Optimal. Leaf size=182 \[ \frac {5958887 \sqrt {1-2 x} \sqrt {3+5 x}}{2048000}+\frac {541717 (1-2 x)^{3/2} \sqrt {3+5 x}}{614400}+\frac {49247 (1-2 x)^{5/2} \sqrt {3+5 x}}{153600}-\frac {4477 (1-2 x)^{7/2} \sqrt {3+5 x}}{5120}-\frac {407}{960} (1-2 x)^{7/2} (3+5 x)^{3/2}-\frac {37}{240} (1-2 x)^{7/2} (3+5 x)^{5/2}-\frac {3}{70} (1-2 x)^{7/2} (3+5 x)^{7/2}+\frac {65547757 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{2048000 \sqrt {10}} \]
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Rubi [A]
time = 0.04, antiderivative size = 182, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {81, 52, 56, 222}
\begin {gather*} \frac {65547757 \text {ArcSin}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{2048000 \sqrt {10}}-\frac {3}{70} (5 x+3)^{7/2} (1-2 x)^{7/2}-\frac {37}{240} (5 x+3)^{5/2} (1-2 x)^{7/2}-\frac {407}{960} (5 x+3)^{3/2} (1-2 x)^{7/2}-\frac {4477 \sqrt {5 x+3} (1-2 x)^{7/2}}{5120}+\frac {49247 \sqrt {5 x+3} (1-2 x)^{5/2}}{153600}+\frac {541717 \sqrt {5 x+3} (1-2 x)^{3/2}}{614400}+\frac {5958887 \sqrt {5 x+3} \sqrt {1-2 x}}{2048000} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 56
Rule 81
Rule 222
Rubi steps
\begin {align*} \int (1-2 x)^{5/2} (2+3 x) (3+5 x)^{5/2} \, dx &=-\frac {3}{70} (1-2 x)^{7/2} (3+5 x)^{7/2}+\frac {37}{20} \int (1-2 x)^{5/2} (3+5 x)^{5/2} \, dx\\ &=-\frac {37}{240} (1-2 x)^{7/2} (3+5 x)^{5/2}-\frac {3}{70} (1-2 x)^{7/2} (3+5 x)^{7/2}+\frac {407}{96} \int (1-2 x)^{5/2} (3+5 x)^{3/2} \, dx\\ &=-\frac {407}{960} (1-2 x)^{7/2} (3+5 x)^{3/2}-\frac {37}{240} (1-2 x)^{7/2} (3+5 x)^{5/2}-\frac {3}{70} (1-2 x)^{7/2} (3+5 x)^{7/2}+\frac {4477}{640} \int (1-2 x)^{5/2} \sqrt {3+5 x} \, dx\\ &=-\frac {4477 (1-2 x)^{7/2} \sqrt {3+5 x}}{5120}-\frac {407}{960} (1-2 x)^{7/2} (3+5 x)^{3/2}-\frac {37}{240} (1-2 x)^{7/2} (3+5 x)^{5/2}-\frac {3}{70} (1-2 x)^{7/2} (3+5 x)^{7/2}+\frac {49247 \int \frac {(1-2 x)^{5/2}}{\sqrt {3+5 x}} \, dx}{10240}\\ &=\frac {49247 (1-2 x)^{5/2} \sqrt {3+5 x}}{153600}-\frac {4477 (1-2 x)^{7/2} \sqrt {3+5 x}}{5120}-\frac {407}{960} (1-2 x)^{7/2} (3+5 x)^{3/2}-\frac {37}{240} (1-2 x)^{7/2} (3+5 x)^{5/2}-\frac {3}{70} (1-2 x)^{7/2} (3+5 x)^{7/2}+\frac {541717 \int \frac {(1-2 x)^{3/2}}{\sqrt {3+5 x}} \, dx}{61440}\\ &=\frac {541717 (1-2 x)^{3/2} \sqrt {3+5 x}}{614400}+\frac {49247 (1-2 x)^{5/2} \sqrt {3+5 x}}{153600}-\frac {4477 (1-2 x)^{7/2} \sqrt {3+5 x}}{5120}-\frac {407}{960} (1-2 x)^{7/2} (3+5 x)^{3/2}-\frac {37}{240} (1-2 x)^{7/2} (3+5 x)^{5/2}-\frac {3}{70} (1-2 x)^{7/2} (3+5 x)^{7/2}+\frac {5958887 \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx}{409600}\\ &=\frac {5958887 \sqrt {1-2 x} \sqrt {3+5 x}}{2048000}+\frac {541717 (1-2 x)^{3/2} \sqrt {3+5 x}}{614400}+\frac {49247 (1-2 x)^{5/2} \sqrt {3+5 x}}{153600}-\frac {4477 (1-2 x)^{7/2} \sqrt {3+5 x}}{5120}-\frac {407}{960} (1-2 x)^{7/2} (3+5 x)^{3/2}-\frac {37}{240} (1-2 x)^{7/2} (3+5 x)^{5/2}-\frac {3}{70} (1-2 x)^{7/2} (3+5 x)^{7/2}+\frac {65547757 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{4096000}\\ &=\frac {5958887 \sqrt {1-2 x} \sqrt {3+5 x}}{2048000}+\frac {541717 (1-2 x)^{3/2} \sqrt {3+5 x}}{614400}+\frac {49247 (1-2 x)^{5/2} \sqrt {3+5 x}}{153600}-\frac {4477 (1-2 x)^{7/2} \sqrt {3+5 x}}{5120}-\frac {407}{960} (1-2 x)^{7/2} (3+5 x)^{3/2}-\frac {37}{240} (1-2 x)^{7/2} (3+5 x)^{5/2}-\frac {3}{70} (1-2 x)^{7/2} (3+5 x)^{7/2}+\frac {65547757 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{2048000 \sqrt {5}}\\ &=\frac {5958887 \sqrt {1-2 x} \sqrt {3+5 x}}{2048000}+\frac {541717 (1-2 x)^{3/2} \sqrt {3+5 x}}{614400}+\frac {49247 (1-2 x)^{5/2} \sqrt {3+5 x}}{153600}-\frac {4477 (1-2 x)^{7/2} \sqrt {3+5 x}}{5120}-\frac {407}{960} (1-2 x)^{7/2} (3+5 x)^{3/2}-\frac {37}{240} (1-2 x)^{7/2} (3+5 x)^{5/2}-\frac {3}{70} (1-2 x)^{7/2} (3+5 x)^{7/2}+\frac {65547757 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{2048000 \sqrt {10}}\\ \end {align*}
