Optimal. Leaf size=144 \[ -\frac {639565 \sqrt {1-2 x}}{1176 \sqrt {3+5 x}}+\frac {\sqrt {1-2 x}}{3 (2+3 x)^3 \sqrt {3+5 x}}+\frac {81 \sqrt {1-2 x}}{28 (2+3 x)^2 \sqrt {3+5 x}}+\frac {14101 \sqrt {1-2 x}}{392 (2+3 x) \sqrt {3+5 x}}+\frac {1463447 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{392 \sqrt {7}} \]
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Rubi [A]
time = 0.03, 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 = {101, 156, 157,
12, 95, 210} \begin {gather*} \frac {1463447 \text {ArcTan}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{392 \sqrt {7}}-\frac {639565 \sqrt {1-2 x}}{1176 \sqrt {5 x+3}}+\frac {14101 \sqrt {1-2 x}}{392 (3 x+2) \sqrt {5 x+3}}+\frac {81 \sqrt {1-2 x}}{28 (3 x+2)^2 \sqrt {5 x+3}}+\frac {\sqrt {1-2 x}}{3 (3 x+2)^3 \sqrt {5 x+3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 95
Rule 101
Rule 156
Rule 157
Rule 210
Rubi steps
\begin {align*} \int \frac {\sqrt {1-2 x}}{(2+3 x)^4 (3+5 x)^{3/2}} \, dx &=\frac {\sqrt {1-2 x}}{3 (2+3 x)^3 \sqrt {3+5 x}}-\frac {1}{3} \int \frac {-\frac {41}{2}+30 x}{\sqrt {1-2 x} (2+3 x)^3 (3+5 x)^{3/2}} \, dx\\ &=\frac {\sqrt {1-2 x}}{3 (2+3 x)^3 \sqrt {3+5 x}}+\frac {81 \sqrt {1-2 x}}{28 (2+3 x)^2 \sqrt {3+5 x}}-\frac {1}{42} \int \frac {-\frac {7621}{4}+2430 x}{\sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{3/2}} \, dx\\ &=\frac {\sqrt {1-2 x}}{3 (2+3 x)^3 \sqrt {3+5 x}}+\frac {81 \sqrt {1-2 x}}{28 (2+3 x)^2 \sqrt {3+5 x}}+\frac {14101 \sqrt {1-2 x}}{392 (2+3 x) \sqrt {3+5 x}}-\frac {1}{294} \int \frac {-\frac {899407}{8}+\frac {211515 x}{2}}{\sqrt {1-2 x} (2+3 x) (3+5 x)^{3/2}} \, dx\\ &=-\frac {639565 \sqrt {1-2 x}}{1176 \sqrt {3+5 x}}+\frac {\sqrt {1-2 x}}{3 (2+3 x)^3 \sqrt {3+5 x}}+\frac {81 \sqrt {1-2 x}}{28 (2+3 x)^2 \sqrt {3+5 x}}+\frac {14101 \sqrt {1-2 x}}{392 (2+3 x) \sqrt {3+5 x}}+\frac {\int -\frac {48293751}{16 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{1617}\\ &=-\frac {639565 \sqrt {1-2 x}}{1176 \sqrt {3+5 x}}+\frac {\sqrt {1-2 x}}{3 (2+3 x)^3 \sqrt {3+5 x}}+\frac {81 \sqrt {1-2 x}}{28 (2+3 x)^2 \sqrt {3+5 x}}+\frac {14101 \sqrt {1-2 x}}{392 (2+3 x) \sqrt {3+5 x}}-\frac {1463447}{784} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=-\frac {639565 \sqrt {1-2 x}}{1176 \sqrt {3+5 x}}+\frac {\sqrt {1-2 x}}{3 (2+3 x)^3 \sqrt {3+5 x}}+\frac {81 \sqrt {1-2 x}}{28 (2+3 x)^2 \sqrt {3+5 x}}+\frac {14101 \sqrt {1-2 x}}{392 (2+3 x) \sqrt {3+5 x}}-\frac {1463447}{392} \text {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )\\ &=-\frac {639565 \sqrt {1-2 x}}{1176 \sqrt {3+5 x}}+\frac {\sqrt {1-2 x}}{3 (2+3 x)^3 \sqrt {3+5 x}}+\frac {81 \sqrt {1-2 x}}{28 (2+3 x)^2 \sqrt {3+5 x}}+\frac {14101 \sqrt {1-2 x}}{392 (2+3 x) \sqrt {3+5 x}}+\frac {1463447 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{392 \sqrt {7}}\\ \end {align*}
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Mathematica [A]
time = 0.23, size = 79, normalized size = 0.55 \begin {gather*} \frac {-\frac {7 \sqrt {1-2 x} \left (1646704+7502166 x+11385261 x^2+5756085 x^3\right )}{(2+3 x)^3 \sqrt {3+5 x}}+1463447 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{2744} \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.
\(2(111)=222\).
time = 0.13, size = 250, normalized size = 1.74
method | result | size |
default | \(-\frac {\left (197565345 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}+513669897 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}+500498874 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}+80585190 x^{3} \sqrt {-10 x^{2}-x +3}+216590156 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +159393654 x^{2} \sqrt {-10 x^{2}-x +3}+35122728 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+105030324 x \sqrt {-10 x^{2}-x +3}+23053856 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}}{5488 \left (2+3 x \right )^{3} \sqrt {-10 x^{2}-x +3}\, \sqrt {3+5 x}}\) | \(250\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 211, normalized size = 1.47 \begin {gather*} -\frac {1463447}{5488} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {639565 \, x}{588 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {222589}{392 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {7}{9 \, {\left (27 \, \sqrt {-10 \, x^{2} - x + 3} x^{3} + 54 \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + 36 \, \sqrt {-10 \, x^{2} - x + 3} x + 8 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} + \frac {235}{36 \, {\left (9 \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + 12 \, \sqrt {-10 \, x^{2} - x + 3} x + 4 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} + \frac {13777}{168 \, {\left (3 \, \sqrt {-10 \, x^{2} - x + 3} x + 2 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.89, size = 116, normalized size = 0.81 \begin {gather*} \frac {1463447 \, \sqrt {7} {\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\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 (5756085 \, x^{3} + 11385261 \, x^{2} + 7502166 \, x + 1646704\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{5488 \, {\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\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. 374 vs.
\(2 (111) = 222\).
time = 1.21, size = 374, normalized size = 2.60 \begin {gather*} -\frac {1}{54880} \, \sqrt {5} {\left (1463447 \, \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 )} + 686000 \, \sqrt {2} {\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 )} + \frac {27720 \, \sqrt {2} {\left (11747 \, {\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} + 5216960 \, {\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 {615675200 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} - \frac {2462700800 \, \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 )}^{3}}\right )} \end {gather*}
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 \begin {gather*} \int \frac {\sqrt {1-2\,x}}{{\left (3\,x+2\right )}^4\,{\left (5\,x+3\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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