Optimal. Leaf size=173 \[ -\frac {7986105 \sqrt {5 x+3}}{845152 \sqrt {1-2 x}}+\frac {698295 \sqrt {5 x+3}}{21952 \sqrt {1-2 x} (3 x+2)}+\frac {6621 \sqrt {5 x+3}}{1568 \sqrt {1-2 x} (3 x+2)^2}+\frac {263 \sqrt {5 x+3}}{392 \sqrt {1-2 x} (3 x+2)^3}+\frac {3 \sqrt {5 x+3}}{28 \sqrt {1-2 x} (3 x+2)^4}-\frac {24922335 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{153664 \sqrt {7}} \]
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Rubi [A] time = 0.06, antiderivative size = 173, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {103, 151, 152, 12, 93, 204} \[ -\frac {7986105 \sqrt {5 x+3}}{845152 \sqrt {1-2 x}}+\frac {698295 \sqrt {5 x+3}}{21952 \sqrt {1-2 x} (3 x+2)}+\frac {6621 \sqrt {5 x+3}}{1568 \sqrt {1-2 x} (3 x+2)^2}+\frac {263 \sqrt {5 x+3}}{392 \sqrt {1-2 x} (3 x+2)^3}+\frac {3 \sqrt {5 x+3}}{28 \sqrt {1-2 x} (3 x+2)^4}-\frac {24922335 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{153664 \sqrt {7}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 93
Rule 103
Rule 151
Rule 152
Rule 204
Rubi steps
\begin {align*} \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^5 \sqrt {3+5 x}} \, dx &=\frac {3 \sqrt {3+5 x}}{28 \sqrt {1-2 x} (2+3 x)^4}+\frac {1}{28} \int \frac {\frac {103}{2}-120 x}{(1-2 x)^{3/2} (2+3 x)^4 \sqrt {3+5 x}} \, dx\\ &=\frac {3 \sqrt {3+5 x}}{28 \sqrt {1-2 x} (2+3 x)^4}+\frac {263 \sqrt {3+5 x}}{392 \sqrt {1-2 x} (2+3 x)^3}+\frac {1}{588} \int \frac {\frac {14787}{4}-11835 x}{(1-2 x)^{3/2} (2+3 x)^3 \sqrt {3+5 x}} \, dx\\ &=\frac {3 \sqrt {3+5 x}}{28 \sqrt {1-2 x} (2+3 x)^4}+\frac {263 \sqrt {3+5 x}}{392 \sqrt {1-2 x} (2+3 x)^3}+\frac {6621 \sqrt {3+5 x}}{1568 \sqrt {1-2 x} (2+3 x)^2}+\frac {\int \frac {\frac {1180305}{8}-695205 x}{(1-2 x)^{3/2} (2+3 x)^2 \sqrt {3+5 x}} \, dx}{8232}\\ &=\frac {3 \sqrt {3+5 x}}{28 \sqrt {1-2 x} (2+3 x)^4}+\frac {263 \sqrt {3+5 x}}{392 \sqrt {1-2 x} (2+3 x)^3}+\frac {6621 \sqrt {3+5 x}}{1568 \sqrt {1-2 x} (2+3 x)^2}+\frac {698295 \sqrt {3+5 x}}{21952 \sqrt {1-2 x} (2+3 x)}+\frac {\int \frac {-\frac {21066255}{16}-\frac {73320975 x}{4}}{(1-2 x)^{3/2} (2+3 x) \sqrt {3+5 x}} \, dx}{57624}\\ &=-\frac {7986105 \sqrt {3+5 x}}{845152 \sqrt {1-2 x}}+\frac {3 \sqrt {3+5 x}}{28 \sqrt {1-2 x} (2+3 x)^4}+\frac {263 \sqrt {3+5 x}}{392 \sqrt {1-2 x} (2+3 x)^3}+\frac {6621 \sqrt {3+5 x}}{1568 \sqrt {1-2 x} (2+3 x)^2}+\frac {698295 \sqrt {3+5 x}}{21952 \sqrt {1-2 x} (2+3 x)}-\frac {\int -\frac {5757059385}{32 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{2218524}\\ &=-\frac {7986105 \sqrt {3+5 x}}{845152 \sqrt {1-2 x}}+\frac {3 \sqrt {3+5 x}}{28 \sqrt {1-2 x} (2+3 x)^4}+\frac {263 \sqrt {3+5 x}}{392 \sqrt {1-2 x} (2+3 x)^3}+\frac {6621 \sqrt {3+5 x}}{1568 \sqrt {1-2 x} (2+3 x)^2}+\frac {698295 \sqrt {3+5 x}}{21952 \sqrt {1-2 x} (2+3 x)}+\frac {24922335 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{307328}\\ &=-\frac {7986105 \sqrt {3+5 x}}{845152 \sqrt {1-2 x}}+\frac {3 \sqrt {3+5 x}}{28 \sqrt {1-2 x} (2+3 x)^4}+\frac {263 \sqrt {3+5 x}}{392 \sqrt {1-2 x} (2+3 x)^3}+\frac {6621 \sqrt {3+5 x}}{1568 \sqrt {1-2 x} (2+3 x)^2}+\frac {698295 \sqrt {3+5 x}}{21952 \sqrt {1-2 x} (2+3 x)}+\frac {24922335 \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{153664}\\ &=-\frac {7986105 \sqrt {3+5 x}}{845152 \sqrt {1-2 x}}+\frac {3 \sqrt {3+5 x}}{28 \sqrt {1-2 x} (2+3 x)^4}+\frac {263 \sqrt {3+5 x}}{392 \sqrt {1-2 x} (2+3 x)^3}+\frac {6621 \sqrt {3+5 x}}{1568 \sqrt {1-2 x} (2+3 x)^2}+\frac {698295 \sqrt {3+5 x}}{21952 \sqrt {1-2 x} (2+3 x)}-\frac {24922335 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{153664 \sqrt {7}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 95, normalized size = 0.55 \[ \frac {-7 \sqrt {5 x+3} \left (1293749010 x^4+1998242055 x^3+482249808 x^2-491393004 x-205593328\right )-274145685 \sqrt {7-14 x} (3 x+2)^4 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{11832128 \sqrt {1-2 x} (3 x+2)^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.88, size = 131, normalized size = 0.76 \[ -\frac {274145685 \, \sqrt {7} {\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (1293749010 \, x^{4} + 1998242055 \, x^{3} + 482249808 \, x^{2} - 491393004 \, x - 205593328\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{23664256 \, {\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 4.03, size = 394, normalized size = 2.28 \[ \frac {4984467}{4302592} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {64 \, \sqrt {5} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{924385 \, {\left (2 \, x - 1\right )}} + \frac {99 \, \sqrt {10} {\left (4411181 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{7} + 2388710520 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} + 506212728000 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} + \frac {38676680000000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} - \frac {154706720000000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{537824 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 305, normalized size = 1.76 \[ \frac {\left (44411600970 \sqrt {7}\, x^{5} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+96225135435 \sqrt {7}\, x^{4} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+18112486140 \sqrt {-10 x^{2}-x +3}\, x^{4}+59215467960 \sqrt {7}\, x^{3} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+27975388770 \sqrt {-10 x^{2}-x +3}\, x^{3}-6579496440 \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+6751497312 \sqrt {-10 x^{2}-x +3}\, x^{2}-17545323840 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-6879502056 \sqrt {-10 x^{2}-x +3}\, x -4386330960 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-2878306592 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {5 x +3}\, \sqrt {-2 x +1}}{23664256 \left (3 x +2\right )^{4} \left (2 x -1\right ) \sqrt {-10 x^{2}-x +3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {5 \, x + 3} {\left (3 \, x + 2\right )}^{5} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {1}{{\left (1-2\,x\right )}^{3/2}\,{\left (3\,x+2\right )}^5\,\sqrt {5\,x+3}} \,d x \]
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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