Optimal. Leaf size=180 \[ \frac {3 (5 x+3)^{3/2} (1-2 x)^{7/2}}{35 (3 x+2)^5}+\frac {251 (5 x+3)^{3/2} (1-2 x)^{5/2}}{280 (3 x+2)^4}+\frac {2761 (5 x+3)^{3/2} (1-2 x)^{3/2}}{336 (3 x+2)^3}+\frac {30371 (5 x+3)^{3/2} \sqrt {1-2 x}}{448 (3 x+2)^2}-\frac {334081 \sqrt {5 x+3} \sqrt {1-2 x}}{6272 (3 x+2)}-\frac {3674891 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{6272 \sqrt {7}} \]
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Rubi [A] time = 0.05, antiderivative size = 180, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {96, 94, 93, 204} \[ \frac {3 (5 x+3)^{3/2} (1-2 x)^{7/2}}{35 (3 x+2)^5}+\frac {251 (5 x+3)^{3/2} (1-2 x)^{5/2}}{280 (3 x+2)^4}+\frac {2761 (5 x+3)^{3/2} (1-2 x)^{3/2}}{336 (3 x+2)^3}+\frac {30371 (5 x+3)^{3/2} \sqrt {1-2 x}}{448 (3 x+2)^2}-\frac {334081 \sqrt {5 x+3} \sqrt {1-2 x}}{6272 (3 x+2)}-\frac {3674891 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{6272 \sqrt {7}} \]
Antiderivative was successfully verified.
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Rule 93
Rule 94
Rule 96
Rule 204
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{(2+3 x)^6} \, dx &=\frac {3 (1-2 x)^{7/2} (3+5 x)^{3/2}}{35 (2+3 x)^5}+\frac {251}{70} \int \frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{(2+3 x)^5} \, dx\\ &=\frac {3 (1-2 x)^{7/2} (3+5 x)^{3/2}}{35 (2+3 x)^5}+\frac {251 (1-2 x)^{5/2} (3+5 x)^{3/2}}{280 (2+3 x)^4}+\frac {2761}{112} \int \frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{(2+3 x)^4} \, dx\\ &=\frac {3 (1-2 x)^{7/2} (3+5 x)^{3/2}}{35 (2+3 x)^5}+\frac {251 (1-2 x)^{5/2} (3+5 x)^{3/2}}{280 (2+3 x)^4}+\frac {2761 (1-2 x)^{3/2} (3+5 x)^{3/2}}{336 (2+3 x)^3}+\frac {30371}{224} \int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{(2+3 x)^3} \, dx\\ &=\frac {3 (1-2 x)^{7/2} (3+5 x)^{3/2}}{35 (2+3 x)^5}+\frac {251 (1-2 x)^{5/2} (3+5 x)^{3/2}}{280 (2+3 x)^4}+\frac {2761 (1-2 x)^{3/2} (3+5 x)^{3/2}}{336 (2+3 x)^3}+\frac {30371 \sqrt {1-2 x} (3+5 x)^{3/2}}{448 (2+3 x)^2}+\frac {334081}{896} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^2} \, dx\\ &=-\frac {334081 \sqrt {1-2 x} \sqrt {3+5 x}}{6272 (2+3 x)}+\frac {3 (1-2 x)^{7/2} (3+5 x)^{3/2}}{35 (2+3 x)^5}+\frac {251 (1-2 x)^{5/2} (3+5 x)^{3/2}}{280 (2+3 x)^4}+\frac {2761 (1-2 x)^{3/2} (3+5 x)^{3/2}}{336 (2+3 x)^3}+\frac {30371 \sqrt {1-2 x} (3+5 x)^{3/2}}{448 (2+3 x)^2}+\frac {3674891 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{12544}\\ &=-\frac {334081 \sqrt {1-2 x} \sqrt {3+5 x}}{6272 (2+3 x)}+\frac {3 (1-2 x)^{7/2} (3+5 x)^{3/2}}{35 (2+3 x)^5}+\frac {251 (1-2 x)^{5/2} (3+5 x)^{3/2}}{280 (2+3 x)^4}+\frac {2761 (1-2 x)^{3/2} (3+5 x)^{3/2}}{336 (2+3 x)^3}+\frac {30371 \sqrt {1-2 x} (3+5 x)^{3/2}}{448 (2+3 x)^2}+\frac {3674891 \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{6272}\\ &=-\frac {334081 \sqrt {1-2 x} \sqrt {3+5 x}}{6272 (2+3 x)}+\frac {3 (1-2 x)^{7/2} (3+5 x)^{3/2}}{35 (2+3 x)^5}+\frac {251 (1-2 x)^{5/2} (3+5 x)^{3/2}}{280 (2+3 x)^4}+\frac {2761 (1-2 x)^{3/2} (3+5 x)^{3/2}}{336 (2+3 x)^3}+\frac {30371 \sqrt {1-2 x} (3+5 x)^{3/2}}{448 (2+3 x)^2}-\frac {3674891 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{6272 \sqrt {7}}\\ \end {align*}
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Mathematica [A] time = 0.20, size = 164, normalized size = 0.91 \[ \frac {1}{70} \left (\frac {6 (5 x+3)^{3/2} (1-2 x)^{7/2}}{(3 x+2)^5}+\frac {251 \left (2352 (5 x+3)^{3/2} (1-2 x)^{5/2}+55 (3 x+2) \left (392 (1-2 x)^{3/2} (5 x+3)^{3/2}+33 (3 x+2) \left (7 \sqrt {1-2 x} \sqrt {5 x+3} (37 x+20)-121 \sqrt {7} (3 x+2)^2 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )\right )\right )\right )}{9408 (3 x+2)^4}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.77, size = 131, normalized size = 0.73 \[ -\frac {55123365 \, \sqrt {7} {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (390269835 \, x^{4} + 1058136330 \, x^{3} + 1076423732 \, x^{2} + 487066088 \, x + 82697568\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{1317120 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 3.45, size = 426, normalized size = 2.37 \[ \frac {3674891}{878080} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {14641 \, \sqrt {10} {\left (753 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{9} - 1524880 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{7} - 503767040 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} - 77139328000 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} - \frac {4628359680000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {18513438720000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{9408 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 298, normalized size = 1.66 \[ \frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (13394977695 \sqrt {7}\, x^{5} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+44649925650 \sqrt {7}\, x^{4} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+5463777690 \sqrt {-10 x^{2}-x +3}\, x^{4}+59533234200 \sqrt {7}\, x^{3} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+14813908620 \sqrt {-10 x^{2}-x +3}\, x^{3}+39688822800 \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+15069932248 \sqrt {-10 x^{2}-x +3}\, x^{2}+13229607600 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+6818925232 \sqrt {-10 x^{2}-x +3}\, x +1763947680 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+1157765952 \sqrt {-10 x^{2}-x +3}\right )}{1317120 \sqrt {-10 x^{2}-x +3}\, \left (3 x +2\right )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.35, size = 198, normalized size = 1.10 \[ \frac {3674891}{87808} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {151855}{4704} \, \sqrt {-10 \, x^{2} - x + 3} + \frac {7 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{15 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac {73 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{40 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac {2573 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{336 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {91113 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{3136 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac {1123727 \, \sqrt {-10 \, x^{2} - x + 3}}{18816 \, {\left (3 \, x + 2\right )}} \]
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 \[ \int \frac {{\left (1-2\,x\right )}^{5/2}\,\sqrt {5\,x+3}}{{\left (3\,x+2\right )}^6} \,d x \]
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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