Optimal. Leaf size=288 \[ -\frac {2}{57} (2 x+3)^{5/2} \left (3 x^2+5 x+2\right )^{7/2}+\frac {430}{969} (2 x+3)^{3/2} \left (3 x^2+5 x+2\right )^{7/2}+\frac {2350 \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{7/2}}{2907}+\frac {25 \sqrt {2 x+3} (86493 x+72737) \left (3 x^2+5 x+2\right )^{5/2}}{1247103}-\frac {125 \sqrt {2 x+3} (79583 x+64006) \left (3 x^2+5 x+2\right )^{3/2}}{52378326}+\frac {25 \sqrt {2 x+3} (216603 x+749099) \sqrt {3 x^2+5 x+2}}{942809868}+\frac {142149125 \sqrt {-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{1885619736 \sqrt {3} \sqrt {3 x^2+5 x+2}}-\frac {16503475 \sqrt {-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{269374248 \sqrt {3} \sqrt {3 x^2+5 x+2}} \]
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Rubi [A] time = 0.22, antiderivative size = 288, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 6, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {832, 814, 843, 718, 424, 419} \[ -\frac {2}{57} (2 x+3)^{5/2} \left (3 x^2+5 x+2\right )^{7/2}+\frac {430}{969} (2 x+3)^{3/2} \left (3 x^2+5 x+2\right )^{7/2}+\frac {2350 \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{7/2}}{2907}+\frac {25 \sqrt {2 x+3} (86493 x+72737) \left (3 x^2+5 x+2\right )^{5/2}}{1247103}-\frac {125 \sqrt {2 x+3} (79583 x+64006) \left (3 x^2+5 x+2\right )^{3/2}}{52378326}+\frac {25 \sqrt {2 x+3} (216603 x+749099) \sqrt {3 x^2+5 x+2}}{942809868}+\frac {142149125 \sqrt {-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{1885619736 \sqrt {3} \sqrt {3 x^2+5 x+2}}-\frac {16503475 \sqrt {-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{269374248 \sqrt {3} \sqrt {3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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Rule 419
Rule 424
Rule 718
Rule 814
Rule 832
Rule 843
Rubi steps
\begin {align*} \int (5-x) (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{5/2} \, dx &=-\frac {2}{57} (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{7/2}+\frac {2}{57} \int (3+2 x)^{3/2} \left (490+\frac {645 x}{2}\right ) \left (2+5 x+3 x^2\right )^{5/2} \, dx\\ &=\frac {430}{969} (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{7/2}-\frac {2}{57} (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{7/2}+\frac {4 \int \sqrt {3+2 x} \left (\frac {74475}{4}+\frac {52875 x}{4}\right ) \left (2+5 x+3 x^2\right )^{5/2} \, dx}{2907}\\ &=\frac {2350 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{7/2}}{2907}+\frac {430}{969} (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{7/2}-\frac {2}{57} (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{7/2}+\frac {8 \int \frac {\left (\frac {2145375}{4}+\frac {2948625 x}{8}\right ) \left (2+5 x+3 x^2\right )^{5/2}}{\sqrt {3+2 x}} \, dx}{130815}\\ &=\frac {25 \sqrt {3+2 x} (72737+86493 x) \left (2+5 x+3 x^2\right )^{5/2}}{1247103}+\frac {2350 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{7/2}}{2907}+\frac {430}{969} (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{7/2}-\frac {2}{57} (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{7/2}-\frac {4 \int \frac {\left (\frac {53845875}{8}+\frac {38370375 x}{8}\right ) \left (2+5 x+3 x^2\right )^{3/2}}{\sqrt {3+2 x}} \, dx}{11223927}\\ &=-\frac {125 \sqrt {3+2 x} (64006+79583 x) \left (2+5 x+3 x^2\right )^{3/2}}{52378326}+\frac {25 \sqrt {3+2 x} (72737+86493 x) \left (2+5 x+3 x^2\right )^{5/2}}{1247103}+\frac {2350 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{7/2}}{2907}+\frac {430}{969} (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{7/2}-\frac {2}{57} (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{7/2}+\frac {2 \int \frac {\left (\frac {119471625}{4}+\frac {81226125 x}{8}\right ) \sqrt {2+5 x+3 x^2}}{\sqrt {3+2 x}} \, dx}{707107401}\\ &=\frac {25 \sqrt {3+2 x} (749099+216603 x) \sqrt {2+5 x+3 x^2}}{942809868}-\frac {125 \sqrt {3+2 x} (64006+79583 x) \left (2+5 x+3 x^2\right )^{3/2}}{52378326}+\frac {25 \sqrt {3+2 x} (72737+86493 x) \left (2+5 x+3 x^2\right )^{5/2}}{1247103}+\frac {2350 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{7/2}}{2907}+\frac {430}{969} (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{7/2}-\frac {2}{57} (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{7/2}-\frac {\int \frac {\frac {13798609875}{8}+\frac {15595783875 x}{8}}{\sqrt {3+2 x} \sqrt {2+5 x+3 x^2}} \, dx}{31819833045}\\ &=\frac {25 \sqrt {3+2 x} (749099+216603 x) \sqrt {2+5 x+3 x^2}}{942809868}-\frac {125 \sqrt {3+2 x} (64006+79583 x) \left (2+5 x+3 x^2\right )^{3/2}}{52378326}+\frac {25 \sqrt {3+2 x} (72737+86493 x) \left (2+5 x+3 x^2\right )^{5/2}}{1247103}+\frac {2350 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{7/2}}{2907}+\frac {430}{969} (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{7/2}-\frac {2}{57} (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{7/2}-\frac {16503475 \int \frac {\sqrt {3+2 x}}{\sqrt {2+5 x+3 x^2}} \, dx}{538748496}+\frac {142149125 \int \frac {1}{\sqrt {3+2 x} \sqrt {2+5 x+3 x^2}} \, dx}{3771239472}\\ &=\frac {25 \sqrt {3+2 x} (749099+216603 x) \sqrt {2+5 x+3 x^2}}{942809868}-\frac {125 \sqrt {3+2 x} (64006+79583 x) \left (2+5 x+3 x^2\right )^{3/2}}{52378326}+\frac {25 \sqrt {3+2 x} (72737+86493 x) \left (2+5 x+3 x^2\right )^{5/2}}{1247103}+\frac {2350 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{7/2}}{2907}+\frac {430}{969} (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{7/2}-\frac {2}{57} (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{7/2}-\frac {\left (16503475 \sqrt {-2-5 x-3 x^2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+\frac {2 x^2}{3}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {6+6 x}}{\sqrt {2}}\right )}{269374248 \sqrt {3} \sqrt {2+5 x+3 x^2}}+\frac {\left (142149125 \sqrt {-2-5 x-3 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {2 x^2}{3}}} \, dx,x,\frac {\sqrt {6+6 x}}{\sqrt {2}}\right )}{1885619736 \sqrt {3} \sqrt {2+5 x+3 x^2}}\\ &=\frac {25 \sqrt {3+2 x} (749099+216603 x) \sqrt {2+5 x+3 x^2}}{942809868}-\frac {125 \sqrt {3+2 x} (64006+79583 x) \left (2+5 x+3 x^2\right )^{3/2}}{52378326}+\frac {25 \sqrt {3+2 x} (72737+86493 x) \left (2+5 x+3 x^2\right )^{5/2}}{1247103}+\frac {2350 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{7/2}}{2907}+\frac {430}{969} (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{7/2}-\frac {2}{57} (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{7/2}-\frac {16503475 \sqrt {-2-5 x-3 x^2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{269374248 \sqrt {3} \sqrt {2+5 x+3 x^2}}+\frac {142149125 \sqrt {-2-5 x-3 x^2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{1885619736 \sqrt {3} \sqrt {2+5 x+3 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.38, size = 228, normalized size = 0.79 \[ -\frac {-30234850 \sqrt {5} \sqrt {\frac {x+1}{2 x+3}} \sqrt {\frac {3 x+2}{2 x+3}} (2 x+3)^2 F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {2 x+3}}\right )|\frac {3}{5}\right )+115524325 \sqrt {5} \sqrt {\frac {x+1}{2 x+3}} \sqrt {\frac {3 x+2}{2 x+3}} (2 x+3)^2 E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {2 x+3}}\right )|\frac {3}{5}\right )+2 \left (64309557312 x^{11}+311460012864 x^{10}-694795413312 x^9-9445976815968 x^8-34294970344572 x^7-69684837178068 x^6-90580760151282 x^5-78460508136978 x^4-45255052994607 x^3-16735272462363 x^2-3595384785664 x-341519551612\right ) \sqrt {2 x+3}}{5656859208 (2 x+3) \sqrt {3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (36 \, x^{7} + 48 \, x^{6} - 551 \, x^{5} - 2151 \, x^{4} - 3381 \, x^{3} - 2717 \, x^{2} - 1104 \, x - 180\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int -{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} {\left (2 \, x + 3\right )}^{\frac {5}{2}} {\left (x - 5\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 176, normalized size = 0.61 \[ \frac {\sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}\, \left (-257238229248 x^{11}-1245840051456 x^{10}+2779181653248 x^{9}+37783907263872 x^{8}+137179881378288 x^{7}+278739348712272 x^{6}+362323040605128 x^{5}+313842032547912 x^{4}+181020211978428 x^{3}+66942476141352 x^{2}+14383849629156 x +23104865 \sqrt {2 x +3}\, \sqrt {15}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, \EllipticE \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+5324960 \sqrt {2 x +3}\, \sqrt {15}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, \EllipticF \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+1367002401048\right )}{67882310496 x^{3}+214960649904 x^{2}+214960649904 x +67882310496} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\int {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} {\left (2 \, x + 3\right )}^{\frac {5}{2}} {\left (x - 5\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ -\int {\left (2\,x+3\right )}^{5/2}\,\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{5/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - \int \left (- 180 \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 1104 x \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 2717 x^{2} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 3381 x^{3} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 2151 x^{4} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 551 x^{5} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int 48 x^{6} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}\, dx - \int 36 x^{7} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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