Optimal. Leaf size=160 \[ -\frac {1}{12} (5 x+3)^{5/2} (1-2 x)^{7/2}-\frac {11}{48} (5 x+3)^{3/2} (1-2 x)^{7/2}-\frac {121}{256} \sqrt {5 x+3} (1-2 x)^{7/2}+\frac {1331 \sqrt {5 x+3} (1-2 x)^{5/2}}{7680}+\frac {14641 \sqrt {5 x+3} (1-2 x)^{3/2}}{30720}+\frac {161051 \sqrt {5 x+3} \sqrt {1-2 x}}{102400}+\frac {1771561 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{102400 \sqrt {10}} \]
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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 = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {50, 54, 216} \[ -\frac {1}{12} (5 x+3)^{5/2} (1-2 x)^{7/2}-\frac {11}{48} (5 x+3)^{3/2} (1-2 x)^{7/2}-\frac {121}{256} \sqrt {5 x+3} (1-2 x)^{7/2}+\frac {1331 \sqrt {5 x+3} (1-2 x)^{5/2}}{7680}+\frac {14641 \sqrt {5 x+3} (1-2 x)^{3/2}}{30720}+\frac {161051 \sqrt {5 x+3} \sqrt {1-2 x}}{102400}+\frac {1771561 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{102400 \sqrt {10}} \]
Antiderivative was successfully verified.
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Rule 50
Rule 54
Rule 216
Rubi steps
\begin {align*} \int (1-2 x)^{5/2} (3+5 x)^{5/2} \, dx &=-\frac {1}{12} (1-2 x)^{7/2} (3+5 x)^{5/2}+\frac {55}{24} \int (1-2 x)^{5/2} (3+5 x)^{3/2} \, dx\\ &=-\frac {11}{48} (1-2 x)^{7/2} (3+5 x)^{3/2}-\frac {1}{12} (1-2 x)^{7/2} (3+5 x)^{5/2}+\frac {121}{32} \int (1-2 x)^{5/2} \sqrt {3+5 x} \, dx\\ &=-\frac {121}{256} (1-2 x)^{7/2} \sqrt {3+5 x}-\frac {11}{48} (1-2 x)^{7/2} (3+5 x)^{3/2}-\frac {1}{12} (1-2 x)^{7/2} (3+5 x)^{5/2}+\frac {1331}{512} \int \frac {(1-2 x)^{5/2}}{\sqrt {3+5 x}} \, dx\\ &=\frac {1331 (1-2 x)^{5/2} \sqrt {3+5 x}}{7680}-\frac {121}{256} (1-2 x)^{7/2} \sqrt {3+5 x}-\frac {11}{48} (1-2 x)^{7/2} (3+5 x)^{3/2}-\frac {1}{12} (1-2 x)^{7/2} (3+5 x)^{5/2}+\frac {14641 \int \frac {(1-2 x)^{3/2}}{\sqrt {3+5 x}} \, dx}{3072}\\ &=\frac {14641 (1-2 x)^{3/2} \sqrt {3+5 x}}{30720}+\frac {1331 (1-2 x)^{5/2} \sqrt {3+5 x}}{7680}-\frac {121}{256} (1-2 x)^{7/2} \sqrt {3+5 x}-\frac {11}{48} (1-2 x)^{7/2} (3+5 x)^{3/2}-\frac {1}{12} (1-2 x)^{7/2} (3+5 x)^{5/2}+\frac {161051 \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx}{20480}\\ &=\frac {161051 \sqrt {1-2 x} \sqrt {3+5 x}}{102400}+\frac {14641 (1-2 x)^{3/2} \sqrt {3+5 x}}{30720}+\frac {1331 (1-2 x)^{5/2} \sqrt {3+5 x}}{7680}-\frac {121}{256} (1-2 x)^{7/2} \sqrt {3+5 x}-\frac {11}{48} (1-2 x)^{7/2} (3+5 x)^{3/2}-\frac {1}{12} (1-2 x)^{7/2} (3+5 x)^{5/2}+\frac {1771561 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{204800}\\ &=\frac {161051 \sqrt {1-2 x} \sqrt {3+5 x}}{102400}+\frac {14641 (1-2 x)^{3/2} \sqrt {3+5 x}}{30720}+\frac {1331 (1-2 x)^{5/2} \sqrt {3+5 x}}{7680}-\frac {121}{256} (1-2 x)^{7/2} \sqrt {3+5 x}-\frac {11}{48} (1-2 x)^{7/2} (3+5 x)^{3/2}-\frac {1}{12} (1-2 x)^{7/2} (3+5 x)^{5/2}+\frac {1771561 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{102400 \sqrt {5}}\\ &=\frac {161051 \sqrt {1-2 x} \sqrt {3+5 x}}{102400}+\frac {14641 (1-2 x)^{3/2} \sqrt {3+5 x}}{30720}+\frac {1331 (1-2 x)^{5/2} \sqrt {3+5 x}}{7680}-\frac {121}{256} (1-2 x)^{7/2} \sqrt {3+5 x}-\frac {11}{48} (1-2 x)^{7/2} (3+5 x)^{3/2}-\frac {1}{12} (1-2 x)^{7/2} (3+5 x)^{5/2}+\frac {1771561 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{102400 \sqrt {10}}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 84, normalized size = 0.52 \[ \frac {5314683 \sqrt {20 x-10} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )-10 \sqrt {5 x+3} \left (10240000 x^6-2560000 x^5-11091200 x^4+3408320 x^3+4538680 x^2-1703014 x-96003\right )}{3072000 \sqrt {1-2 x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.91, size = 82, normalized size = 0.51 \[ \frac {1}{307200} \, {\left (5120000 \, x^{5} + 1280000 \, x^{4} - 4905600 \, x^{3} - 748640 \, x^{2} + 1895020 \, x + 96003\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {1771561}{2048000} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.66, size = 356, normalized size = 2.22 \[ \frac {1}{76800000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (4 \, {\left (16 \, {\left (100 \, x - 311\right )} {\left (5 \, x + 3\right )} + 46071\right )} {\left (5 \, x + 3\right )} - 775911\right )} {\left (5 \, x + 3\right )} + 15385695\right )} {\left (5 \, x + 3\right )} - 99422145\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 220189365 