Optimal. Leaf size=160 \[ -\frac {1}{20} (1-2 x)^{5/2} (5 x+3)^{7/2}-\frac {17}{80} (1-2 x)^{5/2} (5 x+3)^{5/2}-\frac {187}{256} (1-2 x)^{5/2} (5 x+3)^{3/2}-\frac {2057 (1-2 x)^{5/2} \sqrt {5 x+3}}{1024}+\frac {22627 (1-2 x)^{3/2} \sqrt {5 x+3}}{20480}+\frac {746691 \sqrt {1-2 x} \sqrt {5 x+3}}{204800}+\frac {8213601 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{204800 \sqrt {10}} \]
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Rubi [A] time = 0.05, antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {80, 50, 54, 216} \begin {gather*} -\frac {1}{20} (1-2 x)^{5/2} (5 x+3)^{7/2}-\frac {17}{80} (1-2 x)^{5/2} (5 x+3)^{5/2}-\frac {187}{256} (1-2 x)^{5/2} (5 x+3)^{3/2}-\frac {2057 (1-2 x)^{5/2} \sqrt {5 x+3}}{1024}+\frac {22627 (1-2 x)^{3/2} \sqrt {5 x+3}}{20480}+\frac {746691 \sqrt {1-2 x} \sqrt {5 x+3}}{204800}+\frac {8213601 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{204800 \sqrt {10}} \end {gather*}
Antiderivative was successfully verified.
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Rule 50
Rule 54
Rule 80
Rule 216
Rubi steps
\begin {align*} \int (1-2 x)^{3/2} (2+3 x) (3+5 x)^{5/2} \, dx &=-\frac {1}{20} (1-2 x)^{5/2} (3+5 x)^{7/2}+\frac {17}{8} \int (1-2 x)^{3/2} (3+5 x)^{5/2} \, dx\\ &=-\frac {17}{80} (1-2 x)^{5/2} (3+5 x)^{5/2}-\frac {1}{20} (1-2 x)^{5/2} (3+5 x)^{7/2}+\frac {187}{32} \int (1-2 x)^{3/2} (3+5 x)^{3/2} \, dx\\ &=-\frac {187}{256} (1-2 x)^{5/2} (3+5 x)^{3/2}-\frac {17}{80} (1-2 x)^{5/2} (3+5 x)^{5/2}-\frac {1}{20} (1-2 x)^{5/2} (3+5 x)^{7/2}+\frac {6171}{512} \int (1-2 x)^{3/2} \sqrt {3+5 x} \, dx\\ &=-\frac {2057 (1-2 x)^{5/2} \sqrt {3+5 x}}{1024}-\frac {187}{256} (1-2 x)^{5/2} (3+5 x)^{3/2}-\frac {17}{80} (1-2 x)^{5/2} (3+5 x)^{5/2}-\frac {1}{20} (1-2 x)^{5/2} (3+5 x)^{7/2}+\frac {22627 \int \frac {(1-2 x)^{3/2}}{\sqrt {3+5 x}} \, dx}{2048}\\ &=\frac {22627 (1-2 x)^{3/2} \sqrt {3+5 x}}{20480}-\frac {2057 (1-2 x)^{5/2} \sqrt {3+5 x}}{1024}-\frac {187}{256} (1-2 x)^{5/2} (3+5 x)^{3/2}-\frac {17}{80} (1-2 x)^{5/2} (3+5 x)^{5/2}-\frac {1}{20} (1-2 x)^{5/2} (3+5 x)^{7/2}+\frac {746691 \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx}{40960}\\ &=\frac {746691 \sqrt {1-2 x} \sqrt {3+5 x}}{204800}+\frac {22627 (1-2 x)^{3/2} \sqrt {3+5 x}}{20480}-\frac {2057 (1-2 x)^{5/2} \sqrt {3+5 x}}{1024}-\frac {187}{256} (1-2 x)^{5/2} (3+5 x)^{3/2}-\frac {17}{80} (1-2 x)^{5/2} (3+5 x)^{5/2}-\frac {1}{20} (1-2 x)^{5/2} (3+5 x)^{7/2}+\frac {8213601 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{409600}\\ &=\frac {746691 \sqrt {1-2 x} \sqrt {3+5 x}}{204800}+\frac {22627 (1-2 x)^{3/2} \sqrt {3+5 x}}{20480}-\frac {2057 (1-2 x)^{5/2} \sqrt {3+5 x}}{1024}-\frac {187}{256} (1-2 x)^{5/2} (3+5 x)^{3/2}-\frac {17}{80} (1-2 x)^{5/2} (3+5 x)^{5/2}-\frac {1}{20} (1-2 x)^{5/2} (3+5 x)^{7/2}+\frac {8213601 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{204800 \sqrt {5}}\\ &=\frac {746691 \sqrt {1-2 x} \sqrt {3+5 x}}{204800}+\frac {22627 (1-2 x)^{3/2} \sqrt {3+5 x}}{20480}-\frac {2057 (1-2 x)^{5/2} \sqrt {3+5 x}}{1024}-\frac {187}{256} (1-2 x)^{5/2} (3+5 x)^{3/2}-\frac {17}{80} (1-2 x)^{5/2} (3+5 x)^{5/2}-\frac {1}{20} (1-2 x)^{5/2} (3+5 x)^{7/2}+\frac {8213601 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{204800 \sqrt {10}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 84, normalized size = 0.52 \begin {gather*} \frac {10 \sqrt {5 x+3} \left (10240000 x^6+11776000 x^5-5536000 x^4-9933120 x^3-211240 x^2+3335698 x-555399\right )+8213601 \sqrt {20 x-10} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )}{2048000 \sqrt {1-2 x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.25, size = 157, normalized size = 0.98 \begin {gather*} -\frac {161051 \sqrt {1-2 x} \left (\frac {159375 (1-2 x)^5}{(5 x+3)^5}+\frac {361250 (1-2 x)^4}{(5 x+3)^4}+\frac {336600 (1-2 x)^3}{(5 x+3)^3}+\frac {152080 (1-2 x)^2}{(5 x+3)^2}-\frac {23120 (1-2 x)}{5 x+3}-1632\right )}{204800 \sqrt {5 x+3} \left (\frac {5 (1-2 x)}{5 x+3}+2\right )^6}-\frac {8213601 \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}} \sqrt {1-2 x}}{\sqrt {5 x+3}}\right )}{204800 \sqrt {10}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.36, size = 82, normalized size = 0.51 \begin {gather*} -\frac {1}{204800} \, {\left (5120000 \, x^{5} + 8448000 \, x^{4} + 1456000 \, x^{3} - 4238560 \, x^{2} - 2224900 \, x + 555399\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {8213601}{4096000} \, \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 = 1.43, size = 356, normalized size = 2.22 \begin {gather*} -\frac {1}{51200000} \, \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 {59}{38400000} \, \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 {157}{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 {51}{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 {243}{2000} \, \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 {27}{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 = 0.86 \begin {gather*} \frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (-102400000 \sqrt {-10 x^{2}-x +3}\, x^{5}-168960000 \sqrt {-10 x^{2}-x +3}\, x^{4}-29120000 \sqrt {-10 x^{2}-x +3}\, x^{3}+84771200 \sqrt {-10 x^{2}-x +3}\, x^{2}+44498000 \sqrt {-10 x^{2}-x +3}\, x +8213601 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-11107980 \sqrt {-10 x^{2}-x +3}\right )}{4096000 \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 = 1.42, size = 99, normalized size = 0.62 \begin {gather*} -\frac {1}{4} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} x - \frac {29}{80} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} + \frac {187}{128} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + \frac {187}{2560} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {67881}{10240} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {8213601}{4096000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {67881}{204800} \, \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 {\left (1-2\,x\right )}^{3/2}\,\left (3\,x+2\right )\,{\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 [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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