Optimal. Leaf size=180 \[ \frac {694229 \sqrt {1-2 x} \sqrt {5 x+3}}{921984 (3 x+2)}+\frac {6107 \sqrt {1-2 x} \sqrt {5 x+3}}{65856 (3 x+2)^2}-\frac {73 \sqrt {1-2 x} \sqrt {5 x+3}}{11760 (3 x+2)^3}-\frac {367 \sqrt {1-2 x} \sqrt {5 x+3}}{5880 (3 x+2)^4}+\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{105 (3 x+2)^5}-\frac {2664057 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{307328 \sqrt {7}} \]
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Rubi [A] time = 0.06, antiderivative size = 180, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {98, 151, 12, 93, 204} \[ \frac {694229 \sqrt {1-2 x} \sqrt {5 x+3}}{921984 (3 x+2)}+\frac {6107 \sqrt {1-2 x} \sqrt {5 x+3}}{65856 (3 x+2)^2}-\frac {73 \sqrt {1-2 x} \sqrt {5 x+3}}{11760 (3 x+2)^3}-\frac {367 \sqrt {1-2 x} \sqrt {5 x+3}}{5880 (3 x+2)^4}+\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{105 (3 x+2)^5}-\frac {2664057 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{307328 \sqrt {7}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 93
Rule 98
Rule 151
Rule 204
Rubi steps
\begin {align*} \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^6} \, dx &=\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{105 (2+3 x)^5}-\frac {1}{105} \int \frac {-\frac {991}{2}-835 x}{\sqrt {1-2 x} (2+3 x)^5 \sqrt {3+5 x}} \, dx\\ &=\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{105 (2+3 x)^5}-\frac {367 \sqrt {1-2 x} \sqrt {3+5 x}}{5880 (2+3 x)^4}-\frac {\int \frac {-\frac {14169}{4}-5505 x}{\sqrt {1-2 x} (2+3 x)^4 \sqrt {3+5 x}} \, dx}{2940}\\ &=\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{105 (2+3 x)^5}-\frac {367 \sqrt {1-2 x} \sqrt {3+5 x}}{5880 (2+3 x)^4}-\frac {73 \sqrt {1-2 x} \sqrt {3+5 x}}{11760 (2+3 x)^3}-\frac {\int \frac {-\frac {254625}{8}-7665 x}{\sqrt {1-2 x} (2+3 x)^3 \sqrt {3+5 x}} \, dx}{61740}\\ &=\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{105 (2+3 x)^5}-\frac {367 \sqrt {1-2 x} \sqrt {3+5 x}}{5880 (2+3 x)^4}-\frac {73 \sqrt {1-2 x} \sqrt {3+5 x}}{11760 (2+3 x)^3}+\frac {6107 \sqrt {1-2 x} \sqrt {3+5 x}}{65856 (2+3 x)^2}-\frac {\int \frac {-\frac {15748215}{16}+\frac {3206175 x}{4}}{\sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}} \, dx}{864360}\\ &=\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{105 (2+3 x)^5}-\frac {367 \sqrt {1-2 x} \sqrt {3+5 x}}{5880 (2+3 x)^4}-\frac {73 \sqrt {1-2 x} \sqrt {3+5 x}}{11760 (2+3 x)^3}+\frac {6107 \sqrt {1-2 x} \sqrt {3+5 x}}{65856 (2+3 x)^2}+\frac {694229 \sqrt {1-2 x} \sqrt {3+5 x}}{921984 (2+3 x)}-\frac {\int -\frac {839177955}{32 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{6050520}\\ &=\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{105 (2+3 x)^5}-\frac {367 \sqrt {1-2 x} \sqrt {3+5 x}}{5880 (2+3 x)^4}-\frac {73 \sqrt {1-2 x} \sqrt {3+5 x}}{11760 (2+3 x)^3}+\frac {6107 \sqrt {1-2 x} \sqrt {3+5 x}}{65856 (2+3 x)^2}+\frac {694229 \sqrt {1-2 x} \sqrt {3+5 x}}{921984 (2+3 x)}+\frac {2664057 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{614656}\\ &=\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{105 (2+3 x)^5}-\frac {367 \sqrt {1-2 x} \sqrt {3+5 x}}{5880 (2+3 x)^4}-\frac {73 \sqrt {1-2 x} \sqrt {3+5 x}}{11760 (2+3 x)^3}+\frac {6107 \sqrt {1-2 x} \sqrt {3+5 x}}{65856 (2+3 x)^2}+\frac {694229 \sqrt {1-2 x} \sqrt {3+5 x}}{921984 (2+3 x)}+\frac {2664057 \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{307328}\\ &=\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{105 (2+3 x)^5}-\frac {367 \sqrt {1-2 x} \sqrt {3+5 x}}{5880 (2+3 x)^4}-\frac {73 \sqrt {1-2 x} \sqrt {3+5 x}}{11760 (2+3 x)^3}+\frac {6107 \sqrt {1-2 x} \sqrt {3+5 x}}{65856 (2+3 x)^2}+\frac {694229 \sqrt {1-2 x} \sqrt {3+5 x}}{921984 (2+3 x)}-\frac {2664057 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{307328 \sqrt {7}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 84, normalized size = 0.47 \[ \frac {\frac {7 \sqrt {1-2 x} \sqrt {5 x+3} \left (93720915 x^4+253769850 x^3+257531412 x^2+115804328 x+19437408\right )}{(3 x+2)^5}-13320285 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{10756480} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.10, size = 131, normalized size = 0.73 \[ -\frac {13320285 \, \sqrt {7} {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (93720915 \, x^{4} + 253769850 \, x^{3} + 257531412 \, x^{2} + 115804328 \, x + 19437408\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{21512960 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 3.37, size = 426, normalized size = 2.37 \[ \frac {2664057}{43025920} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {121 \, \sqrt {10} {\left (22017 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{9} + 28768880 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{7} - 9856573440 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} - 2123818368000 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} - \frac {133530503680000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {534122014720000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{153664 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 298, normalized size = 1.66 \[ \frac {\sqrt {5 x +3}\, \sqrt {-2 x +1}\, \left (3236829255 \sqrt {7}\, x^{5} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+10789430850 \sqrt {7}\, x^{4} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+1312092810 \sqrt {-10 x^{2}-x +3}\, x^{4}+14385907800 \sqrt {7}\, x^{3} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+3552777900 \sqrt {-10 x^{2}-x +3}\, x^{3}+9590605200 \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+3605439768 \sqrt {-10 x^{2}-x +3}\, x^{2}+3196868400 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+1621260592 \sqrt {-10 x^{2}-x +3}\, x +426249120 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+272123712 \sqrt {-10 x^{2}-x +3}\right )}{21512960 \sqrt {-10 x^{2}-x +3}\, \left (3 x +2\right )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.39, size = 184, normalized size = 1.02 \[ \frac {2664057}{4302592} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {\sqrt {-10 \, x^{2} - x + 3}}{105 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} - \frac {367 \, \sqrt {-10 \, x^{2} - x + 3}}{5880 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} - \frac {73 \, \sqrt {-10 \, x^{2} - x + 3}}{11760 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {6107 \, \sqrt {-10 \, x^{2} - x + 3}}{65856 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac {694229 \, \sqrt {-10 \, x^{2} - x + 3}}{921984 \, {\left (3 \, x + 2\right )}} \]
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 \[ \int \frac {{\left (5\,x+3\right )}^{3/2}}{\sqrt {1-2\,x}\,{\left (3\,x+2\right )}^6} \,d x \]
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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