Optimal. Leaf size=151 \[ \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{4 (2+3 x)^4}+\frac {81 \sqrt {1-2 x} \sqrt {3+5 x}}{56 (2+3 x)^3}+\frac {14145 \sqrt {1-2 x} \sqrt {3+5 x}}{1568 (2+3 x)^2}+\frac {1479375 \sqrt {1-2 x} \sqrt {3+5 x}}{21952 (2+3 x)}-\frac {16925425 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{21952 \sqrt {7}} \]
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Rubi [A]
time = 0.04, 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 = {101, 156, 12,
95, 210} \begin {gather*} -\frac {16925425 \text {ArcTan}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{21952 \sqrt {7}}+\frac {1479375 \sqrt {1-2 x} \sqrt {5 x+3}}{21952 (3 x+2)}+\frac {14145 \sqrt {1-2 x} \sqrt {5 x+3}}{1568 (3 x+2)^2}+\frac {81 \sqrt {1-2 x} \sqrt {5 x+3}}{56 (3 x+2)^3}+\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{4 (3 x+2)^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 95
Rule 101
Rule 156
Rule 210
Rubi steps
\begin {align*} \int \frac {\sqrt {1-2 x}}{(2+3 x)^5 \sqrt {3+5 x}} \, dx &=\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{4 (2+3 x)^4}-\frac {1}{4} \int \frac {-\frac {41}{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}}{4 (2+3 x)^4}+\frac {81 \sqrt {1-2 x} \sqrt {3+5 x}}{56 (2+3 x)^3}-\frac {1}{84} \int \frac {-\frac {7665}{4}+2430 x}{\sqrt {1-2 x} (2+3 x)^3 \sqrt {3+5 x}} \, dx\\ &=\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{4 (2+3 x)^4}+\frac {81 \sqrt {1-2 x} \sqrt {3+5 x}}{56 (2+3 x)^3}+\frac {14145 \sqrt {1-2 x} \sqrt {3+5 x}}{1568 (2+3 x)^2}-\frac {\int \frac {-\frac {913575}{8}+\frac {212175 x}{2}}{\sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}} \, dx}{1176}\\ &=\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{4 (2+3 x)^4}+\frac {81 \sqrt {1-2 x} \sqrt {3+5 x}}{56 (2+3 x)^3}+\frac {14145 \sqrt {1-2 x} \sqrt {3+5 x}}{1568 (2+3 x)^2}+\frac {1479375 \sqrt {1-2 x} \sqrt {3+5 x}}{21952 (2+3 x)}-\frac {\int -\frac {50776275}{16 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{8232}\\ &=\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{4 (2+3 x)^4}+\frac {81 \sqrt {1-2 x} \sqrt {3+5 x}}{56 (2+3 x)^3}+\frac {14145 \sqrt {1-2 x} \sqrt {3+5 x}}{1568 (2+3 x)^2}+\frac {1479375 \sqrt {1-2 x} \sqrt {3+5 x}}{21952 (2+3 x)}+\frac {16925425 \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}}{4 (2+3 x)^4}+\frac {81 \sqrt {1-2 x} \sqrt {3+5 x}}{56 (2+3 x)^3}+\frac {14145 \sqrt {1-2 x} \sqrt {3+5 x}}{1568 (2+3 x)^2}+\frac {1479375 \sqrt {1-2 x} \sqrt {3+5 x}}{21952 (2+3 x)}+\frac {16925425 \text {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}}{4 (2+3 x)^4}+\frac {81 \sqrt {1-2 x} \sqrt {3+5 x}}{56 (2+3 x)^3}+\frac {14145 \sqrt {1-2 x} \sqrt {3+5 x}}{1568 (2+3 x)^2}+\frac {1479375 \sqrt {1-2 x} \sqrt {3+5 x}}{21952 (2+3 x)}-\frac {16925425 \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 [C] Result contains complex when optimal does not.
time = 0.55, size = 101, normalized size = 0.67 \begin {gather*} \frac {25 \left (\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} \left (12696112+55729116 x+81668520 x^2+39943125 x^3\right )}{25 (2+3 x)^4}-677017 i \sqrt {7} \tanh ^{-1}\left (\frac {1}{7} \left (2 \sqrt {70}+3 \sqrt {70} x+3 i \sqrt {7-14 x} \sqrt {3+5 x}\right )\right )\right )}{153664} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(249\) vs.
\(2(118)=236\).
time = 0.15, size = 250, normalized size = 1.66
method | result | size |
risch | \(-\frac {\sqrt {3+5 x}\, \left (-1+2 x \right ) \left (39943125 x^{3}+81668520 x^{2}+55729116 x +12696112\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{21952 \left (2+3 x \right )^{4} \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right )}\, \sqrt {1-2 x}}+\frac {16925425 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{307328 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(129\) |
default | \(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (1370959425 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}+3655891800 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}+3655891800 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}+559203750 x^{3} \sqrt {-10 x^{2}-x +3}+1624840800 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +1143359280 x^{2} \sqrt {-10 x^{2}-x +3}+270806800 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+780207624 x \sqrt {-10 x^{2}-x +3}+177745568 \sqrt {-10 x^{2}-x +3}\right )}{307328 \sqrt {-10 x^{2}-x +3}\, \left (2+3 x \right )^{4}}\) | \(250\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.53, size = 143, normalized size = 0.95 \begin {gather*} \frac {16925425}{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}}{4 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac {81 \, \sqrt {-10 \, x^{2} - x + 3}}{56 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {14145 \, \sqrt {-10 \, x^{2} - x + 3}}{1568 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac {1479375 \, \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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Fricas [A]
time = 0.93, size = 116, normalized size = 0.77 \begin {gather*} -\frac {16925425 \, \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 (39943125 \, x^{3} + 81668520 \, x^{2} + 55729116 \, x + 12696112\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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Sympy [F(-1)] Timed out
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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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 373 vs.
\(2 (118) = 236\).
time = 1.01, size = 373, normalized size = 2.47 \begin {gather*} \frac {55}{614656} \, \sqrt {5} {\left (61547 \, \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 )} + \frac {280 \, \sqrt {2} {\left (157973 \, {\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} + 83743800 \, {\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} + 17691640512 \, {\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 {1351079744000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} - \frac {5404318976000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{{\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}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 18.15, size = 1509, normalized size = 9.99 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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