Optimal. Leaf size=113 \[ \frac {7 (3 x+2)^3}{11 \sqrt {1-2 x} \sqrt {5 x+3}}-\frac {37 \sqrt {1-2 x} (3 x+2)^2}{605 \sqrt {5 x+3}}+\frac {3 \sqrt {1-2 x} \sqrt {5 x+3} (72060 x+173063)}{96800}-\frac {35451 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{800 \sqrt {10}} \]
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Rubi [A] time = 0.03, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {98, 150, 147, 54, 216} \[ \frac {7 (3 x+2)^3}{11 \sqrt {1-2 x} \sqrt {5 x+3}}-\frac {37 \sqrt {1-2 x} (3 x+2)^2}{605 \sqrt {5 x+3}}+\frac {3 \sqrt {1-2 x} \sqrt {5 x+3} (72060 x+173063)}{96800}-\frac {35451 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{800 \sqrt {10}} \]
Antiderivative was successfully verified.
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Rule 54
Rule 98
Rule 147
Rule 150
Rule 216
Rubi steps
\begin {align*} \int \frac {(2+3 x)^4}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx &=\frac {7 (2+3 x)^3}{11 \sqrt {1-2 x} \sqrt {3+5 x}}-\frac {1}{11} \int \frac {(2+3 x)^2 \left (152+\frac {519 x}{2}\right )}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx\\ &=-\frac {37 \sqrt {1-2 x} (2+3 x)^2}{605 \sqrt {3+5 x}}+\frac {7 (2+3 x)^3}{11 \sqrt {1-2 x} \sqrt {3+5 x}}-\frac {2}{605} \int \frac {(2+3 x) \left (\frac {5487}{2}+\frac {18015 x}{4}\right )}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {37 \sqrt {1-2 x} (2+3 x)^2}{605 \sqrt {3+5 x}}+\frac {7 (2+3 x)^3}{11 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {3 \sqrt {1-2 x} \sqrt {3+5 x} (173063+72060 x)}{96800}-\frac {35451 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{1600}\\ &=-\frac {37 \sqrt {1-2 x} (2+3 x)^2}{605 \sqrt {3+5 x}}+\frac {7 (2+3 x)^3}{11 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {3 \sqrt {1-2 x} \sqrt {3+5 x} (173063+72060 x)}{96800}-\frac {35451 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{800 \sqrt {5}}\\ &=-\frac {37 \sqrt {1-2 x} (2+3 x)^2}{605 \sqrt {3+5 x}}+\frac {7 (2+3 x)^3}{11 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {3 \sqrt {1-2 x} \sqrt {3+5 x} (173063+72060 x)}{96800}-\frac {35451 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{800 \sqrt {10}}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 87, normalized size = 0.77 \[ \frac {10 \sqrt {2 x-1} \left (-392040 x^3-1992870 x^2+2323271 x+2026687\right )-4289571 (2 x-1) \sqrt {50 x+30} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )}{968000 \sqrt {-(1-2 x)^2} \sqrt {5 x+3}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 1.40, size = 92, normalized size = 0.81 \[ \frac {4289571 \, \sqrt {10} {\left (10 \, x^{2} + x - 3\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 20 \, {\left (392040 \, x^{3} + 1992870 \, x^{2} - 2323271 \, x - 2026687\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{1936000 \, {\left (10 \, x^{2} + x - 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.30, size = 131, normalized size = 1.16 \[ -\frac {35451}{8000} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + \frac {{\left (6534 \, {\left (12 \, \sqrt {5} {\left (5 \, x + 3\right )} + 197 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} - 21456431 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{12100000 \, {\left (2 \, x - 1\right )}} - \frac {\sqrt {10} {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{151250 \, \sqrt {5 \, x + 3}} + \frac {2 \, \sqrt {10} \sqrt {5 \, x + 3}}{75625 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 137, normalized size = 1.21 \[ -\frac {\sqrt {-2 x +1}\, \left (-7840800 \sqrt {-10 x^{2}-x +3}\, x^{3}+42895710 \sqrt {10}\, x^{2} \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-39857400 \sqrt {-10 x^{2}-x +3}\, x^{2}+4289571 \sqrt {10}\, x \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+46465420 \sqrt {-10 x^{2}-x +3}\, x -12868713 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+40533740 \sqrt {-10 x^{2}-x +3}\right )}{1936000 \left (2 x -1\right ) \sqrt {-10 x^{2}-x +3}\, \sqrt {5 x +3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.14, size = 75, normalized size = 0.66 \[ -\frac {81 \, x^{3}}{20 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {1647 \, x^{2}}{80 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {35451}{16000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {2323271 \, x}{96800 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {2026687}{96800 \, \sqrt {-10 \, x^{2} - x + 3}} \]
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 \[ \int \frac {{\left (3\,x+2\right )}^4}{{\left (1-2\,x\right )}^{3/2}\,{\left (5\,x+3\right )}^{3/2}} \,d x \]
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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