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {1}{2400000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (12 \, {\left (80 \, x - 203\right )} {\left (5 \, x + 3\right )} + 19073\right )} {\left (5 \, x + 3\right )} - 506185\right )} {\left (5 \, x + 3\right )} + 4031895\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 10392195 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} - \frac {47}{1920000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (60 \, x - 119\right )} {\left (5 \, x + 3\right )} + 6163\right )} {\left (5 \, x + 3\right )} - 66189\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 184305 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} - \frac {69}{40000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (40 \, x - 59\right )} {\left (5 \, x + 3\right )} + 1293\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 4785 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {27}{2000} \, \sqrt {5} {\left (2 \, {\left (20 \, x - 23\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 143 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {27}{50} \, \sqrt {5} {\left (11 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + 2 \, \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 136, normalized size = 0.85 \[ \frac {1771561 \sqrt {\left (-2 x +1\right ) \left (5 x +3\right )}\, \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )}{2048000 \sqrt {5 x +3}\, \sqrt {-2 x +1}}+\frac {\left (-2 x +1\right )^{\frac {5}{2}} \left (5 x +3\right )^{\frac {7}{2}}}{30}+\frac {11 \left (-2 x +1\right )^{\frac {3}{2}} \left (5 x +3\right )^{\frac {7}{2}}}{300}+\frac {121 \left (5 x +3\right )^{\frac {7}{2}} \sqrt {-2 x +1}}{4000}-\frac {1331 \left (5 x +3\right )^{\frac {5}{2}} \sqrt {-2 x +1}}{48000}-\frac {14641 \left (5 x +3\right )^{\frac {3}{2}} \sqrt {-2 x +1}}{76800}-\frac {161051 \sqrt {-2 x +1}\, \sqrt {5 x +3}}{102400} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.38, size = 99, normalized size = 0.62 \[ \frac {1}{6} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} x + \frac {1}{120} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} + \frac {121}{192} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + \frac {121}{3840} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {14641}{5120} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {1771561}{2048000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {14641}{102400} \, \sqrt {-10 \, x^{2} - x + 3} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (1-2\,x\right )}^{5/2}\,{\left (5\,x+3\right )}^{5/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 24.94, size = 357, normalized size = 2.23 \[ \begin {cases} \frac {500 i \left (x + \frac {3}{5}\right )^{\frac {13}{2}}}{3 \sqrt {10 x - 5}} - \frac {1925 i \left (x + \frac {3}{5}\right )^{\frac {11}{2}}}{3 \sqrt {10 x - 5}} + \frac {40535 i \left (x + \frac {3}{5}\right )^{\frac {9}{2}}}{48 \sqrt {10 x - 5}} - \frac {73205 i \left (x + \frac {3}{5}\right )^{\frac {7}{2}}}{192 \sqrt {10 x - 5}} - \frac {14641 i \left (x + \frac {3}{5}\right )^{\frac {5}{2}}}{7680 \sqrt {10 x - 5}} - \frac {161051 i \left (x + \frac {3}{5}\right )^{\frac {3}{2}}}{30720 \sqrt {10 x - 5}} + \frac {1771561 i \sqrt {x + \frac {3}{5}}}{102400 \sqrt {10 x - 5}} - \frac {1771561 \sqrt {10} i \operatorname {acosh}{\left (\frac {\sqrt {110} \sqrt {x + \frac {3}{5}}}{11} \right )}}{1024000} & \text {for}\: \frac {10 \left |{x + \frac {3}{5}}\right |}{11} > 1 \\\frac {1771561 \sqrt {10} \operatorname {asin}{\left (\frac {\sqrt {110} \sqrt {x + \frac {3}{5}}}{11} \right )}}{1024000} - \frac {500 \left (x + \frac {3}{5}\right )^{\frac {13}{2}}}{3 \sqrt {5 - 10 x}} + \frac {1925 \left (x + \frac {3}{5}\right )^{\frac {11}{2}}}{3 \sqrt {5 - 10 x}} - \frac {40535 \left (x + \frac {3}{5}\right )^{\frac {9}{2}}}{48 \sqrt {5 - 10 x}} + \frac {73205 \left (x + \frac {3}{5}\right )^{\frac {7}{2}}}{192 \sqrt {5 - 10 x}} + \frac {14641 \left (x + \frac {3}{5}\right )^{\frac {5}{2}}}{7680 \sqrt {5 - 10 x}} + \frac {161051 \left (x + \frac {3}{5}\right )^{\frac {3}{2}}}{30720 \sqrt {5 - 10 x}} - \frac {1771561 \sqrt {x + \frac {3}{5}}}{102400 \sqrt {5 - 10 x}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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