Optimal. Leaf size=144 \[ \frac {415 \sqrt {5 x+3}}{22638 \sqrt {1-2 x}}+\frac {5 \sqrt {5 x+3}}{196 \sqrt {1-2 x} (3 x+2)}-\frac {\sqrt {5 x+3}}{14 \sqrt {1-2 x} (3 x+2)^2}+\frac {2 \sqrt {5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)^2}-\frac {765 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{1372 \sqrt {7}} \]
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Rubi [A] time = 0.05, 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 = {99, 151, 152, 12, 93, 204} \begin {gather*} \frac {415 \sqrt {5 x+3}}{22638 \sqrt {1-2 x}}+\frac {5 \sqrt {5 x+3}}{196 \sqrt {1-2 x} (3 x+2)}-\frac {\sqrt {5 x+3}}{14 \sqrt {1-2 x} (3 x+2)^2}+\frac {2 \sqrt {5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)^2}-\frac {765 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{1372 \sqrt {7}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 93
Rule 99
Rule 151
Rule 152
Rule 204
Rubi steps
\begin {align*} \int \frac {\sqrt {3+5 x}}{(1-2 x)^{5/2} (2+3 x)^3} \, dx &=\frac {2 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^2}-\frac {2}{21} \int \frac {-\frac {53}{2}-45 x}{(1-2 x)^{3/2} (2+3 x)^3 \sqrt {3+5 x}} \, dx\\ &=\frac {2 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^2}-\frac {\sqrt {3+5 x}}{14 \sqrt {1-2 x} (2+3 x)^2}-\frac {1}{147} \int \frac {-\frac {595}{4}-210 x}{(1-2 x)^{3/2} (2+3 x)^2 \sqrt {3+5 x}} \, dx\\ &=\frac {2 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^2}-\frac {\sqrt {3+5 x}}{14 \sqrt {1-2 x} (2+3 x)^2}+\frac {5 \sqrt {3+5 x}}{196 \sqrt {1-2 x} (2+3 x)}-\frac {\int \frac {-\frac {3955}{8}+\frac {525 x}{2}}{(1-2 x)^{3/2} (2+3 x) \sqrt {3+5 x}} \, dx}{1029}\\ &=\frac {415 \sqrt {3+5 x}}{22638 \sqrt {1-2 x}}+\frac {2 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^2}-\frac {\sqrt {3+5 x}}{14 \sqrt {1-2 x} (2+3 x)^2}+\frac {5 \sqrt {3+5 x}}{196 \sqrt {1-2 x} (2+3 x)}+\frac {2 \int \frac {176715}{16 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{79233}\\ &=\frac {415 \sqrt {3+5 x}}{22638 \sqrt {1-2 x}}+\frac {2 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^2}-\frac {\sqrt {3+5 x}}{14 \sqrt {1-2 x} (2+3 x)^2}+\frac {5 \sqrt {3+5 x}}{196 \sqrt {1-2 x} (2+3 x)}+\frac {765 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{2744}\\ &=\frac {415 \sqrt {3+5 x}}{22638 \sqrt {1-2 x}}+\frac {2 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^2}-\frac {\sqrt {3+5 x}}{14 \sqrt {1-2 x} (2+3 x)^2}+\frac {5 \sqrt {3+5 x}}{196 \sqrt {1-2 x} (2+3 x)}+\frac {765 \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{1372}\\ &=\frac {415 \sqrt {3+5 x}}{22638 \sqrt {1-2 x}}+\frac {2 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^2}-\frac {\sqrt {3+5 x}}{14 \sqrt {1-2 x} (2+3 x)^2}+\frac {5 \sqrt {3+5 x}}{196 \sqrt {1-2 x} (2+3 x)}-\frac {765 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{1372 \sqrt {7}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 95, normalized size = 0.66 \begin {gather*} -\frac {7 \sqrt {5 x+3} \left (14940 x^3+19380 x^2-8633 x-6708\right )-25245 \sqrt {7-14 x} (2 x-1) (3 x+2)^2 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{316932 (1-2 x)^{3/2} (3 x+2)^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.22, size = 122, normalized size = 0.85 \begin {gather*} \frac {(5 x+3)^{3/2} \left (-\frac {25245 (1-2 x)^3}{(5 x+3)^3}+\frac {48475 (1-2 x)^2}{(5 x+3)^2}+\frac {67424 (1-2 x)}{5 x+3}+3136\right )}{45276 (1-2 x)^{3/2} \left (\frac {1-2 x}{5 x+3}+7\right )^2}-\frac {765 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{1372 \sqrt {7}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.33, size = 116, normalized size = 0.81 \begin {gather*} -\frac {25245 \, \sqrt {7} {\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\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 (14940 \, x^{3} + 19380 \, x^{2} - 8633 \, x - 6708\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{633864 \, {\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 3.06, size = 291, normalized size = 2.02 \begin {gather*} \frac {153}{38416} \, \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 {8 \, {\left (524 \, \sqrt {5} {\left (5 \, x + 3\right )} - 3267 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{1980825 \, {\left (2 \, x - 1\right )}^{2}} - \frac {297 \, \sqrt {10} {\left (19 \, {\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 {840 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {3360 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{4802 \, {\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 )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 257, normalized size = 1.78 \begin {gather*} \frac {\left (908820 \sqrt {7}\, x^{4} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+302940 \sqrt {7}\, x^{3} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-209160 \sqrt {-10 x^{2}-x +3}\, x^{3}-580635 \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-271320 \sqrt {-10 x^{2}-x +3}\, x^{2}-100980 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+120862 \sqrt {-10 x^{2}-x +3}\, x +100980 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+93912 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {-2 x +1}\, \sqrt {5 x +3}}{633864 \left (3 x +2\right )^{2} \left (2 x -1\right )^{2} \sqrt {-10 x^{2}-x +3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.19, size = 172, normalized size = 1.19 \begin {gather*} \frac {765}{19208} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {2075 \, x}{22638 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {4415}{45276 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {125 \, x}{294 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} - \frac {1}{126 \, {\left (9 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} + 12 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 4 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} + \frac {23}{252 \, {\left (3 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 2 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} - \frac {5}{1764 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} \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 {5\,x+3}}{{\left (1-2\,x\right )}^{5/2}\,{\left (3\,x+2\right )}^3} \,d x \end {gather*}
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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