Optimal. Leaf size=151 \[ \frac {(5 x+3)^{3/2} (1-2 x)^{5/2}}{4 (3 x+2)^4}+\frac {55 (5 x+3)^{3/2} (1-2 x)^{3/2}}{24 (3 x+2)^3}+\frac {605 (5 x+3)^{3/2} \sqrt {1-2 x}}{32 (3 x+2)^2}-\frac {6655 \sqrt {5 x+3} \sqrt {1-2 x}}{448 (3 x+2)}-\frac {73205 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{448 \sqrt {7}} \]
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Rubi [A] time = 0.04, antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {94, 93, 204} \[ \frac {(5 x+3)^{3/2} (1-2 x)^{5/2}}{4 (3 x+2)^4}+\frac {55 (5 x+3)^{3/2} (1-2 x)^{3/2}}{24 (3 x+2)^3}+\frac {605 (5 x+3)^{3/2} \sqrt {1-2 x}}{32 (3 x+2)^2}-\frac {6655 \sqrt {5 x+3} \sqrt {1-2 x}}{448 (3 x+2)}-\frac {73205 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{448 \sqrt {7}} \]
Antiderivative was successfully verified.
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Rule 93
Rule 94
Rule 204
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{(2+3 x)^5} \, dx &=\frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{4 (2+3 x)^4}+\frac {55}{8} \int \frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{(2+3 x)^4} \, dx\\ &=\frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{4 (2+3 x)^4}+\frac {55 (1-2 x)^{3/2} (3+5 x)^{3/2}}{24 (2+3 x)^3}+\frac {605}{16} \int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{(2+3 x)^3} \, dx\\ &=\frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{4 (2+3 x)^4}+\frac {55 (1-2 x)^{3/2} (3+5 x)^{3/2}}{24 (2+3 x)^3}+\frac {605 \sqrt {1-2 x} (3+5 x)^{3/2}}{32 (2+3 x)^2}+\frac {6655}{64} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^2} \, dx\\ &=-\frac {6655 \sqrt {1-2 x} \sqrt {3+5 x}}{448 (2+3 x)}+\frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{4 (2+3 x)^4}+\frac {55 (1-2 x)^{3/2} (3+5 x)^{3/2}}{24 (2+3 x)^3}+\frac {605 \sqrt {1-2 x} (3+5 x)^{3/2}}{32 (2+3 x)^2}+\frac {73205}{896} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=-\frac {6655 \sqrt {1-2 x} \sqrt {3+5 x}}{448 (2+3 x)}+\frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{4 (2+3 x)^4}+\frac {55 (1-2 x)^{3/2} (3+5 x)^{3/2}}{24 (2+3 x)^3}+\frac {605 \sqrt {1-2 x} (3+5 x)^{3/2}}{32 (2+3 x)^2}+\frac {73205}{448} \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )\\ &=-\frac {6655 \sqrt {1-2 x} \sqrt {3+5 x}}{448 (2+3 x)}+\frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{4 (2+3 x)^4}+\frac {55 (1-2 x)^{3/2} (3+5 x)^{3/2}}{24 (2+3 x)^3}+\frac {605 \sqrt {1-2 x} (3+5 x)^{3/2}}{32 (2+3 x)^2}-\frac {73205 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{448 \sqrt {7}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 79, normalized size = 0.52 \[ \frac {\frac {7 \sqrt {1-2 x} \sqrt {5 x+3} \left (518715 x^3+1059032 x^2+723428 x+164688\right )}{(3 x+2)^4}-219615 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{9408} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.86, size = 116, normalized size = 0.77 \[ -\frac {219615 \, \sqrt {7} {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (518715 \, x^{3} + 1059032 \, x^{2} + 723428 \, x + 164688\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{18816 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 2.50, size = 368, normalized size = 2.44 \[ \frac {14641}{12544} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {73205 \, \sqrt {10} {\left (3 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{7} - 4088 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} - 862400 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} - \frac {65856000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {263424000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{672 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 250, normalized size = 1.66 \[ \frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (17788815 \sqrt {7}\, x^{4} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+47436840 \sqrt {7}\, x^{3} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+7262010 \sqrt {-10 x^{2}-x +3}\, x^{3}+47436840 \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+14826448 \sqrt {-10 x^{2}-x +3}\, x^{2}+21083040 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+10127992 \sqrt {-10 x^{2}-x +3}\, x +3513840 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+2305632 \sqrt {-10 x^{2}-x +3}\right )}{18816 \sqrt {-10 x^{2}-x +3}\, \left (3 x +2\right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.15, size = 157, normalized size = 1.04 \[ \frac {73205}{6272} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {3025}{336} \, \sqrt {-10 \, x^{2} - x + 3} + \frac {7 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{12 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac {17 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{8 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {1815 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{224 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac {22385 \, \sqrt {-10 \, x^{2} - x + 3}}{1344 \, {\left (3 \, x + 2\right )}} \]
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 \[ \int \frac {{\left (1-2\,x\right )}^{5/2}\,\sqrt {5\,x+3}}{{\left (3\,x+2\right )}^5} \,d x \]
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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