Optimal. Leaf size=151 \[ \frac {32735 \sqrt {1-2 x} \sqrt {5 x+3}}{21952 (3 x+2)}+\frac {305 \sqrt {1-2 x} \sqrt {5 x+3}}{1568 (3 x+2)^2}+\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{56 (3 x+2)^3}-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{28 (3 x+2)^4}-\frac {375265 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{21952 \sqrt {7}} \]
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Rubi [A] time = 0.05, antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {99, 151, 12, 93, 204} \begin {gather*} \frac {32735 \sqrt {1-2 x} \sqrt {5 x+3}}{21952 (3 x+2)}+\frac {305 \sqrt {1-2 x} \sqrt {5 x+3}}{1568 (3 x+2)^2}+\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{56 (3 x+2)^3}-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{28 (3 x+2)^4}-\frac {375265 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{21952 \sqrt {7}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 93
Rule 99
Rule 151
Rule 204
Rubi steps
\begin {align*} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^5} \, dx &=-\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{28 (2+3 x)^4}+\frac {1}{28} \int \frac {\frac {47}{2}+30 x}{\sqrt {1-2 x} (2+3 x)^4 \sqrt {3+5 x}} \, dx\\ &=-\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{28 (2+3 x)^4}+\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{56 (2+3 x)^3}+\frac {1}{588} \int \frac {\frac {1575}{4}-210 x}{\sqrt {1-2 x} (2+3 x)^3 \sqrt {3+5 x}} \, dx\\ &=-\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{28 (2+3 x)^4}+\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{56 (2+3 x)^3}+\frac {305 \sqrt {1-2 x} \sqrt {3+5 x}}{1568 (2+3 x)^2}+\frac {\int \frac {\frac {143745}{8}-\frac {32025 x}{2}}{\sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}} \, dx}{8232}\\ &=-\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{28 (2+3 x)^4}+\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{56 (2+3 x)^3}+\frac {305 \sqrt {1-2 x} \sqrt {3+5 x}}{1568 (2+3 x)^2}+\frac {32735 \sqrt {1-2 x} \sqrt {3+5 x}}{21952 (2+3 x)}+\frac {\int \frac {7880565}{16 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{57624}\\ &=-\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{28 (2+3 x)^4}+\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{56 (2+3 x)^3}+\frac {305 \sqrt {1-2 x} \sqrt {3+5 x}}{1568 (2+3 x)^2}+\frac {32735 \sqrt {1-2 x} \sqrt {3+5 x}}{21952 (2+3 x)}+\frac {375265 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{43904}\\ &=-\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{28 (2+3 x)^4}+\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{56 (2+3 x)^3}+\frac {305 \sqrt {1-2 x} \sqrt {3+5 x}}{1568 (2+3 x)^2}+\frac {32735 \sqrt {1-2 x} \sqrt {3+5 x}}{21952 (2+3 x)}+\frac {375265 \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{21952}\\ &=-\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{28 (2+3 x)^4}+\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{56 (2+3 x)^3}+\frac {305 \sqrt {1-2 x} \sqrt {3+5 x}}{1568 (2+3 x)^2}+\frac {32735 \sqrt {1-2 x} \sqrt {3+5 x}}{21952 (2+3 x)}-\frac {375265 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{21952 \sqrt {7}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 79, normalized size = 0.52 \begin {gather*} \frac {\frac {7 \sqrt {1-2 x} \sqrt {5 x+3} \left (883845 x^3+1806120 x^2+1230876 x+278960\right )}{(3 x+2)^4}-375265 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{153664} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 3.59, size = 208, normalized size = 1.38 \begin {gather*} \frac {5 \sqrt {11-2 (5 x+3)} \left (176769 \sqrt {5} (5 x+3)^{7/2}+215199 \sqrt {5} (5 x+3)^{5/2}+90423 \sqrt {5} (5 x+3)^{3/2}-6823 \sqrt {5} \sqrt {5 x+3}\right )}{21952 (3 (5 x+3)+1)^4}-\frac {375265 \tan ^{-1}\left (\frac {\sqrt {\frac {2}{34+\sqrt {1155}}} \sqrt {5 x+3}}{\sqrt {11}-\sqrt {11-2 (5 x+3)}}\right )}{21952 \sqrt {7}}-\frac {375265 \tan ^{-1}\left (\frac {\sqrt {68+2 \sqrt {1155}} \sqrt {5 x+3}}{\sqrt {11}-\sqrt {11-2 (5 x+3)}}\right )}{21952 \sqrt {7}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.17, size = 116, normalized size = 0.77 \begin {gather*} -\frac {375265 \, \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 (883845 \, x^{3} + 1806120 \, x^{2} + 1230876 \, x + 278960\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{307328 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 3.23, size = 368, normalized size = 2.44 \begin {gather*} \frac {75053}{614656} \, \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 {55 \, \sqrt {10} {\left (6823 \, {\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} - 7629720 \, {\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} - 1915892160 \, {\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 {149136243200 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {596544972800 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{10976 \, {\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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 250, normalized size = 1.66 \begin {gather*} \frac {\sqrt {5 x +3}\, \sqrt {-2 x +1}\, \left (30396465 \sqrt {7}\, x^{4} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+81057240 \sqrt {7}\, x^{3} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+12373830 \sqrt {-10 x^{2}-x +3}\, x^{3}+81057240 \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+25285680 \sqrt {-10 x^{2}-x +3}\, x^{2}+36025440 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+17232264 \sqrt {-10 x^{2}-x +3}\, x +6004240 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+3905440 \sqrt {-10 x^{2}-x +3}\right )}{307328 \sqrt {-10 x^{2}-x +3}\, \left (3 x +2\right )^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.33, size = 143, normalized size = 0.95 \begin {gather*} \frac {375265}{307328} \, \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}}{28 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac {\sqrt {-10 \, x^{2} - x + 3}}{56 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {305 \, \sqrt {-10 \, x^{2} - x + 3}}{1568 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac {32735 \, \sqrt {-10 \, x^{2} - x + 3}}{21952 \, {\left (3 \, x + 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 18.54, size = 1509, normalized size = 9.99
result too large to display
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 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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