Optimal. Leaf size=144 \[ \frac {5 \sqrt {5 x+3}}{42 \sqrt {1-2 x}}-\frac {5 \sqrt {5 x+3}}{28 \sqrt {1-2 x} (3 x+2)}-\frac {3 \sqrt {5 x+3}}{14 \sqrt {1-2 x} (3 x+2)^2}+\frac {11 \sqrt {5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)^2}-\frac {5 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{28 \sqrt {7}} \]
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Rubi [A] time = 0.05, antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {98, 151, 152, 12, 93, 204} \[ \frac {5 \sqrt {5 x+3}}{42 \sqrt {1-2 x}}-\frac {5 \sqrt {5 x+3}}{28 \sqrt {1-2 x} (3 x+2)}-\frac {3 \sqrt {5 x+3}}{14 \sqrt {1-2 x} (3 x+2)^2}+\frac {11 \sqrt {5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)^2}-\frac {5 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{28 \sqrt {7}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 93
Rule 98
Rule 151
Rule 152
Rule 204
Rubi steps
\begin {align*} \int \frac {(3+5 x)^{3/2}}{(1-2 x)^{5/2} (2+3 x)^3} \, dx &=\frac {11 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^2}-\frac {1}{21} \int \frac {-134-\frac {465 x}{2}}{(1-2 x)^{3/2} (2+3 x)^3 \sqrt {3+5 x}} \, dx\\ &=\frac {11 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^2}-\frac {3 \sqrt {3+5 x}}{14 \sqrt {1-2 x} (2+3 x)^2}-\frac {1}{294} \int \frac {-\frac {1435}{2}-1260 x}{(1-2 x)^{3/2} (2+3 x)^2 \sqrt {3+5 x}} \, dx\\ &=\frac {11 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^2}-\frac {3 \sqrt {3+5 x}}{14 \sqrt {1-2 x} (2+3 x)^2}-\frac {5 \sqrt {3+5 x}}{28 \sqrt {1-2 x} (2+3 x)}-\frac {\int \frac {-\frac {11515}{4}-3675 x}{(1-2 x)^{3/2} (2+3 x) \sqrt {3+5 x}} \, dx}{2058}\\ &=\frac {5 \sqrt {3+5 x}}{42 \sqrt {1-2 x}}+\frac {11 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^2}-\frac {3 \sqrt {3+5 x}}{14 \sqrt {1-2 x} (2+3 x)^2}-\frac {5 \sqrt {3+5 x}}{28 \sqrt {1-2 x} (2+3 x)}+\frac {\int \frac {56595}{8 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{79233}\\ &=\frac {5 \sqrt {3+5 x}}{42 \sqrt {1-2 x}}+\frac {11 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^2}-\frac {3 \sqrt {3+5 x}}{14 \sqrt {1-2 x} (2+3 x)^2}-\frac {5 \sqrt {3+5 x}}{28 \sqrt {1-2 x} (2+3 x)}+\frac {5}{56} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=\frac {5 \sqrt {3+5 x}}{42 \sqrt {1-2 x}}+\frac {11 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^2}-\frac {3 \sqrt {3+5 x}}{14 \sqrt {1-2 x} (2+3 x)^2}-\frac {5 \sqrt {3+5 x}}{28 \sqrt {1-2 x} (2+3 x)}+\frac {5}{28} \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )\\ &=\frac {5 \sqrt {3+5 x}}{42 \sqrt {1-2 x}}+\frac {11 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^2}-\frac {3 \sqrt {3+5 x}}{14 \sqrt {1-2 x} (2+3 x)^2}-\frac {5 \sqrt {3+5 x}}{28 \sqrt {1-2 x} (2+3 x)}-\frac {5 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{28 \sqrt {7}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 95, normalized size = 0.66 \[ -\frac {7 \sqrt {5 x+3} \left (180 x^3+60 x^2-91 x-36\right )-15 \sqrt {7-14 x} (2 x-1) (3 x+2)^2 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{588 (1-2 x)^{3/2} (3 x+2)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.05, size = 116, normalized size = 0.81 \[ -\frac {15 \, \sqrt {7} {\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 14 \, {\left (180 \, x^{3} + 60 \, x^{2} - 91 \, x - 36\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{1176 \, {\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 3.03, size = 291, normalized size = 2.02 \[ \frac {1}{784} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {8 \, {\left (157 \, \sqrt {5} {\left (5 \, x + 3\right )} - 1056 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{180075 \, {\left (2 \, x - 1\right )}^{2}} - \frac {33 \, \sqrt {10} {\left (83 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} + \frac {41720 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} - \frac {166880 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{4802 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 257, normalized size = 1.78 \[ \frac {\left (540 \sqrt {7}\, x^{4} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+180 \sqrt {7}\, x^{3} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-2520 \sqrt {-10 x^{2}-x +3}\, x^{3}-345 \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-840 \sqrt {-10 x^{2}-x +3}\, x^{2}-60 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+1274 \sqrt {-10 x^{2}-x +3}\, x +60 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+504 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {-2 x +1}\, \sqrt {5 x +3}}{1176 \left (3 x +2\right )^{2} \left (2 x -1\right )^{2} \sqrt {-10 x^{2}-x +3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.30, size = 172, normalized size = 1.19 \[ \frac {5}{392} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {25 \, x}{42 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {5}{84 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {125 \, x}{126 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} + \frac {1}{378 \, {\left (9 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} + 12 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 4 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} - \frac {43}{756 \, {\left (3 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 2 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} + \frac {205}{252 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} \]
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 \[ \int \frac {{\left (5\,x+3\right )}^{3/2}}{{\left (1-2\,x\right )}^{5/2}\,{\left (3\,x+2\right )}^3} \,d x \]
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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