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Mathematica [A]
time = 0.25, size = 93, normalized size = 0.51 \begin {gather*} \frac {10 \sqrt {1-2 x} \left (-74704869+1497221625 x+3564415220 x^2-3365562400 x^3-11818608000 x^4-723200000 x^5+14924800000 x^6+9216000000 x^7\right )-1376502897 \sqrt {30+50 x} \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{430080000 \sqrt {3+5 x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 155, normalized size = 0.85
method | result | size |
risch | \(-\frac {\left (1843200000 x^{6}+1879040000 x^{5}-1272064000 x^{4}-1600483200 x^{3}+287177440 x^{2}+540576580 x -24901623\right ) \sqrt {3+5 x}\, \left (-1+2 x \right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{43008000 \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right )}\, \sqrt {1-2 x}}+\frac {65547757 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{40960000 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(118\) |
default | \(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (36864000000 \sqrt {-10 x^{2}-x +3}\, x^{6}+37580800000 x^{5} \sqrt {-10 x^{2}-x +3}-25441280000 x^{4} \sqrt {-10 x^{2}-x +3}-32009664000 x^{3} \sqrt {-10 x^{2}-x +3}+5743548800 x^{2} \sqrt {-10 x^{2}-x +3}+1376502897 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+10811531600 x \sqrt {-10 x^{2}-x +3}-498032460 \sqrt {-10 x^{2}-x +3}\right )}{860160000 \sqrt {-10 x^{2}-x +3}}\) | \(155\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 113, normalized size = 0.62 \begin {gather*} -\frac {3}{70} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {7}{2}} + \frac {37}{120} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} x + \frac {37}{2400} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} + \frac {4477}{3840} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + \frac {4477}{76800} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {541717}{102400} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {65547757}{40960000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {541717}{2048000} \, \sqrt {-10 \, x^{2} - x + 3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.55, size = 87, normalized size = 0.48 \begin {gather*} \frac {1}{43008000} \, {\left (1843200000 \, x^{6} + 1879040000 \, x^{5} - 1272064000 \, x^{4} - 1600483200 \, x^{3} + 287177440 \, x^{2} + 540576580 \, x - 24901623\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {65547757}{40960000} \, \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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Sympy [F(-1)] Timed out
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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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 446 vs.
\(2 (131) = 262\).
time = 0.72, size = 446, normalized size = 2.45 \begin {gather*} \frac {1}{3584000000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (4 \, {\left (16 \, {\left (20 \, {\left (120 \, x - 443\right )} {\left (5 \, x + 3\right )} + 94933\right )} {\left (5 \, x + 3\right )} - 7838433\right )} {\left (5 \, x + 3\right )} + 98794353\right )} {\left (5 \, x + 3\right )} - 1568443065\right )} {\left (5 \, x + 3\right )} + 8438816295\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 17534989395 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {11}{192000000} \, \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 {19}{192000000} \, \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 {1091}{9600000} \, \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 {111}{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 {27}{400} \, \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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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (1-2\,x\right )}^{5/2}\,\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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