Optimal. Leaf size=84 \[ \frac {3 (1-2 x)^{3/2}}{7 (3 x+2) \sqrt {5 x+3}}-\frac {103 \sqrt {1-2 x}}{7 \sqrt {5 x+3}}+\frac {103 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{\sqrt {7}} \]
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Rubi [A] time = 0.02, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {96, 94, 93, 204} \[ \frac {3 (1-2 x)^{3/2}}{7 (3 x+2) \sqrt {5 x+3}}-\frac {103 \sqrt {1-2 x}}{7 \sqrt {5 x+3}}+\frac {103 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{\sqrt {7}} \]
Antiderivative was successfully verified.
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Rule 93
Rule 94
Rule 96
Rule 204
Rubi steps
\begin {align*} \int \frac {\sqrt {1-2 x}}{(2+3 x)^2 (3+5 x)^{3/2}} \, dx &=\frac {3 (1-2 x)^{3/2}}{7 (2+3 x) \sqrt {3+5 x}}+\frac {103}{14} \int \frac {\sqrt {1-2 x}}{(2+3 x) (3+5 x)^{3/2}} \, dx\\ &=-\frac {103 \sqrt {1-2 x}}{7 \sqrt {3+5 x}}+\frac {3 (1-2 x)^{3/2}}{7 (2+3 x) \sqrt {3+5 x}}-\frac {103}{2} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=-\frac {103 \sqrt {1-2 x}}{7 \sqrt {3+5 x}}+\frac {3 (1-2 x)^{3/2}}{7 (2+3 x) \sqrt {3+5 x}}-103 \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )\\ &=-\frac {103 \sqrt {1-2 x}}{7 \sqrt {3+5 x}}+\frac {3 (1-2 x)^{3/2}}{7 (2+3 x) \sqrt {3+5 x}}+\frac {103 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{\sqrt {7}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 65, normalized size = 0.77 \[ \frac {103 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{\sqrt {7}}-\frac {\sqrt {1-2 x} (45 x+29)}{(3 x+2) \sqrt {5 x+3}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.24, size = 86, normalized size = 1.02 \[ \frac {103 \, \sqrt {7} {\left (15 \, x^{2} + 19 \, x + 6\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (45 \, x + 29\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (15 \, x^{2} + 19 \, x + 6\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.67, size = 257, normalized size = 3.06 \[ -\frac {1}{140} \, \sqrt {5} {\left (103 \, \sqrt {70} \sqrt {2} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} + 70 \, \sqrt {2} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )} + \frac {9240 \, \sqrt {2} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{{\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 154, normalized size = 1.83 \[ -\frac {\left (1545 \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+1957 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+630 \sqrt {-10 x^{2}-x +3}\, x +618 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+406 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {-2 x +1}}{14 \left (3 x +2\right ) \sqrt {-10 x^{2}-x +3}\, \sqrt {5 x +3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.37, size = 92, normalized size = 1.10 \[ -\frac {103}{14} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {30 \, x}{\sqrt {-10 \, x^{2} - x + 3}} - \frac {47}{3 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {7}{3 \, {\left (3 \, \sqrt {-10 \, x^{2} - x + 3} x + 2 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} \]
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 \[ \int \frac {\sqrt {1-2\,x}}{{\left (3\,x+2\right )}^2\,{\left (5\,x+3\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {1 - 2 x}}{\left (3 x + 2\right )^{2} \left (5 x + 3\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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