3-16472168 x^2-12064222 x+6023169\right )}{2457600 \sqrt {1-2 x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.25, size = 157, normalized size = 0.95 \begin {gather*} -\frac {14641 \sqrt {1-2 x} \left (\frac {10021875 (1-2 x)^5}{(5 x+3)^5}+\frac {22716250 (1-2 x)^4}{(5 x+3)^4}+\frac {21166200 (1-2 x)^3}{(5 x+3)^3}+\frac {10285008 (1-2 x)^2}{(5 x+3)^2}+\frac {2560240 (1-2 x)}{5 x+3}-102624\right )}{245760 \sqrt {5 x+3} \left (\frac {5 (1-2 x)}{5 x+3}+2\right )^6}-\frac {15651229 \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}} \sqrt {1-2 x}}{\sqrt {5 x+3}}\right )}{81920 \sqrt {10}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.60, size = 82, normalized size = 0.50 \begin {gather*} \frac {1}{245760} \, {\left (9216000 \, x^{5} + 28108800 \, x^{4} + 32887680 \, x^{3} + 16507936 \, x^{2} + 17884 \, x - 6023169\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {15651229}{1638400} \, \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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.16, size = 356, normalized size = 2.16 \begin {gather*} \frac {3}{102400000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (4 \, {\left (16 \, {\left (100 \, x - 311\right )} {\left (5 \, x + 3\right )} + 46071\right )} {\left (5 \, x + 3\right )} - 775911\right )} {\left (5 \, x + 3\right )} + 15385695\right )} {\left (5 \, x + 3\right )} - 99422145\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 220189365 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {47}{12800000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (12 \, {\left (80 \, x - 203\right )} {\left (5 \, x + 3\right )} + 19073\right )} {\left (5 \, x + 3\right )} - 506185\right )} {\left (5 \, x + 3\right )} + 4031895\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 10392195 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {883}{1920000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (60 \, x - 119\right )} {\left (5 \, x + 3\right )} + 6163\right )} {\left (5 \, x + 3\right )} - 66189\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 184305 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {921}{40000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (40 \, x - 59\right )} {\left (5 \, x + 3\right )} + 1293\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 4785 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {54}{125} \, \sqrt {5} {\left (2 \, {\left (20 \, x - 23\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 143 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {54}{25} \, \sqrt {5} {\left (11 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + 2 \, \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 138, normalized size = 0.84 \begin {gather*} \frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (184320000 \sqrt {-10 x^{2}-x +3}\, x^{5}+562176000 \sqrt {-10 x^{2}-x +3}\, x^{4}+657753600 \sqrt {-10 x^{2}-x +3}\, x^{3}+330158720 \sqrt {-10 x^{2}-x +3}\, x^{2}+357680 \sqrt {-10 x^{2}-x +3}\, x +46953687 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-120463380 \sqrt {-10 x^{2}-x +3}\right )}{4915200 \sqrt {-10 x^{2}-x +3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.46, size = 104, normalized size = 0.63 \begin {gather*} -\frac {15}{4} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{3} - \frac {177}{16} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} - \frac {17153}{1280} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x - \frac {133567}{15360} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {129349}{4096} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {15651229}{1638400} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {129349}{81920} \, \sqrt {-10 \, x^{2} - x + 3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \sqrt {1-2\,x}\,{\left (3\,x+2\right )}^2\,{\left (5\,x+3\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 121.61, size = 694, normalized size = 4.21 \begin {gather*} - \frac {5929 \sqrt {2} \left (\begin {cases} \frac {121 \sqrt {5} \left (- \frac {\sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6} \left (20 x + 1\right )}{121} + \operatorname {asin}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}\right )}{200} & \text {for}\: x \leq \frac {1}{2} \wedge x > - \frac {3}{5} \end {cases}\right )}{32} + \frac {1309 \sqrt {2} \left (\begin {cases} \frac {1331 \sqrt {5} \left (- \frac {5 \sqrt {5} \left (1 - 2 x\right )^{\frac {3}{2}} \left (10 x + 6\right )^{\frac {3}{2}}}{7986} - \frac {\sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6} \left (20 x + 1\right )}{1936} + \frac {\operatorname {asin}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{16}\right )}{125} & \text {for}\: x \leq \frac {1}{2} \wedge x > - \frac {3}{5} \end {cases}\right )}{4} - \frac {3467 \sqrt {2} \left (\begin {cases} \frac {14641 \sqrt {5} \left (- \frac {5 \sqrt {5} \left (1 - 2 x\right )^{\frac {3}{2}} \left (10 x + 6\right )^{\frac {3}{2}}}{7986} - \frac {\sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6} \left (20 x + 1\right )}{3872} - \frac {\sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6} \left (12100 x - 2000 \left (1 - 2 x\right )^{3} + 6600 \left (1 - 2 x\right )^{2} - 4719\right )}{1874048} + \frac {5 \operatorname {asin}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{128}\right )}{625} & \text {for}\: x \leq \frac {1}{2} \wedge x > - \frac {3}{5} \end {cases}\right )}{16} + \frac {255 \sqrt {2} \left (\begin {cases} \frac {161051 \sqrt {5} \left (\frac {5 \sqrt {5} \left (1 - 2 x\right )^{\frac {5}{2}} \left (10 x + 6\right )^{\frac {5}{2}}}{322102} - \frac {5 \sqrt {5} \left (1 - 2 x\right )^{\frac {3}{2}} \left (10 x + 6\right )^{\frac {3}{2}}}{7986} - \frac {\sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6} \left (20 x + 1\right )}{7744} - \frac {3 \sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6} \left (12100 x - 2000 \left (1 - 2 x\right )^{3} + 6600 \left (1 - 2 x\right )^{2} - 4719\right )}{3748096} + \frac {7 \operatorname {asin}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{256}\right )}{3125} & \text {for}\: x \leq \frac {1}{2} \wedge x > - \frac {3}{5} \end {cases}\right )}{4} - \frac {225 \sqrt {2} \left (\begin {cases} \frac {1771561 \sqrt {5} \left (\frac {5 \sqrt {5} \left (1 - 2 x\right )^{\frac {5}{2}} \left (10 x + 6\right )^{\frac {5}{2}}}{161051} + \frac {5 \sqrt {5} \left (1 - 2 x\right )^{\frac {3}{2}} \left (10 x + 6\right )^{\frac {3}{2}} \left (20 x + 1\right )^{3}}{170069856} - \frac {5 \sqrt {5} \left (1 - 2 x\right )^{\frac {3}{2}} \left (10 x + 6\right )^{\frac {3}{2}}}{7986} - \frac {\sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6} \left (20 x + 1\right )}{15488} - \frac {13 \sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6} \left (12100 x - 2000 \left (1 - 2 x\right )^{3} + 6600 \left (1 - 2 x\right )^{2} - 4719\right )}{14992384} + \frac {21 \operatorname {asin}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{1024}\right )}{15625} & \text {for}\: x \leq \frac {1}{2} \wedge x > - \frac {3}{5} \end {cases}\right )}{32} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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